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An analysis of an interlock mechanism for small running skyline yarders Pendlebury, Ian 1980

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AN ANALYSIS OF AN INTERLOCK MECHANISM FOR SMALL RUNNING SKYLINE YARDERS by © IAN PENDLEBURY B.Sc., U n i v e r s i t y of Wales, 1976 THESIS SUBMITTED IN PARTIAL FULFILMENT THE REQUIREMENTS FOR THE DEGREE OF MASTER OF FORESTRY THE FACULTY OF GRADUATE STUDIES (FACULTY OF FORESTRY) We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA October 1980 (c) Ian Pendlebury, 1980 i n In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the 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 l y a v a i l a b l e 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 the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . It 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 not 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 . Department o f The 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 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date 6* 6 ABSTRACT T h i s t h e s i s p r o p o s e s a n e w i n t e r l o c k m e c h a n i s m f o r a r u n n i n g s k y l i n e y a r d e r w h i c h u t i l i z e s a n a u t o m o t i v e - t y p e b e v e l g e a r d i f f e r e n t i a l . I t p r o p o s e s t o p l a c e t h i s d i f f e r e n t i a l i n s i d e t h e h a u l b a c k d r u m , w h i c h a l s o a c t s a s a g e a r b o x / o i l b a t h . A h y d r o s t a t i c t r a n s m i s s i o n i s u s e d t o c o n t r o l t h e s p e e d r a t i o a c r o s s t h e d i f f e r e n t i a l . T h e r e s u l t i n g i n t e r l o c k p e r m i t s t h e r e l a t i v e d r u m s p e e d s t o b e c o n t i n u o u s l y v a r i e d t o c o m p e n s a t e f o r t h e e f f e c t s o f c h a n g i n g d r u m w r a p - r a d i i a s l i n e i s s p o o l e d o n a n d o f f t h e d r u m s . I t a l s o p r o v i d e s a n e f f i c i e n t p a t h w a y f o r t h e c i r c u l a t i o n o f p o w e r b e t w e e n t h e d r u m s . A g e a r f o r c e a n a l y s i s w a s u s e d t o d e t e r m i n e t h e t o r q u e - s p e e d r e l a t i o n s h i p s o f t h e d i f f e r e n t i a l . T h e r e s u l t i n g r e l a t i o n s h i p s w e r e u s e d t o c o n s t r u c t a m a t h e m a t i c a l m o d e l o f t h e p o w e r f l o w s i n t h e s i m p l i f i e d c a b l e s y s t e m , i n t e r l o c k e d w i n c h s e t a n d h y d r o s t a t i c c o n t r o l c i r c u i t . A c o m p u t e r p r o g r a m , w r i t t e n f o r t h e H e w l e t t P a c k a r d 9 8 4 5 A d e s k t o p c o m p u t e r , i n B A S I C l a n g u a g e , w a s u s e d t o c o m b i n e a p a r a b o l i c c a b l e m e c h a n i c s m o d e l , f o r t h e d e t e r m i n a t i o n o f l i n e t e n s i o n s , a n d t h e p o w e r f l o w m o d e l ; t o m o d e l t h e y a r d i n g o f l o g s w i t h t h e i n t e r l o c k e d w i n c h s e t . An examination of the torque loading imposed on the d i f f e r e n t i a l , when modelling the yarding of thinnings/ smallwood-sized materials, indicates that a large truck d i f f e r e n t i a l unit would have s u f f i c i e n t torque capacity to handle t h i s loading. TABLE OF CONTENTS Page ABSTRACT i i LIST OF TABLES v i LIST OF FIGURES V 1 1 ACKNOWLEDGEMENTS X INTRODUCTION x i CHAPTER I. THE INTERLOCK PROBLEM 1 CHAPTER I I . INTERLOCK MECHANISMS 17 1) The Regenerative Brake or S l i p p i n g -C l u t c h I n t e r l o c k 17 2) The H y d r o s t a t i c a l l y - C o n t r o l l e d P l a n e t a r y D i f f e r e n t i a l I n t e r l o c k 19 3) The A l l - H y d r a u l i c I n t e r l o c k 24 4) The E l e c t r i c I n t e r l o c k 29 CHAPTER I I I . A NEW INTERLOCK MECHANISM: THE BEVEL GEAR DIFFERENTIAL 30 1) The Automotive D i f f e r e n t i a l U n i t : Advantages and Disadvantages 30 2) The Torque/Speed R e l a t i o n s h i p s W i t h i n the D i f f e r e n t i a l 38 3) Power-flow W i t h i n the D i f f e r e n t i a l 42 4) Power-flow i n the I n t e r l o c k e d Winch Set 4 3 CHAPTER IV. A COMPUTER MODEL OF THE DIFFERENTIAL-INTERLOCKED WINCH 4 7 CHAPTER V. THE RESULTS OF THE COMPUTER MODEL 52 CHAPTER VI. CONCLUSIONS AND RECOMMENDATIONS 77 BIBLIOGRAPHY 80 APPENDICES: A The Cable Mechanics Model 82 i v APPENDICES Page B The Dragging Load Model 90 C The Line Length Model 9 3 D The Drum Wrap-Radius Model 96 E L i s t i n g of the Computer Program 100 F Specifications of the Winch Drums and Lines Used i n the Computer Model 109 G Imperial and Metric Units, and the Conversion Factors used i n the Computer Model 111 v LIST OF TABLES TABLE Page 1 Yarding Conditions and Summarized Supporting Data for the System Modelled i n Figure 19 55 2 Yarding Conditions and Summarized Supporting Data for the System Modelled i n Figure 20 57 3 Yarding Conditions and Summarized Supporting Data for the System Modelled i n Figure 23 63 4 Yarding Conditions and Summarized Supporting Data for the System Modelled i n Figure 24 67 5 Yarding Conditions and Summarized Supporting Data for the System Modelled i n Figure 25 69 F l Table of Winch Drum and Line Specifications Used i n the Computer Model 110 Gl Table of Imperial and Metric Units and the Conversion Factors Used i n the Computer 112 Model v i LIST OF FIGURES  FIGURE Page 1 Power, Torque and Speed R e l a t i o n s h i p s f o r a Winch Drum 3 2 Power Flow to and From a Winch Drum: a) Power Flow From a Winding Drum b) Power Flow to an Unwinding Drum 4 3 The E s s e n t i a l Components of an I n t e r l o c k e d Running S k y l i n e Yarder and i t s Rigging 5 4 The S i m p l i f i e d Running S k y l i n e System Used f o r the I n t e r l o c k A n a l y s i s 7 5 The Meshing B u l l Gear I n t e r l o c k 11 6 The M a i n l i n e : Haulback Drum Radius R a t i o vs. the P o s i t i o n of the C a r r i a g e i n the Span 12 7 The V a r i a t i o n o f M a i n l i n e and Haulback L i n e Speed vs. The P o s i t i o n of the C a r r i a g e i n the Span, f o r the Meshing B u l l Gear I n t e r l o c k 13 8 Schematic of an E p i c y c l i c P l a n e t a r y Gear T r a i n 20 9 The A p p l i c a t i o n of an E p i c y c l i c P l a n e t a