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Theory and design of a wave generator for a short flume Chappell, Eric Reginald 1969

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THEORY AND DESIGN OF A WAVE GENERATOR FOR A SHORT FLUME by ERIC REGINALD CHAPPELL Gradua t ion D ip loma ( C i v i l ) , Roya l M i l i t a r y C o l l e g e o f Canada, 1953 B . S c , Queen's U n i v e r s i t y , K i n g s t o n , O n t a r i o , 1951* A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE In the Department o f C i v i l E n g i n e e r i n g We accept t h i s t h e s i s as conforming t o the r e q u i r e d s t anda rd THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1969 In p r e s e n t i n g . t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e 1 y a v a i 1ab 1e f o r r e f e r e n c e and S t u d y . . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by, t h e Head o f my De p a r t m e n t o r by h iis r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . (E.R. Chappe 7 ! ! ) D e p a r t m e n t o f C i v i l E n g i n e e r i n g The U n i v e r s j t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8, Canada Date 9 September, 1969 i i ABSTRACT This thesis describes the design of a proposed, new hydraulic laboratory, wave generator for use i n a 3 9 ' - W long, 30" wide and 36" deep flume, and the re-design of a small wave generator previously b u i l t f o r a 21'-3 5/8" long, 8 3/4" wide and 10 5/8" deep flume. These r e l a t i v e l y short flumes are i n s t a l l e d i n the Hydraulics Laboratory of the Department of C i v i l Engineering, U n i v e r s i t y of B r i t i s h Columbia, and the i n s t a l l a t i o n of the proposed, new wave generator would augment the present l i m i t e d wave research f a c i l i t i e s . The project i s supported by an operating grant from the National Research Council of Canada. The preparatory study of laboratory wave generators i n use, presented herein, was made to determine how they function and th e i r design problems. I t was concluded that a r i g i d paddle, double a r t i c u l a t i o n type would be best f o r generating deep-water, t r a n s i t i o n and shallow-water waves i n a flume of r e l a t i v e l y short length. Bie s e l ' s wave generator theory i s outlined and was used i n estimating wave heights and i n determining power and strength require-ments . The e x i s t i n g wave generator for the "small" flume i s a r i g i d paddle, double a r t i c u l a t i o n type. I t did not function s a t i s -f a c t o r i l y due to a very i r r e g u l a r paddle motion. The causes were i i i i s o l a t e d and a new drive system designed and i n s t a l l e d , with good r e s u l t s . The r e s u l t i n g new operating s p e c i f i c a t i o n s are: power = 1.. Hp (D.C.) wave period range = 0.34 to 2.1 sees. design water depth = 6.5" estimated maximum wave height = 4" ' The proposed, new wave generator for the "large " flume i s a r i g i d paddle, double a r t i c u l a t i o n type designed around the ad-just a b l e paddle concept of G.D. Ransford (1949) as modified by L t . C B . Coyer (1953). The designed operating c h a r a c t e r i s t i c s are: power wave period range design water depth estimated maximum wave = 10-Hp (D.C.) = 0.68 to 4.28 sees. = 25" height = 14" CONTENTS IV 1. INTRODUCTION - defining the problem 1 1.1 Background 1 1.2 Thesis Objective 2 2. WATER WAVES - defining the waves to be produced by 5 the laboratory wave generator. 2.1 The Spectrum of Water-Surface Waves 5 2.2 Phy s i c a l C h a r a c t e r i s t i c s of Gravity Waves 7 2.3 Theory of Gravity Wave Motion 10 3. WAVE GENERATORS - gaining an understanding of the types i n 16 use and t h e i r design problems. 3.1 Wave Generator Basic Design Requirements 16 3.1.1 Kinematics of the Wave Generating Member 16 3.1.2 Mechanical Requirements 17 3.1.3 Wave R e f l e c t i o n 18 3.1.4 Control Adjustments 18 3.1.5 M o b i l i t y 19 3.1.6 Water Leakage 19 3.2 Types of Wave Generators 19 3.2.1 Scope 19 3.2.2 The Movable-Wall Type 20 3.2.3 Plunger Types 26 3.2.4 Pneumatic Types 26 3.3 Wave Channel Problems A f f e c t i n g Wave 28 Generator Design 3.4 S p e c i f i c Problems of Wave Generator Design 36 3.4.1 Minimum Wave Period 36 3.4.2 Maximum Wave Period 38 3.4.3 Design Water Depth 39 3.4.4 I n e r t i a 40 3.4.5 Wave Generator Motion 41 3.4.6 Wave Period Control 43 3.4.7 Wave Height Control 45 3.4.8 Anticipated Performance 49 V 3.5 Choice of Wave Generator for a Short 50 Channel 4. WAVE GENERATOR THEORY - a means of determining 53 design forces 4.1 T h e o r e t i c a l Analysis of a Wave Generator 53 4.2 Bie"sel's Theory Applied to a Ri g i d Paddle, 59 Double A r t i c u l a t i o n , Wave Generator 4.2 .1 General 59 4.2 .2 Piston Motion 59 4 .2 .3 Hinged-Flap Motion 63 4 .2 .4 Intermediate Motion 65 4 .2 .5 Discussion 67 4.3 Design Graphs 69 5. RE-DESIGN OF A WAVE GENERATOR - i s o l a t i n g and r e c t i f y i n g 78 the operating problems of the wave generator i n the "Small" flume 5.1 Background , 78 5.2 I s o l a t i n g the Operating Problems 80 5.3 Re-design of the "Small" Flume waye Generator 89 Drive System 5.3.1 Data on the E x i s t i n g Wave Generator and Channel 89 5.3.2 Maximum Motor-Gear-Box Output RPM 89 5.3.3 Minimum Motor-Gear-Box Output RPM 90 5.3.4 Estimated Maximum Wave Heights f o r 91 the Generator 5.3.5 Maximum Paddle Forces Due to Water 92 5.3.6 Maximum Paddle Force Due to 93 Mechanism I n e r t i a 5.3.7 C a l c u l a t i n g Motor-Gear-Box Output Torque 94 5.3.8 Protective High Speed Crank Throw L i m i t a t i o n 97 5.3.9 Crank Wheel Mounting Height for Optimum 101 Sinusoidal Paddle Motion VI 5.3.10 New Drive Unit • 102 5.3.11 Re-design of Connecting Rod 103 5.3.12 Summary of New Operating S p e c i f i c a t i o n s 103 6. DESIGN OF A WAVE GENERATOR - designing the proposed new wave generator for the "large" flume 108 6.1 Scope 108 6.2 Flume Data 108 6.3 Fundamental Design Decisions 109 6.4 E f f e c t of Limited Flume Length Behind Paddle 115 6.5 Design of the Proposed, New Wave Generator Drive 116 System 6.5.1 Summary of Basic Design Data 116 6.5.2 Maximum Crank RPM 116 6.5.3 Minimum Crank RPM 118 6.5.4 Maximum Water Forces Acting on Paddle 119 6.5.5 Maximum Paddle Force Due to Mechanism I n e r t i a 119 6.5.6 C a l c u l a t i n g Motor-Gear-Box Output Torque 120 6.5.7 Discussions of Torque Requirements 120 6.5.8 Selection of a Drive System 125 6.5.9. Estimated Wave Height C a p a b i l i t y 127 6.5.10 High Speed Crank Throw L i m i t a t i o n 128 6.6 Design of Paddle Mechanism Geometry 131 6.6.1 Basic Mechanism Geometry 131 6.6.2 Paddle End-Clearance Plate 132 6.6.3 Paddle S t r u c t u r a l Geometry 134 6.7 Wave Generator S t r u c t u r a l Design Loads 134 6.7.1 General 134 6.7.2 Crank and Crankpin Design Loads 135 6.7.3 Connecting Rod Design Load 135 v 6.7.4 Paddle Design Loads 138 6.7.5 Base Plate Reaction Loads 145 6.7.6 Other Component Design Loads 145 6.7.7 Bearing Loads 145 VII 6.8 Corrosion and I n s t a l l a t i o n Problems 146 6.9 Design Drawings 146 6.10 Summary of Operating S p e c i f i c a t i o n s 146 BIBLIOGRAPHY 148 A. Wave Theory 148 B. Wave Generator Theory 150 C. Wave Generators 152 D. Mechanisms 153 E. Related Subjects 154 F. Wave Generator Correspondence 155 APPENDICES Appendix A: Variable-Speed Drive Unit for the Proposed New Wave Generator f or the Large Flume Appendix B: Corrosion Protection for the Proposed New Wave Generator for the Large Flume Appendix C: Problems Relative to the I n s t a l l a t i o n of the Proposed New Wave Generator i n the Large Flume Appendix D: Design Drawings for the Proposed New Wave Generator for the Large Flume IX LIST OF TABLES Table Page I C a l c u l a t i o n of Maximum Paddle Forces for Small Flume Wave Generator, for Piston Motion, a Crank Throw of 4.3" and Water Depth of 6.5" 96 II Calculations f or Maximum Gear-Box Output Torque Required at various Speeds by the Small Flume Wave Generator for Piston Motion, Maximum Crank Throw of 4.3" and Water Depth of 6.5" 100 III Calculations of Maximum Paddle Forces at various Crank RPM for the Large Flume Wave Generator for Piston Motion, Crank Throw of 10.5" and Water Depth of 25" 122 IV Calculations for Maximum Motor-Gear-Box Output Torque Required by the Large Flume Wave Generator for Pi s t o n Motion, Maximum Crank Throw of 10.5" and Water Depth of 25".123 V Summary of Maximum Torque Requirements for the Large Flume Wave Generator for Generating a Wave Having L/d = 4, Using Piston Motion and D i f f e r e n t Limits of Crank Throw and Water Depth 126 X LIST OF ILLUSTRATIONS Figure Page 1 Three Views of the "Large", 39 ' - 4 V Long, 30" Wide and 36" Deep, Fixed Steel and Glass Flume 3 2 Two Views of the "Small", 21'-3 5/8" Long, 8 3/8" Wide and 10 5/8" Deep, T i l t i n g , Steel and Glass Flume 4 3 Wave P r o f i l e and Associated Terminology 9 4 Deep and Shallow-Water Wave P a r t i c l e Motion 9 5 E f f e c t of Surface Tension on Deep Water Wave V e l o c i t y i n Fresh Water at 70°F. (Wiegel) 13 "a L . 6 Relationship of Ratio ~ — to Ratio — j - at Proportional Z Water Depths - j - f o j . ^ & ± n W a v e G e n e r a t o r C a l i b r a t i o n 15 7 Synoptic Table of Various Wave Generators -. 22 8 Schematic of a Pneumatic Wave Generator Designed to Eliminate Wave R e f l e c t i o n 29 9 Schematic of a T y p i c a l Wave Channel Layout 29 10 Methods of Dealing with Transverse Waves i n a Channel 33 11 P r i n c i p l e of Wave Height Control by Crank Throw Adjustment 47 12 Mechanism for Automatic Crank Throw Adjustment 47 13 P r i n c i p l e of Wave Height Control by Link Throw Adjustment.. 48 XI Figure Page 14 Mechanism for Automatic Adjustment of Link Throw 48 15 P r i n c i p l e of Ransford's Pendulum-Type Wave Generator 51 16 Coyer's Modified Version of Ransford's Pendulum-Type Wave Generator, I l l u s t r a t i n g the Operating P r i n c i p l e . 51 17 D e f i n i t i o n of Co-Ordinate System and Terms Used i n Biesel's Theory ..... 56 18 Graphical Determination of Values of m^d for various values of — ( B i e " s e l ) 60 d 19 V a r i a t i o n of C o e f f i c i e n t K (Piston Motion) and K' (Hinged Flap Motion) for Various Values of —~-(Bi£sel) 61 20 Paddle Displacements for the Three Operating M6des of a Rigid Paddle Wave Generator with Double A r t i c u l a t i o n 64 21 Values of Wave Height C o e f f i c i e n t K (Piston Motion) for Various Values of — 74 22 Maximum Values of Wave Force Fn per Foot of Paddle Width for Various Values of — ~ 75 a 23 Maximum Values of Water I n e r t i a Force F. per Foot of Paddle L 1 Width for Various Values of 76 a 24 D i r e c t i o n of Forces Acting on Wave Generator Paddle During 1 Cycle of O s c i l l a t i o n 77 25 The "Small" Flume Wave Generator Adjusted f o r Piston Motion 79 XII Figure Page 26 Wave Surface I r r e g u l a r i t i e s Due to Uneven Drive Motion.... 79 27 The "small" Flume Wave Generator Adjusted for Hinged-Flap Motion 79 28 The "Small" Flume Wave Generator Crank Wheel Bolted i n t o P o s i t i o n on New Flange . 82 29 Three Views of "small" Flume Wave Generator Drive System Which Had an Output Speed Range of 270 to 30 RPM 84 30 Gear Arrangement i n the Shop-Manufactured Gear Box, C.E. Dept. U.B.C 85 31 "Small" Flume Wave Generator Motor F i t t e d with a Fixed Diameter Drive Pulley to Check Motion Improvements 85 32 Estimated Wave Making C a p a b i l i t y of "Small" Flume Wave Generator for a 6.5" Water Depth and Piston Paddle Motion. 92 a 33 Maximum Torque Required by the "Small" Flume Wave Generator Crank, and Torque a v a i l a b l e from Proposed Drive Unit, at Various RPM 105 34 A Plot of Maximum Values of Wave Force F and I n e r t i a n Forces F. and F. to Show Relative Magnitudes 106 x i m 35 Determining the Crank Wheel Location for Optimum Sinusoidal Displacement of the Paddle at the Piston Motion Setting.... 107 36 I n t e r i o r D e t a i l of the 39'-4V Long, 30" Wide and 36" Deep Flume i n the Region Chosen for I n s t a l l a t i o n of the Proposed New Wave Generator 110 XIII Figure Page 37 E x t e r i o r D e t i a l of the 39 ' - 4 V Long 30" Wide and 36" Deep Flume i n the Region Chosen for I n s t a l l a t i o n of the Proposed New Wave Generator I l l 38 Basic Layout of the "Large" Steel and Glass Flume 112 39 Maximum Torque Required by the Proposed Wave Generator for the Large Flume at Various RPM 124 40 Theoretical Wave Making C a p a b i l i t y of the Proposed Wave Generator for the Large Flume, for a Water Depth of 25" and Maximum Crank Throw of 10.5" 129 41 Developing the Geometry of the Paddle Mechanism 133 42 Side View of the Proposed Wave Generator Paddle S t r u c t u r a l Geometry 136 43 Diagram of the Case for Maximum Loading by External Forces Acting on the Paddle of the Proposed New Wave Generator. 144 XIV ACKNOWLEDGEMENT The author i s indebted to his supervisor, Professor E.S. Pretious, for the s e l e c t i o n of thesis t o p i c , for the con-siderable amount of personal l i t e r a t u r e on wave generators which he made a v a i l a b l e and for h i s encouragement and guidance throughout the preparation of t h i s t h e s i s . Professor Pretious also arranged v i s i t s to the La S a l l e Hydraulic Laboratory, North Vancouver, B.C., and to the C i v i l Engineering Hydraulics Laboratory at the U n i v e r s i t y of Washington, Seattle, Washington, U.S.A., and the U n i v e r s i t y of C a l i f o r n i a , Berkley, C a l i f o r n i a , U.S.A., which proved very informat-ive . The author wishes to express hi s appreciation to the thesis readers for t h e i r constructive c r i t i c i s m s and suggestions on thesis presentation; to the s t a f f of the C i v i l Engineering Workshop for assistance i n making modifications to the e x i s t i n g wave generator, and to the National Research Council of Canada for providing f i n a n -c i a l support for t h i s project. LIST OF SYMBOLS XV WAVE THEORY C = Wave c e l e r i t y ( v e l o c i t y of a wave). L = Wave length. H = Wave height. d = Water depth from bottom to s t i l l water l e v e l , x, y = Coordinates. T = Wave period (sees.). 6 = Wave steepness. g = Acceleration due to gravit y . 2 - 4 p = Water mass density i n slug s / c u . f t . i e . (lb.sec. f t ) a = Water surface tension ( l b s . / f t . ) a = F u l l h o r i z o n t a l amplitude of water p a r t i c l e d i s p l a c e -ment i n a wave form. 3 = F u l l v e r t i c a l amplitude of water p a r t i c l e d i s p l a c e -ment i n a wave form, z = Depth below s t i l l water surface (measured p o s i t i v e downwards). WAVE GENERATOR THEORY see F i g . 17 for l i s t of symbols and d e f i n i t i o n s . WAVE GENERATOR - RECIPROCATING DRIVE THEORY See diagram accompanying TABLE I I or TABLE IV for symbols and d e f i n i t i o n s . THEORY AND DESIGN OF A WAVE GENERATOR FOR A SHORT FLUME 1. INTRODUCTION - defining the problem. 1.1. Background The Department of C i v i l Engineering, University of B r i t i s h Columbia, plans to increase the f a c i l i t i e s of the present hydraulics laboratory to permit undergraduate and graduate wave experiments and research. Laboratory space i s not presently a v a i l a b l e f o r the i n s t a l l a t i o n of new wave channels or tanks of r e l a t i v e l y large s i z e . Therefore, i t was decided to provide wave research f a c i l i t i e s u t i l i z i n g e x i s t i n g flume i n s t a l l a t i o n s . The hydraulics laboratory contains a large, f i x e d , s t e e l and glass flume (Fig. 1) measuring 30" i n width, 33" i n usable depth and 39'-4V i n length, and a small, t i l t i n g , s t e e l and glass flume (Fig. 2) measuring 8 3/4" i n width, 10 5/8" i n usable depth and 21'-3 5/8" i n length. The large flume could be used as a wave channel f o r engineering model studies i f the flume was equipped with a sui t a b l e type of wave generator. The small flume i s equipped with a wave generator which was fabricated i n the C i v i l Engineering Department workshop and i n s t a l l e d during the years 1965 and 1966. Unfortunately, t h i s wave generator has 2 . operating problems which need correct i n g . 1 . 2 Thesis Objective The object of t h i s thesis i s : (a) to i s o l a t e and r e c t i f y the operating problems of the small flume, wave generator; and (b) to design a wave generator for the large flume that w i l l produce waves sui t a b l e for engineering model study purposes. No s i n g l e , complete, source of wave generator design information could be found i n the l i t e r a t u r e . Therefore, a preparatory study was made to determine what wave forms the wave generator must produce, which wave generator type would be' best suited for the large, r e l a t i v e l y short, flume, the associated design problems, and the hydraulic operating s p e c i f i c a t i o n s and to present pertinent theory. The f i n a l f u l f i l l m e n t of the two objectives i s covered i n paragraphs 5 and 6, re s p e c t i v e l y , of the thesi s . Since a complete design procedure f o r designing a wave generator from s t a r t to f i n i s h could not be found the basic pro-cedure the author has used i s h i s own. ( E x i s t i n g Wave Generator i n Foreground) F i g . 2. Two Views of the "Small" 21'-3 5/8" Long, 8 3/8" Wide and 10 5/8" Deep, T i l t i n g , Steel and Glass Flume. 2. WATER WAVES - defining the waves to be produced by the laboratory wave generator. 2.1 The Spectrum of Water-Surface Waves On commencing the design of a water-wave generator i t was f i r s t necessary to examine the water-surface wave forms e x i s t i n g i n nature and to decide which of these forms were to be reproduced i n miniature i n the wave channel. Surface waves (Ref. 7,16 and 18 - Wave Theory) can be c l a s s i f i e d on the basis of wave period as follows: C l a s s i f i c a t i o n C a p i l l a r y waves Ul t r a - g r a v i t y waves Ordinary gravity waves Infr a - g r a v i t y waves Long-period waves Ordinary tides T r a n s - t i d a l waves Period less than 0.1 sees, from 0.1 sec. to 1 sec. from 1 sec. to 30 sees, from 30 sees, to 5 min. from 5 min. to 12 hrs. from 12 hrs. to 24 hrs. 24 hrs. and up. The c l a s s i f i c a t i o n s of i n t e r e s t are those of c a p i l l a r y waves, u l t r a - g r a v i t y waves and ordinary gravity waves. 6. C a p i l l a r y waves i n nature are generally caused by the wind but t h e i r mode of o r i g i n i s i n question. They are influenced more by surface tension than by gravity and hence are greatly affected by surface-active agents, such as o i l s and detergents. These waves are subject to rapid damping by viscous forces. Wave lengths are always shorter than about 0.44 inches while the corresponding wave v e l o c i t y of about 0.95 f t . per sec. i s always exceeded since c a p i l l a r y wave v e l o c i t y increases with decreasing period and length. The•behaviour of these waves i s considerably d i f f e r e n t from that of u l t r a - g r a v i t y and ordinary gravity waves. Ul t r a - g r a v i t y waves i n nature are wind-generated waves l y i n g i n the t r a n s i t i o n zone between c a p i l l a r y waves, which are more influenced by surface tension than gravity, and ordinary gravity waves where gravity i s the predominant influence and surface tension can be neglected. Ordinary g r a v i t y waves i n nature are wind-generated waves for which gravity i s the primary r e s t o r i n g force. The wave generator to be'constructed w i l l be used to pro-duce scaled down r e p l i c a s of ordinary gravity waves for use i n hydraulic model studies. Due to the reduction i n scale, the equivalent laboratory waves w i l l l i e i n the short-period end of  the ordinary gravity wave band and the long-period end of the  u l t r a - g r a v i t y wave band. The lower end of the u l t r a - g r a v i t y wave band, and e s p e c i a l l y the c a p i l l a r y wave band, should be avoided 7. due to undesirable surface tension e f f e c t s . 2.2 Physical C h a r a c t e r i s t i c s of Gravity Waves Since the laboratory wave generator w i l l be designed to produce short-period, ordinary gravity waves and long-period, u l t r a - g r a v i t y waves, as miniature r e p l i c a s of larger gravity waves i n nature, i t i s desirable to review the observed p h y s i c a l c h a r a c t e r i s t i c s of the l a t t e r . The p r o f i l e of a gra v i t y wave with i t s associated term-inology i s shown i n F i g . 3 . Each wave form moves over the s t i l l water surface with a v e l o c i t y or c e l e r i t y C. The distance from crest to crest i s the wave length L. The wave height H i s measured from crest to trough. The depth of water d i s measured from the s t i l l water l e v e l to the ocean bed. The o r i g i n of coordinates x and y i s a point below mid-crest on the s t i l l water l e v e l l i n e . The time for two successive wave crests to pass a fixed point i s defined as the wave period T. An important, p h y s i c a l , wave c l a s s i f i c a t i o n depends upon the length of the wave r e l a t i v e to the depth of the water expressed as ^j-. When a wave moves along the surface i t causes water part-i c l e s beneath i t to move up and down as w e l l as to and f r o . This motion dies out with depth. I f the bottom i s f a r enough away from the s t i l l water l e v e l , r e l a t i v e to the wave length, i t does not i n t e r f e r e with the wave motion and the water p a r t i c l e motion i s c i r -cular of decreasing diameter with depth ( F i g .4 ) . If the bottom i s near 8. the surface, i n s u f f i c i e n t room i s l e f t for the water p a r t i c l e v e r t i c a l motion to develop and the p a r t i c l e motion becomes e l l i p t i c a l (Fig.4). P h y s i c a l l y then, the water i s e i t h e r deep or shallow when compared with the wave length. I f the value of ^ ~ i s small (usually taken as = 2), the wave i s a deep-water wave. If — i s large (— = 20) the wave i s a shallow-water wave. This r a t i o shows that a tsunami with a wave-length of hundreds of miles i s a shallow-water wave i n the deepest parts of the ocean, while r i p p l e s may be deep-water waves i n a pond a foot deep. Since other aspects of wave motion remain s i m i l a r for a given ^  r a t i o , t h i s c l a s s i f i c a t -ion assumes importance i n hydraulic model wave studies and i s used to r e l a t e the motion of the waves i n nature to the scaled down r e p l i c a s produced i n a laboratory. Another s i g n i f i c a n t p h y s i c a l feature of waves i s steepness. This i s the r a t i o of the wave height to the wave length, 6 = —. Waves have a l i m i t i n g value of steepness where they commence to break, which occurs when the c e n t r i p e t a l a c c e l e r a t i o n of the water p a r t i c l e s at the crest i s g. This makes sense p h y s i c a l l y , since, i f the acc e l e r a t i o n was greater than g then some sort of downward pressure to supplement gravity would be required to keep the wave from f l y i n g apart. Steepness and the ^ j- r a t i o n are the two factors used i n es t a b l i s h i n g geometric s i m i l i t u d e between n a t u r a l waves and the scaled down laboratory waves. Wave Form Velocity c 1 Fig. 3 Wave P r o f i l e and Associated Terminology S h a l l o w - W a t e r Wave Wave Crest d I — a 20 J v O ( + ) z o(.W-f- Ot-»l yyyyyyyyyyyyyyyyyyyyyy%& vyyyyyyyyyyyyyjmM. Bottom d* D e e p - W a t e r Wave d i / J Tyyyyyyyyyyyyyyyyyy/yyyyyyyyyy/yyyyyyyyyy. Bottom Fig. 4 Deep and Shallow-Water Wave P a r t i c l e Motion 10. 2.3 Theory of Gravity Wave Motion Mathematical theories describing the behavious of gravity waves i n water have been developed by Laplace (1776), Gerstner (1802), A i r y (1845), Stokes (1847), Froude (1862), Rankine (1863), Havelock (1918), L e v i - C i v i t a (1925), Struik (1926), Lamb (1932) , B i e s e l (1952), Crapper (1957) and others. Unfortunately, which theories most c l o s e l y describe the p h y s i c a l c h a r a c t e r i s t i c s of wind driven gravity waves i n nature, are as yet unknown, and the accuracy with which these theories describe waves i n a laboratory wave channel changes s l i g h t l y for waves of d i f f e r e n t periods. Obviously, the choice of theory to describe gravity wave motion i s a r b i t r a r y . The r e s u l t s of the theories of A i r y and Lamb are outlined below since they are the most widely known and are mathematically easy If waves are of small amplitude compared to t h e i r length and to the depth of the water, the wave p r o f i l e c l o s e l y app-roximates a sine curve. Terminology for t h i s theory i s i l l u s t r a t -ed i n F i g . 