r y Gear T r a i n as an I n t e r l o c k Mechanism 22 10 Schematic o f the Washington Iron Works H y d r a u l i c I n t e r l o c k 25 11 Schematic o f the D i r e c t D r i v e H y d r a u l i c I n t e r l o c k 28 12 Schematic of the A p p l i c a t i o n of a Bevel Gear D i f f e r e n t i a l i n an Automotive Power T r a i n 31 13 The E s s e n t i a l Components of a T y p i c a l Automotive-Type Bevel Gear D i f f e r e n t i a l U n i t 33 14 The Proposed A p p l i c a t i o n of the Automotive-Type Bevel Gear D i f f e r e n t i a l as an I n t e r l o c k Mechanism 34 15 Schematic of the Proposed I n t e r l o c k e d Winch Set 35 16 Gear Dimensions f o r the Gear Force A n a l y s i s 39 v i i FIGURE Page 17 T y p i c a l Load Paths Modelled by the Computer Program 49 18 Flowchart f o r the Computer Program 51 19 I n t e r l o c k Power Flow Components vs. C a r r i a g e P o s i t i o n : 1200 f t span, U p h i l l Yarding and F u l l y Suspended Load 54 20 I n t e r l o c k Power Flow Components vs. C a r r i a g e P o s i t i o n : 1200 f t span, Downhill Yarding and F u l l y Suspended Load 56 21 Schematic of the Power Flows i n the Winch Set when the C a r r i a g e i s Near the T a i l h o l d 59 22 Schematic of the Power Flows i n the Winch Set When the C a r r i a g e i s Near the Headspar 60 23 I n t e r l o c k Power Flow Components vs. The C a r r i a g e P o s i t i o n ; 600 f t span, U p h i l l Y arding and F u l l y Suspended Load 62 24 I n t e r l o c k Power Flow Components vs. The C a r r i a g e P o s i t i o n : 1200 f t Span, U p h i l l Yarding and Dragging Log Load 6 6 25 I n t e r l o c k Power Flow Components vs. The C a r r i a g e P o s i t i o n : 1200 f t span, Downhill Yarding and Dragging Log Load 6 8 26 Yarding Power Requirement vs. C a r r i a g e P o s i t i o n f o r Span Slopes Ranging from -100% to 100%: 2000 l b Log Load 71 27 Yarding Power Requirement vs. C a r r i a g e P o s i t i o n f o r Load Weights Ranging from 1000 l b to 5000 l b ; 50% Span Slope (Downhill Yarding) 72 28 Yardi n g Power Requirement vs. C a r r i a g e P o s i t i o n f o r Load Weights Ranging from 1000 l b to 5000 l b ; 0% Span Slope 7 3 29 Yarding Power Requirement vs. C a r r i a g e P o s i t i o n f o r Load Weights Ranging from 1000 l b to 5000 l b ; -50% Span Slope ( U p h i l l Yarding) 74 A l The Running S k y l i n e System Dimensions and Components used i n the P a r a b o l i c Cable Mechanics Model 84 v i i i FIGURE Page Bl The Dragging Load Geometry 91 Cl The Geometry of a Suspended Cable 94 Dl The Geometry of a Line-Wound Drum 9 7 D2 Flowchart for the Drum Wrap-Radius Algorithm 99 ix ACKNOWLEDGEMENT I would l i k e to acknowledge the patient assistance and guidance given by G. Glen Young, M.ASc, and Daniel Y. Guimier, M.ASc, throughout the preparation of t h i s t h e s i s . I would also l i k e to thank Dr. P h i l l i p L. C o t t e l l for his c r i t i c a l review of the f i r s t draft of the thesis. x I N T R O D U C T I O N T h i s t h e s i s e x a m i n e s t h e s u i t a b i l i t y o f a n a u t o m o t i v e - t y p e b e v e l g e a r d i f f e r e n t i a l u n i t a s a w i n c h d r u m - i n t e r l o c k i n g d e v i c e . T h i s a u t o m o t i v e - t y p e u n i t i s r e a d i l y a v a i l a b l e a s a c o m p l e t e l y e n g i n e e r e d u n i t w h i c h i s m a n u f a c t u r e d i n l a r g e q u a n t i t i e s a n d a t l o w c o s t . T h e p r o p o s e d a p p l i c a t i o n o f t h e b e v e l g e a r d i f f e r e n t i a l s h o u l d r e s u l t i n a c o m p a c t , e a s i l y c o n s t r u c t e d a n d r e l a t i v e l y i n e x p e n s i v e w i n c h d r u m s e t w h i c h i s s u i t a b l e f o r t h e o p e r a t i o n o f a r u n n i n g s k y l i n e s y s t e m . T h e r u n n i n g s k y l i n e h a s f o u n d w i d e a c c e p t a n c e i n t h e P a c i f i c N o r t h w e s t l o g g i n g i n d u s t r y . T h e r i g g i n g o f t h i s t y p e o f s k y l i n e r a n g e s i n c o m p l e x i t y f r o m t h e t w o - l i n e G r a b i n s k i t o t h e t h r e e - l i n e g r a p p l e a n d s l a c k - p u l l i n g c a r r i a g e s y s t e m s ; i t p r o v i d e s t h e u p h i l l a n d d o w n h i l l y a r d i n g c a p a b i l i t i e s o f t h e s l a c k l i n e s y s t e m , w i t h t h e a d v a n t a g e o f s i m p l e r r i g g i n g a t t h e t a i l h o l d . I n t h e r u n n i n g s k y l i n e s y s t e m , t h e c a r r i a g e r i d e s o n t h e h a u l b a c k l i n e : t h i s l i n e i s t h e " r u n n i n g s k y l i n e " . T o p u l l t h e c a r r i a g e i n t o t h e l a n d i n g , t h e m a i n l i n e i s w o u n d o n t o t h e m a i n l i n e d r u m a n d l i n e i s w o u n d o f f t h e h a u l b a c k d r u m . T o m a i n t a i n t h e c a r r i a g e i n a s u s p e n d e d s t a t e , t h e r a t e a t w h i c h t h e l i n e s a r e w o u n d o n a n d o f f t h e i r r e s p e c t i v e d r u m s m u s t b e e q u a l . T h e p r o b l e m o f e q u a l i z a t i o n x i of the l i n e speeds i s complicated by the fact that as l i n e i s wound on and off the drums, t h e i r r e l a t i v e spooling r a d i i change; thus, the r e l a t i v e r o t a t i o n a l speeds of the drums must be constantly changed to make allowance for these changes in spooling r a d i i . The application of a retarding brake to the drum from which l i n e i s being unwound i s a simple solution to t h i s problem, but i s very wasteful of power. To r e - c i r c u l a t e the power which would otherwise be wasted, a drum-interlocking mechanism i s required. For t h i s interlock mechanism to permit control of l i n e speed, l i n e tension and v e r t i c a l carriage position, i t must provide an i n f i n i t e l y variable speed r a t i o between the drums. An outline of the s i g n i f i c a n t types of interlock mechanism which have been, or are currently, i n use i s presented i n Chapter I I . The most e f f i c i e n t type of interlock i n terms of power transfer e f f i c i e n c y and ease of control of the r e l a t i v e drum speeds, employs an e p i c y c l i c planetary gear t r a i n , controlled by an hydrostatic transmission, between the drums. This interlock mechanism i s commonly ca l l e d a split-torque drive. In t h i s thesis, a new form of split-torque drive i s proposed as an interlock mechanism. The proposed interlock replaces the e p i c y c l i c planetary with an automotive-type, bevel-gear d i f f e r e n t i a l unit, retaining the hydrostatic control x i i c i r c u i t . In addition, i t i s proposed to place t h i s unit inside the haulback drum i t s e l f . The thesis determines the torque loadings imposed on such a d i f f e r e n t i a l unit when used as an interlock mechanism and examines the required torque-speed c h a r a c t e r i s t i c s of the control c i r c u i t . This examination i s f a c i l i t a t e d by the use of a computer program which models the es s e n t i a l components of a running skyline system and the interlocked drum set required for i t s operation. This thesis concerns i t s e l f only with the analysis of the proposed interlock mechanism when applied to two conventional winch drums. The exis t i n g interlock mechanisms are reviewed only i n s u f f i c i e n t d e t a i l so that the reader can gain an insight into the interlock design problem and an appreciation of the requirements of an e f f i c i e n t i n t e r l o c k . The thesis does not purport to offer a design of a complete working drum set; rather, i t provides a computational aid to the determination of the basic requirements of the interlock components as determined by the proposed operating conditions of a running skyline winch set. x i i i CHAPTER I THE INTERLOCK PROBLEM T h i s chapter i s intended to be a g e n e r a l i n t r o d u c t i o n to the requirements o f an i n t e r l o c k e d y arder winch s e t . I t i s presented i n the form of a comprehensively annotated l i t e r a t u r e review which o u t l i n e s the problems a s s o c i a t e d w i t h the c o u p l i n g of the drums : w h i l s t m a i n t a i n i n g c o n t r o l of both l i n e t e n s i o n and l i n e v e l o c i t y . An i n t e r l o c k mechanism can be d e f i n e d as"...a means of c o u p l i n g the m a i n l i n e and haulback drums so as to m a i n t a i n r u n n i n g - l i n e t e n s i o n " (U.S.D.A. F o r e s t S e r v i c e , 1969). I t can be a p p l i e d to any type of running s k y l i n e , c a b l e l o g g i n g system which employs (at l e a s t ) a separate m a i n l i n e and haulback drum. A w e l l designed i n t e r l o c k mechanism i s c o n s e r v a t i v e of power and serves as a means of c o n t r o l l i n g the flow of power between these drums. Before examining the d e s i r a b i l i t y of an i n t e r l o c k mechanism on a yarder and problems f a c i n g the i n t e r l o c k designer, i t i s necessary t o understand the way i n which power can flow through a c a b l e system and i t s a s s o c i a t e d winch drum s e t . T h i s understanding i s f a c i l i t a t e d by employing the concept of (winch) drum power and by adopting a s e t of conven-t i o n s r e l a t i n g l i n e t e n s i o n , l i n e speed, drum torque, drum speed and drum r a d i u s . These conventions are based on the r i g h t - h a n d 1 rule of physics (Beer and Johnston, 1977). The conventions adopted are: 1) When a cable i s unwinding from a drum, the cable v e l o c i t y i s p o s i t i v e ; 2) Cable tension always acts away from the drum and i s always p o s i t i v e ; 3) When the torque and angular v e l o c i t y of the drum are i n the same d i r e c t i o n , power flows into the drum, i . e . , power int o a system i s always p o s i t i v e ; 4) Power cannot be stored on a drum, i t must always leave by either the shaft or the l i n e . Figure 1 shows a winch drum wound with l i n e and outlines the relationships among the above parameters. Figures 2a and 2b show the two possible cases of drum rotation; l i n e winding onto the drum and l i n e being unwound from the drum. In Figure 2a the cable i s being wound onto the drum. The torque on the shaft and the angular v e l o c i t y of the shaft are i n the same d i r e c t i o n and are both p o s i t i v e ; thus, power i s flowing into the drum. Power cannot be stored on the drum and so must leave v i a the l i n e . Line tension i s acting away from the drum and i s p o s i t i v e , but the cable i s winding onto the drum so i t s v e l o c i t y i s negative. The product of l i n e tension and l i n e speed (line power) i s negative. In Figure 2b the cable i s being wound o f f the drum. The l i n e tension i s away from the drum and i s p o s i t i v e . Line 2 Figure 1. Power, Torque and Speed Relationships for a Winch Drum. LINE TENSION ( T.) LINE VELOCITY (V) TORQUE m ANGULAR VELOCITY OF THE DRUM (W) Line v e l o c i t y Drum torque y * Drum power P cX Angular v e l o c i t y of the drum x drum radius \J x R Line tension x drum radius T * R Drum torque x l i n e v e l o c i t y 3 gure 2 . Power Flow to and from a Winch Drum: (a) Power Flow from a Winding Drum L I N E V E L O C I T Y ( - v e ) (b) Power Flow to an Unwinding Drum L I NE V E L O C I T Y ( + v e ) V E L O C I T Y F i g u r e 3 . The E s s e n t i a l Components of an I n t e r l o c k e d Running Sk y l i n e Yarder and i t s Rigging. TA I LB L O C K V A R I A B L E R A T I O ClNTERLOCK) T R A N S M I S S I O N F I X E D ( 1:1 ) R A T I O T R A N S M I S S I O N T A G L I N E (OR G R A P P L E - C L O S I N G L I N E ) i s unwinding from the drum and i t s v e l o c i t y i s p o s i t i v e . The product of l i n e t e n s i o n and l i n e speed ( l i n e power) i s p o s i t i v e . Power i s f l o w i n g i n t o the drum and must leave on the s h a f t . The torque on the s h a f t i s p o s i t i v e , the angular v e l o c i t y i s negati v e , so t h e i r product, drum power, i s n e g a t i v e . F i g u r e 3 shows the e s s e n t i a l components of a running s k y l i n e system and the drum s e t r e q u i r e d . f o r i t s o p e r a t i o n . For the purposes of a n a l y s i s of the i n t e r l o c k problem, the system shown i n F i g u r e 4 w i l l s u f f i c e . The s i n g l e m a i n l i n e can be c o n s i d e r e d to pr o v i d e the same t e n s i o n and power requirements as