3. The rate of t r a v e l or c e l e r i t y (c) of an i n d i v i d u a l wave form (Lamb, 1932), considering both gravity and surface tension, i s : to use. c + 2TTO P L tanh 2?rd L (1) 11. The relative effects on velocity, of the gravity and the surface tension components for deep-water waves, are presented in Fig.5. Since the calculation of the percent difference in Fig.5 is not affected by the wave being either a deep-water or a shallow-water wave type, i t is evident that, for any wave more than one foot in length, the effect of surface tension may be safely omitted when calculating the wave form velocity. Neglecting surface tension effects, the equation for velocity of propagation of gravity waves (Airy, 1854; Lamb, 1932) becomes c - I ^ . tanh -j^ • (2) For water deeper than one-half the wave length, that is = 2, the term tanh is almost equal to 1 and the equation reduces to the "cleep-water" wave equation c = / . (3) For water depths less than of the wave length, that is h = 20, the term tanh approaches the value of — A and equation (2) reduces to the "shallow-water" wave equation c =./gd- (4) Actually, there is no abrupt change from "deep" to "shallow" water waves and the depths for which these simplified equations are no longer applicable depends upon the degree of accuracy desired in calculations. But for most engineering studies i t has become custom to c a l l waves having ^=~ 2 "deep water" waves (equation 3) and waves having — g 20 "shallow-water" waves (equation 4). Waves 12. i n between are known as " t r a n s i t i o n " waves; where equation (2) applies. The r e l a t i o n s h i p between length, period and v e l o c i t y i s defined by L = CT. (5) The surface p r o f i l e i s given by the s i n u s o i d a l equation y = 2 c o s I?" ~ ' where x and y are measured as i n F i g . 3. The p r o f i l e and c e l e r i t y equations given above are, mathematically, only f i r s t approximations to the t h e o r e t i c a l s o l u t i o n s , but are quite s a t i s f a c t o r y f o r most p r a c t i c a l purposes. Although the c e l e r i t y equations only apply s t r i c t l y to waves of small amplitude, they are s u f f i c i e n t l y accurate for most cases except when the wave i s becoming large and steep enough to break. In F i g . 4, <* and 3 are the f u l l h o r i z o n t a l and v e r t i c a l displacements, r e s p e c t i v e l y , of a water p a r t i c l e as i t describes an o r b i t about i t s average depth (Z) below the s t i l l water surface. Values for = and g can be calculated using the following equations . . 2T.(d-z) H cosh z  B i n h 2 I L _ ( 7 ) Fig. 5 Effect of Surface Tension on Deep Water Wave Velocity in Fresh Water at 70° F. (Wiegel) 14. = H s i n h I J . , 2iTd (8) smh Waves have a limiting value of steepness. Michell (1893), found found the maximum possible steepness for a Stokes deep-water wave to be 6 max. = f = 0.142 = y. (9) Theory suggests that shallow water waves w i l l break when 5. = 781 = 1 d - / 8 1 4 (10) A graphical presentation of theoretical information useful in calibration of wave generators of the double articulation type to be designed, is given in Fig. 6. Note: ^7-cosh ££T (d-z) sinh 32s1 0.4 d WHERE Z c average depth of orbiting p a r t i c l e below s t i l l water surface and measured positive downwards. d= depth of water *c= f u l l horizontal amplitude of p a r t i c l e osci 1 l a t i o n H«= wave height L= wave length H (O'Brien and Mason) Fig. 6 Relationship of Ratio — to Ratio -*r at Proportional Water Depths - j for use in Wave n a a Generator Calibration. 0 1 16. 3. WAVE GENERATORS - gaining an understanding of the types i n use and t h e i r design problems. 3.1- Wave Generator Basic Design Requirements 3.1.1 Kinematics of the Wave Generating Member There are a number of basic requirements to be considered when designing a laboratory wave generator. The f i r s t of these, the kinematics of the wave generating member, i s of p a r t i c u l a r importance when the wave generator i s to be designed for use i n a flume of r e l a t i v e l y short length. Any p e r i o d i c disturbance i n one end of a wave channel, i f of s u f f i c i e n t strength, w i l l o r i g i n a t e a t r a i n of waves which runs along the channel to the opposite end; but, i f the generat-ing member f a i l s to conform with c e r t a i n kinematic requirements, the distance along the channel at which the wave t r a i n s t a b i l i z e s may be quite long, and the type of wave which r e s u l t s may be wholly undesirable. The deformation of a v e r t i c a l plane i n water, upon passage of e i t h e r a shallow-water wave or a deep-water wave, corresponds to the wave's respective envelope of water p a r t i c l e motion (Fig.4). The wave generator must approximate, as c l o s e l y as possib l e , t h i s deformation appropriate to the wave desired, i f stable waves with correct p a r t i c l e motion are to be produced i n the immediate v i c i n i t y of the generating member. 17. Conformance to t h i s kinematic requirement allows a r e -l a t i v e l y short wave channel to be u t i l i z e d with r e s u l t a n t compact-ness and space economy. Also i t conserves power required to generate the wave by forming only the wave which i s desired. On the other hand, the accurate simulation of the o r b i t a l motion of the l i q u i d p a r t i c l e s i n a wave increases the mechanical complexity of a wave generator, n e c e s s i t a t i n g a degree of compromise. (Ref. 1 -Wave Generator Theory). 3.1.2 Mechanical Requirements Most wave generators u t i l i z e o s c i l l a t i n g members to i n i t i a t e waves. Since the o s c i l l a t i o n rate may amount to 2 or 3 cycles per second i t i s important to keep the i n e r t i a of the o s c i l l a t i n g members r e l a t i v e l y low. High i n e r t i a gives r i s e to two detrimental e f f e c t s : f i r s t , the motor power required to operate the generator i s greatly increased, and second, the motion of the generator i s made more i r r e g u l a r f or any given power input. In the l a t t e r case departures from the assigned motion produce unwanted i r r e g u l a r -i t i e s i n the generated waves. R i g i d i t y of drive members has proved very important i n wave generator operation. Any form of f l e x i n g a l t e r s the drive motion s l i g h t l y with undesirable e f f e c t s on the wave forms produced. Therefore, although the weight of moving parts must be kept reasonably l i g h t to reduce i n e r t i a l e f f e c t s , i t i s imperative that lightness not be achieved at the expense of r i g i d i t y . 18. Materials used i n constructing the wave generator should be r e s i s t a n t to corrosion i n water. Also, maintenance w i l l be re-duced i f moving parts such as bearings, gears and r o l l e r s are positioned clear of the water to avoid the damaging e f f e c t s of water borne sediment. 3.1.3 Wave R e f l e c t i o n In nature, waves r e f l e c t e d from a structure depart seaward where they are f i n a l l y d i s s i p a t e d . In a laboratory wave channel, waves r e f l e c t e d from the model of a structure propagate back to-wards the wave generator. I d e a l l y , the wave-generator should not again r e f l e c t these waves but unfortunately, most generators are good r e f l e c t o r s . Two solutions e x i s t for t h i s problem. The f i r s t , and most common, i s the use of a f i l t e r i n front of the generator to re-duce the amplitude of the r e f l e c t e d wave to an acceptable height. The second i s the use of s p e c i a l wave generators producing e i t h e r l i t t l e , or no r e f l e c t i o n s ; or s p e c i a l devices to deal with r e f l e c t -ed waves. Examples of such s p e c i a l wave generators are the pneumatic types, to be described l a t e r , and those used at the D e l f t Hydraulic Laboratory i n Holland which employ a strong a i r stream to increase the s i z e of the generated wave and attenuate the r e -f l e c t e d waves. 3.1.4 Control Adjustments Model tests i n a wave channel may require the use of eit h e r deep-water, t r a n s i t i o n , or shallow-water waves of d i f f e r e n t heights. 19. Therefore, the generator must be capable of adjustment to produce waves of d i f f e r e n t lengths and heights i n a p a r t i c u l a r depth of water. The adjustments should be simple to make and scales whould be provided which ensure the accuracy of the s e t t i n g . 3.1.5 M o b i l i t y M o b i l i t y i s not usually a requirement for a wave generator designed f o r use i n a wave channel. 3.1.6 Water Leakage Most wave generators use a paddle of some form to generate waves. Leakage of water around the bottom and sides of the paddle can account f o r the following poor, wave c h a r a c t e r i s t i c s : a) decrease i n wave amplitude; b) i n s t a b i l i t y i n the wave form; and c) i n c o r r e c t wave p r o f i l e . On the other hand, i f leakages are e f f e c t i v e l y sealed, the increased f r i c t i o n introduced by gaskets or other sealing devices may r e s u l t i n i r r e g u l a r generator motion. Most frequently a com-promise i s required. 3.2 Types of Wave Generators 3.2.1 Scope Wave generators presently used i n hydraulic laboratory wave channels assume a wide v a r i e t y of forms. Fortunately, they may be loosely grouped i n a few markedly d i f f e r e n t categories. Within each 20. category there are as many d i f f e r e n t v a r i a t i o n s i n the basic p r i n c i p l e as there are designers. Some of these v a r i a t i o n s have been employed to emphasize one or more of the basic generator r e -quirements already mentioned, e s p e c i a l l y kinematic requirements. A comprehensive review of d i f f e r e n t types of wave generators i s given i n the p u b l i c a t i o n "Laboratory Wave-Generating Apparatus", (R e f . l - Wave Theory), which i s a t r a n s l a t i o n of a serie s of four French a r t i c l e s i n La H o u i l l e Blanche by F. B i e s e l , F. Suquet and a group of engineers at the Laboratoire Dauphinois d' Hydraulique (Neyrpic), i n France. A synoptic table from t h i s p u b l i c a t i o n l i s t -ing types of water wave generators and showing t h e i r p r i n c i p l e s of operation has been reproduced i n Fig.7. 3.2.2 The Movable-Wall Type This general category includes those types whose way,e-generating members are immersed i n water and o s c i l l a t e d back and fo r t h i n accordance with some established law. These machines, although wave r e f l e c t i n g , are very v e r s a t i l e , since by imposing an appropriate law of motion they w i l l generate waves of any desired c h a r a c t e r i s t i c . The f i r s t machine of t h i s category shown i n Fig.7 i s a f l e x i b l e f l a p type. I t i s capable of cl o s e l y approximating water p a r t i c l e motion r e s u l t i n g i n waves with stable and correct motions being produced immediately i n front of the f l a p . This i s an import-ant advantage i n channels of l i m i t e d length. Also the machine's i n e r t i a i s small, contributing to r e g u l a r i t y of o s c i l l a t i o n ; The 21. disadvantage i s the complicated mechanism which i s not r e a d i l y adjustable, together with high maintenance due to cranks and bush-ings being located underwater. Next shown i n F i g . 7 are three versions of r i g i d f l a p , wave generators with s i n g l e a r t i c u l a t i o n . These machines are of simple construction being hinged at one point and driven by a r o t a t i n g crank and connecting rod. There i s the undesirable feature of an under-water hinge i n only two cases. They have low i n e r t i a , and are e a s i l y adjusted. The motion imparted to the water i n the f i r s t two examples i s that f o r deep-water waves. Therefore, i f t r a n s i t i o n or shallow-water waves are required, a long channel i s necessary to permit the correct wave p r o f i l e and p a r t i c l e motion to develop, and wave height w i l l be l i m i t e d . Locating the hinge above the channel bed, as i n the second example, may be necessary i n the case of a model with a movable sand bed. The example showing a r t i c u l a t i o n above the free surface i s not very s a t i s f a c t o r y since i t ignores the prime kinematic requirement of wave generation - namely the decrease i n si z e of water p a r t i c l e o r b i t s with depth. Water leakage under the paddle r e s u l t s i n low wave heights and may r e s u l t i n wave i n -s t a b i l i t i e s . The t h i r d wave generator type i n the movable w a l l category i s the pist o n . I t i s a simple machine actuated by a r o t a t i n g crank and connecting rod. The piston moves back and for t h i n the channel's l o n g i t u d i n a l d i r e c t i o n with usually a simple harmonic 22 Fig. 7 SYNOPTIC TABLE OF VARIOUS WAVE GENERATORS (Bi6sel, Suquet and Others) Type of Have Generator Diagram of Principle Amplitude Flexible flap ;</ >7 -77777777777-Rigid flap with single articulation Articulation at channel bed ^ \ j U i m m v M / w w w / , Calculable Articulation above channel bed Articulation above free surface ^ Hot oalculable by Biesel's theory Piston Calculable Fig. 7 SYNOPTIC TABLE OF VARIOUS WAVE GENERATORS (Cont'd) 23 Type of Wave Generator Diagram of Principle Amplitude Rigid paddle with double articulation Suspended paddle Paddle in contact with bed by means of rollers or slides //WW/WW/WW) www Calculable i f the paddle is plane Plunger Generally not calculable by Biesel's theory 24 F i g . 7 SYNOPTIC TABLE OF VARIOUS WAVE GENERATORS (Cont'd) Type of Wave Generator Diagram of Principle Amplitude Serpent Calculable Eccentric roller Elliptical cylinder Various ' Devices Cylinder with paddles ©u. Hot calculable by Biesel's theory / Paddle wheel Surface paddle WWW77////7//S//7, Pneumatic device 25. motion. The piston type i s p a r t i c u l a r l y suited f or generating shallow-water waves, which means, that, when deep-water waves are. required, a r e l a t i v e l y long channel, to permit the waves to s t a b i l i z e , i s needed. The fourth group i n the movable w a l l category i s composed of r i g i d paddle generators with double a r t i c u l a t i o n , s i x examples of which are shown on the second page of Fig.7. The lower end of the paddle of t h i s group can be adjusted to traverse a l o n g i t u d i n a l distance d i f f e r e n t to that traversed by the top end. Assuming that the top and bottom ends of the paddle are adjusted to t r a v e l the same distance, the generator then operates as a p i s t o n type and i s p a r t i c u l a r l y suited f or generating shallow-water waves. If the bottom end of the paddle i s adjusted for zero t r a v e l distance, the generator operates as a f l a p with s i n g l e a r t i c u l a t i o n and i s p a r t i c u l a r l y suited for generating deep-water waves. I f the bottom end of the paddle i s adjusted for a t r a v e l distance somewhere between zero and the distance t r a v e l l e d by the top end, the motion i s then suited to the generation of a transition.wave. Because these machines with double a r t i c u l a t i o n c l o s e l y approximate water p a r t i c l e motion at the paddle, they are p a r t i c -u l a r l y u seful when the wave channel i s of short length. Further-more, maintenance i s s l i g h t because moving parts are out of the water and i n e r t i a can be kept reasonably low. 26. For a l l machines i n the movable-wall category, the length of the generated wave, i s c o n t r o l l e d by the period of. o s c i l l a t i o n of the movable-wall or paddle while the wave height i s c o n t r o l l e d by the amplitude of paddle o s c i l l a t i o n . 3.2.3 Plunger Types This second general category comprises those types which operate on the p r i n c i p l e of a plunger.being p e r i o d i c a l l y thrust into and withdrawn from the water, forming waves by water dis p l a c e -ment. The plunger may have various c r o s s - s e c t i o n a l forms, the most common being e i t h e r a wedge-shape, or one side p a r a b o l i c a l l y con-toured with a v e r t i c a l back. Wave length and height may be c o n t r o l l e d by varying the period of o s c i l l a t i o n and stroke length (displacement) of the plunger. Plunger-type machines do not c l o s e l y approximate water p a r t i c l e motion over the f u l l range from deep to shallow-water waves and therefore r e l a t i v e l y long wave channels are required. Two examples of wave generators i n the plunger category are shown i n Fig.7. 3.2.4 Pneumatic Types The t h i r d general category i s made up of wave generators which produce waves by a s p i r a t i n g the water i n t o a chamber and then l e t t i n g i t f a l l f r e e l y downwards. The chamber i s e s s e n t i a l l y a con-t r o l l e d surge tank. The p r i n c i p l e of pneumatic generators i s i l l u s t r a t e d at the end of F i g . 7. These generators' require adequate channel length, i n front of them to f a c i l i t a t e s t a b i l i z a t -ion of the wave motion. This type i s used almost e x c l u s i v e l y i n the David Taylor Model Basin (U.S. Navy). In operation, there are l i m i t s to the wave periods which can be obtained i f the water i s allowed to j u s t f a l l f r e e l y i n the surge chamber under gravity. To achieve shorter periods, the chamber i s connected a l t e r n a t e l y to the suction and pressure' sides of a compressor instead of just the suction side.. This procedure also permits higher wave amplitudes to be obtained. These generators have had a reputation of being d i f f i c u l t to adjust for a p a r t i c u l a r wave form, but this may no longer be true i n view of increased i n t e r e s t and work on t h i s type i n the United States. Water flow out the discharge tubes generally gives an i n i t i a l water p a r t i c l e motion approximating that of shallow-water waves. The b i g advantage of pneumatic generators i s that they can be designed i n configurations ( F i g . 8) which do not r e f l e c t waves r e f l e c t e d back along the channel by the model. Various other forms of generators e x i s t which f a l l outside of the three general categories mentioned. Some of these machines are shown on the l a s t page of F i g . 7. The serpent generator i s designed 28. for use i n large, rectangular, wave basins rather than i n channels. The time- at which small sections along i t s length.are h o r i z o n t a l l y displaced can be adjusted so as to l i m i t the l a t e r a l length, of a wave crest and thereby, i n coastal studies simulate waves advancing to the beach along a st r e t c h of c o a s t l i n e . The serpent generator i s complex and. d i f f i c u l t to adjust. The e c c e n t r i c r o l l e r , e l l i p t i c a l c y l i n d e r , cylinder with paddles, paddle wheel and surface paddle generators, ignore to varying de-grees the kinematic requirements of wave motion thereby n e c e s s i t a t -ing the use of long wave channels. However, the wave generators equipped with paddles do have a very low r e f l e c t i o n c o e f f i c i e n t . 3.3 Wave Channel Problems A f f e c t i n g Wave Generator Design There are other aspects of wave generator design to be con-sidered besides the basic design requirements of kinematics, mechanics, r e f l e c t i o n , adjustment, mobility and water leakage already discussed. A t y p i c a l wave channel layout i s shown i n F i g . 9. If the wave generator produces waves i n two d i r e c t i o n s , such as i n the case of a r i g i d f l a p generator with si n g l e a r t i c u l a t i o n , then a length of channel behind the fl a p must be u t i l i z e d for a wave absorber. The length required may run i n the region of 5' to 15' (Ref. 5 -Correspondence) depending on the s i z e of channel and absorber e f f i c i e n c y . 29 suction line •pressure line reflected-wave absorber fflr/7//s//yy/7//. Fig. 8 Schematic of a Pneumatic Wave Generator Designed to Eliminate Wave Reflection. wave absorber - for waves from back of flap generator flap parallel plate wave filter wave absorbing beach U— model test - — * i » reach > \yy/yy/s////yy/yy/y////y^^ Fig. 9 Schematic of a Typical Wave Channel Layout 30. When, s e l e c t i n g the type of generator to be b u i l t , channel length i s important. Long channels permit many i r r e g u l a r i t i e s - i n water-particle motion to correct themselves, allowing a choice from a wide v a r i e t y of generator types. I f the channel i s of r e l a t i v e l y short length, v i z . shorter than about 50', then i t becomes de-s i r a b l e to produce waves which are i n i t i a l l y as perfect as possi b l e . This requires the use of a generator which c l o s e l y approximates the desired water-particle o r b i t a l motion. If a wave channel i s to be designed along with the generator i t should be as b i g as space and finances allow, since the greater the width, length and depth of the channel, the closer i t approximates actual prototype conditions. Generators u t i l i z i n g f l a t plates for the wave-generating member may experience d i f f i c u l t i e s from transverse waves being formed i n . f r o n t of the pla t e , and causing i r r e g u l a r i t i e s i n the wave pro-f i l e . Transverse wave formation, when due to eit h e r wave d i f f r a c t -ion (Ref.7 - Wave Generator Theory) or l o c a l surges induced by water leakage around the sides of the pla t e , becomes an increasing problem with increased plate width. Ransford (Ref.l - Wave Generators) r e -commends the use of unequally spaced " c l a p o t i s " plates (Fig.10) to hinder the formation of these waves. The unequal spacing prevents any secondary transverse waves formed i n the spaces from r e i n -f o r c i n g each other. Other designers also have frequently u t i l i z e d these transverse o s c i l l a t i o n b a f f l e s on t h e i r wave generator paddles, but there does not appear to be any design information a v a i l a b l e 3 1 . which would allow the c a l c u l a t i o n of optimum s i z e and shape, beyond an empirical t r i a l and error procedure. These b a f f l e s are pa r t i c u l a r - , l y u s eful on very wide wave generating members used i n large rectangular wave basins where, the use of wave f i l t e r s i s d i f f i c u l t . In wave channels, however, many designers omit the b a f f i e • p l a t e s altogether and r e l y e n t i r e l y on a system of wave f i l t e r s to smooth, out the wave p r o f i l e . Transverse waves may create an even bigger problem i n a channel when the wave period i s such that resonance occurs for wave modes across the channel width. Kravtschenko and Santon (Ref. 6 -Wave Generator Theory) mention that these transverse waves or p r i n c i p l e phenomena can produce parasite phenomena, composed of waves with s i g n i f i c a n t harmonics, whose basic period i s 2/3 that of the generator paddle. The net r e s u l t of transverse waves i s an i r r e g u l a r wave p r o f i l e and water p a r t i c l e motion. To r e c t i f y t h i s s i t u a t i o n i n a wave channel, a f i l t e r i s used i n front of the wave generator. Many types of f i l t e r s have been t r i e d and t h e i r design involves a separate study i n i t s e l f . However, a series of t h i n v e r t i c a l plates p a r a l l e l to the l o n g i t u d i n a l exis of the channel have proven an e f f e c t i v e f i l t e r f o r removing i n i t i a l d i f f r a c t i o n r i p p l e s and transverse waves ( F i g . 