the m a i n l i n e and s l a c k p u l l i n g l i n e s shown i n F i g u r e 3. (Appendix A d e t a i l s the cable mechanics of the running s k y l i n e system and c o n t a i n s a f u l l e x p l a n a t i o n o f t h i s statement.) In the process o f b r i n g i n g the c a r r i a g e t o the l a n d i n g , the m a i n l i n e i s wound onto the m a i n l i n e drum and the haulback l i n e unwound from the haulback drum (corresponding to F i g u r e s 2a and 2b r e s p e c t i v e l y ) . Power from the engine e n t e r s the m a i n l i n e drum and i s p r o p o r t i o n a l t o the product of the main-l i n e t e n s i o n and the m a i n l i n e v e l o c i t y . T h i s power leaves the drum and i s t r a n s f e r r e d , v i a the m a i n l i n e and haulback l i n e , t o the haulback drum. Power e n t e r i n g the haulback drum i s again p r o p o r t i o n a l t o the product of haulback l i n e v e l o c i t y and t e n s i o n . T h i s power w i l l not be equal to the power i n p u t from the engine due p r i m a r i l y t o the changes i n the p o t e n t i a l energy of the l o a d ( n e g l e c t i n g f r i c t i o n between the l o g and the ground). The haulback drum power leaves the drum on the s h a f t ; t h i s • 6 F i g u r e 4 . The S i m p l i f i e d Running S k y l i n e System used f o r the I n t e r l o c k A n a l y s i s . TAILHOLD I NTERLOCK 7 haulback drum power i s said to be "regenerated". If the haulback drum were allowed to turn f r e e l y under the action of the haulback l i n e , the rigging and carriage would sag to the ground. The t r a d i t i o n a l method of r a i s i n g and main-taining the l i n e s and carriage o f f the ground i s to increase the tension of the haulback l i n e by the application of a torque i n opposition to the rotation of the haulback drum. This i s usu-a l l y accomplished by the use of f r i c t i o n s on the drum or i t s shaft. In increasing the tension i n the haulback l i n e , the tension requirement of the mainline i s increased because the tension of the haulback l i n e acts i n opposition to that of the mainline. This increases the power requirements of the main-l i n e drum and requires that the braking system on the haulback drum be capable of d i s s i p a t i n g the (increased) power regener-ated there. This regenerated power i s dissipated as heat and depen-ding upon the tensions maintained i n the l i n e s , may be as great as 90% of the power entering the mainline drum (see page 11). Dissipation of t h i s heat frequently requires the use of water-cooled brakes (e.g., "Wichitas"), due to the large amounts of heat generated at the surface of the f r i c t i o n materials. Failu r e to provide for the d i s s i p a t i o n of th i s heat results • i n rapid degeneration of the f r i c t i o n s with the r e s u l t i n g loss of braking power and ultimately loss of v e r t i c a l carriage control. 8 An a t t r a c t i v e a l t e r n a t i v e to the d i s s i p a t i o n of the regenerated power i s to somehow r e c i r c u l a t e t h i s power back t o the m a i n l i n e drum. I f t h i s t r a n s f e r process were one hundred percent e f f i c i e n t , then the power r e q u i r e d to b r i n g i n the c a r r i a g e and l o a d would simply be the product of the d i f f e r e n c e i n m a i n l i n e and haulback l i n e t e n s i o n s , and the v e l o c i t y o f the l i n e s (assuming the m a i n l i n e and haulback l i n e v e l o c i t i e s to be e q u a l ) . For example:"'" M a i n l i n e t e n s i o n = 14338.25 l b (6505.56 kg) Haulback l i n e t e n s i o n = 13000.00 l b (5898.37 kg) Haulback l i n e v e l o c i t y = M a i n l i n e v e l o c i t y = 500 f t m i n - 1 (2.54 ms" 1) = t e n s i o n (lb) x l i n e speed ( f t min "*") 2 x T f x 5252 M a i n l i n e power = 14338.25 x 500 = 217.22 hp (291.09kW) 2 x T f x 5252 Haulback l i n e power = 13000 x 500 = 199.27 hp (267.03kW) 2 x l f x 5252 To m a i n t a i n the above l i n e t e n s i o n s and to s i m u l t a n -e o u s l y move the l i n e s a t 500 f t min "^  (2.54 ms "*") w i l l r e q u i r e a power in p u t o f 217.22 hp (291.09 kW) and w i l l r e q u i r e t h a t the haulback brake d i s s i p a t e 199.27 hp (267.03 kW). I f , however, 1 The l i n e v e l o c i t i e s and t e n s i o n s are taken from the r e s u l t s of the computer model o u t l i n e d i n Chapter IV, see a l s o Table 1. 9 the haulback drum power were to be r e c i r c u l a t e d back t o the m a i n l i n e drum a t 1 0 0 % e f f i c i e n c y , the power requirement would be reduced t o : 2 1 7 . 2 2 hp - 1 9 9 . 2 7 hp = 1 7 . 9 5 hp ( 2 4 . 0 5 kW). T h i s r e p r e s e n t s a s a v i n g i n power i n p u t o f 9 2 % . Although the example makes no allowance f o r the l o s s e s i n the power r e c i r c u l a t i o n p r o c e s s , the magnitude of the poten-t i a l r e d u c t i o n i n power requirements i s obvious. The s i m p l e s t method o f power r e c i r c u l a t i o n i n v o l v e s g e a r i n g the haulback and m a i n l i n e drums d i r e c t l y t o gether u s i n g a f i x e d r a t i o ( 1 : 1 ) gear t r a i n . F i g u r e 5 shows two drums d i r e c t l y geared t o g e t h e r , or i n t e r l o c k e d , by the meshing of the two b u l l gears. D i s r e g a r d i n g the power l o s s e s at the gear f a c e s , a l l of the power regenerated a t the haulback drum would be r e c i r c u l a t e d to the m a i n l i n e drum. While t h i s method i s simple, i t has one major drawback i n t h a t a c curate v e r t i c a l c o n t r o l o f the c a r r i a g e p o s i t i o n i s not p o s s i b l e due to the v a r i a t i o n s i n the drum r a d i i with the amount of l i n e s t o r e d on the drums: t h i s i n t u r n a l t e r s the r e l a t i v e v e l o c i t y of the l i n e s as they are wound on and o f f the drums. F i g u r e 6 shows how the r a d i i of the haulback and main-l i n e drums vary as the c a r r i a g e i s brought from the t a i l s p a r to the y a r d e r . F i g u r e 7 shows a p l o t o f m a i n l i n e and haulback l i n e v e l o c i t i e s f o r the case where the two b u l l gears are meshed toge t h e r , i . e . , the angular v e l o c i t i e s of both drums are equal 10 Figure 5 . The Meshing B u l l Gear Interlock. 