1 0 ) . The required length of most f i l t e r systems, for a given 32. attenuation i n wave i r r e g u l a r i t i e s , increases with increased wave length. Therefore the f i l t e r space required i n the channel w i l l depend on the maximum wave length to be used. Other types of wave f i l t e r s may be required to reduce the height of waves r e f l e c t e d by the model i f the wave generator i s r e f l e c t i v e . Other transverse wave problems, not e a s i l y cured, can r e -s u l t from f l e x i b l e channel walls. This points to the need f o r considerable r i g i d i t y i n the channel structure. Likewise, misalign-ment of the channel f l o o r and walls can produce an i r r e g u l a r wave form. Achieving and maintaining large wave heights poses some problems. The use of wave f i l t e r s to smooth out the waves r e s u l t s i n a loss of wave height. In addition, the i n i t i a l wave height may" have been l i m i t e d by the maximum amplitude of the wave generator plate motion and i t s kinematics. To i l l u s t r a t e the e f f e c t of poor kinematics, which may not be r e a d i l y apparent, consider a normal deep-water wave. As i t steepens i t becomes incr e a s i n g l y unstable u n t i l i t breaks. This occurs when the wave height (H) - y L. I f the wave generator kinematics do not approximate the wave motion c l o s e l y , then the i n s t a b i l i t y of the wave form i s increased, causing the wave to break sooner, l i m i t i n g the wave height. This loss of wave amplitude reduces the steepness and may pose a problem i n Connecting Rod Generator flap hinged at channel bed. Fig. 10. Methods of Dealing with Transverse Waves i n a Channel 3.4. in certain tests ? usually ones of a theoretical nature. Adequate wave amplitude may be maintained in two ways. The f i r s t method i s to have the channel walls converge in plan, downstream of the f i l t e r s , which w i l l increase the wave ampli-tude. The second method consists of raising the channel bed by a floor sloping gradually upward away from the wave generator. This has l i t t l e direct effect on the wave amplitude (the amplitude may even decrease in certain cases) but shortens the wave length (the period remaining the same) which in turn increases the wave steep-ness . Since this second method leads to appreciable intensificat-ion of steepness only when the f i n a l ratio of wave length to depth is sufficiently large (shallow water waves), the f i r s t method i s more generally used. Moreover, the rate of convergence should be low to avoid any wave reflection from the sides. A successful, long-length, channel designed by the National Research Council of Canada, converges from 6' to 4' in width over a length of 40' (Ref.2 - Wave Generator Correspondence). When wave steepness is of prime importance, consideration must be given to height attenuation due to channel boundary f r i c t i o n . This i s especially so i f the channel is relatively narrow - under about 2', and sand or pebble bottom layers are to be used. For example, a channel 1.5' wide, 0.75' deep and 60' long with a sand bottom caused a 21% reduction in wave height over i t s length (Ref.10 -Wave Generators). •35, At the f a r end of the channel ( F i g . 9), opposite to the generator, a spending beach i s required to absorb the energy of incoming waves to avoid r e f l e c t i o n . The spending beach comprises some form of wave absorber. Although the design of wave absorbers does not l i e within the scope of t h i s t h e s i s , i t i s worth noting that t h e i r e f f i c i e n c y must be high (Ref. 4 - Related Subjects). For an incident wave s t r i k i n g the beach, 99% absorption of i t s energy s t i l l r e s u l t s i n a r e f l e c t e d wave going back to the generator with a height 10% of the incident wave height. Fortunately, absorbers can be de-signed with r e f l e c t i o n c o e f f i c i e n t s ( r a t i o of r e f l e c t e d wave height to incident wave height) i n the range of 0.04 to 0.08. When tests involve sand beaches, the length of channel r e -quired f o r the beach alone may be considerable. Many beach sands are unstable under wave action on slopes steeper than 1 on 10, and equilibrium may require a mean slope of 1 on 20, or f l a t t e r . For a 1 on 20 slope, the channel length required by the beach w i l l have to exceed 20 feet for a 1 foot water depth i f space i s to be l e f t for wave runup on the beach. From the preceding discussion i t can be seen that, when designing a wave generator f o r a r e l a t i v e l y short channel, the length of channel assigned to the generator, the wave absorber behind the generator, and the wave f i l t e r , must be held to a minimum i n order to maximize the length of the channel for wave s t a b i l i z a t i o n and t e s t s , between the wave f i l t e r and the beach. In s p i t e of the great amount of research that has been done, i t seems that no p a r t i c u l a r type of generator has yet been generally 36. accepted as o f f e r i n g the optimum so l u t i o n to these wave channel problems. 3.4 S p e c i f i c Problems of Way;e Generator Design 3.4.1 Minimum Wave Period An important decision when designing a wave generator i s the range of wave periods i t must produce. The shorter periods w i l l apply to deep-water waves while the longer periods w i l l apply to shallow-water waves. The wave length associated with a p a r t i c u l a r wave period can be calculated using equations (2) and (5) as follows: ib" , 2nd wave c e l e r i t y C = / . tanh —-— y 2TT* and L = CT / g 2 2-r.d from which L = T tanh —-— For deep-water waves the term tanh ZlA. equals 1, and expressing units i n feet and seconds, equation (11) reduces to L = 5.12T 2 f t . (12) Using a t h e o r e t i c a l approach the minimum wave period i s governed by surface tension. Waves with lengths under 2" are cont r o l l e d by surface tension and should d e f i n i t e l y be avoided. To reduce surface tension e f f e c t s to a n e g l i g i b l e l e v e l (Fig.5) i t 3*7. i s desirable to avoid waves with lengths under about 1 foot. In deep water a 2 inch long wave has a period of about 0.18 sec. while the period of a 1 foot wave i s 0.44 sec. From the p r a c t i c a l viewpoint Galvin (Ref. 8 - Correspondence) recommends a minimum wave period of about 0.5 sec. and preferably 0.75 sec. These periods y i e l d corresponding wave lengths i n deep water of 15.4 i n . and 34.5 i n . The use of Galvin's recommendations, instead of the t h e o r e t i c a l minimum wave period, i s supported by two p r a c t i c a l considerations. F i r s t , f o r wave lengths of 2, 12, 15.4 and 34.5 inches the t h e o r e t i c a l , maximum, deep-water wave heights are 0.29, 1.7, 2.2 and 4.9 inches r e s p e c t i v e l y (eqn.9). Due to wave generator imperfections these maximum heights w i l l not be c l o s e l y approached without using a converging channel. In model work, measurement errors assume increasing s i g n i f i c a n c e with decreasing wave s i z e . Therefore, due to t h e i r l i m i t e d height, the generation of very short waves with periods le s s than 0.5 sec. i s not normally j u s t i f i e d . The second point supporting Galvin's recommendations concerns the high o s c i l l a t i o n rate at which wave generator paddles must operate to produce these short waves. Periods of 0.18, 0.44, 0.50 and 0.75 sec. require o s c i l l a t i o n rates (crank speeds) of 38. 313, 136, 120 and 80 R.P.M. respectively. As shown later in this thesis, mechanical and water inertia loads rise rapidly beyond roughly 100 R.P.M. Hence short wave lengths require wave generat-ors having considerably increased structural strength and motive power, the cost of which is not normally just i f i e d by usage due to the very limited wave height and extreme departure in wave size from prototype conditions. 3.4.2 Maximum Wave Period There are a number of opinions as to the maximum wave period which should be designed into a generator for a laboratory flume of given length. Some researchers suggest that distances of 3 to 5 wave lengths should be allowed between generator and model, to insure stable shape and uniformity of the waves, but have not stated the type of generators used. One commercial hydraulic laboratory visited, which uses a ri g i d flap generator with single articulation at the channel bed to generate deep-water, transition and shallow-water waves, likes to have at least 10 to 11 wave lengths between generator and model. Galvin (Ref. 8 - Correspondence) suggests a maximum wave length of no more than half the distance between generator and s t i l l water line on the beach. Theory developed by Havelock (1929) and util i z e d by Biesel (1951) Ref 1 and 2 - Wave Generator Theory) states that i f a plate oscillates sinusoidally, then a length along the channel equal to three times the water depth should suffice to provide natural compensation for the defective wave form. This theory would imply 39. no l i m i t to the maximum design wave length. Unfortunately, f i l t e r s required to smooth out the wave form, become incr e a s i n g l y long with increased wave length, f o r the same eff e c t i v e n e s s . Also, t h i s theory i s a f i r s t order approximation and i s s t i l l unproven. Therefore, designing to Calvin's l i m i t s would appear to be a reasonable compromise. The period T for the longest wave length can be computed using the shallow-water form of equation (11) which gives T = -L (13) /gd~ The wave length L w i l l be f i x e d by the distance from the wave generator to the beach s t i l l - w a t e r l i n e , assuming Galvin's method. From equation (13) i t i s seen that, lowering the water depth (d) w i l l increase the period, n e c e s s i t a t i n g the choice of a minimum design value f o r depth. I t appears that experiments are r a r e l y done i n water depths of less than 6n, consequently t h i s value seems reasonable i n c a l c u l a t i n g the maximum period. Galvin f e e l s that, i n general, a maximum period of at le a s t 3 sees, i s desirable i n a wave channel, i f s u f f i c i e n t wave v a r i e t y i s to be achieved. 3.4.3 Design Water Depth Maximum design depth i s an important factor i n determining the s i z e of waves generated - deep water waves excepted, since t h e i r l i m i t i n g height i s dependent only on wave length. For a shallow-water wave of maximum design height (H), the required 40. t h e o r e t i c a l depth (d) can be calculated from equation (10). which rearranged gives d = 1.33 H. However, from p r a c t i c a l observations Galvin suggests using a design water depth of 1.5 to 2 times the maximum desired wave height, because the waves tend to become unstable and break when the depth to height r a t i o i s l e s s . Used i n the converse sense t h i s r u l e of thumb is.handy when estimating shallow-water wave heights to be expected from a wave generator. When deciding the design water depth and maximum wave height some channel freeboard must be allowed to control splash. 3.4.4 I n e r t i a Generators of the movable-wall and plunger types produce waves using members which o s c i l l a t e . Because of the e f f e c t s of i n e r t i a on the assigned motion the weights of the moving parts must be minimized, commensurate with adequate r i g i d i t y , n e c e s s i t a t i n g an adequate knowledge of the forces involved when designing the moving parts. Snyder, Wiegel and Bermel (Ref. 9 - Wave Generators) give a method for estimating the forces acting on movable-wall generators from wave energy u t i l i z i n g experimental r e s u l t s of the U.S. Beach Erosion Board (1949). Force c a l c u l a t i o n s can also bes made using the theories of B i e s e l (Ref. 1 - Wave Generator Theory). Forces for plunger-type generators, using plunger bodies of various cross sections, can be deduced from the theories of Schuler (Ref 9 - Wave Generator Theory), a resume of which i s a v a i l a b l e i n 41. "Laboratory Surface Wave Equipment" (Ref. 10 - Wave Generator Theory). 3.4.5 Wave Generator Motion Departures from the motion assigned to a wave generator create wave problems which can be divided into two grpups - f i r s t , i r r e g u l a r i t i e s i n the wave form and second, changes i n the wave length. The f i r s t problem, i r r e g u l a r i t i e s i n the wave form, i s due to momentary departures from the assigned motion, caused by load f l u c t u a t i o n s . The main load on the wave generator paddle breaks down in t o wave forces and i n e r t i a forces (water and mechanical), both of which fluctuate and are 90° out of phase. E l e c t r i c motors are the usual source of power i n laboratory wave generation and are subject to small speed changes r e s u l t i n g from these load changes. Another source of f l u c t u a t i n g load i s f r i c t i o n , due to seals sometimes employed for reducing leakages between the plates of movable-wall type generators and the channel bed and side w a l l s . These seals rub against walls that may be rough or s l i g h t l y out of alignment, so that the f r i c t i o n forces can be both appreciable and i r r e g u l a r . Most designers omit seals and are content to reduce clearances to a value s u f f i c i e n t to avoid large f r i c t i o n forces. These fl u c t u a t i o n s of load a f f e c t the choice of power source and associated speed-changing mechanisms. Hydraulic drives and f l u i d couplings have been t r i e d but some of these systems were 42. found unsatisfactory due to v e l o c i t y changes1 as the load came on and o f f . Backlash must be considered when u t i l i z i n g gear-type speed reducers. When using an e l e c t r i c motor drive the output R.P.M. w i l l drop a c e r t a i n amount as the load goes.from no load to f u l l load. By using a more powerful motor than required, the maximum wave-generator load becomes a smaller percentage of motor f u l l - l o a d c a p a b i l i t y , with a corresponding reduction i n range bf speed f l u c t u a t i o n . For t h i s reason many wave generators are overpowered i n s p i t e of the increased c a p i t a l costs and s l i g h t loss of motor e f f i c i e n c y . The second problem, changes i n wave length, r e s u l t s from v a r i a t i o n s i n wave generator o s c i l l a t i o n period. This problem i s caused by long-period f l u c t u a t i o n s i n l i n e voltage which a f f e c t motor operating R.P.M. Some laboratories have used e l e c t r i c motor-generator sets to overcome t h i s problem. The Dauphin Hydraulic Laboratory (Neyrpic), Grenoble, France (Ref. 1 - Wave Generator Theory) r e l i e s on the r e l a t i v e l y constant power-line frequencies, c e r t a i n l y more constant than voltage, by using synchronous motors of excessive power. In t h i s way, motor slippage i s reduced and the motor revolves at nearly i t s nominal speed. The seriousness of t h i s problem depends on the rate of voltage f l u c t u a t i o n , the magnitude of the f l u c t u a t i o n and the type of test being done. Many wave 43. generators are powered by e l e c t r i c motors connected d i r e c t l y , to the normal mains without any s p e c i a l voltage regulating devices. When considering the problem of i r r e g u l a r motion, the use of a flywheel would appear to be a s o l u t i o n and a number of wave generators have been so designed. Unfortunately, the users have found that flywheels are slow to accelerate and decelerate which re s u l t s i n the channel being f i l l e d with a confusion of waves of varying lengths whenever the generator i s st a r t e d or stopped, or whenever a speed change i s made. Not only must the model be pro-tected from this confused sea, whose e f f e c t s are much greater than the wave e f f e c t s at uniform regime, but also considerable time i s l o s t waiting f o r s t a b i l i t y to be achieved. T h e o r e t i c a l studies show, that as a f i r s t approximation, wave-generator motion should be s i n u s o i d a l (simple harmonic motion). Ex-periments i n d i c a t e that i t would be preferable, to assign a more complex pattern to the motion, but as yet this pattern i s not know exactly. The generally used s i n u s o i d a l motion i s usually obtained by actuating the wave generator paddle with a long connecting rod connected to a r o t a t i n g crank. 3.4.6 Wave Period Control Wave period i s the same as the period of o s c i l l a t i o n of the wave generator which i n turn i s determined by the output R.P.M. of the drive system. From the explanations given i n sections 3.4.1 and 3.4.2 on how to determine minimum and maximum wave periods, i t i s 44. obvious that c o n t r o l of the wave period also gives c o n t r o l of the wave length. There are a number of ways by which the output R.P.M. of the drive system can be var i e d . F i r s t i s the use of a manually s h i f t e d gearbox, which i s not recommended due to the rather l i m i t e d speed s e l e c t i o n . Next i s the use of a va r i a b l e drive pulley system. These are of two types, b e l t and chain. The belt-types employing p o s i t i v e l y c o n t r o l l e d sheaves to give speed v a r i a t i o n s may prove s a t i s f a c t o r y , but any units employing spring c o n t r o l l e d sheaves, f o r reasons ex-plained l a t e r i n Section 5.2 of th i s t h e s i s , should be avoided. Chain types using mechanically co n t r o l l e d sheaves give p o s i t i v e speed c o n t r o l and are popular. Direct-current motors give c o n t r o l over a wide speed range and are used i n conjunction with a gear box. Immediate speed changes are obtained at the turn of.a rheostat. Synchronous motors require reduction gears and a speed-control device which permits adjustment of the frequency to any desired value between two d e f i n i t e l i m i t s . Hydraulic devices and other means of v a r i a b l e speed control are commercially a v a i l a b l e but when s e l e c t i n g one i t must be p o s i t i v e i n i t s operation and reasonably free of backlast i n order to handle the f l u c t u a t i n g load of a wave generator, without contribut-ing i r r e g u l a r i t i e s to the motion. 45. I t i s desirable to be able to e f f e c t a change i n period quickly while the wave generator i s operating. For example, changing the wave generator o s c i l l a t i o n rate d i r e c t l y from 60 to 80 R.P.M., without stopping the generator and r e s t a r t i n g i t , reduces the i n -te n s i t y and duration of the confused sea produced by the change. Also i t permits f i n e period adjustments to obtain a desired wave length. 3.4.7 Wave Height Control. Wave height i s c o n t r o l l e d by the amplitude of o s c i l l a t i o n of the wave generator. The o s c i l l a t i o n amplitude of machines driven by a r o t a t i n g crank and connecting rod can be e a s i l y c o n t r o l l e d by adjustment of the crank throw ( F i g . 11). Some wave generators are designed with a d d i t i o n a l l i n k s between the drive system and the fla p and the fl a p amplitude i s con t r o l l e d by changing the length of the l i n k ( F i g . 13). These two examples of amplitude co n t r o l can be designed for e i t h e r manual adjustment with the generator stopped,.or automatic adjustment when eit h e r running or stopped. Schematics for the design of automatic adjustment controls are shown i n F i g . 12 and F i g . 14. A study of F i g . 11 and F i g . 13 show that i n the case of manual c o n t r o l , the adjustable crank i s best, the use of an adjustable l i n k making the drive system unnecessarily complex. However, i n the case of automatic controls, F i g . 12 and F i g . 14, the use of a l i n k adjustment can be j u s t i f i e d by the simpler automatic mechanism involved. 46. With reference to Fig. 12, the automatic control of crank throw adjustment functions as follows. The fixed differential causes outer shaft #2 to rotate in the opposite direction to the outer main drive shaft #1. The second differential causes the inner shaft to rotate in the same direction and at the same speed as the outer main shaft #1. By using a reversible control motor to rotate the casing of the moveable differential the inner shaft can be caused to rotate either faster or slower than the outer drive shaft #1. This speed, difference causes the threaded guide rod to be ; rotated in one direction when the inner shaft i s rotating faster than the main outer shaft #1, and in the other direction when slower, pro-ducing a shift in crank pin position. Either limit switches or a s l i p -ping clutch must be incorporated in the system to prevent damage by over adjustment. The act of changing the wave-generator oscillation amplitude while the wave generator is operating, can produce irregularities in the wave form detrimental to the model; requiring that the model be protected. Since the rate at which these automatic amplitude changers work i s relatively slow, especially compared to the time required for a change in motor speed, i t is usual to stop the wave generator for amplitude changes. Once the wave generator is stopped, the automatic system offers only a convenience over the manual adjustment. Galvin mentions that in his experience the automatic amplitude control i s rarely used with the wave generator running. 47 Fig. 11 Principle of Wave Height Control by Crank Throw Adjustment Fig. 12 Mechanism for Automatic Crank Throw Adjustment. 48 Amplitude of flap oscillation controlled by adjusting length of link throw Fig. 13 P r i n c i p a l of Wave Height Control by Link Throw Adjustment. Connecting rod from drive wheel Overrun limit switch Rotation of threaded rod moves axle block along guide slot / — ^Connecting rod to flap Overrun limit switch Electric gear motor Fig. 14 Mechanism for Automatic Adjustment of Link Throw. 49. In summary, automatic amplitude controls complicate the wave-generator mechanism and are expensive, p a r t i c u l a r l y i n the case of the adjustable crank type. A high degree of accuracy, i s necessary i n t h e i r construction since some parts must be able to move smoothly and yet not have enough free play to produce i r r e g u l a r i t i e s i n the f l a p motion. Therefore, the p r o v i s i o n of automatic amplitude con-t r o l s over manual adjustment can not normally be j u s t i f i e d unless the wave generator i s to be used to s p e c i f i c a l l y generate i r r e g u l a r waves for c e r t a i n studies in v o l v i n g surf, seiches and ship r o l l i n g . 3.4.8 Anticipated Performance Experience indicates that any wave generator can be ex-pected to produce f i n e r waves of small steepness than of large steepness and better short waves than long waves for the same steepness. In f a c t , generated waves of large steepness are d e f i n i t e l y not of uniform q u a l i t y , even i f observed a f t e r they have traversed a great length. B i e s e l ( R e f . l - Wave Generator Theory), o f f e r s a b r i e f p l a u s i b l e i n t e r p r e t a t i o n of these findings which would suggest that the best wave generator i s always the piston-type machine. I t i s i n t e r e s t i n g to note that Galvin (Ref. 8 - Corres-pondence) states that, the Coastal Engineering Research Centre, Washington, D.C, has found the piston-type generator to be the most s a t i s f a c t o r y . (Their channels are long and the other types of generators used were not mentioned). In any case, there i s no t h e o r e t i c a l analysis yet developed which gives a rigorous explanat-ion for these performance' r e s u l t s . 50. 3.5 Choice of Wave Generator for a Short Channel The key problem when generating waves i n a channel of short length i s the lack of spare channel length to enable i r r e g u l a r i t i e s i n water p a r t i c l e motion and wave form to correct themselves. I t now becomes important to have water p a r t i c l e motion f o r a p a r t i c u l a r wave form established r i g h t at the wave generator paddle. This kinematic requirement appears to be t h e o r e t i c a l l y best s a t i s f i e d by a wave generator of the r i g i d paddle, double a r t i c u l a t i o n type. Examination of the schematics of r i g i d paddle generators with double a r t i c u l a t i o n shown i n F i g . 7, suggested consideration of the pendulum type due to i t s s i m p l i c i t y . This type was developed by Ransford (Ref. 1 - Wave Generators) and i s shown i n F i g . 15. A modification of the geometry of Ransford's generator was made by Coyer (Ref. 3 - Wave Generators) and the arrangement i s shown i n F i g . 16. I n e r t i a l e f f e c t s due to water and wave generator mass are a maximum at the extremes of paddle o s c i l l a t i o n . In Ransford's de-sign, the weight of the paddle parts works against i n e r t i a l e f f e c t s and helps the motion, whereas i n Coyer's arrangement, the weight adds to the i n e r t i a l e f f e c t s . On the other hand, Coyer's arrange-ment off e r s advantages of compactness and s l i g h t l y less water leak-age under the paddle (due to the geometry of the paddle motion). Experimental data presented by Coyer shows well-established deep and shallow-water wave p r o f i l e s and correct p a r t i c l e motions e x i s t i n g at a test point only 12' from the generator paddle when the motion of 51 Setting for piston motion for shallow-water waves'— Connecting rod A d j u s t m e n t a r c Maximum setting for generating deep-water waves \ \ / / — — \ \ V -j j — ,, ! 777 Rigid paddle j Paddle motion not symmetrical Fig. 15 . Principle of Ransford's Pendulum-Type Wave Generator. Extreme shallow-water wave setting ( piston motion ) Extreme deep-water wave setting ( hinged-flap motion ) 777777777 Fig. 16 Coyer's Modified Version of Ransford's Pendulum-Type Wave Generator, Illustrating the OPERATING PRINCIPLE. 52 . the paddle i s adjusted to s u i t the water p a r t i c l e motion of the generated wave. The v a r i e t y of wave generator designs and the d i v e r s i t y of opinions, in d i c a t e that wave generator design has not yet been optimized. The l i t e r a t u r e leads one to conclude that most e x i s t i n g wave generators are i n the movable-wall category. I t i s i n t e r e s t -ing to note that from the p r a c t i c a l aspect of actual performance, the opinions of numerous researchers appear to be favouring p i s t o n -type wave generators for a l l types of waves. These researchers were using long channels but i t brings up the point of ju s t where i s the d i v i d i n g l i n e between "long" and "short" channels? Since the double a r t i c u l a t i o n arrangements permit the wave generator paddle to function as a piston, use of such a wave generator would permit a future research opportunity, v i z . , to compare the per-formance of a piston-type wave generator, producing a l l wave types i n a "short" length channel, with r e s u l t s from a supposedly kinematically superior double a r t i c u l a t i o n type u t i l i z i n g the same channel, drive mechanism and recording equipment. For reasons stated, i t was decided to design f o r the large flume -aawave generator of the r i g i d paddle, double a r t i c u l a t i o n type using the geometrical arrangement of Coyer ( F i g . 16). 53. 4. WAVE GENERATOR THEORY - a means of determining design forces. 4.1 The o r e t i c a l Analysis of a Wave Generator When designing a wave generator i t i s necessary to know the water forces involved i n i t s operation, i n order to cal c u l a t e power requirements and to give the mechanism adequate strength and r i g i d i t y without unnecessarily increasing the i n e r t i a of the o s c i l l a t i n g members. Also, some means i s required f o r determining the amplitude of paddle motion required to produce a wave of a cer t a i n height. This information can be calculated f o r r i g i d paddle wave generators with double a r t i c u l a t i o n , using the theory of Biesel(Ref. 1 - Wave Generator Theory), which i s the most "usuable" of a v a i l a b l e mathematical theories. Biesel's theory (1951) i s based on e a r l i e r work by Havelock (1929) (Ref. 2 - Wave Generator Theory) . The wave generators considered by B i e s e l are e s e n t i a l l y such that t h e i r action i s equivalent to that of a membrane or a f l e x i b l e . blade completely obstructing the channel and o s c i l l a t i n g with a si n u s o i d a l (simple harmonic) motion about a mean v e r t i c a l p o s i t i o n . The theory i s based upon the assumptions that: the water motion i s i r r o t a t i o n a l ; i . e . the water i s f r i c t i o n l e s s and incom-p r e s s i b l e ( i d e a l f l u i d ) ; the equations of motion s a t i s f y a l l the hydrodynamic laws to the f i r s t order of approximation and; motion 54. is two-dimensional occuring parallel to the side walls of the channel. The formulae derived are rigorously valid only for i n ^ f i n i t e l y small wave heights and in cases where viscosity and turbulence may be neglected. Water leakage around the flap (paddle), which occurs in an actual machine, is not considered. Summarizing this theory briefly (Ref. 10 - Wave Generator Theory) boundary conditions are: <ai)) - 0 . for y = 0, x ^ O d y where the origin of the coordinates i s placed on the channel bed with the ox-axis extending along the bed in the direction of wave propagation (Fig. 17) and the oy-axis designating the mean position of the oscillating member; 2 J 2 a y b) — + g — = o 3 t for the surface condition in a first-order theory! and C ) (y) cos kt 3 x J for the boundary condition at the generator flap where k = — and the motion of the generator i s x = §(y) sin kt. Biesel's solution for this problem is velocity potential i> T - — c cosh my sin (kt-mx) - E , c — cos m y . e m n X . cos kt (14) m • -•• n=l n m nJ n ni •u o ° § (oc)cosh o <* (d <*) / 1 C . where c = c = 2m — r — r 3 r X T 1~ (15) o o sxnh m d. cosh m d + m d 55. and G = 2m d § (<*) cos m K (d<*) b',: n (16) n n s i n m d. cos m d + m d n n n In these expressions, mQ i s the p o s i t i v e s o l u t i o n of the equation 2 k = ^g. tanh ^d and m^  represents the p o s i t i v e solutions of the equation k« = - c c g . tan <*d (18) Using the v e l o c i t y p o t e n t i a l cj>, the displacements of the p a r t i c l e s whose mean p o s i t i o n i s x and y are found to be oo —H1 X X=c cosh my. s i n (kt-mx) +. E, c .cos m y.e n .sin kt (19) J 'n=l n nJ oo —III X and Y=c sinh my.cos (kt-mx) +.. E c .sin m y.e n .sin kt (20) n=l n n The magnitude of the semi-height (a) of the generated wave can be deduced as being a = c sinh md. (21) o Neglecting terms of the second order i t can be proven that Substituting for $ i t s value from equation (14) y i e l d s the general equation for pressure i n front of the wave generator f l a p which i s P = -pg <d-y> + "PS a e°sh md ' C O S ( k t " m x ) + 0 ^ 7 gc tan m. d. cos m y.e m n X . s i n kt (22) n=l ° n -n n^ 56. Mean position of wave generating flap Sti l l water level a m L d §<y) x, y p * n p i P g k T t n e X, Y e n — height oi the. geneAoted wave. L m F. -nw.ve length. mean uoateA depth mexut>uK.eA iiom bottom. denote* the. maximum (Lupla.cejne.nt oi the. va/Uou* point* oi the. wave geneAaton. ilap at a iunction oi theiA depth y. coordinate*. total p/ie*4>un.e. hydno&tatic pn.e*tuAt. wave pfied&uJie. ineAtia pAe*&un.e oi wateA. wateA density, acceleration oi gtuwity. 2L T period oi ilap oscillation, time, {£ea>.) 1, 2, 3 . . . maximum displacement oi generator, ilap iiom mean position, horizontal and vertical particle cLcsptacementi, irom mean position, a numbeA. amplitude oi initial water, disturbance. wave jJoA.ce per. unit width ion. water, on both sides oi ilap. water ineAtia iorce peA unit width ion. water, on both tide* oi ilap. F i g . 17 D e f i n i t i o n of Coordinate System and Terms used i n Biesel's Theory. Setting x=0 i n formula (22) y i e l d s the expression for the pressure acting on the generator f l a p . This pressure breaks down int o three terms. a) The f i r s t term i s hydrostatic pressure P h = p g(d-y) (23) b) The second term i s wave pressure cosh my , , 0 / * p = p ga r — j r cos kt (24) r n r cosh md and represents pressure required to form the actual wave. This pressure i s i n phase with the o s c i l l a t i o n speed of the wave generator f l a p . Energy expended to overcome t h i s pressure i s recovered i n the wave. c) The t h i r d term i s i n e r t i a pressure p_^  = pg c^ tan m^d. cos n^y. s i n kt (25) I t i s the pressure required to overcome water i n e r t i a and i s i n phase quadrature with the wave generator f l a p speed. This com-ponent acts as an augmentation to the i n e r t i a of the moving parts of the generator. When a generator f l a p emits waves i n both d i r e c t i o n s the hydrostatic pressures e x i s t i n g on e i t h e r side of the f l a p balance each other out while the wave pressures and i n e r t i a pressures on each side are resp e c t i v e l y additive doubling t h e i r i n d i v i d u a l e f f e c t . 58. When using Biesels equations values of the c o e f f i c i e n t m^, or preferably of the dimensionless product m^d, are most e a s i l y determined graphically using F i g . 18. Analysis of equations (19) and (20), which give water p a r t i c l e displacements X and Y, shows that the motion imparted by the wave generator to the water i s composed of an ordinary wave (such as would be produced by an i d e a l wave generator) and an i n i t i a l disturbance, or t r a n s i t o r y i n space, which decreases ex-ponen'tially as e n .with increasing distance x from the generator f l a p . B i e s e l shows that,at distance x=d from the generator f l a p mean position,the maximum amplitude of t h i s disturbance i s 21 per cent of the i n i t i a l amplitude, while at x=2d the amplitude i s only 4.3 per cent and at x=3d i t i s reduced to 1.0 per cent. B i e s e l and Havelock both concluded that i f the motion ef a wave generator f l a p (paddle) reasonably approximates the wave form, water p a r t i c l e motion, then a length equal to three times the depth of the channel w i l l s u f f i c e to provide natural compensation f o r defects i n the wave form. The v e r t i c a l amplitude of the transient o s c i l l a t i o n at the generator f l a p (x=0) from equation (20) i s n = c s i n m d (26) n n n With regard to the height of the generated wave, B i e s e l suggests an e f f i c i e n c y c o e f f i c i e n t of about 70 per cent for r e -l a t i v e l y large laboratory wave generators. E f f i c i e n c y w i l l i n 59. fact'vary widely depending on the r a t i o of wave length.to water depth, the amount of water leakage past the wave generator f l a p and fla p proportions. 4,2 Biesels Theory Applied to a Rigid Paddle, Double A r t i c u l a t i o n Wave  Generator 4.2.1 General o A r i g i d paddle wave generator with double a r t i c u l a t i o n can be adjusted to make the paddle operatevwith a piston motion, with a hinged-flap motion, or with an intermediate motion (Fig. 20). The a p p l i c a t i o n of Biesels formulae f o r force c a l c u l a t i o n s w i l l now be examined for each one of these operating modes. 4.2.2 Piston Motion For piston motion the fl a p displacement function §(y) has the very simple form §(y) = e where e i s a constant (Fig. 20a). From equation (21) the semi-height of the generated wave i s jd e cosh my (dy) a = c sinh md = 2m^ sinh md sinh md. cosh md + md 2 = 2 sinh md e sinh md. cosh md + md = Ke (27) Fig. 19. Variation of Coefficient K (Piston Motion) and K* (Hinged L Flap Motion) for Various Values of -j- (Bi6sel) . ON 62. 2 2 sinh md where K = sinh md. cosh md + md Figure 19 shows the v a r i a t i o n s of c o e f f i c i e n t K as a function of the r a t i o L_ . d Water pressures acting on one side of the f l a p can be computed. The hydrostatic pressure P , given by equation (23) remains as P = pg (d-y). (28) ri The wave pressure from equation (24) becomes T, cosh my. cos kt P n = p g K e - cosh md (29) The water i n e r t i a pressure from equation (25) i s P:". = p>e E C . tan m d. cos m y. s i n kt. (30) l H & , n n n n=l where from equation (16) C = ^ e s i n rn^d  s i n m d. cos m d +1 in d n n n When c a l c u l a t i n g i n e r t i a pressure i t i s s u f f i c i e n t to compute only the f i r s t three terms of the s e r i e s . The resultant forces per unit width on the f l a p , with water  on both sides, can be determined from the pressure equations. For water on both sides of the f l a p the hydrostatic pressures balance out and can be neglected. The wave force for water on both sides of the f l a p i s ;d „ F = 2 1 P (dy) = 2 (^2-S- . tanh md) cos kt (32) n o n J m and w i l l be a maximum when cos kt = 1, that i s , when kt = 0. 63. The water i n e r t i a force for water on both sides of the f l a p i s from equation (30) F. = 2 / d' p. (dy) i o x c c = 2pg {( _1 . tan m^d. s i n m^d) + ( _2. tan n^d.sin n^d) m l m2 + ( 3. tan m^d. s i n m-jd) ^ s^-n (.33) where values f o r c^, and c^ can be obtained from equation (31). The i n e r t i a force w i l l be a maximum when s i n kt = 1, that i s , when kt = \ . When using the force equations, values such as 1 can be d m i put into the f o r m — r and values f or m..d read from figure 18. i . a i 4.2.3 Hinged-Flap Motion For hinged f l a p motion (Fig. 20b) the f l a p displacement function § (y) has the form § (y) = y. (34) From equation (21) the semi-height (amplitude), of the generated wave i s J d e a = C sinh md = 2 m o -r. y. cosh my (dy) . , , o d } 3 J . sxnh md sinh md. cosh md + md _ 1-cosh md + md. sinh md . . , = 2 e : : . sinh md md (sinh md. cosh md+ md) = Ke (35) (a) Piston motion Mean paddle position (b) Hmged-flap motion Mean paddle position '////////////////// (c) Intermediate motion Mean paddle position Fig, 20 - Paddle Displacements for the Three Operating Modes o Rigid Paddle Wave Generator with Double A r t i c u l a t i o n 65, where Yi - 2 s :*- n n m d ( x ~ 'cosh md + md. sinh md ) md (sinh md. cosh md + md) Values of K for various values of ^  are given i n F i g . 19. The value of i s given i n equation (16). Substituting f o r § (y) and i n t e g r a t i n g by parts gives _ m d. s i n m d + cos m d - 1 (36)' C = 2 n n r, Y\ Q U • m d ( s i n m d. cos m d + m d) n n n n Force equations can now be determined i f required using the same procedure as for piston motion (Ref. sub-section 4.2.2, equations 32 and 33). 4.2.4 Intermediate Motion For intermediate motion ( F i g . 20c) the f l a p displacement function § (y) has the form § ( y ) = e 1 + ( e 2 - ep J (37) where e^ i s the maximum displacement of the bottom of the f l a p from mean p o s i t i o n and e.^ i s the maximum displacement of the top of the f l a p at s t i l l water l e v e l from mean p o s i t i o n . From equation (21) the semi-height (amplitude) of the generated wave i s jd , n ° { e i + (e-> - e i ) } cosh my (dy) . , , a = C sinh md = 2m 1 2 V d J } sinh md sinh md. cosh md + md 66. 2m ° e r C O s h m y ( d y ) . sinh md sinh md. cosh md + md jd + 2m° ( £2 - el> d * C ° s h m y ( d y ) sinh md sinh md. cosh md 4- md = 2me.. o cosh my (dy) . sinh- md sinh md. cosh md + md jd % + 2m (e 2 - e.) o 2d' ( e m y + e " m y ) . dy sinh md. cosh md 4- md 2en {sinh my}d . , , 1 o . sinh md sinh md. cosh md + md. . sinh md 1 e m y e~ m y d + 2m (e 2 - v ~2~ (my-1) H ^—(-my-1) } . sinh md m_ m sinh md. cosh md + md _ sinh^ md 1" sinh md. cosh md + md (1 - cosh md + md sinh md) + 2(e 0-e,) 2 1 md (sinh md. cosh md + md) Ke. + K (e 2-e 1) (38) where. K is as previously defined in equation (27) and.K is as defined in equation (35). From figure 19 i t can be seen that when the wave generator is operating with- intermediate motion the heights of the generated waves w i l l l i e between the heights achieved using piston motion and f l a p motion, assuming other conditions equal. The value of C for intermediate motion i s obtained by n s u b s t i t u t i n g for § (y) i n equation (16) and i n t e g r a t i n g : rd--C = 2m .: n n-o/ {e^ + ( e 2 _ e P ^ c o s m n y « ( d y) s i n m d. cos m d + m d n n n = 2 m n o j d e x cos rn^y (dy) +- 0J d ( ^ 2 ~ e ] ? ^ m c o s m n v ( dy) si n m d. cos m d + m d n n n s i n m d . 0 , ,m d.sin m d + cos m d-1 = 2e n + 2 ( e 2 - e 1 ) _ n n n ( 3 9 ) s i n m,n.cos m d+m d m d(sin m d.cos m d+m d)* d n n n n n n Force equations can now be determined using the same pro-cedure as for piston motion. (Ref. sub-section 4.2.2, equations 32 and 33). 4.2.5 Discussion. A r i g i d paddle, double a r t i c u l a t i o n , wave generator, of the type to be designed, i s normally adjusted to operate with an intermediate paddle motion (Fig.20c) suited to the water particle, motions of the generated wave. At one. extreme of adjustment this motion becomes a piston motion suited to generating shallow-water waves and at the other extreme, a hinged-flap motion suited to generating deep-water waves. 68. B i e s e l (Ref.1 - Wave Generator Theory) presents sample cal c u l a t i o n s which show that the tra n s i t o r y wave deformation (Eqn.26) occuring i n front of the paddle i s minimized when the paddle motion c l o s e l y approximates the water p a r t i c l e motion of the generated wave. Therefore, i n a flume of short length, i t would appear p a r t i c u l a r l y important that the wave generator paddle motion always be c o r r e c t l y adjusted. With t h i s type of wave generator i t i s s t i l l p ossible f o r the operator to generate a l l types of waves using e i t h e r a piston motion or a hinged-flap motion. For a deep-water wave having L = 0.5, B i e s e l shows that for piston motion the r a t i o of d F water i n e r t i a to wave force i s __L = _35_, whereas f or hinged-flap F 1 F n motion £i = 13.6 . I t becomes evident then, that more power i s F 1 n required to drive the wave generator when the paddle motion i s not suited to the water p a r t i c l e motion of the generated wave. The freedom which the wave generator operator has to ad-ju s t the paddle motion, brings up the question as to which operat-ing mode (Fig.20) w i l l produce the greatest loads on the paddle. From. F i g . 19, i t i s evident that the wave amplitude c o e f f i c i e n t K i s always greatest for piston motion, r e s u l t i n g i n the wave pressure (P ) acting on. the paddle-, being greatest for th i s case (Eqn. 24). Wave i n e r t i a pressure (P Eqn.25) increases with decreasing wave period (length). I f the water depth i s s u f f i c i e n t 69. to make the short-length wave a deep-water wave,'and i f the motion of the paddle i s not suited to the water p a r t i c l e motion of the generated wave, then the water i n e r t i a pressure increases even more, and i s greatest for piston motion. Therefore the wave forces acting on the paddle w i l l reach a maximum when piston motion i s used to generate short period waves. When using Bi e s e l ' s theory to ca l c u l a t e forces, i t should be remembered that t h i s theory i s only of the f i r s t order of approximation and has not yet been f u l l y proven. 4.3 Design Graphs The behaviour of wave forces ( n) and water i n e r t i a forces F ( i ) i s not r e a d i l y apparent from Biesel's equations, so the author calculated values of F and F. per unit f l a p width f o r n 1 various values of L_ and plo t t e d them i n graph form. Because of the d lengthy c a l c u l a t i o n s involved, F n and F^ curves were prepared only for the case of piston motion. This case i s of prime i n t e r e s t as i t y i e l d s the maximum forces to which a r i g i d paddle generator with double a r t i c u l a t i o n could be subjected. For ease of design, the values of F and F. were calculated i n terms of e.d where e n i i s the maximum displacement of the generator paddle i n feet from i t s mean p o s i t i o n ( F ig. 2 0 a ) and d i s the water depth i n feet measured from the s t i l l - w a t e r l e v e l . 70. In preparing the graphs, the wave-amplitude c o e f f i c i e n t K for piston motion was calculated for various values of L_ using d equation (27), the r e s u l t s being p l o t t e d i n fig u r e 21. A sample c a l c u l a t i o n f o r K for L_ = 20 i s as follows: d K = 2 2 sinh md sinh md. cosh md + md 2 s i n h 2 2TT J-LI sinh 2TT J - " . cosh 2IT + 2TT J-LI JL_I J_J 1 . u2 2TT 2 sinh . , 2TT 2TT , 2TT sinn 20_.. cosh 2 0 + 2 0 = 2 sinh . 314 sinh .314 cosh .314 + .314 2 (.319) 2 .319 X 1.050 + .314 .315 The maximum wave force F^ per foot width of paddle for various values of L_ was calculated for piston motion and water on d both sides of the paddle using equation (32) and p l o t t e d i n figur e 22. A sample c a l c u l a t i o n f o r F f o r L = 4 i s as follows: n d 71. F = 2(^££ tanh md) n m 9 /1.44 ed 62.4 X 32.2 tanh 2TT d . K 2^d L ; 1.44 X 62.4 X 32.2 ed ^ . 2TT , = 2( r tanh —r ) ZTT 4 ?r 1.44 X 62.4 X 32.2 X .917 ed) 1.57 = 105.2 ed l b s . / f t . flap width The maximum water i n e r t i a l force F_^  per foot width of paddle for various values of L_ was calculated for piston motion and d water on both sides of the paddle using equation (33) and plotted in figure 23. A sample calculation for F. for L_ = 4 is as follows: 1 d F. = 2pg {( j . tan m..d. sin m,d) + ( T tan m„d.sin m.d) l m^ u 1 1 m^ Q 2. L c^d + ( v • tan m„d. sin m.d)} m^ d 3 3 from,Fig. 27 m^ d = . 85TT = 2.67 (radians) m2d = 1.93* = 6.06 m3d = 2.96TT = 9.29 72. from equation (32) c 1 = 2e s i n rn^d 1 s i n m^d. cos m^d + m^d = 2e (.455)  (.455) (-.891) + 2.67 ,402 e s i m i l a r l y c2 = ~ and c 3 = .0296e therefore F. = 2X62.4{ ( - | ° | ^ X X . 4 5 5 ) + ( ^ | l e d X=JM x ( _ > 2 2 2 ) ) } , ,.0296ed .1356 , + ( 9.29 X r ^ 9 l X - 1 3 5 6 > > = 2 X 62. !4 (-.0349ed -.0006 ed -.0001 ed) = -4.44 ed. The negative sign applies to the d i r e c t i o n of action of force F. r e l a t i v e to force F as i l l u s t r a t e d i n F i g . 24. x n In studying F i g . 24 i t becomes evident that there i s con-siderable f l u c t u a t i o n i n the load and hence i n the required d r i v i n g torque during one paddle o s c i l l a t i o n . As the drive wheel rotates through quadrant 1 and 111 the load i s r e l a t i v e l y l i g h t . In quadrant 11 and IV the load w i l l be the same and reach a maximum 73. value although in quadrant 11 the connecting rod is in tension while in quadrant IV the connecting rod is in compression. Use of the graphs presented in Fig. 21, 22 and 23 greatly simplifies the design procedure when calculating wave heights and water forces, for a wave generator having a piston motion. lOOed Maximum wave f o r m i n g f o r c e 1 , a c t i n g on a wave g e n e r a t o r p a d d l e w i t h w a t e r on b o t h s i d e s and moving w i t h a p i s t o n motion i s : L p _ p 9K e tanh md where m = n m L d" 2 TT tanh L y ^ ] ed 80ed Q < i— o 60ed :40ed CO 20ed 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 i d F i g . 22 - Maximum V a l u e s o f Wave Force F p e r Foot o f Paddle Width f o r V a r i o u s V a l u e s o f L/d . 76 -1600ed -1400ed -1200ed -lOGOed o o < •-800ed -600ed -400ed -200ed 0: 0i ;Maximum w a t e r i n e r t i a l f o r c e ( a c t i n g o n a wave g e n e r a t o r •paddle moving w i t h a p i s t o n ; mot i o n w i t h w a t e r o n bo t h s i d e s , o f t h e p a d d l e i s : . . . F f = 2 Pg [ ( c i ' ^ j . t a n m.d.sin m xd) d ( c 2 j f j j j . tan m 2 d.s in m2d) +j(c3 jjj^Tptan m 3 d.s in m 3d)] l b . / f t . of paddle wi dth. 0 n s in m d Ze n s in m d.cos m d + rrid n n n values read from F i g . 27 s t i l l water depth in feet i tota l amplitude of paddle os c i11at i on i n fee t . 2 "L~ d 4 P i g . 23 - Maximum V a l u e s o f Water I n e r t i a F_ p e r Foot o f Paddle Width f o r V a r i o u s V a l u e s o f L/d . 77 1V: 1 si n - sin+ COS + i C0S + iii n s i n - sin+ cos- cos-| Sinusoidal, I Paddle i Motion Oc 270° | Rigid Paddle 180° Low Load Using Graph lvalues Fn (pos.) Fi (neg.) Fn=(+) cos kt -(+) ; c F H - ) sin kt F n =(- )— Paddle Decelerating High Load Time t-i 2 .4 1 1 ^—t F n - U -~-r~ i .— i h F i l Time t= % 1 2 i - 1 i High Load T ime t - % 1 Fn=(+) cos kt =(-) F i=( - )_s in kt '•(-) Paddle Accelerating "FrH+) cos kt =(-) F i=( - )_s in kt Paddle Decelerating Fn=(+)_ cos kt 3+T Fi-(-) sin kt =(+) Paddle Accelerating Fig. 24 - Direction of Forces Acting on Wave Generator Paddle During 1 Cycle o f Oscillation. 78. 5. RE-DESIGN OF A WAVE GENERATOR - i s o l a t i n g and r e c t i f y i n g the operating problems of the wave generator i n the "small" flume. 5.1 Background The wave generator i n s t a l l e d i n the "small", 21'-3 5/8" long, 8 3/8" wide and 10 5/8" deep, t i l t i n g , s t e e l and glass flume (F i g . 2) i n the Un i v e r s i t y of B r i t i s h Columbia, C i v i l Engineering Department's Hydraulics Laboratory, i s of the r i g i d paddle type with double a r t i c u l a t i o n . The geometry of the paddle actuating mechanism was that developed by Coyer (Ref. 3 - Wave Generators). F i g . 25 shows the wave generator paddle mechanism adjusted to give a piston motion. F i g 27 shows the paddle mechanism adjusted to give a hinged-flap motion. F u l l d e t a i l s of t h i s generator are given i n a report by Pretious (Ref. 4 - Wave Generators). The problem with t h i s wave generator was that the drive system produced a very uneven paddle motion, r e s u l t i n g i n i r r e g u l a r -i t i e s i n the wave form. Departures from the desired s i n u s o i d a l motion were extensive and v i s u a l l y evident. Correction of the operating problems of t h i s wave generator were undertaken before the design of the proposed new wave generator for the large flume, so that use could be made of the experience gained. 80. 5.2 I s o l a t i n g the Operating Problems The predominant i r r e g u l a r i t i e s i n motion consisted of two perceptible points of slowing down and two separate points of jerkiness i n each r o t a t i o n of the crank d i s c . The motion problem was previously assumed to be backlash only and a f r i c t i o n brake had been i n s t a l l e d which.rubbed on the crank d i s c circumference. Use of the brake reduced the jerkiness somewhat, but the slowing down s t i l l occurred. To i s o l a t e the causes, a methodical examinat-ion of the whole wave generator system was made, with the following r e s u l t s . I f v i b r a t i o n was a cause, paddle motion i r r e g u l a r i t i e s would increase for c e r t a i n paddle o s c i l l a t i o n frequencies, due to resonance, and would change s i g n i f i c a n t l y with changes i n paddle mass. The r e s u l t s achieved by varying the paddle o s c i l l a t i o n frequency and by changing the paddle mass about 30%, indicated that v i b r a t i o n was not responsible. The crank d i s c was found to be s l i p p i n g on i t s drive shaft due to an inadequate connection. . This problem was r e c t i f i e d by machining a flange and welding i t to the gear-box drive shaft. The crank di s c was recessed s l i g h t l y on the back to take the flange and the two units were bolted together ( F i g . 28). The drive system ( F i g . 