1 1 Figure 6. The Mainline: Haulback Drum Radius Ratio v. the Positi o n of the Carriage i n the Span. 1.5 r JC 3 JO i X 0 s • 1 o i 8 DRUM RRDIU8 RATIO v*. CflRRXflGE POSITION L0CKP01NT .3 i . | i . r . T . r • , t i 288 488 688 888 1888 CflRRIRGE POSITION [Dltttwoe from the U l U p t r tft.] 1 2 Figure 7. The V a r i a t i o n of Mainline and Haulback Line Speed v. the P o s i t i o n of the Carriage i n the Span for the Meshing B u l l Gear Interlock 788 t LINE SPEED v. CflRRIRGE POSITION F 858 3688 e N «» «-S58 8 K U 5 8 8 M I , "L0CKP01... -J [ . \ Haulbmok r / i 458 I- J 1 NT \ \ CflRRIRGE POSITION [Dlttmoe from the tmtUpsr ift] 488 1 ' ' ' ' ' ' ' ' ' 1 ' ' ' 1 ' ' ' ' ' 1 ' ' ' 1 ' i . . . . t t , t , i 288 488 688 888 1888 1 3 (Figures 6 and 7 were a l s o produced by the computer model d e t a i l e d i n Chapter I V ) . In t h i s case the common angular v e l o -c i t y has been chosen so t h a t the l i n e s have the same v e l o c i t y at approximately midspan of 500 f e e t per minute (2.54 ms 1 ) . The p o i n t or range of p o i n t s i n the span where the l i n e v e l o c i -t i e s are equal i s commonly c a l l e d the " l o c k - p o i n t " and i n t h i s p a r t i c u l a r case, corresponds to the p o r t i o n of the span where the drum r a d i i are e q u a l . Examination of F i g u r e 7 r e v e a l s t h a t t h e r e are t h r e e d i s t i n c t p o r t i o n s o f the span, i n terms of r e l a t i v e l i n e speeds: 1) The p o r t i o n adjacent to the t a i l s p a r : the haulback drum has a l a r g e r e f f e c t i v e r a d i u s than the m a i n l i n e drum and pays out l i n e at a f a s t e r r a t e than the m a i n l i n e drum r e e l s the l i n e i n ; 2) The midspan p o r t i o n : the drum r a d i i are equal and l i n e i s payed out and r e e l e d i n a t equal r a t e s ; 3) The p o r t i o n a djacent to the y a r d e r : the m a i n l i n e drum has a l a r g e r e f f e c t i v e r a d i u s than the haulback drum and r e e l s i n l i n e f a s t e r than the haulback can pay i t out. The results o f the r e l a t i v e v a r i a t i o n s i n the drum r a d i i are t h a t as the c a r r i a g e moves across the span from t a i l h o l d to headspar, the s k y l i n e w i l l i n i t i a l l y drop due to a decrease i n t e n s i o n s , and then, as i t approaches the headspar, i t w i l l be r a i s e d as the l i n e t e n s i o n s i n c r e a s e . A d d i t i o n a l l y , due to v a r i a t i o n s i n the drum r a d i i , t h e r e w i l l be f l u c t u a t i o n s i n the 14 torque and hence power requirements across the span. So, whilst a fix e d r a t i o gear t r a i n i n t e r l o c k i n g the drums would r e c i r c u l a t e power very e f f i c i e n t l y , over p r a c t i c a l yarding distances, i t i s an unworkable solution i n terms of the control of r e l a t i v e l i n e v e l o c i t i e s , l i n e tensions and control of the v e r t i c a l carriage p o s i t i o n . What i s required i s some method of r e c i r c u l a t i n g the haulback drum power to the mainline drum, whilst simultaneously permitting a variable-speed r a t i o between the drums. This w i l l permit the haulback l i n e and mainline v e l o c i t i e s to be equalized (or maintained within a d e f i n i t e r e l a t i v e range as prescribed by the desired load path or t e r r a i n p r o f i l e ) for a l l positions across the span. There are several possible solutions to the problem of providing the variable-speed r a t i o i n t e r l o c k . The old steam-powered cable skidders employed a series of change gears, which were successively engaged by a clutching mechanism, to maintain the l i n e s at the desired r e l a t i v e v e l o c i t i e s (Anon. 1905). The major drawbacks to t h i s method were that the carriage had to be stopped whilst gears were changed and that the gear changes resulted i n d e f i n i t e "steps" i n the r e l a t i v e l i n e speeds, as the carriage crossed the span, allowing only r e l a t i v e l y poor v e r t i c a l carriage control. I t was recognized that i n order to obtain complete v e r t i c a l control, the int e r l o c k gear r a t i o had to be i n f i n i t e l y variable, at least within the required range of r a t i o s . 15 In summary, the conservative nature of the interlocked drum set makes i t an a t t r a c t i v e proposition due to the reduced power input required when compared with the conventional brake-on-drum methods of maintaining l i n e tensions. The root of the interlock problem l i e s i n the changing r e l a t i v e wrap-radius of the drums as l i n e i s wound on and o f f . In order to maintain accurate v e r t i c a l carriage control, i d e a l l y the interlock must provide an i n f i n i t e l y variable r a t i o between the drums whilst furnishing an e f f i c i e n t pathway for the re c i r c u l a t e d power. Chapter II discusses the ess e n t i a l features of, and the advantages and disadvantages of, the major types of currently employed interlock mechanisms. 16 CHAPTER I I INTERLOCK MECHANISMS Th i s chapter o u t l i n e s the e s s e n t i a l f e a t u r e s of s e v e r a l types of i n t e r l o c k e d winch drum s e t s of rec e n t or c u r r e n t manufacture. The l a s t s e c t i o n o f t h i s chapter d i s c u s s e s o t h e r p o s s i b l e i n t e r l o c k i n g mechanisms. 