29) comprised three u n i t s . The prime mover was a 1/2-Hp., 1720 R.P.M., 110-volt, 60-cycle, 81. single-phase, Brook E l e c t r i c motor f i t t e d with a v a r i - d r i v e attachment. This attachment was a Boston, variable-speed drive (VDF6-1) consisting of an adjustable motor base and a spring-loaded, cone-belt sheave capable of a 3:1 speed reduction. The b e l t was connected to a Morton Engineering, worm gear, speed reducer, model T118, with a 20:1 speed reduction. This reducer was d i r e c t l y connected to. a shop-manufactured, 2 speed, gear-box which allowed the s e l e c t i o n of ei t h e r a d i r e c t - d r i v e slow s e t t i n g , or a 1:3 speed increase, f a s t s e t t i n g . The slow-setting output speed range was 90 to 30 R.P.M. while the fast s e t t i n g range was 270 to 90 R.P.M. The choice of a variable-speed drive unit was unfortunate, as the basic design of these units makes them unsuited f o r d r i v i n g a wave generator. V a r i a t i o n i n the diameter of the spring-loaded, cone-belt sheave on the motor drive shaft i s achieved by moving the motor along i t s base, using the adjustment wheel shown i n Fig.29(a) on the extreme r i g h t of the base. Moving the motor away from the pulley being driven, increases b e l t tension which forces the spring-loaded sides of the motor-mounted cone sheave apart, thus reducing the sheave diameter. This increases the r a t i o of speed reduction, which i n e f f e c t reduces the output RPM of the system. When running, the diameter of the cone-sheave w i l l be governed by i n i t i a l b e l t tension and by tension due to load. For a constant load t h i s unit gives good s e r v i c e , but for f l u c t u a t i n g loads the diameter of the cone sheave correspondingly f l u c t u a t e s , r e s u l t i n g i n v a r i a t i o n s i n drive output speeds. For increasing load the output speed slows 82. 83. down and v i c e versa. As i l l u s t r a t e d i n F i g . 24, a wave generator load f l u c t u a t e s . Obviously, i t i s p h y s i c a l l y impossible for this type of v a r i - d r i v e to give the smooth, s i n u s o i d a l , paddle motion desired. To check out this l i n e of reasoning, the v a r i - d r i v e cone-sheave was replaced by a f i x e d diameter p u l l y shown i n F i g . 31. The slowing down of the drive motion showed noticeable improvement but the jerkiness was s t i l l present. This jerkiness was traced to the shop-manufactured gear box, the gear arrangement of which i s shown i n F i g . 30. A drive gear on the input shaft (the lower shaft barely v i s i b l e i n F i g . 30) i s moved from one p o s i t i o n to another along the shaft when a change i s made from "low" to "high" speed. To allow the gear to s l i d e over i t s key there has to be some clearance, and i n t h i s case the clearance was excessive. Because of the heavy #90 g e a r - o i l bath this free-play was not evident unless firm pressure was applied. This was considered as the major source of the backlash and the re s u l t a n t , jerky, drive motion. Further examination revealed a second cause of uneven motion. Bevelled wear was noticed on the ends of the teeth of the small gear on the r i g h t end of the intermediate shaft (Fig. 30) i n -d i c a t i n g that the teeth of t h i s gear were not remaining p a r a l l e l with the teeth of the gear d r i v i n g i t . The cause was assumed to be 84. System Which Had an Output Speed Range of 270 to 30 R.P.M. 85. F i g . 30. Gear Arrangement i n the Shop-Manufactured Gear Box, C.E.Dept., U.B.C. 31. "Small" Flume Wave Generator Motor F i t t e d with a Fixed Diameter Drive Pulley to Check Motion Improvements. 86. due to the h e l i c a l gear, on the l e f t end of the intermediate shaft, "walking" on the output shaft h e l i c a l gear during move-ments of peak load, causing the intermediate shaft to bow out and then whip back as the load decreased. Dismantling of the i n t e r -mediate shaft assembly, revealed an undersized inner shaft surrounded by spacing c o l l a r s which gave i t the appearance of having a larger diameter. Obviously, t h i s shaft was not r i g i d enough for the job. Since i n e r t i a l forces produced by ac c e l e r a t i o n and de-ce l e r a t i o n of the water and paddle mass w i l l produce forces acting on the connecting rod and crank d i s c , which reverse d i r e c t i o n , i t would appear necessary to minimize normal gear back-lash i n the drive system. This wave generator drive t r a i n employs 3 sets of mating gears. To reduce backlash i t appears desirable to reduce t h i s number to 1 gear set i f possible. The paddle of t h i s wave generator has a brass front face which i s free to move up and down (Fig. 28). The weight of th i s face causes i t to ride on a f r i c t i o n - r e d u c i n g , white, t e f l o n sheet on the bottom of the flume (Fig.29), thereby stopping water leakage under the f l a p . The face formed a close f i t i n the channel and was found to be rubbing on parts of the glass walls of the flume evidenced by deposits of brass. On occasion, the f r i c t i o n with the glass appeared to induce a l a t e r a l v i b r a t i o n of the face 87. causing a d d i t i o n a l momentary drag. Increasing the clearance between the face and flume sidewalls, cured t h i s problem. How-ever, the paddle face s t i l l makes a close f i t i n the flume and care should be taken to keep sand or g r i t away from the face during experiments. The drive wheel (crank disc) was found to be s l i g h t l y o f f from a precise p a r a l l e l alignment with the connecting rod causing binding between the crank pin and the c l o s e - f i t t i n g sleeve bearing on the end of the connecting rod. Also, the aluminium base plate upon which the drive assembly was mounted (Fig. 29) flexed s l i g h t l y with high load outputs, which would momentarily change the crank p i n alignment. To cure t h i s problem the drive assembly was re-aligned and the old connecting rod ( F i g . 25) was replaced by a temporary connecting rod of the same length f i t t e d with a s e l f - a l i g n i n g bearing which worked w e l l . This p r a c t i c a l aspect of maintaining precise alignment of the drive system, led to the use of a s e l f - a l i g n i n g bearing on the connecting rod of the proposed, new wave generator designed for use i n the large flume. The r a t i o of connecting rod length to maximum crank throw was 3.3:1. An improved s i n u s o i d a l motion would be obtained by increasing t h i s r a t i o to about 6:1, or greater. The two v e r t i c a l arms supporting the rear of the paddle were each f i t t e d at the top end with a large-diameter, sleeve-type bearing 88. adjustable for f r i c t i o n ( F i g. 28). The reason for these bearings was not c l e a r , but i t was assumed they were used to give more l a t e r a l s t a b i l i t y to the wave generator and to provide l i m i t e d f r i c t i o n as a counter to gear back lash. The adjustment nuts were slackened o f f to reduce the f r i c t i o n of these bearings, which was found to be considerable at the time of examination. Care w i l l have to be  taken i n the future to see that these bearings are riot t i g h t l y  adjusted. The lower ends of the v e r t i c a l , rocker arms are positioned on a graduated arc mounted on plates located on each side of the flume (F i g . 28). One of these plates was t i l t e d s l i g h t l y out of v e r t i c a l p o s i t i o n . As the arm ends were f i t t e d with sleeve type bearings th i s produced a s l i g h t binding. Re-alignment of t h i s p l a t e solved the binding problem. The p r a c t i c a l s o l u t i o n of t h i s problem suggest-ed the use of s e l f - a l i g n i n g bearings on the lower ends of the rocker arms of wave generators of this d o u b l e - a r t i c u l a t i n g type. Because t h i s flume i s a t i l t i n g type, the drive assembly must be supported on the flume making weight an item of consideration. The e l e c t r i c motor and i t s base ( F i g . 29) weighed 86 l b s . , the worm-drive gear box 20 l b s . , and the shop-manufactured gear box 67 l b s . , for a t o t a l of 173 l b s . , not i n c l u d i n g the aluminium base pl a t e upon which the whole drive system was mounted. This weight should be reduced. 89. Improved, wave-generator motion, w i l l be obtained i f the i n e r t i a of the moving parts i s f a i r l y low. This wave generator paddle was fabricated from s o l i d aluminium p l a t e , r e s u l t i n g i n i t being r e l a t i v e l y heavy. Also the paddle face was made of brass. By d r i l l i n g holes i n the material, the weight of the paddle assembly was reduced from 22 to 18.5 l b s . and the brass face from 8.5 to 7.4 l b s . The t o t a l weight of 25.9 l b s . was s t i l l f e l t to be high, but no further weight reduction was done l e s t r i g i d i t y be s a c r i f i c e d . The best s o l u t i o n would be to r e b u i l d t h i s paddle mechanism using t h i n aluminium tubing for r i g i d i t y and making the paddle face of aluminium plate instead of brass. Besides the changes already made, i t was concluded that a new drive system should be designed f o r t h i s wave generator, i n -cluding a longer connecting rod. 5.3 Re-Design of the "Small" Flume Wave Generator Drive System 5.3.1 Data on E x i s t i n g Wave Generator and Channel Maximum crank throw e = 4.3" Useful channel depth = 10 5/8" Operating water depth d = 6.5" Channel width w = 8 3/8" Channel length =21' - 3 5/8" 5.3.2 Maximum Motor-Gear-Box Output RPM To avoid s i g n i f i c a n t surface tension e f f e c t s ( F ig. 5) choose the minimum wave length L = 0.578 X 12 = 6.94". 90. For s t i l l water depth d =6.5", L_ = 6.94 = 1.068 which i s < 2, giving a deep-water wave, d 6.50 From equation (3), the deep-water wave v e l o c i t y i s /g L = / 32.2 X 6.94" = \/ 2 IT y 2TT. X 12 From equation (5) wave period T Therefore, maximum crank speed Note: Galvin recommends a minimum wave period (T) of 0.5 sec. ( 120 RPM ), with 0.75 sees. ( 80 RPM) preferred. The figu r e of 178 RPM was chosen to give a wider -^ range for i n s t r u c t i o n a l purposes. Wave heights w i l l be low for very short period waves. A lower RPM would reduce paddle stresses. 1.721 f t . / s e c . = L = 6.94 ^ 12 X 1.721 = .336 sec. 60 .336 = 178 RPM 5.3.3 Minimum Motor-Gear-Box Output RPM Galvin recommends at l e a s t two f u l l wave lengths between paddle and spending beach, s t i l l - w a t e r l i n e . Hence, maximum wave length L = = 100". — = = 15.4, which i s < 20, giving a t r a n s i t i o n wave. From equation ( 2 ) C = f .tanh M 2 T T - L 32.2 X 1 0 0 _ , 2 T T X 6 . 5 tanh 2 T T X 1 2 1 0 0 4 . 0 6 f t . / s e c . From equation ( 5 ) T = 1 0 0 4 . 0 6 X 1 2 = 2 . 0 5 sees. Hence, minimum crank speed = 6 0 . 0 5 2 9 RPM 5 . 3 . 4 Estimated Maximum Wave Heights for Generator: a) Piston motion y i e l d s maximum wave heights. From equation ( 2 1 ) semi-height (amplitude) of generated wave i s a = Ke. Hence wave height (H) = 2 K e. For ^ = 1 5 . 4 and reading K from F i g . 2 1 , maximum H = 2 x . 4 0 8 x . 4 3 = 3 . 5 1 " Maximum wave heights f o r other values of ^j- were calculated and r e s u l t s p l o t t e d i n F i g . 3 2 . (For intermediate motion the t h e o r e t i c a l wave heights w i l l be less.) b) Maximum deep-water wave height i s H - y L from equation ( 9 ) . For ^ = 1 . 0 7 , L = 6 . 9 4 " and H ^ 4^- = . 9 6 " . a / He ight l i m i t s for other values of -j- i n the deep-92. water wave range were calculated and p l o t t e d i n Fig.32, c) Maximum shallow-water wave height from equation (10) i s H = = 3 X 6 . 5 = 4.87" 4 -, d) Galvin's rule-of-thumb range, where waves lose s t a b i l i t y due to depth, runs between H - —— = = 43" 1.5 1.5 ' and H = | = = 3.25" 5.3.5 Maximum Paddle Forces Due to Water At 178 RPM 3 - = 1.07 . a Using F i g . 22 F = 42.2 edw = 42.2 X ^ ~ X ^ x ~ r i ~ = 5.73 l b s . n 12 12 12 Using F i g . 23 F. = 190 edw = 190 X ^ X X = 25.8 l b s . i 12 12 12 Values of F and F. were also calculated for n i crank RPMS of 151, 121, 88.5, 53 and 29. Results are recorded i n Table I. WAVE HEIGHT H 5 (inches) 4 ow-wo :Maximum wave height from Biesels theory Using piston motion, water depth of 6.5" and crank throw of 4.3". i M r 11111; 1111111111 Shallow-water waves break when H= ^  d. er waves s -able Galvin's Observed Umi Channel waves become _uns and break when H is between H-,- c = 4.3" andH=T^ 3.3" 1.5 -or generating waves in this range as Max. L -100". 21 4 6 (1-0 rpm) (178 rpm) 20 22 24 26 28 30 32 Fig. 32 - Estimated Wave Making Capability of "Small" Flume Wave Generator for a 6.5" Water Depth and Piston Paddle Motion. IO 9 3 . 5.3.6 Maximum Flap Force Due to Mechanism I n e r t i a Assume the motion of the paddle mechanism mass i s that of a true piston. (The actu a l motion of the paddle mass i s slightly-curved and the l i g h t , paddle support arms move i n arcs, but the assumption of a true piston motion for a l l parts i s s u f f i c i e n t l y accurate for design purposes and errs s l i g h t l y on the safe s i d e ) . From Kent's Mechanical Engineer's Handbook (Ref. 4 -Mechanical Mechanisms) the i n e r t i a force of the paddle mechanism 12 WVc2 ( . , R . fl s i s F. = (cos & + -p cos 2 0 ) lm g R JL where W = t o t a l weight of rec i p r o c a t i n g parts of paddle i n l b s . = 26 l b . 2 g = acc e l e r a t i o n due to gravity i n f t . / s e c . Vc = v e l o c i t y of crank pin i n f t . / s e c . JL = length of connecting rod i n inches, R = maximum crank throw radius i n inches, 0 = angle i n degrees defined on the diagram on Table I I , (cos 0 + £- cos 20 ) values are given i n Kent, Table - 2, page 7-38. For a speed of 178 RPM Vc = RPS X circumference'of crankpin c i r c l e !78 2 x 4.3 , , Q _ , Vc = —p- x IT x — z — = 6.68 f t . / s e c . 60 12 94 . Maximum F. = 1 2 * I6 X . 6 : 6 8 (1.167) = 117.3 l b s . im 32.2 x 4.3 Values for F. were also calculated f o r crank speeds im of 151, 121, 88.5, 53 and 29 RPM. Results are recorded i n Table I. 5.3.7 C a l c u l a t i n g Motor-Gear-Box Output Torque Calculations f o r the maximum motor-gear-box output torque at various crank RPMS are presented i n Table I I . Terms used r e -l a t i n g to the mechanism are defined on a diagram included on the Table. The r a t i o between connecting rod length and crank throw was I set at — = 6 : 1. This r a t i o gives a connecting rod length of R t ; -£ = 6 x 4 . 3 = 25.8 inches and represents a compromise between an improved s i n u s o i d a l motion, due to a longer connecting rod, and a c c e s s i b i l i t y to the space i n front of the paddle for i n s t a l l i n g wave f i l t e r s . The f i r s t two columns i n Table II give the crank angle 0 and the corresponding paddle angle kt from Bi e s e l ' s theory. The paddle angle has been designated "nominal" as the geometry of the mechanism a f f e c t s the r e l a t i o n s h i p between crank angle G and the paddle angle. Their r e l a t i o n s h i p was approximated by considering the paddle as moving with true piston motion. (Actually the paddle 95. moves along a s l i g h t " h o r i z o n t a l arc" as can be seen by studying F i g . 16 where the mechanism i s shown adjusted f o r shallow-water waves (piston motion) ). The approximated values of paddle angle . (18(T - kt) are designated " a c t u a l " and were found with the aid of Table I, page 7-G4, Kent's Mechanical Engineer's Handbook (Ref. 4 - Mechanical Mechanisms) which gives piston(paddle) p o s i t i o n for various crank angles. The values of cos (180 -kt) were then computed for use l a t e r on. Values for the "tangential e f f o r t and v e l o c i t y f a c t o r " sec 8 s i n ( 0 +$), corresponding to chosen values of crank angle I 6 for — = 6 : 1 , were obtained from Table I, P. 7-36 i n Kent and R l i s t e d for use. S i m i l a r l y , values for the " i n e r t i a and ac c e l e r a t -ion f a c t o r " cos 6 + cos 2 9 were obtained from Table 2, P.7-38 i n Kent and l i s t e d . The f i r s t set of c a l c u l a t i o n s was done for a crank speed of 178 RPM. The maximum value of F was read from Table I. n Values of F n cos (180-kt) were then computed for the l i s t e d crank angles.- S i m i l a r l y , values of F^ were calculated f o r the same crank angles. The maximum value of the mechanism i n e r t i a force F. was read from Table I and the applicable value calculated f o r im the required crank angles. I t should be noted that the maximum value of R I cos 6 + -j cos 2 0 for — = 6 : 1 i s 1.167 and X, R therefore the maximum value of F. must be divided by 1.167 i m J t-3 > CO i 2 O O CO rt- |—' H> O O C ct P H-o O j3 *1 SO O t-3 p o 3 0 o >->> T) Co Xrfa . a 3 fD o a: fo (a co ct fD •1 O o fD 0 0 ct p D* t—1 O t—1 CTv C . 3 UI fD a: < a> a fD 3 fD *i co ct o 1 o CO ct O 3 ro UI U) 88.5 H ro M M UI M y CO CRANKWHEEL RPM 1 o o • o UI ro • o ro cn o UI o CO vu O cn • vo Xr L ( Inches) H UI • Xr CO o Xr • O ro UJ ro • Xr • H O -<l L j ar j Xr O cn UJ • CO ro U) • M VO ro • u i UJ ro • o ro M —J ro (—i C - / g ^ . tanh ^ f t . / s e c . 2.05 • t—1 U) ro CO • ' XT vo CT\ UJ vo UJ U) cn T " gf ( s e c . ) | CD • Xr CO M • ro M o UI ro CO UJ • o u i cn xr to ro P n ( l b s . ) - T h i s i s max. edw <» s-0 < fD fD a> a, v a l u e . ( F i g . 31) ii o tl o Xr Xr Xr fD Cb t. U) CTv O fD s: H O UI ro fD o. 1—' vo o fD £ F± ( l b s . ) - T h i s i s max.' ' v a l u e . ( F i g . 32) 3- = •'••3 3 = M UJ UI cn r... _ 4.3 x 6i;5 x 8.375 e d w ° 12 x 12 x 12 (Un i t s i n U f e e t y ON UI OS t—1 M • O o H. XT i> . CO M 1 ro -j —j CO H UI • U) MAX. P n (lbs.)£: o o •- • ' cn o u> Xr CO CO t—1 Xr • ro CO ro u i • CO MAX. F ± ( l b s . ) 3 3 3 3 3 • cn —3 cos 9+2 cos 20 Kent Handbook-Table 2 p.7-38 Max. Va lue f o r R/£ = 1/6 1.088 M VO CO vo UJ UJ ro Xr • UI Xr UI • cn cn cn CO CRANK VELOCITY V c ( f t . / s e c . ) ro 12W 12 x 26 gR 32.2 x 4.3 • M ro H' O XT o ro vo • o UI Xr ro CO Xr ro M —] UJ MAX. F . _ im 2 * 1 2 W V c ( c o s 9+1 cos 20) g R *• l b s . *96 91:. before multiplying by the next value of cos 9 + -£ cos 2 . 6 . Values of F , F. and F. for a p a r t i c u l a r crank angle were n 1 lm r • summed to obtain P, the t o t a l force acting on the paddle. Then using the equation F = P sec <f> s i n ( 0 * cf>) the e f f e c t i v e tangential force, i n pounds, acting on the crank pin was obtained. For a crank speed of 1 7 8 RPM the maximum value of F was - 7 4 . 4 l b . The negative sign r e s u l t s from the r e l a t i o n s h i p of forces as ex-plained i n F i g . 2 4 and can be ignored beyond t h i s point. The maximum gear-box output torque required i s 7 4 . 4 x 4 . 3 = 3 2 0 i n . l b . The maximum torque required at crank speeds of 1 5 1 , 1 2 1 , 8 8 . 5 , 5 3 and 2 9 RPM was also computed and the ca l c u l a t i o n s recorded i n Table I I . The torque requirements were p l o t t e d i n F i g . 3 3 for various RPMSs and values of ^j. a Maximum values of F , F., and F. occuring at various speeds n' i ' lm ° • r are p l o t t e d i n F i g . 3 4 to show v i s u a l l y t h e i r r e l a t i v e s i g n i f i c a n c e for t h i s wave generator, p a r t i c u l a r l y that of mechanism i n e r t i a at higher.speeds. 5 . 3 . 8 Protective High Speed Crank Throw L i m i t a t i o n Because of the poor motion of the o r i g i n a l drive system, the wave generator had not been operated at f u l l crank throw at speeds much above 1 2 0 RPM. This was rather fortunate as some of the fastenings, and the axle supporting the ends of each of the 98. four vertical-paddle-support arms, appear of inadequate strength f o r sustained high loadings. The o r i g i n a l \ Hp. e l e c t r i c motor was of inadequate power r a t i n g f o r the load at high speed and f u l l crank throw, but could e a s i l y have appeared adequate. E l e c t r i c motors w i l l operate at up to about 250% overload f o r short periods of time u n t i l a protective thermal switch cuts the power. Since the peak wave generator load i s i ntermittent, occurring only b r i e f l y twice i n each crank cycle (Fig.24), the motor would have driven the paddle f o r an appreciable time at higher RPMS and crank throws than those used, before motor overheating occurred. Rather than rebuild, parts of the paddle to guard against overstressing, i t was f e l t that overstressing could be prevented by requesting the wave generator operator to l i m i t the crank throw at high speed. This l i m i t a t i o n would not l i m i t the wave making c a p a b i l i t y of the wave generator. The l i m i t a t i o n was computed as follows: At 178 R.P.M. L = 6.94" and L = 1.07 d The maximum deep-water wave height i s H = 1 L = = .99" (Table I) (Fig. 32) 9 8 . For piston motion • K s 2 (Fig.21) From equation (27) H = 2a = 2Ke where e i s as defined i n F i g . 20a. The t h e o r e t i c a l crank throw required to achieve maximum wave height for piston motion i s e = H_. = _99 = 2K 2X2 For hinged-flap operation / ~ - 1.7 (Fig.19) H 99 and e = ^ r r = 7 = where e i s as defined i n F i g . 20 B. But e i s located 6.5" above the channel bottom whereas the connecting rod connection to the fl a p i s 20.25" above the channel bottom, (Fig. 16: Deep-water Wave Setting ) . Therefore the re-10.21 "oTT 20 25 quired crank throw i s * X .29 = .91" . As the calculated values of e are approached, the waves become unstable and break. For higher values of e, splash r e s u l t s (assuming 100% f l a p e f f i c i e n c y ) . At 130 RPM where ^j- = 1185, the maximum required crank throw 100. CJ <D Mz. CD w tt <D 0C o i-4 . > 14 !S H S o 03 ^ 0) - P CD X u O CD o T4 C O w O ! l I [— <*\K ' (X PC O fe rH w w i-4 m < < > EH I H EH •4 S 04 W < W EH * H Co i-4 > W i-4 CQ < EH I o o in CO > H !=> i-4 CQ ra < > CD EH 1 « i-4 re in •3 M EH De li O *—* .—. < -p •Hk) - P o II C O 1 H o • °*|CC o C O ' CC 1-•I H O 00 FOR £/R=l/6 •p i o oo CO O o 178 RPM oo I S i a + • « rH CD H CO M i-4 0Q EH ca EH O H W « CO * CM C O ro l CO C\J o O H , i-4 < EH I EH CO W O W O * + CD 04 1-4 oo - P I o 00 fe to o O X fe w 04 i-4 C O - P X in I CM i 'O | CO II fe fe < •ri S C P C M CO o o CCl<=* 4-CD CO O o CMO > CM H PQ t-4 oo r-rH rH I fe II X £ < •H S fe S •H fe fe' 4-c fe II OH PQ i-4 CS. 4-G> H CO CS. o w co -p fe 151 RPM PQ - P H ^ co I o t— C O H II ^ C fe CO O o x fe w PQ t-Q co CM - P I rH O I CO || rH fe CO * •rl S fe w o CM o ^ o PC|«? PQ 4- i-4 CD CO CM O O ^ C O I CMO > CM rH CC bO £ •ri fe X fe s •H fe ' + •H fe + fe II fe PQ >-4 CS. + CD H CO C S o W CO PH II - P fe 121 RPM -p i o oo H PQ i-4 c — C M r-i rH CO O O fe' PQ i-4 ^ - N C O +J C O o I C O || fe H CO X < •H S fe w CD CM CO ^ O Q PQ piT°* h4 + CD CM CO O -=r O LTV 1 CMO > II cc e hO-H fe II fe •rl S S •rl fe + •r fe + fe PQ CS. + CD ^> H CO CS. o w CO II fe 88.5 RPM p> !^ I o C O PQ i-4 CO CM II fe CO o C S fe ^ PQ .-4 -^s ro - P o X vo o co II . rH ^ -rl fe CO « •rl S fe CD CM CO O ^ O CCl°« PQ 4- 4^ CD CO o o O O A v_> CM CMO II CC 6 &D-H .fe CM fe fe + •r fe + II cu PQ i-4 CS. CD H CO CS. o w CO fe -p fe 53 RPM 04 i-Q o o • •p I o C O rH fe fe X < -p ^ ! I o C O 04 t-4 O 11 fe co x •H S fe ^ N O > CM H O CM CO O O K M + CD CO o CJ P4 i-4 o o rH K S bO •H . f e •rl fe X fe 4-•r fe • 4-fe II PH PQ i-4 CS. 4-CD H CO CS. o w CO PH II •p fe 29 RPM -p i o oo rH OQ 1-4 VO LTV VO fe co o O X d s fe w -p i o C O H H CO T fe 04 i-4 o w fe x o CM CO o cc?«* 4-CD CO O O 04 i-4 CM rH O O I CM O > •H^ CM rH S CC T H bDfe II X e < •H s fe ^ s •H fe 4-TH fe fe PH PQ i-4 cs 4-CD rH CO CS. o w CO fe -p fe (90) 500 1.000 90 1.167 -25.8 -117. 3h1 43. Ij 0 10 (80) .009 .491 .982 79.1 -.189 .202 1.142 -1.082-25.3 -114.8 141.21-28.5 20 (70) 035 465 .930 68.5 ••367 .396 1.067 -2.10 -24.0 -107.2-133.3 -52.8 -4.13 -4.54 -49.6 •58.3 -23.1 -5.24 -.561 -26.5 -32'..3 -12.79' 30 (60) 077 .432 .846 57.8 -.533 .572 .949 -3.05 -21.8 -95.4 j-120.3 -68.8 -4.16 12.08L68.5 -84.7 48.4 -6.00 -4.13 -44.1 -54.2, r31-0 -7.61 •.510 •23.6 -31.72 -18.1Q -5.86 -8,46. -14.32 •8.2 -3.50 -2.54 r-6.04 -3.46 40 (50) 134 .366 732 47.1 682 .725 .795 ,3.90 -l8,88i-79.8 1-102.6 -74.4 -5.33 -10. 451-57.4' 1 •73.2 -53.0 -7.68 -3.57 -36.9 -48.2 -34.9 •9.72 -.441 -19. 8C •30.0 -21.8 -7.50 -7.09 -14.59 -10.57 -4.47 -2.13 • 6.60 -4.78 50 (40) 203 .297 .594 36.4 -.805 849 .614 4.61 |-15.3^-6l.7 i-81.6 ! • I -.69.3 •6.29 -8.48 U 4.3 -59.1 -50.2 -9.06 2.90 -28.5 4.0.5 -34.4 -11.5C -.358 -15.30-27.2: -23.1 •8.85 5.48 -14.33 -12.18 -5.28 -1.643 •6.92 •7.02 -5.88 60 70 (30) .281 (20) .219 .438 26.0 -.899 939 417 -5.15 -11.3Q-4l.9 1-58.4 -54.8 .366 .134 .268 15.6 -.963 .994 214 -5.51 •6.91 -21.5 -33.9 -33.7 -7.02 -6.26 •7.51 r 3 0 . 1 •3.83 -43.4 -15.45 -26.8 •40.7 -10.12 •2.14 -19.39-31.7 -29.7 •26.6 -12.83 -.264 -10.383-23.5 •22.1 -9.89 -3.72 -13.'61 -12'.79 -5.90 -1.118 -6.58 -10.59 0 -I.908 -12.50 -12 ..42 -6.31 -.574 -6.88 -6:. 84 80 90 (10) .454 046 (0) .542 -.042 .092 5.28 -.996 1.014 ,015 -5.70 -2.38 -l.508j-9.59 -.084 -4.82 -.996 1.000 -.167 -5.70 4-2.16 +16.75+13.21+13.21 •9.74 WAX.TORQUE = 34.9 X 4. 3=1151 "Inlb, MAX.TORQUE =23.1X4.3=99 i n . lb MAX. TORQUE=74:4X4.'3=320in/lb MAX. TORQUE=53X4.3 =228 I n / l b * C a r m l c h a l l , C o l i n - KENT'S MECHANICAL ENGINEERS MAX.T0RQUE=12.79X4.3=55 I n . l b . Low Force S e c t o r . 51 sac. High Force Sector HANDBOOK, Design and P r o d u c t i o n Volume, 12th E d i t i o n , John Wiley and Sons, New York, 1950 TABLE II: C a l c u l a t i o n s f o r Maximum Gear-box Output Torque Required at Vario u s Speeds by the 21' - 3 5/8" X 8 3/8" x 10 5/8" Flume Wave Generator f o r P i s t o n Motion, Maximum Crank Throw o f 4.3" and Water Depth o f 6.5". High Force Sector -.04.02 -6.58 -6.68 rORQUE=6 .84X4.3=29. i n . l b 270* 0 sec Low Force Sector 1Q«L. i s 0.5" for piston motion and 1.8" for "hinge d r -flap motion., I t i s evident that at crank speeds above 130 RPM the "crank throw r e -quired to achieve maximum t h e o r e t i c a l wave height i s le s s than \ maximum crank throw, i . e . % x 4 . 3 =2.15". The torque demand for a crank throw s e t t i n g of % maximum, or 2.15", was computed for crank speeds above 130 RPM and p l o t t e d i n Fig.33. In view of the drive unit f i n a l l y selected i t was concluded that the crank throw should be l i m i t e d to % maximum (i.e.2.15")  for crank speeds above 130 RPM, i r r e s p e c t i v e of the paddle motion  involved. This l i m i t a t i o n should r e s t r i c t the maximum applied torque to about 170 i n . l b . without, as shown, r e s t r i c t i n g the wave making c a p a b i l i t y . 5.3.9 Crank Wheel Mounting Height for Optimum Sinusoidal  Paddle Motion. The geometry of the paddle mechanism a f f e c t s the desired s i n u s o i d a l motion of the paddle. Since t r a n s i t i o n a l and shallow-water waves were f e l t to be of greater i n t e r e s t - f o r model tests than deep-water waves, the best l o c a t i o n of the crank wheel was determined for the paddle adjusted for p.iston-motion. This work was done gra p h i c a l l y and the f i n a l c a l c u l a t i o n i s shown i n F i g . 35. I t was decided that the crank wheel should be mounted 0.34" above the h o r i z o n t a l l i n e passing through the centre of the j o i n t where the connecting rod i s attached to the paddle. 102. 5.3.1Q New Drive Unit A f t e r i n v e s t i g a t i n g a number of drive systems, i t was decided to use a 1 Hp. Winspeed SCR drive u n i t , with an RPM range from 180 to 9, to replace the o r i g i n a l components. The torque curve i s shown i n F i g . 33. The torque a v a i l a b l e from the new unit s u b s t a n t i a l l y ex-ceeds the calculated value of torque required by the wave generator crank. This situation-was considered desirable for three reasons: a) Biesel's theory, used to compute the wave and water i n e r t i a forces, i s only f i r s t order theory and i s not yet f u l l y proven; b) bearing f r i c t i o n losses and f r i c t i o n losses from the paddle face dragging over the t e f l o n pad (Fig. 28) and along the flume's glass sides are unknown; and c) better speed regulations w i l l be obtained i f a unit having surplus power i s used. The 1 Hp. motor and speed reducer are flanged coupled pro-v i d i n g a " t i d y " i n s t a l l a t i o n . Their combined weight of 96 l b . i s less than 60% of the weight of the previous drive system. The Winspeed MCTR (2). worm-gear speed reducer, was checked for overhung, output-shaft, load and torque r a t i n g and found s a t i s -factory. 103.. 5.3.11 Re-design of Connecting:Rod The length of the new connecting rod was previously s p e c i f i e d at 25.8" (Section 5.3.7). Ca l c u l a t i o n of the maximum load was as follows: Maximum load acting along the centre l i n e of the connect-ing rod i s Kent's Mechanical Engineer's Handbook P. 7-37. Using values of P and 6 from Table II for a crank speed of 178 RPM., the maximum value of F occurred for P=143.1 l b . A new connecting rod was designed of aluminium a l l o y 6061-T6 to carry t h i s load. The crank end of the connecting rod was f i t t e d with a s e l f - a l i g n i n g , b a l l bearing. A new crankpin was fabricated to accommodate th i s bearing and to f i t the e x i s t i n g crank d i s c . 5.3.12 Summary of New Operating S p e c i f i c a t i o n s P from c and 6 = 0 ° , that i s F c = 143 l b . i n compression. design wave period range 0.34 to 2.1 sees. design water depth (d) 6.5 it minimum wave length 6.9 I I maximum wave length (d =6.5") 100 I I range for a depth of 6.5 d I I 1.07 to 15.4 maximum crank throw '. =4.3" (estimated) maximum wave height - 4" when 5 325 300 [ 2751 250 [ ~ 225F ca | z- 200 LU 175 ZD t cy f 0 150 1 125 < F Cxi \ ° 100 75 50 25. 180 170 160 150 140 130 120 110 100 90 R P M 80 70 60 50 40 30 20 10 0 o ! R P M "; Fig. 34. A Plot of Maximum Values of Wave Forces F and I n e r t i a Forces Fj and F, to Show n i im Relative Magnitudes. CrnnK Angle ^  o- True Sinusoidal Motion I 1—I.-True Piston Motion V HI - Paddle Motion - Centi n A i 1—1 - Paddle Motion - Centre B I 1—I - Paddle Motion - Centre C Ratio of Connecting Rod length to crank throw - 6:1 A and C are alternate positions of the crank wheel centre . 34 inches above and below B respectively along the arc shown. Ilume floor Fig. 35 Determining the Crank Wheel Location for Optimum Sinusoidal Displacement of the Paddle at the Piston Motion Setting. 108. 6. DESIGN OF A WAVE.GENERATOR - designing the proposed new wave generator for the "large" flume. 6.1 Scope The large, 39'-4%" long, 30" wide and 36" deep, f i x e d , s t e e l and glass flume i n the U.B.C. Hydraulics Laboratory i s , i n wave-channel p r a c t i c e , r e l a t i v e l y short i n length, and as a r e s u l t r e -quires a wave generator which establishes the correct wave form and water p a r t i c l e motion r i g h t at the paddle. I t was therefore de-cided to design a wave generator f o r th i s flume of the r i g i d paddle, double a r t i c u l a t i o n type (Section 3.6), using the geometry arrange-ment of Coyer (Ref. 3 - Wave Generators). The design of th i s wave generator i s presented i n t h i s section. 6.2 Flume Data The 39'-4V. long, 30" wide and 36" deep flume i s fabricated of s t e e l , r i g i d l y attached to the -laboratory f l o o r , and has glass panels on both sides along the middle t h i r d of i t s length. Views of the large flume are presented i n F i g . 1. I n t e r i o r d e t a i l , i n the region chosen for i n s t a l l a t i o n of the proposed new wave generator, i s shown i n F i g . 36, while corresponding e x t e r i o r d e t a i l i s shown i n F i g . 37. Basic flume layout and dimensions are given i n Fig.38. For design purposes: usable flume length t o t a l flume depth = 39*-4V = 36 3/4" 1 0 9 . usable flume depth = 3 3 " i n t e r i o r flume width = 30" and extreme ex t e r i o r width = 38 1/8" Fundamental Design Decisions The f i r s t decision concerned the paddle motion upon which force c a l c u l a t i o n s should be based^ The wave forces ^ acting' on--the paddle of a dou b l e - a r t i c u l a t i o n type wave generator w i l l be greatest should piston motion be used to generate shallow-water waves (Section 4.2.5). Also, mechanism i n e r t i a forces (F. ) are highest i m ° for piston motion due to the manner i n which the material masses move. Since the operator i s free to adjust the wave generator mechanism f o r e i t h e r piston, intermediate or hinged-flap motion (F i g . 20), i t was decided that the proposed new wave generator should be designed on the basis of piston motion being used throughout the operating range. Due to obstructions at the i n l e t end of the flume i t was decided to i n s t a l l the wave generator at the t a i l - g a t e end, as i n -dicated i n F i g . 38. The distance from t h i s end of the flume to the end of the l a s t glass panel i s only 25'-4". For purposes of observat-ion, most tests would be conducted i n the region of the l a s t set of glass panels. To allow a maximum reach between the wave generator paddle and the test area, f o r the i n s t a l l a t i o n of wave f i l t e r s and for the water-particle motion to s t a b i l i z e , i t was necessary to minimize the length of flume behind the paddle for the i n s t a l l a t i o n of wave absorbers. The volume of t h i s space behind the paddle then 110. F i g . 3 6 . I n t e r i o r D e t a i l of the 3 9 ' - 4 V Long, 3 0 " Wide and 3 6 " Deep Flume i n the Region Chosen f o r I n s t a l l a t i o n of the Proposed New Wave Generator. 111. F i g . 37. E x t e r i o r D e t a i l of the 39'-4V Long, 30" Wide and 36" Deep Flume i n the Region Chosen for I n s t a l l a t i o n of the Proposed New Wave Generator. Usable Length 139' -41" i !30" * iTaJI Gate 43": •10' - 6"; ± 3 6 | " _ i _ ;i4' - io" / / 14' - J 354 " r I ft •—:10s l iV Usable Interior Flume Depth 33" WAVE GENERATOR I LOCATION Fig. 38 Basic Layout of the "Large" Steel and Glass Flume. 113.. becomes r e l a t i v e l y small re l a t e d to the volume of water displaced by the paddle, r e s u l t i n g i n an appreciable f l u c t u a t i o n of the mean water l e v e l i n i t as the paddle o s c i l l a t e s . Because of t h i s f l u c t u a t i o n , the decision on maximum crank throw, design water depth and generator paddle l o c a t i o n became interdependent, n e c e s s i t a t i n g some compromises. The crank throw chosen f o r the drive mechanism w i l l govern the wave heights obtainable. A maximum wave height i n the range of 10" to 12" was desired. An i n i t i a l estimate of wave heights for piston motion and d i f f e r e n t crank throw lengths was obtained from equation (27), rearranged to give wave height H = 2a = 2 Ke. Si m i l a r -l y , wave heights for hinged-flap motion were estimated, using the equation H = 2a = 2 K'e. Values of K and K' were read from Fig.19. For hinged-flap operation allowance was made for the value of f l a p displacement at the water l e v e l being only about h a l f the f l a p d i s -placement at the connecting-rod connection. Normally, the wave generator paddle w i l l be adjusted to operate with intermediate motion, suited to the water p a r t i c l e motion of the wave being de-veloped. This w i l l give a wave height l y i n g between that of piston motion and hinged-flap motion. When deciding the crank throw to be used, another point was kept i n mind. Large paddle displacements r a p i d l y increase the drive unit power requirements and costs, e s p e c i a l l y i n the case of piston motion. Yet, when operating i n the hinged-flap mode, a large crank throw i s desirable to compensate for the reduced f l a p displacement 114. at the s t i l l - w a t e r l e v e l , due to mechanism geometry. Water depth a f f e c t s the maximum wave height which can be obtained. Galvin's observation that waves become unstable i n wave channels when the wave height H i s i n the region of - j — - to means that large paddle displacements w i l l only r e s u l t i n breaking waves, unless adequate water depth i s a v a i l a b l e . Since water depth sets a l i m i t on the maximum wave height, and since deep-water wave heights are also l i m i t e d by wave length (H = y L ) , the use of large paddle displacements, beyond a c e r t a i n l i m i t , serves only to increase the shallow-water wave heights. In s e t t i n g the design water depth, wave height and channel freeboard were taken into account. Due to the short reach between the paddle and the tes t area, i t i s desirable that the paddle motion should be as close t o . s i n u s o i d a l as f e a s i b l e . Therefore.it was decided to use a connecting rod length JL 7 to crank-throw r a t i o of. - -r. For a 10.5" crank throw, t h i s means — _L use of a 73.5" long connecting rod, which i s about as long as the av a i l a b l e space w i l l permit without having the motor mount extend-ing beyond the end of the flume. T r i a l c a l c u l a t i o n s were made to investigate the i n t e r l o c k i n g requirements of crank throw, water depth and paddle l o c a t i o n i n the flume. I t was decided to.use the following design l i m i t s : a) maximum crank throw. = 10.5", b) maximum water depth = 25", c) the paddle face positioned 5' from the i n t e r i o r t a i l -115. gate end of the flume; and d) a connecting rod length to crank-throw r a t i o of I I R 1 ' 6.4 E f f e c t of Limited Flume Length Behind Paddle The e f f e c t of the flume length behind the paddle being l i m i t e d to 5 f t . was investigated. In i t s rearward stroke the ,„ ,. , 30 25 10.5 . ,., ft_ f paddle displaces x "J2" x 12 = * c u . f t . of water causing a r i s e i n mean water l e v e l behind the f l a p of 5.3". For a 12" wave, 12 the wave crests behind the f l a p w i l l be 25 + 5.3 + — = 36.3" above the flume f l o o r . With a t o t a l flume depth of 36.75", i t i s apparent that splash boards w i l l be required along with an e f f i c i e n t wave absorber and possibly a cover p l a t e , to prevent water s p i l l a g e . This s i t u a t i o n indicates that f u l l crank throw w i l l be excessive when used with the piston-type motion at the design depth of 25 i n . However, the extra crank throw i s desirable f or intermediate or hinged-flap operations, where the amount of water displaced by the paddle i s less than that for a piston-type operation and the e f f e c t i v e paddle displacement at the water l e v e l i s less than the crank throw due to mechanism geometry. The f l u c t u a t i n g water l e v e l behind the paddle may serve to improve the paddle motion at higher crank speeds by reducing water and mechanism i n e r t i a e f f e c t s . With the aid of F i g . 24 i t can be seen that as the paddle moves forward, the water l e v e l behind the paddle drops, producing an unbalanced hydrostatic pressure on the 116,. front of the paddle which.works against, or absorbs, the i n e r t i a energy of the water and paddle mechanism. As the paddle s t a r t s to move backwards, the differe n c e i n water l e v e l w i l l a s s i s t i n accelerating the paddle. In the t h i r d diagram the pressure differ e n c e due to the higher water l e v e l behind the paddle absorbs some of the i n e r t i a energy.... In the l a s t diagram the pressure differ e n c e w i l l a i d i n accelerating the paddle. 6.5 Design of the Proposed, New Wave Generator Drive System 6.5.1 Summary of Basic Design Data Usable flume length = 39'-4V To t a l flume depth = 36 3/4" Usable flume depth = 33" I n t e r i o r flume width = 30" Maximum crank throw = 10.5" Maximum water depth = 25" Mean p o s i t i o n of paddle face from i n t e r i o r t a i l - g a t e end of flume = 5' Connecting rod length to crank I 1 throw ratxon — = K / Connecting rod length = 73.5" 6.5.2 Maximum Crank RPM To avoid surface tension e f f e c t s , F i g . 5, choose a minimum wave length of L = 12". Then at the design depth of d = 25" 111. -j - .48 which i s less than 2, i n d i c a t i n g deep-water wave. From equation (3) the deep-water wave v e l o c i t y i s / 32.2 ~uT V 2TT X 12 ' = 2.26 "/sec. From equation (5) T " £ = 12 x 2.26 " « 4 4 3 s e c -requi r i n g a maximum crank speed = = 135.5 RPM. Galvin recommends a minimum wave period of 0.5 sec.(120 RPM) with 0.75 sec (80 RPM) preferred. These slower crank speeds reduce i n e r t i a l forces considerably, with only a s l i g h t loss i n deep-water wave making c a p a b i l i t y . Therefore two other values of crank speed, wave length (L) and wave length to water depth r a t i o s 0~jj-) were com-puted, for use i n s e l e c t i n g the maximum speed of the power u n i t : L = 20" J = 0.80 d C = 2.92 f t . / s e c . T = .571 sec. Assume minimum then 118. requiring a crank speed = 105 RPM. Assume a crank speed =-••' 88 RPM then T = .683 sec. C = 5 . 1 2 T = 3 . 5 0 f t . / s e c . L = C T = 2 8 . 7 " and L = 1.15 d These ca l c u l a t i o n s were summarized i n Table I I I . F i n a l s e l e c t i o n of maximum crank speed w i l l be done l a t e r , when the drive system i s selected, but the speed w i l l l i e somewhere between 80 and 135.5 RPM. 6 .5 .3 Minimum Crank RPM Galvin recommends having at le a s t two f u l l wave lengths be-tween the mean paddle p o s i t i o n and the beach, s t i l l - w a t e r l i n e . Therefore, the maximum wave length i s L = (34' - 4V) = 17V - 2 1/8" = 206" For d = 25" L 206 25 = 8.24 which i s a t r a n s i t i o n wave giving n £k >- - 2ird /32.2 x 206 ^  . 2TT 7 _ , C = / -f— tanh -rr— = -z zrz— tanh Q -~-r = 7.52 f t . / s e c 2TT L \ 2IT x 12 8.24 T = |S|r ^ = 2.28 sec. 7.52 x 12 60 and a minimum crank speed = 2~28 = Assuming a minimum, usable water depth of d = 6" gives 11.9. ]± - 206 _ 34^2 which i s greater than 20 and i s a shallow-d o water wave. From equation (4) C = J g d = / 32.2 x 1^ = 4*01 f t . / s e c . From equation (5) T = 7 ^ r- = 4.28 sec. 4.01 x 12 60 and minimum crank speed = ^r~2g = 14 RPM 6.5.4 Maximum Water Forces Acting on Paddle Maximum values of the wave forces (F ) and the water i n e r t i a n forces (F^) acting on the paddle during piston motion at speeds of 135.5, 105 and 88 RPM, were calculated with the aid of F i g . 22 and 23 and l i s t e d i n Table I I I . Since the maximum wave forces for d i f f e r e n t ^j- values occur when \ = 4, Table I II was extended to i n -d d elude values of — equal to 2, 3 and 4. Values of F and F. for L d ^ 5 n 1 -r d greater than 4 were not considered, as these forces both decrease beyond t h i s point f o r any fi x e d water depth and crank throw. 6.5.5 Maximum Paddle Force Due to Mechanism I n e r t i a To ca l c u l a t e forces due to the I n e r t i a of the paddle mechanism, i t was assumed that the piston-type motion of the paddle mechanism was that of a true piston. This assumption i s s u f f i c i e n t -l y accurate f o r design purposes and errs on the safe sid e . The 120. ca l c u l a t i o n s f or mechanism i n e r t i a at various speeds were c a r r i e d out as previously explained i n Section 5.3.6 and the r e s u l t s l i s t e d i n Table I I I . For these c a l c u l a t i o n s the weight of the paddle mechanism was estimated at 160 l b s . A sample c a l c u l a t i o n for paddle mechanism i n e r t i a at 88 RPM i s as follows: 12 WVc2 • _ , R . F. = r - (cos 6 + -s- cos 2 0 ) im g R L where V = yyr x 77 x = 8.06 f t . / s e c . c 60 12 A • 12 x 160 x (8.06) 2 1 / r . and maximum F. = 0 0 ^ n c — (1.143) im 32. 2 x 10.5 = 422 l b . 6.5.6 C a l c u l a t i n g Motor-Gear-Box Output Torque Calculations for the maximum motor-gear-box output torque required to drive the crank at various RPMS are presented i n Table IV and were c a r r i e d out as previously described i n s e c t i o n 5.3.7. These values of torque were computed for f u l l crank throw of 10.5" and a 25" depth of water. The r e s u l t s are p l o t t e d i n F i g . 39. 6.5.7 Discussion of Torque Requirements F i g . 39 shows a r a p i d l y increasing torque requirement at high crank speeds. When generating deep-water waves, wave steepness H l i m i t s L the' -. wave heights and hence the u s e f u l crank throw begins de-12<1. creasing with.decreasing wave period. Therefore, at high crank speeds the power required to a c t u a l l y generate waves of maximum height i s considerably less than that needed to run the wave generator at maximum crank throw which w i l l only give splash and breaking waves. The peak torque demand at which waves are produced, occurs when the waves being produced have a length to depth r a t i o of ^j- = 4 which i s obtained i n a 25" depth of water at a crank speed of 45.1 RPM. I t i s obvious, then, that the power requirements can be considerably reduced, without reducing the wave making c a p a b i l i t y , i f the crank throw at high RPM i s l i m i t e d . Hinged-flap wave generators most often do not require a crank throw l i m i t a t i o n unless the maximum speeds are very high. This i s due to the lower water i n e r t i a forces and lower mechanism i n e r t i a . When the paddle moves with a piston motion and deep-water waves are to be developed, economics d i c t a t e a l i m i t a t i o n of the torque demand at high speeds. There are a number of ways of l i m i t i n g the high speed torque demand. The operator of the piston-type wave generator at the University of Washington i s required to not exceed a c e r t a i n crank throw at higher speeds. Torque l i m i t e r s or motor-current l i m i t e r s can be used, but at added expense. NRC. describes i t s approach to t h i s problem, for a hinged-flap generator, i n t h e i r report on the >-3 > to to < o CD 3 o *1 >-b o H» CO ct o 3 O 3 O 03 •-3 cr o o o •a & O •a ct cr o o 03 M O c cV H" O 3 O M) 03 X H* 3 c 3 03 & CL M CD o o CD W <! 03 *i H" O c co o 03 fV) 50 T J 3 ! O ct cr re . C ; 03 • i CD *J M £ 3 Ul M Ul LO -Cr ON ON UI 00 oo M o Ul M LO Ul » VJl CRANK RPM h-1 O O —J UI UI O ro oo -<] ro o H ro L (Inches) -Cr LO ro h-1 M Ul OO o -Cr do | f o r d. = 25" ON ro ON UI UI JCr - • ON ro LO Ul o ro vo ro ro ro •ON C = /f| .tanh f t . / s e c . M LO U3 ro M M ro ro vo o ro ON OO LO Ul —3 -Cr -Cr LO T = £ (sec.) : 1—1 o UI ro CD Pi s: vo -a ro a> D. s; LO -Cr CD P-s: -Cr -Cr Ul CD §" LO M ON CD P. s: M VO O CD P. s; P n ( l b . ) : T h i s i s max. value read from P i g . 31. -Cr -Cr CD pi s: ON o CD Pi s: UI ro oo CD h-1 ON OO CD P" LO LO ro CD p. —J ro o CD P-s: F^ ( l b . ) : T h i s i s max/ value read from F i g . 32 = . - --Cr Ul ON e d w . . I f c i x || x 3 0 ( U n i t s i n f e e t ) •'• -Cr OO o -Cr LO LO LO •Cr ro o LO h-1 -Cr -Cr. CO —J Max. F. ( l b . ) n ro o ro ro —3 -Cr ro -Cr h-1 ON ON 1—1 Ul M -Cr LO ro oo o Max. F 1 ( l b . ) 3 = = M • t—1 LO :os Q + j COS, 20 Sent Handbook -Table 2 p.7-38 Vlax. value f o r R/A = 1/7 -Cr H U3 -Cr oo vo ON o oo o ON vo LO M ro -Cr M CRANK VELOCITY v c ( f t . / s e c . ) = = = M ON O Estimated Paddle Wt. W (lb.) 111 M UI UI ro -Cr O -Cr ro ro ON O ro M O O o Max. Fim 2 = -ISW-Vc (cosO+yCOS 20) lb . g R •231, 123. CO CD co SH to CD T 3 W o 55 < 55 < K o 10 20 30 40 50 60 70 80 90 KJ < IH £ O o •P « I O O 00 rH CO CU 0) to CD -a ( 9 0 ) ( 8 0 ) ( 7 0 ) o rH|l- I II «•»!« ' OC O fe W & > PQ < EH fe W PQ < EH EH 55 W . 0 0 9 . 0 3 4 (60) (50) (40) ( 3 0 ) (20) (10) . 0 7 6 w rl < > 3 OQ EH O O i n ,500 .491 .466 .424 .132 .200 . 2 7 7 361 .443 . 3 6 8 .300 . 2 2 3 . 1 3 9 .052 (0) . 5 3 6 - . 0 3 6 » w < En I i n II •p «• I o CO rH 55 H CO iin 1.000 . 9 8 2 .932 .848 • 736 .600 446 .278 .104 -.072 w rl > to CD CD to CD o -o 5 3 EH o -p « rH|C-t l o C O PC rH O w fe 90 79.1 6 8 . 8 5 8 . 0 4 7 . 4 3 6 . 9 26.5 1 6 . 1 5 . 9 0 - 4 . 1 0 -p o 00 rH CO O CJ 0 -.189 - . 3 6 2 - . 5 3 0 . 6 7 7 -.800 - . 8 9 5 - . 9 6 1 - . 9 9 5 . 9 9 8 T JEL on I m as ^ PQ < 55 EH M CO •• EH -9- 55 W O « W CO w . 1 9 8 . 3 8 8 . 5 6 2 . 7 1 3 . 8 3 7 ,928 . 9 8 6 1 . 0 0 9 1.000 h.143 00 r o 1 t — * fe o • CM CM CO W O J O CQ , < «!•» EH + •» EH CD 25 W CO « O O * 1.143 1.119 1.049 . 9 3 8 .791 . 6 1 8 .429 . 2 3 3 • 039 45.1 RPM -P rH O I CO •=r o CO (I fe co o • O X fe - 9 1 -174 -254 -323 -380 -429 -462 -478 X i ri ""N CM •P ' « W O I CM O I CO l i rH « r i fe 55 M CO • X ' < • -•H £ fe — o CM CO o p e f U x i + rH O CO o o — ' > -20 CM rH II |CC bO I II £ J T fe X • • B •< *ri £ fe -111 -20 -19 -17 -15 - 1 0 9 fe + •rt fe fe II fe rH .—N -e-+ CD 55 H CO -e-O w CO fe -P fe . -131 -220 -102 -91 - 2 9 5 - 3 6 2 -77 -12 - 9 - 6 - 6 0 -42 -23 - 2 •479 + 1.4 - 4 +16 -415 -452 -480 -491 -484 -462 - 4 4 -115 -203 - 2 9 6 - 3 7 8 - 4 4 5 53.4 RPM .p CO « -sr o oo 1  fe CO o • c s fe — ' p « f-I CM O I C O || fe 55 H CO X < Ti £ fe " C D CM CO CC l«* -jQ H- rH C D i n c o m O rH O I - 2 3 5 - 2 3 - 3 0 0 - 3 5 4 - 3 9 6 -484 -488 •462 C a r m i c h a i l , C o l i n - KENT'S MECHANICAL ENGINEERS' HANDBOOK. Des ign and P r o d u c t i o n Volume, 12TH E d i t i o n , John Wi ley and Sons , New York , 1950 MAX. r0RQUE=488X10.5=5120 in.lbj MAX. TORQUE - 4820 i n . l b . -425 -440 -20 - 1 6 CM O > CM II II fe ^ X fe — - 1 2 7 - 1 0 7 -84 fe •rt fe fe II fe •©-••Hi-C D 55 H CO o w CO fe II +• fe - 3 8 5 -427 - 4 5 4 -12 - 8 - 3 . - 5 8 - 3 2 - 5 -466 -216 - 3 0 5 - 3 8 0 - 4 3 3 -465 T448 -458 -453 6 6 . 5 RPM 88 RPM X i . 