1) The Regenerative Brake or S l i p p i n g - C l u t c h I n t e r l o c k A c l u t c h and a brake are e s s e n t i a l l y the same mechanism. They c o n t a i n f r i c t i o n s u r f a c e s which are brought i n t o c o n t a c t by v a r i o u s means (mechanical, a i r or h y d r a u l i c p r e s s u r e ) . I f s u f f i c i e n t p r e s s u r e i s maintained on the f r i c t i o n surfaces,no s l i p -page occurs and i n the case of the c l u t c h , a l l the power e n t e r i n g on the i n p u t s h a f t l e a v e s , v i a the "locked up" f r i c t i o n s , on the output s h a f t : w i t h the brake, the i n p u t s h a f t i s prevented from t u r n i n g . Both the brake and the c l u t c h may, however, be allowed to s l i p by re d u c i n g the pre s s u r e h o l d i n g the f r i c t i o n s u r f a c e s t o g e ther. In t h i s s l i p p i n g p r o c e s s , power i s l o s t a t the f r i c t i o n s u r f a c e s . T h i s power, which i s l o s t u l t i m a t e l y as heat, i s p r o p o r t i o n a l to the r e l a t i v e speeds of the f r i c t i o n s u r f a c e s and to the magnitude o f the f r i c t i o n f o r c e s between them. A s l i p p i n g c l u t c h may be used to p r o v i d e a v a r i a b l e r a t i o i n t e r l o c k between the m a i n l i n e and haulback drums, a l b e i t with considerable power loss at the clutch. This method, com-, monly c a l l e d "regenerative braking", requires that the clutch face driven by the haulback drum w i l l , for a l l positions i n the span, be running faster than the speed required on the other, mainline drum-driving face. This i s achieved by the s e l e c t i o n of a suitable gear r a t i o between the haulback drum and i t s associated clutch face. With the correct v e r t i c a l p o s i t i o n of the carriage established at the t a i l h o l d (by tensioning the l i n e s ) , the clutch i s f u l l y engaged and the inhaul process started (in this case, the lock point i s at the t a i l h o l d ) . With the clutch f u l l y engaged the haulback drum w i l l not be paying out s u f f i c i e n t l i n e to match the rate at which l i n e i s being wound i n at the mainline drum: t h i s w i l l cause the l i n e tensions to increase and the carriage w i l l r i s e . To maintain the desired l i n e tensions, and hence carriage control, the clutch must be allowed to s l i p . In the s l i p p i n g process, torque and power are s t i l l e d transmitted to the mainline drum, but power i s also l o s t at the clutch faces. This power i s proportional to the difference i n angular v e l o c i t y of the clutch faces and i s l o s t as heat. The amount of power l o s t at the clutch w i l l increase as the carriage approaches the headspar due to the increasing difference i n the speeds of the clutch faces. Excellent discussions of the slipping-clutch interlock can be found i n Carson and Jorgensen (1974), Jorgensen (1974) and Ryan (1977). Regenerative braking permits control of the v e r t i c a l 18 position of the carriage, but at the expense of power loss at the inter l o c k . It does, however, allow r e c i r c u l a t i o n of power and represents a substantial improvement i n e f f i c i e n c y over the conventional braking method of maintaining l i n e tension. This type of interlock has been successfully employed on the Tyee range of yarders (Anon. 1969) . 2) The Hydrostatically-Controlled Planetary D i f f e r e n t i a l Interlock The most commonly used type of interlock i n the current-ly manufactured running skyline yarders u t i l i z e s a coaxial or e p i c y c l i c planetary gear d i f f e r e n t i a l with a hydrostatic trans-mission used as a control c i r c u i t . i t has three possible input/output shafts and consequently has two degrees of freedom (Cowie, 1961, p. 300): a conventional gear t r a i n has only one. With any one shaft used as an input, another as an output, the t h i r d shaft may be used to add or remove angular v e l o c i t y and hence power. The angular v e l o c i t y relationships of such a planetary gear t r a i n are given by Wilkinson (1960) as: An e p i c y c l i c planetary gear t r a i n i s shown i n Figure 8: W 3 W 1 n n s s where n s i s the r a t i o of the sun gear to the ring gear ( i . e . , i t i s the reduction across the planetary) and WQ, W^  and W^  are the angular v e l o c i t i e s of the planet gear c a r r i e r , the ring gear and the sun gear respectively. Figure 8 . Schematic of an E p i c y c l i c Planetary Gear Train. R l N G G E A R S U N G E A R P L A N E T G E A R C A R R I E R - P L A N E T G E A R ct:: S U N G E A R S H A F T \ 2a P L A N E T G E A R P L A N E T G E A R C A R R I E R S H A F T f2b R I N G G E A R 20 The torque relationships within the planetary are given by the inverse of the angular v e l o c i t y r e l a t i o n s h i p s . Figure 9 shows how t h i s type of planetary may be used to interlock the drums of a yarder. In practice, any combina-t i o n of the shafts can be used to drive the drums and / or the hydrostatic transmission; t h i s p a r t i c u l a r arrangement i s used simply to i l l u s t r a t e the working of th i s type of interlo c k . If one assumes that gears A, B and C have a r a t i o of 1:1:1, then the relationship of the angular v e l o c i t y of the mainline drum to that of the haulback drum i s given by: ^ / ( 2 ) 11 + n \ s n I s / With th i s p a r t i c u l a r arrangement of the planetary in t e r l o c k , the haulback drum w i l l rotate faster than the mainline drum (by a factor of 1 + n ) when the sun gear i s not rotating, s To equalize the speeds of the drums, i t w i l l be neces-sary to turn the sun gear i n the same d i r e c t i o n of rotation as the planet gear c a r r i e r . The required angular v e l o c i t y of the sun gear i s given by equation (1). In practice, due to the changing wrap r a d i i of the drums, simple equalization of the angular v e l o c i t i e s of the drums i s i n s u f f i c i e n t . Considerably more angular v e l o c i t y must be added to, or removed from the haulback drum to maintain the l i n e speeds i n the desired r a t i o as dictated by the required load path. Figure 9 . The Application of an E p i c y c l i c Planetary Gear Train as an Interlock Mechanism. L H Y D R O S T A T I C T R A N S M I S S I O N M f j x = F I X E D D I S P L A C E M E N T H Y D R A U L I C M O T O R P v n r = V A R I A B L E D I S P L A C E M E N T H Y D R A U L I C P U M P 22 The method of determining the r e q u i r e d l i n e speeds and drum torques i s o u t l i n e d i n Chapter I I I . The e s s e n t i a l f e a t u r e of the p l a n e t a r y component of the i n t e r l o c k i s t h a t i t p r o v i d e s a method of v a r y i n g the r e l a t i v e angular v e l o c i t i e s of the drums. A h y d r o s t a t i c t r a n s m i s s i o n s u i t a b l e f o r use as a c o n t r o l c i r c u i t c o n s i s t s e s s e n t i a l l y of a v a r i a b l e displacement h y d r a u l i c pump and a f i x e d displacement h y d r a u l i c motor. F i g u r e 9 shows one p o s s i b l e arrangement o f t h i s type of t r a n s m i s s i o n i n a loop around the d i f f e r e n t i a l . Other l o c a t i o n s f o r the pump are pos-s i b l e and t h e i r r e l a t i v e advantages and disadvantages are d i s -cussed i n Jorgensen (1971, p. 44) and Carson (1972). By the use of a v a r i a b l e displacement pump, the speed of the h y d r a u l i c motor, and hence the sun gear, may be i n f i n i t e l y v a r i e d over a range from zero to a maximum which-is determined by the a c t u a l t r a n s m i s s i o n components s e l e c t e d . In a d d i t i o n , the h y d r o s t a t i c t r a n s m i s s i o n i s f u l l y r e v e r s i b l e and capable o f the same i n f i n i t e l y v a r i a b l e speed c o n t r o l i n t h i s mode a l s o ; though the torque-speed c h a r a c t e r i s t i c s are changed (Holzbok, 1966, p. 244). In t h i s mode, the h y d r a u l i c motor a c t s as a f i x e d displacement h y d r a u l i c pump and the h y d r a u l i c pump ac t s as a v a r i a b l e displacement motor. The use of the h y d r o s t a t i c t r a n s m i s s i o n i n a loop around a p l a n e t a r y allows an i n f i n i t e l y v a r i a b l e speed c o n t r o l of the a u x i l i a r y i n p u t / o u t p u t ( i n the case of F i g u r e 9, the sun g e a r ) , w h i l s t s i m ultaneously p r o v i d i n g a method of r e c i r c u l a t i n g the power a t the a u x i l i a r y i n p u t / o u t p u t . With proper design, the 23 amount of power f l o w i n g through the h y d r a u l i c loop i s small compared wi t h t h a t f l o w i n g " s t r a i g h t - t h r o u g h " the (mechanical) p l a n e t a r y . T h i s means t h a t the l a r g e r p o r t i o n o f the power i s subj e c t e d to the s m a l l e r l o s s e s a s s o c i a t e d with the mechanical t r a n s m i s s i o n ( t y p i c a l l y 95 to 98% e f f i c i e n c y versus 80% f o r a h y d r o s t a t i c t r a n s m i s s i o n ) . M i n i m i s i n g the power r a t i n g of the h y d r o s t a t i c t r a n s m i s s i o n w i l l a l s o reduce the f i r s t c o s t of the i n t e r l o c k . In essence, u s i n g the h y d r o s t a t i c t r a n s m i s s i o n as a c o n t r o l c i r c u i t permits i n f i n i t e c o n t r o l , over a narrow range of speeds, of r e l a t i v e l y l a r g e power flows. T h i s type o f hydro-s t a t i c a l l y c o n t r o l l e d p l a n e t a r y _ d i f f e r e n t i a l i s commonly c a l l e d a s p l i t torque d r i v e (Holzbok, 1966, p. 253). Jorgensen (1974), c o n t a i n s an e x c e l l e n t i n t r o d u c t o r y d e s c r i p t i o n o f t h i s type of i n t e r l o c k . Pease (1977) analysed the power flow through the mechanical and h y d r a u l i c components of the h y d r o s t a t i c a l l y c o n t r o l l e d p l a n e t a r y gear d i f f e r e n t i a l . Chapter I I I c o n t a i n s a s i m i l a r a n a l y s i s and power flow model f o r a h y d r o s t a t i c a l l y c o n t r o l l e d bevel-gear d i f f e r e n t i a l based on the method developed by Pease. 3) The A l l - H y d r a u l i c I n t e r l o c k (a) The Washington Iron Works I n t e r l o c k T h i s i n t e r l o c k mechanism u t i l i z e s a h y d r a u l i c motor i n the gear t r a i n c o u p l i n g the m a i n l i n e drum to the haulback drum (Figure 10). A gear on the motor s h a f t d r i v e s the m a i n l i n e drum and a gear mounted on the motor c a s i n g d r i v e s the haulback 24 Figure 10. Schematic of the Washington Iron Works Hydraulic Interlock. ( From Mann (1977) ) R O T A R Y U N I O N M A I N L I N E D R U M M O T O R S H A F T G E A R MOTOR C A S I N G G E A R H A U L B A C K D R U M 25 drum. A r o t a r y union mounted on the motor permits the h y d r a u l i c pump i n the c i r c u i t t o supply o i l at pre s s u r e ( i . e . , power) w h i l s t the motor assembly i s i t s e l f r o t a t i n g . I f power i s n e i t h e r s u p p l i e d t o or removed from the motor, then the drum s e t would behave i n e x a c t l y the same manner as the meshing b u l l gear i n t e r l o c k d i s c u s s e d i n Chapter I , the r e being no r e l a t i v e motion between the motor s h a f t and c a s i n g when the motor i s "locked-up" h y d r a u l i c a l l y . When power i s s u p p l i e d to the motor ( i . e . , the motor ac t s as a motor) the motor case r o t a t e s r e l a t i v e to the s h a f t and the angular v e l o c i t y o f the haulback drum i s i n c r e a s e d r e l a t i v e to t h a t of the m a i n l i n e drum. P e r m i t t i n g the motor to a c t as a pump w i l l remove angular v e l o c i t y , hence power, from the haulback drum. T h i s i n t e r l o c k behaves i n the same manner as the pl a n e -t a r y d i f f e r e n t i a l and the r e q u i r e d d i r e c t i o n o f r o t a t i o n o f the motor i s the same as t h a t r e q u i r e d o f the sun gear i n F i g u r e 9. The power r e l a t i o n s h i p s of the p l a n e t a r y d i f f e r e n t i a l and t h i s i n t e r l o c k are mathematically e q u i v a l e n t (Mann, 1977) . The patent f o r t h i s type of i n t e r l o c k i s h e l d by Washington Iron Works, S e a t t l e , Washington. A d e s c r i p t i o n of the f i r s t machine to use t h i s i n t e r l o c k mechanism, the Washington 78A, i s g i v e n i n Raven (1974). (b) The D i r e c t - D r i v e H y d r a u l i c I n t e r l o c k The e s s e n t i a l components of t h i s type of i n t e r l o c k are 26 shown i n F i g u r e 11. In t h i s case, the h y d r a u l i c loop formed by the t r a n s m i s s i o n comprises a v a r i a b l e displacement h y d r a u l i c pump coupled d i r e c t l y to the haulback drum and a f i x e d d i s p l a c e -ment motor connected to the m a i n l i n e drum. T h i s t r a n s m i s s i o n must be of such a s i z e as to be able to t r a n s m i t a l l of the power f l o w i n g from the haulback drum d u r i n g the i n h a u l phase. The r e l a t i v e angular v e l o c i t i e s of the drums