4J H p l m 00 o co 1  X i rH CM , 1 K o II fe fe I co 'z o • H c j X i o fe W l i , w X < C D CM CO o ^ C c f « > * X i H- r H C D o co O CM o l > CM rH II II I c e E l tO -r fe X e < •H 2 fe ^ -121 'p224 -177 -204 -226 -177 - 3 6 7 -145 -298 r 1 0 7 -321 J - 6 7 -220 E fe H^ fe fe II fe -565 -197 -166 -130 -90 - 4 9 -578 -569 -542 -495 -437 H> H > O 55' M; co -e-O w CO fe I) •p fe +> rH OO I O CM O CO || rH ^ c fe CO o • C_> X fe -219 - 3 2 5 -405 - 4 5 4 -459 -430 0 - 3 9 - 7 4 ^ rH •P « VO VO t -I u I o CO rH 55 H CO fe •H fe X < s - 7 6 6 • - 7 5 3 -714. O CM CO O O • ccTo + rH C O CM CO ca O O l N O 1  > CM bO ^H H fe SI • . X e < H £ fe -422 -412 -388 fe + •vH fe fe IS fe •©-55 M CO o w CO fe II 4-fe - 1 1 8 0 -1204 -1176 -108 - 1 3 8 - 1 6 2 -182 -195 - 6 5 0 •346 -564 -460 -342 -213 -292 -228 -158 - 8 6 -1104 - 9 9 4 - 8 5 0 -682 •494 MAX. TORQUE = 4830 i n . l b . ' 1 MAX. TORQUE = 7460 i n . - 2 3 9 -456 -620 105 RPM •P o CO rH X3 rH •sr XT rH II fe fe X C £ X3 rH - 5 2 -76 - 7 0 7 -711 - 6 3 3 - 4 8 7 l b . - 9 8 -115 - 1 2 9 - 1 3 8 -1410 -1284 C D CM c o O ^ a CCI«* X5 H-C D CM « rH CO O i n O VO l O 1 0 C M O II CO 1  > rH « E CM to ^ fe rH fe 55 M » II • CO X . X < E < •H £ T 1 £ fe "-- fe - 5 5 2 - 4 9 3 -1113 - 9 0 8 - 6 7 5 •422 fe + •H fe fe u fe -2014 -1853 -416 - 3 2 6 - 2 2 6 -123 -1627 -1349 -1030 - 6 8 3 X i rH *—\ -e-+ C D 55 H CO O w CO fe II Jr-fe 135.5 RPM • -p -X3 . rH 1 t -00 0 CO t l rH c fe CO O • C_> X < c £ fe - 7 8 0 -32 U306O -1042 - 1 1 5 8 -1128 - 9 5 5 -673 MAX. TORQUE - 1 2 , 1 8 0 i n . l b . Low Force Sector High Force Sector -46 - 5 9 - 7 0 - 7 8 X i 1—1 •*-•% • P o « co CM I ro I o C O II rH fe CO X Ti £ fe C D lb CM f—^ H> CO . 0 X i + 0 rH PCl«* C D O ^-^ C D O E O •H 55 CO rH fe H O CO O 1 + •e-CM O II •H > fe O s= cc e W CM bD -rl • f CO rH fe C fe II • X fe II e < II H^ £ •p fe fe fe - 2 7 8 0 - 9 1 8 -820 -4010 -1556 -3646 •2410 Ul968 Ul462 i'ABLE IV. C a l c u l a t i o n s f o r Maximum Motor-Gear-box Output Torque ; Requ i red by the Large Plume Wave Genera to r f o r P i s t o n M o t i o n , Maximum Crank Throw o f 1 0 . 5 " and Water High Force Sector - 6 9 4 -541 - 3 7 5 - 3 1 6 3 -2597 -1915 -2050 -2250 -2180 -1775 MAX. TORQUE = 2 3,600 i n l b Low Force Sector CO 12,000 11,000 110,000 : 9000 8000 7000: 6000 5000 4000 3000 12000 1000 0 Fi e . Maximum Torque Required by the proposed Wave Generator f o r the Large 1 Flume at Various RPM 1 rnnK torque Tor / - U U I I N l l II UVV U I / • i V Note: C r a n k t h r o w i; mi+oH tn ax a e s i g n IIIIIIIGU W i (see water d ept X" ] n 01 ~ sect o n 6.5.10) y _ _ > » _ T o r q u e c u r v e for 10 -In \ k rip. «n<:Tnn -.pnr KminTrn / • — y M i n . u s a b l e =iUn + (88 3 R P M ) y i / c r a n k s peed 4 / / vrax. cruriK bueei I 1 d .48 | 8 n 15 ?- 3 4 1 u • • W # 1— f 1 1 l 1 II 1 1 i II • i i i 140; 130 120 110 100 90 80 70 60 50 40 30 20 C r a n k R P M 10 0 125. "Design of a Wave Generator for the Hydraulics Laboratory" (Ref. 7 - Wave Generators). I t was decided that the drive unit must provide at l e a s t 5120 i n . l b . of torque at 45 RPM and that the crank throw would be li m i t e d f o r high speed operation. The crank throw and water depth used i n the ca l c u l a t i o n s presented i n Tables III."and IV, were only decided upon a f t e r numerous other t r i a l c a l c u l a t i o n s had been made. The r e s u l t s of these other ca l c u l a t i o n s are summarized i n Table V, which l i s t s the maximum torque requirements for generating a wave having rj- =4, using d i f f e r e n t crank throws and water depths. 6.5.8 Selection of a Drive System The s e l e c t i o n of "a variable-speed drive system was narrowed down to a choice between eit h e r a variable-speed, DC motor system or a variable-speed, mechanical system. A f t e r considering speed regulation, costs and compactness of the f i n a l i n s t a l l a t i o n , i t was decided to use a variable-speed, DC motor, " R a t i o t r o l " system, pro-duced by the Boston Gear Co. The R a t i o t r o l system chosen i s a 10-Hp unit d e l i v e r i n g a constant torque of 5290 i n . l b . over a speed range of 88 - 3 RPM. The a v a i l a b l e torque exceeds the maximum required torque of 5120 i n . l b . Addition of the optional tachometer feedback c i r c u i t w i l l give a 1% speed regulation and reduce the e f f e c t s of changing l i n e 126 PISTON MOTION Crank Throw (inches) S t i l l Water Depth d (inches) Max. Crank Torque ( i n .lbs.) Crank RPM f o r Wave Having L/d = 4 9.5 22 3 7 5 0 48 23 3960 -24 4-120 _ 2 5 4200 45.1 10 22 4l40 48 25 4650 4-5.1 10 . 5 22 4540 48 25 5120 45.1 11 22 5040 48 25 5640 45.1 N o t e ; F o r p i s t o n mot ion the padd le d i sp l acement (e) equa l s the crank throw. TABLE V - Summary of Maximum Torque Requirements f o r the 3 9 * - ^ | " x 3 0 " x 3 6 " Flume Wave Genera to r f o r G e n e r a t i n g a Wave Hav ing L/d = 4 U s i ng P i s t o n Mot ion and D i f f e r e n t L i m i t s of Crank Throw and Water Depth . 127.. voltage. The maximum speed of 88 RPM w i l l give a true, deep-water wave i n 25" of water with a length of depth r a t i o of — = 1.15. This wave has a period of 0.68 sec. which i s s t i l l shorter than Galvin's preferred l i m i t of 0.75 sec. The minimum speed of 3 RPM exceeds the minimum usable speed of 14 RPM. Sp e c i f i c a t i o n s for ordering the recommended drive unit are given i n Appendix A. 6.5.9 Estimated Wave Height C a p a b i l i t y The maximum wave height c a p a b i l i t y of the wave generator w i l l be obtained using piston motion for waves of a l l periods and a maximum water depth of 25". Th e o r e t i c a l values of these wave heights were computed using equation (27), rearranged to give H = 2a = 2Ke, and pl o t t e d i n F i g . 40. S i m i l a r l y , wave heights f o r hinged-flap motion were calculated using the equation H = 2a = 2K^2. Values of K and K' were read from F i g . 19. For hinged-flap motion, the value of f l a p displacement (e) at the s t i l l water l e v e l f o r 25 f u l l crank throw, was taken as e = x 10.5 = 5.47", where 48" i s the height of the connecting rod attachment to the paddle, measured above the flume f l o o r . For normal paddle operation ( i . e . intermediate motion) the  wave heights obtained w i l l l i e between those for piston motion and  those f o r hinged-flap motion. 128, Galvin's observed wave height l i m i t a t i o n s due to depth, were plotted along with the height l i m i t a t i o n for deep-water waves due to H I " steepness ( — = if'.). Although B i e s e l ' s theories are mostly unproven, some research work has been done by Galvin concerning the wave height factor K/ (Ref. 3 and 4 - Wave Generator Theory). Using the 635-ft. long, 15 f t . wide, 20 f t . deep, concrete wave channel at the U.S. Army, Coastal Engineering Research Centre i n Washington, D.C, he found that wave heights obtained for hinged-flap motion were within 90% of the t h e o r e t i c a l height. This lends v a l i d i t y to Biesel's work and to h i s statement that wave heights for smaller sized wave generators w i l l be about 70% of the t h e o r e t i c a l height, depending on the s i z e of the paddle, the wave period, and the amount of water leakage past the paddle. Values of wave heights for 70% e f f i c i e n c y were computed and p l o t t e d i n F i g . 40. 6.5.10 High Speed, Crank Throw, L i m i t a t i o n From F i g . 39 i t i s evident that the crank torque at speeds above 72 RPM must be l i m i t e d to protect the e l e c t r i c motor against overload. I t was decided to accomplish t h i s by l i m i t i n g the crank throw. In F i g . 40 i t i s apparent that, even for hinged-flap motion, the crank throw required to generate a wave of maximum height having TJ- = 2, i s small. The actual value can be obtained from the equation H = 2 K'e where e = 25 7-5- x. crank throw. Solving gives a crank throw H 48 x 2K' 25 + - i ~ l i t l | 1 1 1 1 i.I 11; 1111111 Mimimtrnft | 7 0 % E f f i c i e n c y ( P i s t o n ) i i t t t t t t t t r ft 3&: i H i n g e d - ^ j g F l a p M o t i o n iii H = 2 K ' (e) •t-X m H-H •t-H-; i 1 i P i s t o n m o t i o n H = 2 K e 5 F i g . ^0 . T h e o r e t i c a l Wave Mak i rg C a p a b i l i t y o f t he Proposed Wave Genera tor f o r the Large F lume, f o r a Water Depth o f 25" arri Maximum Crark Throw o f 10 .5" . i H e i g h t at w h i c h w a v e s b e c o m e u n s t a b l e i n i w a v e c h a n n e l . i f 7 0 % E f f i c i e n c y ( F l a p ) m 6 6 . 5 A 5 3 . 4 3 4 5 . 1 i l l C R A N K R P M HP E s t i m a t e d w a v e h e i g h t p i s t o n m o t i o n . 2 6 . 3 HHtttW m m i t W a v e s c a n be g e n e r a t e d i n t h i s r e g o n u s n g l o w e r w a t e r d e p t h s . ffFH u e o f —r i s a M a x i m u m d e s i g n v a 34.8 f o r a w a t e r d e p t h o f 6" a n d a c r a n k s p e e d o f 14 R m m 8 10 11 12 13 14 15 16 17 18 130. 7.14 2x1.4 = 4.9 i n . A wave having T = 2 Is produced' i n 25" of water at a crank speed of 66.5 RPM ( F i g . 39). At f u l l crank throw of 10.5" the required crank torque i s 4830 i n . l b s . which i s less than the 5290 i n . l b s . capacity of the drive system. Since, as j u s t shown, a crank throw of 4.9 i n . i s a l l that i s required to get maximum wave height, even with hinged-flap operation, t h i s speed appeared as a good choice to s t a r t a crank throw r e s t r i c t i o n . I f the crank throw i s l i m i t e d to a maximum of 8", at a crank speed above 67 RPM, the crank torque at the maximum speed of 88 RPM r i s e s to only 4350 i n . l b s . g i ving f u l l p r o tection against overload. I t i s evident that the wave-making c a p a b i l i t y of the wave generator i s i n no way r e s t r i c t e d . against overload damage. In any case, the maximum possible over-If the operator forgets to use t h i s r e s t r i c t i o n , the drive system fuzing should protect the e l e c t r i c motor and con t r o l unit load i s 7460 5290 x 100 = 141% at 88 RPM which i s normally not serious f o r an e l e c t r i c a l system. When the wave generator i s b u i l t , both the speed co n t r o l unit and the crank should have the following warning attached: "Maximum crank throw l i m i t e d to 8" at speeds above 67 RPM " 131. 6.6 Design of Paddle Mechanism Geometry 6.6.1 Basic Mechanism Geometry With reference to F i g . 41. length of the arms AB and CD was a r b i t r a r i l y set at 51.000". Increasing the length of these arms reduces the amount of change In the opening between the end of the paddle and the flume bottom as the paddle o s c i l l a t e s , but has the disadvantage of increasing the paddle mass and i n e r t i a . Considering the s i z e of the crank d i s c and the method of attaching the connecting rod to the paddle, the attachment p o s i t i o n G was set 3.00" from A. The length of EC was set at 22.00". Increasing this length increases the separation between arm AB and CD making the paddle motionless s e n s i t i v e to adjustments of point D along the arc BD. The length of EC i s l i m i t e d by the necessity of having to clear the flume top edge when point C i s at i t s extreme p o s i t i o n C*. When the paddle i s i n i t s mean p o s i t i o n , point C must l i e on the arc of point A and therefore AE may be computed using the 2 formula shown i n F i g . 41, which gives EC = AE (2AB - AE) i . e . AE 2 = 2AE.AB + EC 2 = 0 AE 2 = 102.000AE + (22.000) 2 = 0 , A W _ + 102.000 ± V 102.000 - 4 (22.000) 2 =• 132. The required s o l u t i o n i s AE = 4.989" The length of AF i s l i m i t e d by the space a v a i l a b l e when the paddle i s i n the extreme p o s i t i o n for piston motion. Using the same procedure as for f i n d i n g AE, the length of AF must be shorter than AB by the amount h 2 - 102.000 h + 10.50 2 = 0 h = 1.09" To prevent binding when sand i s present i n the water, allow an operating clearance of 0.11". Therefore, AF = 51.00 - 1.09 - 0.11 = 49.80" 6.2.2 Paddle End-Clearance Plate The paddle w i l l c l e a r the bottom of the flume by 1.09 + 0.11 = 1.20" when i t i s i n i t s mean p o s i t i o n . This space w i l l permit considerable water leakage past the paddle with a loss i n wave height. To reduce t h i s leakage, use of an end-clearance plate i s recommended, the p r o f i l e of which i s shown i n F i g . 41. The p r o f i l e of t h i s plate can be s a f e l y approximated by two arcs, representing the extremes of p o s i t i o n of the end of the paddle F. throughout i t s complete range of possible motion. To maintain operating clearance, the length of AF i s taken as 49.91" rather than 49.80" when drawing t h i s p r o f i l e . The f i r s t arc i s F'F" with a radius of 1.09", centre B and running from the mean paddle p o s i t i o n F' to the extreme p o s i t i o n F" for hinged-flap motion. The second arc i s F"F with a radius of 49.91 - 3.00 =46.91", 155 1 0 . 5 0 " — Extreme Position for Piston Motion Extreme Position for Hinged-Flap Motion End-Clearance Plate, 22.000" -/ c> / O O / °* f D C a l c u l a t i o n s Summary AB = 517000" CD = 51.000" AE = 4.989" EC • 22.000" AF - 49.80" AG - 3.00" Clearance Plate Radius of arc F' F" , centre B - 1.09" Radius of arc F" F , centre G =46.91" Fig. 41. Developing the Geometry of the Paddle Mechanism 134. centre G and running from F", the extreme of hinged-flap motion to F, the extreme of piston motion. This p r o f i l e i s mirrored i n the second h a l f of the paddle o s c i l l a t i o n . Although an end-clearance plate with the curved p r o f i l e shown, would be more e f f i c i e n t , the curves may present manufactur-ing d i f f i c u l t i e s , i n which case a wedge shape could be used. In thi s case the side of the wedge would be a s t r a i g h t l i n e from F' tangent to the arc F"E. The end-clearance plate should be made of mild s t e e l and attached to the flume f l o o r with an epoxy r e s i n glue, to avoid d i s t o r t i o n of the flume f l o o r from the heat of welding. 6.6.3 Paddle S t r u c t u r a l Geometry A side view of the paddle s t r u c t u r a l geometry i s shown i n F i g . 42 along with basic dimensions. 6.7 Wave Generator S t r u c t u r a l Design Loads 6.7.1 General The design loads for various parts of the wave generator mechanism are presented i n t h i s section. The loads, as given, do not include load impact f a c t o r , due to the re c i p r o c a t i n g motion, or any safety f a c t o r . The actual s e l e c t i o n of members to carry these loads i s not de t a i l e d i n this t h e s i s , but the selected members are s p e c i f i e d i n the accompanying drawings (Appendix D). 135. To keep the i n e r t i a of the o s c i l l a t i o n members low, the connect-ing rod, paddle assembly and paddle support arms, were designed for construction using aluminium a l l o y 6061 - T6 (Alcan No.65S-T6). Other parts were designed for construction using mild s t e e l . In s e l e c t i n g s t r u c t u r a l components, r i g i d i t y was the guiding f a c t o r , and therefore, some of the chosen components are more than adequate i n strength i n order to ensure a minimum d e f l e c t i o n . 6.7.2 Crank and Crankpin Design Loads The overhung crank was designed i n accordance with pro-cedures outlined i n Kent's Mechanical Engineer's Handbook, page 7T34. The maximum bending moment when crank and connecting rod are at r i g h t angles, was taken as 150% of the output torque of the drive system, which i s 5290 x 1.50 = 7940 i n . l b s . The greatest bending moment on the crank, due to overhand of the crankpin and occurring when the crank i s on dead centre, r e s u l t s from a load of 1188 l b s . at 88 RPM. Maximum bending moment on the overhung crankpin i s 1.10 x 1204 = 1325 i n . l b s . and occurs at 88 RPM. Crankpin proportions were d e t a i l e d as given i n Kent, P.7-35. 6.7.3 Connecting Rod Design Load The maximum load i n pounds, acting along the centre l i n e of the connecting rod, was obtained from the equation 136 LTV i r y • i LTV Scale: size Note: G is position of connecting 1 rod fitting. Fig. 42. Side View of the Proposed Wave Generator Paddle Structural Geometry. 1 L F 137. F = c \ - ( | s i n 0 ) 2 given i n Kent's Mechanical Engineer's Handbook P.7-37. Using values of P and 0 from Table IV, the maximum sustained value of F^ when generating waves, was found to occur for P = 491 l b s . and 0 = 70°, at a crank speed of 45.1 RPM. The value obtained was F c = 495 l b s . acting a l t e r n a t e l y as a t e n s i l e and then compressive load. Should the crank throw l i m i t a t i o n be ignored and the wave generator run at 88 RPM using f u l l crank throw and a water depth of 25", the peak motor overload would be ^"frj x ^ 0 = 141%. Mostly splash and breaking waves would be generated and i t i s assumed that the operator would r e a l i z e h i s error and shut down. Should he not shut down, the motor would probably run for some time before overheating caused the fuzing to shut i t o f f , since the 141% overload occurs during only part of each paddle cycle. Under such conditions, the connecting rod load F c = 1204 l b s . for P = 1204 l b s . and 0 = 10° at a crank speed of 88 RPM. I t was decided to design the connecting rod for an a x i a l load i n compression of 1204 l b s . A short 6.62"-long rod, used to transmit the load from the end of the connecting rod to the frame of the flap, was designed in aluminium to transmit a load of 1204 lbs. The large diameter 138. was based on the need to l i m i t shaft d e f l e c t i o n to a maximum of .010" per foot, between bearings. Although the s p e c i f i e d bearings carrying the rod are s e l f - a l i g n i n g , i n that they a l i g n with the shaft, they are not of a type designed to handle a continually varying misalighment. 6.7.4 Paddle Design Loads Using the l o g i c employed i n determining the connecting rod design load (Section 6.7.3.), the paddle was designed to operate at 88 RPM at f u l l crank throw i n water 25" deep. The water forces acting on the paddle are the wave forces (F ) and the water i n e r t i a forces (F^). The point of a p p l i c a t i o n of F n on the paddle, measured from the flume bottom, was de-termined by f i n d i n g the height of the centre of gravity of the area of the wave pressure (P ) diagram using equation (29) as follows: Y n * p r 4 P gke cosh my cosh md cos kt (dy) = J p gke cosh my cosh md cos kt.y(dy) n o J y cosh my. dy o ^ cosh my. dy = (^- sinh my) .,J-~-~ o sinh my. dy — (sinh my) r . 139. = d sinh md (cosh md - 1 ) m sinh md _ , cosh md - 1 . m sinh md = (d - i tanh fL) m Z and for d = 25", m = ^ , L = 28.7" and ^ = 1.15 (Table III) - 28.7 . , 2 TT . y n = ( 2 5 " 2 t a n h 2 x 1.15 > = 25 - 4.54 = 20.46" Similarly the point of application of F^, measured from the flume bottom, was found using equation (30) as follows: O3C0 P. = pe E , C tan m d. cos m y . sin kt x n=l n n n J y. r d p . dy= rd , , ,. •'x oJ *x. J o y.(dy) y• = 0 J d P± .y-tey> 0;d P r dy Now o ^ p^ y. (dy) = pg o ^ y'(2 n_^ C n tan in^d sin kt cos mny).dy 140. 0 0 r d pg E . C tan m d s i n kt o 1 y cos m y . dy ° n = l n n  1 n J 00 ' V V d r ! pg E • C tan m d s i n kt {( s i n m y): J n=l n n m n o n rd s i n m y . dy j-o J n _ j m n r , 0 0 J . i rd s m m d , 1 , , .., -, = pg E . C tan m d s m kt { n + — o ( c o s m d - 1J> n=l n n { m I n J m n n , s i n m d r d °° j 1 • n and o J p . . d y = p g E , C tan m d s i n kt — * i ^ • n=l n n m n — oo 1 y i e l d i n g y. = E . C — ~ tan m d (m d s i n m d + cos m d -1) -.->: l n = l n m 2 n n n n ii s i n m d 00 jl E , C tan m d —; r n=l n n m n 2 e s i n m d where C = — — j : j — ; T from equation (31) . n s i n m d cos m d + m d n n n The serie s was evaluated f o r the f i r s t three terms using d = 25" = 2.08*, 1 d e = 10.5" = .875 u, and converting — in t o the form —-=— where m m a n n necessary: From F i g . 18 for j = 1.15 m^d = .64TT = 2.01 radians, m^d = 1.76TT = 5.52 radians, 141. m^d = 2.82TT = 8.85 radians, then C2 , - .976 (|^ f)2 (-2.12) { 2.01 (9904) + (- .472) - 1 } and y. = ^.Ul J j_ — ; .976 (-2.12) | ^ (.904) + (-.244) (|^||) 2 (-.951) {5.52 (-.689) + .825 - 1} + (-.244) (-.951) (-.689) .+ .1142 (f^lf) 2 (-.655) {8.85 (.548) + (-.837) - 1} + .1142 (-.655) | ~ | (.548) - .764 - . 1311 - . 0124 - 1.932 - .0603 - .016 .452' = 5.4" Forces due to mechanism i n e r t i a were considered to act 30" above the flume f l o o r . The s i t u a t i o n f o r maximum paddle s t r u c t u r a l loading due to external forces, i s shown i n F i g . 43. 2e (.904) (.904) (-.427)"+ 2.01 = 1.116e = .976 2e (-.689) (-.689) (.825) + 5.52 = -.279e = .244 2e (.548) . , ., . 0 (.548) (-.837). + 8.85" = ' 1 3 0 6 e = ' 1 1 4 2 142. Maximum loading at 88 RPM occurs at 10 of crank motion (Table IV). The values of external load are : P = 1204 l b s . F = 39 l b s . n F. = 753 l b s . x and F. = 412 l b s . x m At a crank angle of 10°, the angle <j> (Table IV) that the connect-ing rod makes with the h o r i z o n t a l , i s very small and P (Fig.43) can be considered a h o r i z o n t a l force. Also i n the f i r s t 10° of crank r o t a t i o n the paddle has moved only 0.091" and therefore the paddle was considered as heing at the extreme p o s i t i o n of crank throw, so that a v a i l a b l e measurements (Fig.41 and 42) could be used. The r e s u l t i n g errors are n e g l i g i b l e . The paddle weight was considered to act through the front of the paddle, giving a small error on the safe side since i t increased the load on the support arm set AB, which gave the design load used for a l l four support arms. The loads acting on the paddle structure, for conditions shown i n Fig.43, were computed as follows: a) Load on DC. Taking moments about point A, the load acting along DC was found to be 1740 l b s . Divided between the DC set 143. of two support arms, the load per arm i s 870 l b s . i n tension. b) Load on AB, Taking moments about point C, the load acting along AB was found to be 1860 l b s . Divided between the AB set of two support arms, the load per arm i s 930 l b s . i n compression. A l l four support arms were designed to carry t h i s load. c) Load on QC. Taking moments about point A, the load acting along QC was found to be 2990 l b s . Divided among 4 members, th i s would be 813 l b s . per member, i n compression. d) Load on AC. The sum of the v e r t i c a l forces at point C i s zero. Therefore, the load acting along AC i s 2460 l b s . Divided among 4 members, t h i s would be 615 l b s . per member, i n tension. e) Bending moment i n AF. Using bending moment and shear diagrams, the maximum bending moment i n AF was found to occur at point Q and to have a value of 25,700 i n . l b s . 144 Paddle Motion r — — 2 2 . 0 0 " S c a l e size Fig. 43. Diagram of the Case for Maximum Loading by External Forces Acting on the Paddle of the Proposed New, Wave Generator. 145. 6.7.5 Base Plate Reaction Loads The two paddle support arms on each side of the wave generator are connected to base p l a t e s , one on each side of the flume, into which a s l o t BD (Fig.43) i s cut. The h o r i z o n t a l separation between the arm ends B and D on the base plates i s 22". Assuming a minimum base p l a t e length of 30" between f l o o r attachments, spaced equal distances on e i t h e r side of B and D, the v e r t i c a l r e action at the B end of the base plate was estimated to be 678 l b s . Therefore, each end of a base p l a t e w i l l have to be anchored to the f l o o r with fasteners capable of r e s i s t i n g an u p l i f t i n g force i n the order of 680 l b s . 6.7.6 Other Component Design Loads Loads•in l a t e r a l members of the paddle assembly were com-puted as required when s e l e c t i n g components using the load data given. 6.7.7 Bearing Loads B a l l bearing units were designed for a minimum l i f e of 2500 hrs., but some units exceed t h i s , since use of large shaft d i a -meters, to l i m i t shaft d e f l e c t i o n , required use of bearings of larger load c a p a c i t i e s . B a l l bearing assemblies at e i t h e r end of the connecting rod were designed for a r a d i a l load of 1204 l b . 146. The b a l l - b e a r i n g assemblies at eit h e r end of the four support arms, were designed f o r r a d i a l loads of 930 l b s . each. The upper arm bearings are double-row b a l l bearings, selected to give the paddle mechanism l a t e r a l r i g i d i t y . The lower arm bearings are screw-on, s e l f - a l i g n i n g , hangar bearings. At 14 threads per inch, one h a l f a turn of these bearing units w i l l allow the support arms to be adjusted 0.036" i n length., i f assembly ad-justments are required. The s e l f - a l i g n i n g bearings w i l l avoid bending moments being applied to the support arms by~ any minor, baser-plate misalignments. 6.8 Corrosion and I n s t a l l a t i o n Problems Information concerning corrosion protection i s presented i n Appendix B. Problems pertaining to the i n s t a l l a t i o n of th i s wave generator i n the large flume are presented i n Appendix C. 6.9 Design Drawings Design drawings f o r the proposed, new wave generator for the 39'-4V1 long, 30" wide and 36" deep flume are included i n Appendix D. 6.10 Summary of Operating S p e c i f i c a t i o n s The summarized operating s p e c i f i c a t i o n s f o r the proposed new wave generator are as follows: maximum crank throw minimum crank throw = 10.5" = 0" usable wave period range design water depth usable wave period range at design water depth (equivalent crank speed range maximum usable.wave period at 6" water depth (equivalent crank speed minimum wave length maximum usable wave length ^ range for a depth of 25" -^ maximum for a depth of 6" estimated maximum wave height at design water depth 0.68 to 4.3 sees 25" 0.68 to 2.3 sees 88 to 26.3 RPM) 4.3 sees. 14 RPM ) 28.7" 206" 1.15 to 8.24 34.8 14" 148. BIBLIOGRAPHY A. Wave Theory 1. 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Anthony F a l l s H y d r a u l i c L a b o r a t o r y , U n i v e r s i t y o f M inneso t a , March 1954. 2. Have l o ck , T . H . -3. G a l v i n , C y r i l J , 4. G a l v i n , C y r i l J . -" F o r c e d Su r f ace Waves on Water " , P h i l o s o p h i c a l Magaz ine , V o l . b, S e r i e s 7, (P. 5 6 9 ) , 1929. "He igh ts o f Waves Generated by a F lap-Type  Wave G e n e r a t o r " , C o a s t a l E n g i n e e r i n g Research Cent re B u l l e t i n and Summary Report o f Research P rogress f o r F i s c a l Years 1965-66, U.S. Army C o a s t a l E n g i n e e r i n g Research C e n t r e , 5201 L i t t l e F a l l s Road, N.W., Wash ington , D .C . "Wave-Height P r e d i c t i o n f o r Wave  Genera tors i n Sha l low Water " , T e c h n i c a l Memorandum No. 4, U.S. Army C o a s t a l E n g i n e e r i n g Research C e n t r e , 5201 L i t t l e F a l l s Road, N.W. Wash ington, D .C . 151. B. Wave Genera tor Theory (Con t ' d ) 5. K r av t chenko , J . "Remarques s u r l e C a l c u l des Ampl i tudes  de l a Houle L i n e a i r e Engendree pa r ~un  B a t t e u r " , P roceed ings o f the F i f t h Conference on C o a s t a l E n g i n e e r i n g , G renob l e , F r a n c e , September 1954 p u b l i s h e d by C o u n c i l on Wave R e s e a r c h , E n g i n e e r i n g F i e l d S t a t i o n , U n i v e r s i t y o f C a l f o r n i a , Richmond 4, C a l i f o r n i a , 1965. 6. K r a v t chenko , J . and San ton , L. - "Sur Les P a r a s i t e s Dans un Cana l a Phenomenes Hou l e " P roceed ings o f the F i f t h Conference on C o a s t a l E n g i n e e r i n g , G renob l e , F r a n c e , September 1954 p u b l i s h e d by C o u n c i l on Wave Resea r ch , E n g i n e e r i n g F i e l d S t a t i o n , U n i v e r s i t y o f C a l i f o r n i a , Richmond 4, C a l i f o r n i a , 1955 7. Bouyssou , R, 8. Ap t e , A . S . -" D i f f r a c t i o n P l a n " , da L a Houle P roceed ings Devant F i f t h un B a t t e u r o f the i  Conference oh C o a s t a l E n g i n e e r i n g , G r e n o b l e , F r a n c e , September 1954 p u b l i s h e d by C o u n c i l on Wave R e s e a r c h , E n g i n e e r i n g F i e l d S t a t i o n , U n i v e r s i t y o f C a l i f o r n i a , Richmond 4, C a l i f o r n i a , 1955. "Harmonics F i f t h o f a Wave", Conference G r e n o b l e , France P roceed ings o f the on C o a s t a l E n g i n e e r i n g , , September 1954 p u b l i s h e d Wave Resea r ch , E n g i n e e r i n g , U n i v e r s i t y o f C a l i f o r n i a , Richmond 4, C a l i f o r n i a , 1955. by C o u n c i l on F i e l d S t a t i o n 9. S c h u l e r , M. -10. Ross , J . and Bowers "E rzengung von O b e r f l a c h e w e l l e n durch schwlngeride Ko rpe r " , Z e i t s c h r i f t f u r Angewandte Mathematik und Mechanik , V o l . 16, No. 2, p p . 65-73, A p r i l , 1936. » C ' E « * " L a b o r a t o r y Su r f ace Wave  Equipment - A Summary o f L i t e r a t u r e " , S t . Anthony F a l l s H y d r a u l i c L a b o r a t o r y , U n i v e r s i t y o f M i n n e s o t a , P r o j e c t Report No. 38, November, 1953-152. C. Wave Genera tors 1. R a n s f o r d , G.D. 2. S m i t h j A l l a n A. 3. L t . C o y e r , C B . 4. P r e t i o u s , E .S . 5. G r i d e l , H. -6. Hsu , E .Y . -7. N a z z e r , D.B. -8. J ohnson , J .W. -"A Wave Machine o f Nove l T y p e " , I n t e r n a t i o n a l A s s o c i a t i o n f o r H y d r a u l i c S t r u c t u r e s Research ( IAHSR), T h i r d Mee t i ng , G r e n o b l e , F r ance , September 1949. " L a b o r a t o r y Wave Gene ra t i on - A Comparison  o f T h e o r e t i c a l and Expe r imen t a l  Per f o rmance " , I n t e r n a t i o n a l Asso c i at i on f o r H y d r a u l i c R e s e a r c h , Tenth C o n g r e s s , London , E n g l a n d , V o l . 1, 1963. "A Mu l t i -Pu rpose Wave Gene ra to r " -P roceed ings o f M innesota I n t e r n a t i o n a l H y d r a u l i c s C o n v e n t i o n , A j o i n t meet ing o f the I n t e r n a t i o n a l A s s o c i a t i o n f o r H y d r a u l i c Research and H y d r a u l i c s D i v i s i o n , American S o c i e t y o f C i v i l E n g i n e e r s , U n i v e r s i t y o f M i n n e s o t a , September 1953. "A Genera l-Purpose Wave Genera to r t o  Produce a l l Types o f Water-Waves i n Short  F l u m e s " , Department o f C i v i l E n g i n e e r i n g , U n i v e r s i t y o f B r i t i s h Co lumb ia , J u l y 1967. " L a R e p r e s e n t a t i o n Pes Phenomenes Mar lns  su r Modeles R e d u l t s " , P roceed ings o f the F i f t h Conference on C o a s t a l E n g i n e e r i n g G renob l e , F r a n c e , September 1954 p u b l i s h e d by C o u n c i l on Wave R e s e a r c h , E n g i n e e r i n g F i e l d S t a t i o n , U n i v e r s i t y o f C a l i f o r n i a , Richmond 4, C a l i f o r n i a 1955. "A Wind, Water-Wave Research F a c i l i t y " , T e c h n i c a l Report No. 57, Department o f C i v i l E n g i n e e r i n g , S t a n f o r d U n i v e r s i t y , October 1965. "Des i gn o f a Wave Genera to r f o r t h e _ H y d r a u l i c s  L a b o r a t o r y " , L abo ra to r y Memorandum EG-9 , D i v i s i o n Of Mechan i ca l E n g i n e e r i n g , N a t i o n a l Research C o u n c i l , Ot tawa , Canada, March 1, 1956. " L a b o r a t o r y F a c i l i t i e s i n the U n i t e d S t a t e s  f o r Research on Water G r a v i t y Waves" i rj C o u n c i l on Wave R e s e a r c h , The E n g i n e e r i n g F o u n d a t i o n , U n i v e r s i t y o f C a l i f o r n i a , B e r k e l e y , C a l i f o r n i a , A p r i l I960. 153. C. Wave Genera tors (Con t ' d ) 9. Snyder , C M . , W i e g e l , R.L. , and Be rme l , K . J . - " L a b o r a t o r y F a c i l i t i e s f o r S t u d y i n g Water G r a v i t y  Wave Phenomena", P roceed ings o f S i x t h Conference on C o a s t a l E n g i n e e r i n g , C o u n c i l on Wave R e s e a r c h , The E n g i n e e r i n g F o u n d a t i o n , B l d g . 159, Richmond F i e l d S t a t i o n , U n i v e r s i t y o f C a l i f o r n i a , Richmond 4, C a l i f o r n i a , 1958. 10. C a l d w e l l , J . M . - "Des ign o f Wave C h a n n e l s " , P roceed ings o f the F i r s t Conference on Sh ips and Waves, C o u n c i l on Wave R e s e a r c h , The E n g i n e e r i n g F o u n d a t i o n , Richmond F i e l d S t a t i o n , U n i v e r s i t y o f C a l i f o r n i a , Richmond 4, C a l i f o r n i a , 1955. 11. Corps o f Eng inee r s - "Summary o f C a p a b i l i t i e s " , M i s c e l l a n e o u s Paper No. 3-6 4, U.S. Army C o a s t a l E n g i n e e r i n g Research C e n t r e , A p r i l 1964. 12. ( S t a f f ) - " P r o v i s i o n a l Notes on a Gene ra l Purpose Pneumatic Wave Mach ine " , Mechan i s a t i on S e c t i o n T e c h n i c a l Memorandum 2/1963, H y d r a u l i c s Research S t a t i o n , W a l l i n g f o r d , B e r k s h i r e , E n g l a n d , November 19.63. D. Mechanisms 1. Schwamb, P. and M e r r i l l , A . L . - E lements o f Mechanism, New York , John Wi ley and Sons, l y i o . 2. Angus , Robert W. - The Theory o f Mach ines , New York , Mc G raw-H i l l Book C o . , 1917. 3. Marks , L i o n e l S. - Mechan i ca l E n g i n e e r s ' Handbook, F i f t h E d i t i o n , New Yo rk , McGraw-Hi l l Book C o . , 1951. 4. C a r m i c h a i l , C o l i n - K e n t ' s Mechan i ca l E n g i n e e r s ' Handbook, Des ign and P r o d u c t i o n Volume, T w e l f t h E d i t i o n , John Wi ley and Sons , New Y o r k , 1950. 154. D. Mechanisms (Con t ' d ) 5. D u r l e y , R.J . - K inemat i cs o f Mach ines , Second E d i t i o n , John Wi ley and Sons , London , 1911. 6. Dunke r l e y , S. - Mechanism, T h i r d E d i t i o n , Longmans, Green and C o . , London , 1914. E. R e l a t ed Sub j ec t s Bowers, C . E . and H e r b i c h , J . B . - "An Expe r imen t a l Study o f Wave A b s o r b e r s " , P r o j e c t Report No. 5 4 , S t . Anthony F a l l s H y d r a u l i c L a b o r a t o r y , U n i v e r s i t y o f M i n n e s o t a , January 1957. H e r b i c h , John B. - " E x p e r i m e n t a l S t ud i e s o f Wave F i l t e r s and A b s o r b e r s " , P r o j e c t Report No. 44, S t . Anthony F a l l s H y d r a u l i c L a b o r a t o r y , U n i v e r s i t y o f M inneso t a , January 1956. L e a n , G.H. -H a m i l l , P.A. -"A S i m p l i f i e d Theory o f Permeable Wave  A b s o r b e r s " , J o u r n a l o f H y d r a u l i c R e s e a r c h , V o l . 5, No. 1, 1967. " Expe r imen t a l Development o f a P e r f o r a t e d  Wave Absorbe r o f S imple C o n s t r u c t i o n and  Minimum L e n g t h " , Mechan i ca l E n g i n e e r i n g Report MB-252, N a t i o n a l Research C o u n c i l o f Canada , Ot tawa, N .R .C . No. 7472, May 1963. Goda, Y. and Ippen , A . T . - " T h e o r e t i c a l and Expe r imen t a l I n v e s t i g a t i o n o f Wave Energy p i s s i p a t o r s  Composed o f Wire Mesh S c r e e n s " , Hydrodynamics Labo ra to r y Report No. 6 0 , Department o f C i v i l E n g i n e e r i n g , Massachuset ts I n s t i t u t e o f Techno logy , August 1963. 6. Z o l l i n g e r , H.A. - "A-C Motor Ad jus tab l e-Speed D r i v e s " , Machine D e s i g n , February 3, 19&6. 155. E. R e l a t e d Sub jec t s (Con t ' d ) 7. Aluminum C o . o f Canada L t d . , - "Handbook o f A luminum", Aluminum Co . o f Canada L t d . , 1260 Vu l can Way, Richmond, B . C . , 1957. 8. Aluminum Co . o f Canada L t d . , - " S t r e n g t h o f A luminum", Aluminum Co. o f Canada L t d . , 1260 Vu l can Way, R ichmond, B . C . , August 1968. 9. Canadian SKF Company L t d . , - "SKF B a l l and R o l l e r B e a r i n g s " , Canadian SKF Co . L t d . , 2201 E g l i n t o n Ave. E a s t , S c a r b o r s , O n t . , Des ign Cata logue No. 551. 10. H o r g e r , J . Oscar - ASME Handbook "Meta l s E n g i n e e r i n g D e s i g n " , F i r s t E d i t i o n , Mc G raw-H i l l Book C o . , T o r o n t o , 1953. 11 . Gray , A . , and W a l l a c e , G. - " P r i n c i p l e s and P r a c t i c e o f E l e c t r i c a l E n g i n e e r i n g " , S i x t h E d i t i o n , Mc G raw-H i l l Book C o . , London, 1947. 12. Dwight , H.B. - "Ma themat i ca l T a b l e s " , T h i r d E d i t i o n , :Dover P u b l i c a t i o n s , I n c . , New Yo rk , 1961. 13. Greenwood, D.C\ - "Mechan i c a l Power T r a n s m i s s i o n - Component S e l e c t i o n and A p p l i c a t i o n " , Mc G raw-H i l l Book C o . , To ron to* 1962. 14. P r e t i o u s , E .S . - " C . E . 560, H y d r a u l i c E n g i n e e r i n g f o r R i v e r s , Harbours and C o a s t s " y Graduate C o u r s e , Department o f C i v i l E n g i n e e r i n g , U n i v e r s i t y o f B r i t i s h Co lumb ia , 1968. F. WaVe Genera tor Correspondence ( F i l e s o f P r o f . E .S . P r e t i o u s ) 1. T i f f a n y , J . B . - U.S. Army Eng inee r Waterways Exper iment S t a t i o n , Corps o f E n g i n e e r s , O f f i c e o f the D i r e c t o r , V i c k s b u r g , M i s s i s s i p p i l e t t e r dated 28 February 1964. 2. T u r n e r , E. -- H y d r a u l i c s S e c t i o n , N a t i o n a l Research C o u n c i l , Ot tawa, l e t t e r s dated 8 October 1964 and 27 J u l y 1964. 156. F. Wave Gene ra to r Correspondence (Con t ' d ) ( F l i e s o f P r o f . E .S . P r e t i o u s ) Numata, E. -Be rme l , K . J . -Head, Sh ip Research D i v i s i o n , Dav idson L a b o r a t o r y , Stevens I n s t i t u t e o f Techno logy , C a s t l e Po in t S t a t i o n , Hoboken, New J e r s e y , 2 June 1964. S en i o r E n g i n e e r , Department o f C i v i l E n g i n e e r i n g , U n i v e r s i t y o f C a l i f o r n i a , B e r k e l e y , 24 F e b r u a r y , 1964. 5. L t . C o l . S t e v e n s , M.E. - D i r e c t o r , C o a s t a l E n g i n e e r i n g Research C e n t r e , Corps o f E n g i n e e r s , U.S. Army, 5201 L i t t l e F a l l s Road, N.W., Washington D . C , 19 February 1964. 6. C o l . D i e r c h s , F .O . - D i r e c t o r , C o a s t a l E n g i n e e r i n g Research C e n t r e , Corps o f E n g i n e e r s , U.S. Army, 5201 L i t t l e F a l l s Road, N.W., Washington D .C . , 4 October 1965. Seddon, A . E . -8. G a l v i n , C . J . -A s s i s t a n t D i r e c t o r , H y d r a u l i c s Research S t a t i o n W a l l i n g f o r d , B e r k s h i r e , E n g l a n d , 27 February 1964. Oceanographer , Research D i v i s i o n , C o a s t a l E n g i n e e r i n g Research C e n t r e , 5201 L i t t l e F a l l s Road N.W., Wash ington , D .C . [Copy o f l e t t e r dated November 17 , 1965 and addressed to P r o f e s s o r B.M. Hand, Department o f Geo logy , Amherst C o l l e g e , Amherst , M a s s a c h u s e t t s , U . S .A . ] 157. Appendix A Variable-Speed Drive Unit f o r the Proposed New Wave  Generator f o r the Large Flume RPM range; 8 8 - 3 Recommended u n i t : 10 Hp. Boston Gear PE Series R a t i o t r o l System. Reference: Boston Gear Catalogue No. 59, page 623. Torque output: 5290 i n . l b . Motor: 10 Hp. foot mounted DC motor, catalogue No. 241001 Reductor: catalogue No.U 152 F - 20, page 286 R a t i o t r o l u n i t : PE s e r i e s , catalogue No. PE 1000 equipped with "Tachometer Feedback", item 24, page 658 Weight of motor and reductor = 470 + 200 = 670 lb s . Manufacturer: Boston Gear Works, Quincy 71, Mass., U.S.A. Vancouver D i s t r i b u t o r : Robert Morse Corp. Ltd. Notes: a) Tachometer feedback on the motor gives an improved speed regulation of 1% and minimizes the e f f e c t s of changing l i n e voltage. b) The worm gear service factor for moderate shock up to 10 hrs./day i s 1.2. The maximum equivalent torque r e -quirement i s 5290 x 1.2 = 6350 i n . l b . i n the speed range of 40 to 50 RPM which i s within the reductors capacity. l5'8 • Appendix B Corrosion Protection for the Proposed New  Wave Generator for the Large Flume The aluminium a l l o y 6061 (65S-T6) has excellent resistance to atmospheric and fresh water corrosion, and f a i r resistance to sea water. I t i s not considered necessary to paint the aluminium parts of the paddle mechanism. A l l f i t t i n g s i n d i r e c t contact with the aluminium should be eit h e r zinc-plated or cadmium-plated, or made of s t a i n l e s s s t e e l . Cadmium-plated fasteners are r e a d i l y a v a i l a b l e and are recommended. The bearing assemblies unfortunately necessitate contact between aluminum, and s t e e l . To avoid crevice corrosion, i t i s recommended that the j o i n t s between the bearing units and the housing, and between bearing units and shafts, be sealed against the ingress of moisture using a polysulphide or b u t y l rubber sealant. Where f e a s i b l e , the s t e e l components should be undercoated and painted to prevent r u s t i n g . 159. Appendix C Problems Relative to the I n s t a l l a t i o n of  the Proposed New Wave Generator i n the  Large Flume The i n s t a l l a t i o n of the wave generator i n the 39'-4V 1 long, 30" wide and 36".deep flume, w i l l require attention to the following problems: a) The flume i n t e r i o r i n the region of the paddle i n s t a l l a t -ion has a l e v e l f l o o r and true sides up as high as the h o r i z o n t a l angle running along the flume sides about 14" up from the flume f l o o r . (Fig.37). Above t h i s height, the sidewalls bulge s l i g h t l y . I t i s recommended that these bulges be f i l l e d with a q u a l i t y a u t o - f i l l e r and sanded true. b) A cross member j o i n s the two sides of the flume at the point of paddle i n s t a l l a t i o n . This must be removed and replaced by two cross members - one. i n front of and one behind the paddle. c) The glass sides of the flume are not l a t e r a l l y braced at the top. Large waves create a f l u c t u a t i n g l a t e r a l pressure as they run down the flume. To avoid f l u c t u a t -ing changes i n flume cross section, two clamp-on cross 160. Appendix G members should be made up and f i t t e d . These members should be removable, so that they do not i n t e r f e r e with instrumentation. d) The threaded gate rod and wheel which r a i s e and lower the t a i l gate, must be moved out of l i n e with the wave generator, connecting rod and crank. e) Pipes running along one side of the flume must be removed to permit i n s t a l l a t i o n of the base plate supports. f) The base plate supports require a firm foundation. The asphalt covering the f l o o r must be removed at the points of support and the supports bedded on poured concrete pads with the anchor b o l t s f i r m l y bedded i n the concrete f l o o r . Because of the r e c i p r o c a t i n g load, the reaction load of 680 l b s . at each support (Section 6.7.6) should be m u l t i p l i e d by an impact factor of 1.5 and each support made capable of r e s i s t i n g a v e r t i c a l p u l l of 680 x 1.5 = 1020 l b s . g) A mount w i l l be required for the drive u n i t . I t must 161. Appendix C be designed for an overturning force, allowing for impact, of 1204 x 1.5 =1800 lbs. This force acts through the crank-wheel drive shaft, alternating fore and aft along a horizontal line parallel to the flume longitudinal axis. This drive-unit mount should, also be bedded into the concrete floor slab. h) The gate slots between the glass panels must be f i l l e d i f a smooth wave profile i s tc be obtained. 162. Appendix D Design Drawings f or the Proposed New Wave  Generator for the Large Flume SHEET 1 Paddle SHEET 2 Connecting Rod, Support Arms and Base Assembly SHEET 3 Adjustable Crank SHEET 4 Wave Generator Assembly and I n s t a l l a t i o n Data (Cj) 3 i x 2 2"x"fe ANGLE (2 as shown, 2 leg other side) 2.50 1 C4> l^x.095 TUBING (2 pes) 3.40 SECTION A-A. SCALE: -£= I" DETAILS OF CONNECTING ROD BEARING INSTALLATION. SECOND BEARING, DETAIL SAME,ON MEMBER OPPOSITE A-A. 3 HOLES EQUALLY SPACED. _ TWO HOLES 1L DRILL. USE - | RD HD BOLT, AMER. STD 6 d PLAIN LT WASHER, HEX NUT, ALL CD PL STEEL. THIRD HOLE AS SHOWN. SEALMASTER BALL BEARING 3-BOLT FLANGE UNIT ~ 2 _ L F - I 6 (2 REQ'D) LET. F DRILL, -|-I8UNC-2B USE -ft-- 18 UNC - 2A xl^-HEX HD MACH SCREW, •ft AMER STD PLAIN LT WASHER. ALL CD PL STEEL. 7.25 x 6.74 x.250 PLATE A 3 I.P.S. AL PIPE TOP VIEW A L L MATL: 6 5 S - T 6 (606I-T6) ALUMINUM WELDING SPEC: PROCESS-SMAC (MIG or TIG) FILLER A L L O Y - 33S (4043) SCALE: I" FRONT VIEW (C9)ARM END BEARING SHAFT (2 req'd) -J--I4UNF-2AX I" 4.50 NUT AND WASHERS - SAME AS ( C 8 ) CIVIL ENGINEERING DEPARTMENT UNIVERSITY OF BRITISH COLUMBIA P A D D L E FOR 39* FLUME WAVE GENERATOR DESIGNED BY: E.R. CHAPPELL DRAWN BY; E.R. CHAPPELL DATE : JULY 69 SCALE: 1/4 = I REF: THESIS "THEORY AND DESIGN OF A WAVE GENERATOR FOR A SHORT FLUME" SHEET NO. I OF 4 i C R A N K SIDE + 3.31 -—-14 N . P . S . M . x .34 1.6264 1.6260 1.615 1.32 JL DRILL AND R E A M FOR .31 NO.O TP (STN STL) x l ' WITH CROSS-PIECE IN POSITION. 1.050 ~ i 69.71 67.70 I 1.00 7T-35-1-4 SEALMASTER HANGER BEARING SEHB^22, S H A F T DIAM. I 8 I^ I.P.S. PIPE KOOOO .9995 1.90 , 6 . 6 2 4-1.32 1.6250 1.6256 CROSS-PIECE PRESS INTO END OF CONNECTING ROD. FACES SHAFT'S SUPPORTED SIDE -X-M NPSM x .94 4 1.050 _ 49.31 44.61 23 CHAMFER ALL AROUND S C A L E : ^ = I I B 1.00 _ L _ J SEALMASTER HANGER BEARING SEHB-16 SHAFT D I A M I" 2 x.083 ROUND TUBING 1.50 CONNECTING ROD MATL:65S-T6(606I-T6) ALUMINUM WELDING: PROCESS SMAC (MIG or TIG) FILLER ALLOY 3 3 S ( 4 0 4 3 ) 2.0452 2.0465 DRIVE FIT SKF DOUBLE ROW ANGULAR CONTACT BALL BEARING UNIT NO. 3205 SUPPORT ARM (4 req'd) .375 1 — > • .0000 . 9 9 9 5 T 1.63--2.88 . 7 5 0 1 1.0013 1.0008 T 4 5 C H A M F E R T C C L E A R ANGLE FILLET SUPPORT ARM LOWER AXLE (4 req'd) SCALE •. l" I" DRILL, 2 HOLES I MED PLAIN WASHER .750 i f . 5 DRILL HOLES FOR TIE-DOWN BOLTS (USE 4/PLATE) _ L 5x3-±-x-|-4-+ -h BASE PLATE (2 req'd, 2nd reversed) ADJUSTABLE„ A X L E (2 req'd) i.oooo D I A I. 0008 SCALE-..^ |" L E V E L AND SET AT SAME E L E V . A S FLUME FLOOR S U R F A C E WHEN I N S T A L L E D . 750 .750 l M - 12 UNF- 2A xi3-T-FIN HEX BOLT WITH FIN HEX NUT. CUT BOLT TO SHORTEN LENGTH TO 2.30. CHECK DIA. WHEN SELECTING 2 REQ'D PER A X L E UNIT. F L U M E FLOOR 4 POSITION A X L E ON OTHER SIDE OF SECOND UNIT -J--l6UNF-2Axl2f-FIN HEX BOLT TO HOLD BASE PLATE, 4- M E D PLAIN WASHERS ' UNDER HD AND NUT, H E X NUT {6 sets req'fi) Z.00 1.0000 " 5 1 7 " 3 6 . 4 0 INSTALLED 5.00 1.64 50 6.25 R . 5 0 13.31 .31 1.63 i 1.50 t 4.82 13.00 4.8 Z T K-.750 SCALE: \" 2 HOLES BASE ASSEMBLY MATL: S T E E L ( 1 0 2 0 H 0 2 5 ) BASE PLATE SUPPORTS (2 sets req'd, 2nd set reversed) 2.000>+U 2.625 2.07 CIVIL ENGINEERING DEPT. UNIVERSITY OF BRITISH COLUMBIA C O N N E C T I N G R O D 8 S U P P O R T A R M S t and B A S E ASSEMBLY FOR 3 9 ' F L U M E W A V E G ENERATOR DESIGNED BY: E .R . C H A P P E L L DRAWN B Y : E.R. CHAPPELL DATE : JULY 69 SCALE : AS SHOWN REF: THESIS "THEORY AND DESIGN OF A WAVE GENERATOR FOR A SHORT FLUME' SHEET NO. 2 OF 4 28.07 13.81 14.26 REF. 13.44 i — r SCREW OMITTED FOR CLARITY — T USE -?>—13 UNC — 3 A x -jsp HEX SOCKET, CUP PT, SETSCREW. — I-Q--I2 UNF-2B FIN HEX NUT. — J H — -SCALE! rjT ~ I NO. 8-32 UNC - 2A x 2T PAN HD MACH SCREW, NO.8 AMER STD LT LOCK WASHER, 3 SCREWS -NO. 4/0 x .75 TAPER PIN "B" NO. 8-32 UNC - 2 A HD MACH SCREW, NO,8°AMER STD LT LOCK WASHER, CUT .090 THICK WASHER AS SPACER BEHIND POINTER, 2 SCREWS ~j CONNECTING ROD <t NO. 10 - 32 UNF-2Ax ^ HEX HD MACH SCREW, NO. 10 AMER STD LT LOCK WASHER, 10 SCREWS. ^ 2 SHIM ADJUSTMENT SCREW WITH WASHERS FOR ZERO LENGTHWISE FREE PLAY. S C A L E : l " = l " ^^^\^TL». N$T"E»E. L> (SAE 1020-1035) 2.50 *.62 **.62 ** .62 -> .25-4.00 -(CHANNEL WEB ) ^ .939 .937 (ASSEMBLED) TWWiferi — — — - f La— -——mm ^ ^^^mi • • ^ J ~ l 1 I I I I USE ~-\3 UNC -3A x l " HEX SOCKET, CUP PT, SETSCREW irnnnnrnn UUUVUUUU nnmrnnnninnmnnnnnnnnmrrinnmnrpnn1 MAX CRANKPIN \TRAVEL 10.50 fA9l C R A N K A R M .375 2.00 .3125 .3165 .19 DIA. NO. 42 DRILL, REAM FOR NO. 4/0 x .75 TAPER PIN ]%• DRILL ,A5)ADJUSTING W H E E L NO. 10 DRILL, 10 HOLES SD P L A T E . 3 6 9 0 4 06 <A3) A D J U S T I N G S C R E W -g- SQ. KEY WAY I .75C0 1.719 DIA. , DRILL 1.75 10 AFTER WELDING . 3 4 11.25 5.62 2.01 NO. 29 DRILL x .45 DEEP, NO. 8 - 3 2 U N C - 2 B x . 3 l DEEP, 3 HOLES NO. 21 DRILL x .50 DEEP, NO. 1 0 - 3 2 UNF - 2 B x .38 DEEP, 10 HOLES SPACED AS IN DETAIL A 4 1.1258 F 1.63 £ CRANKSHAFT HOLE 12.13 4.00 2.00 4 x 2 y l — i 13 .8 (SHIPBUILDING) 27 DRILL, 13 UNC-3 B £ - l 3 U N C - 2 B . 3 0 6 5 2.231 250 1.3750 1.3745 2! • 21 «3 CHAMFER TO .3750 .3790 .375 DEEP 1.625 ISHiNG CUT SLOPE FOR CLOSE FIT IN 4 x2^- L _ l 13.8 3.3 1 .3037 NO. 42 DRILL- SEE DETAIL A 4 | | DRILL, ™ ~ I 3 U N C - ^ 3 B , DRILL AND TAP AFTER WELDING. 3.25 DIA. 3.44 27.25 NO. 29 DRILL x .45 DEEP, NO. 8 - 3 2 U NC — 2 B x .31 DEEP, 2 HOLES 3.2 5 2.50 1.1247 . 19 .50 ,A8) S C A L E ||-DRILL, -j- - 13 UNC-3B, DRILL AND TAP AFTER WELDING. 5 0 .06 THICK 2. 12 .50 0 I I I I I I I I I 11 l I l I I I I M I I I l I 1 .38 -5.62 STEEL SCALE- MAX. .04 THICK L 1 ± L 10 I I I 1 11 I I I 5 . 6 2 12.0 i . 187 .1242 = —4-- o -1.75 g- -12 UNF-2 A x 1.35 (AS) C R A N K P I N .939 9 3 7 DRILL, -L-I3 U N C - 2 A .38 fA9) P O I N T E R NO. 19 DRILL 2 HOLES 1.625 1.62 .50 CIVIL ENGINEERING DEPARTMENT UNIVERSITY OF BRITISH COLUMBIA ADJUSTABLE CRANK 391 FLUME WAVE GENERATOR DESIGNED BY: E.R. CHAPPELL DRAWN BY. E. R. CHAPPELL DATE •. JULY 69 SCALE.l"=l", 1/2"= I' REF: THESIS " THEORY AND DESIGN OF A WAVE GENERATOR FOR A SHORT FLUME' SHEET NO. 3 OF 4 SPEED REDUCTOR OUTPUT SHAFT MAX. CRANK-ARC 14.26" 12.25 MAX. CRANK THROW 10.5" ^ F L U M E TOP CRANK— CRANK POSITIONED BEHIND FLUME GATE. REDUCTOR OUTPUT SHAFT INSERTED 2 .75" INTO CRANK HUB. MOUNT FOR DRIVE MOTOR AND SPEED REDUCTOR TO BE PROVIDED. FLUME GATE M A J O R C O M P O N E N T L I S T PC NAME MATL QUAfs A CRANK STL 1 B CONNECTING ROD AL 1 C PADDLE AL 1 D SUPPORT ARMS AL 4 E BASE ASSEMBLY STL 2 END-CL E ARANCE PLATE OPT SET SCREW TIGHTENING SCREW MIN. TORQUE DIAM. INCH LBS 5/16 126 3/8 2 2 8 7 / 1 6 3 4 8 1/2 5 0 4 (F) PADDLE END-CLEARANCE PLATE SCALE . -~- = I" MATL: ( OPTIONAL ) 4 nA 3=? . SSSSSS i /-THIS SUPPORT V OMITTED IN SECTION VIEW FOR CLARITY. k A k 5 . 0 0 19.06 TRIM PADDLE SIDES TO CLEAR FLUME WALLS, KEEP CLEARANCE AT A MINIMUM. M A X . CLEARANCE . I i IT SECTION A—A CIVIL ENGINEERING DEPARTMENT UNIVERSITY OF BRITISH COLUMBIA WAVE GENERATOR ASSEMBLY AND INSTALLATION DATA FOR 39 ' F L U M E WAVE GENERATOR DESIGNED BY-E.R. CHAPPELL DRAWN B Y - E.R. CHAPPELL DATE - JULY '69 SCA LE: 1/4"= I" REF: THESIS " THEORY AND DESIGN OF A WAVE GENERATOR FOR A SHORT FLUME" S H E E T NO. 4 OF 4 

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