may be c o n t r o l l e d by the v a r i a b l e displacement o f the h y d r a u l i c pump. Th i s type of i n t e r l o c k p r o v i d e s a very s t r a i g h t f o r w a r d s o l u t i o n t o the i n t e r l o c k problem, although there are s e v e r a l drawbacks to t h i s mechanism. The f a c t t h a t the t r a n s m i s s i o n must be s i z e d to handle a l l of the power c i r c u l a t i n g between the drums means t h a t r e l a t i v e l y l a r g e and expensive t r a n s m i s s i o n s w i l l be r e q u i r e d ; indeed, t h i s i s l i k e l y to l i m i t i t s a p p l i c a -t i o n to the s m a l l e r y a r d e r s u n t i l l a r g e r t r a n s m i s s i o n s become a v a i l a b l e . Pumps and motors may, of course, be mounted i n tandem to i n c r e a s e the r a t i n g of a t r a n s m i s s i o n but t h i s i s an expensive s o l u t i o n . In a d d i t i o n , the h y d r o s t a t i c t r a n s m i s s i o n i s l e s s e f f i c i e n t (page 24) than a h y d r o s t a t i c a l l y - c o n t r o l l e d s p l i t - t o r q u e d r i v e . The power l o s s e s i n the t r a n s m i s s i o n r e s u l t i n the g e n e r a t i o n of heat, and some form of o i l c o o l i n g i s r e -q u i r e d . W h i l s t t h i s i n t e r l o c k may appear simple, the a s s o c i a t e d c o n t r o l c i r c u i t s may be complex; a d d i t i o n a l l y , due to the h i g h p r e s s u r e s and l a r g e flow volumes, c o n t r o l v a l v e s w i l l be expensive. 27 Figure 11. Schematic of the Direct Drive Hydraulic Interlock. ( From Carson and Jorgensen (1974)) P R I M E M O V E R 28 Carson and Jorgensen (1974) discussed the d i r e c t drive i n t e r l o c k . A description of a smallwood yarder, the Rosedale Machine Shop Ecologger, which uses a d i r e c t drive interlock of t h i s type was given by Plummer and Koury (1974). 4) The E l e c t r i c Interlock The use of an e l e c t r i c transmission ( i . e . , a generator and motor) i n a s i m i l a r manner to that of the d i r e c t drive hydraulic interlock would seem to be a possible solution to the interlock problem. An e l e c t r i c motor i s capable of acting as a generator and vice versa ( c f . the hydrostatic pump and motor). Again the size of the power flow i n t h i s arrangement w i l l mean that the transmission must be of a high power ra t i n g : i n the case of the e l e c t r i c motor/generator the size and weight of suitable units may be a major drawback. However, the ease of control and the sophistication possible with e l e c t r i c a l control devices could be a major advantage. There i s no l i t e r a t u r e published on the subject of a l l - e l e c t r i c i n t e r l o c k s . Of a l l the interlock types outlined i n t h i s chapter, the h y d r o s t a t i c a l l y - c o n t r o l l e d e p i c y c l i c planetary gear in t e r l o c k i s currently the most commonly used, and also the most e f f i c i e n t i n terms of power transfer. Chapter III proposes a new interlock which uses an h y d r o s t a t i c a l l y controlled bevel gear d i f f e r e n t i a l as the interlocking mechanism.. 29 C H A P T E R I I I A N E W I N T E R L O C K M E C H A N I S M : T H E B E V E L G E A R D I F F E R E N T I A L T h i s c h a p t e r p r o p o s e s a n e w m e t h o d o f i n t e r l o c k i n g t w o w i n c h d r u m s w h i c h u s e s a n a u t o m o t i v e d i f f e r e n t i a l g e a r u n i t l o -c a t e d i n s i d e t h e h a u l b a c k d r u m i t s e l f . I t o u t l i n e s t h e e x p e c -t e d a d v a n t a g e s a n d d i s a d v a n t a g e s o f t h i s a p p r o a c h , e x p l o r e s t h e t o r q u e - s p e e d r e l a t i o n s h p i s o f t h e d i f f e r e n t i a l a n d p r e s e n t s a p o w e r f l o w m o d e l o f t h e p r o p o s e d i n t e r l o c k e d w i n c h s e t . 1) T h e A u t o m o t i v e D i f f e r e n t i a l U n i t : A d v a n t a g e s a n d D i s a d v a n t a g e s  T h e b e v e l o r m i t r e g e a r d i f f e r e n t i a l i s f a m i l i a r t o m o s t p e o p l e > t h r o u g h i t s a p p l i c a t i o n i n t h e " r e a r e n d s " o f a u t o m o -b i l e s a n d t r u c k s . I n t h i s a p p l i c a t i o n , a s i n g l e i n p u t , t h e d r i v e s h a f t , i s a b l e t o p r o d u c e t w o s e p a r a t e a n d d i f f e r i n g s p e e d o u t p u t s t o e a c h o f t h e r e a r w h e e l s . T h i s p e r m i t s t h e v e h i c l e t o c o r n e r w h i l s t p r o v i d i n g t r a c t i v e e f f o r t t o t h e d r i v i n g w h e e l s . F i g u r e 12 i s a s c h e m a t i c s h o w i n g t h e w a y i n w h i c h a b e v e l g e a r d i f f e r e n t i a l i s a p p l i e d i n a v e h i c l e p o w e r t r a i n . P o w e r f r o m t h e e n g i n e , v i a t h e t r a n s m i s s i o n a n d d r i v e s h a f t , a r r i v e s a t t h e i n p u t p i n i o n . T h i s p o w e r i s t r a n s f e r r e d t o t h e d i f f e r e n t i a l u n i t v i a t h e r i n g g e a r w h i c h i s b o l t e d t o t h e d i f -f e r e n t i a l u n i t c a s e a n d w h i c h r o t a t e s a b o u t t h e a x i s o f t h e w h e e l a x l e s . I f t h e v e h i c l e i s t r a v e l l i n g i n a s t r a i g h t l i n e , t h e p l a n e t p i n i o n s r e v o l v e a r o u n d t h e a x i s o f t h e a x l e s , b e i n g 30 Figure 12. Schematic of the Application of a Bevel Gear D i f f e r e n t i a l i n an Automotive Power Train. WHEELS INPUT PI NION DRIVE SHAFT TRANS-MI SSION ENGINE 31 m o v e d b y t h e d i f f e r e n t i a l c a s e . T h e y d o n o t r o t a t e a b o u t t h e i r o w n a x e s . W i t h t h e p i n i o n s m o v i n g i n t h i s f a s h i o n , t h e s i d e g e a r s a r e r o t a t e d a b o u t t h e i r o w n a x e s , t h e s a m e t e e t h o n t h e p l a n e t p i n i o n s a n d t h e s i d e g e a r s r e m a i n i n g i n c o n t a c t t h r o u g h -o u t e a c h r e v o l u t i o n ( i . e . , t h e g e a r s d o n o t m e s h ) , t h u s c a u s i n g t h e a x l e s a n d w h e e l s t o r o t a t e a t t h e s a m e s p e e d . I f , h o w e v e r , t h e v e h i c l e i s c o r n e r i n g , o n e w h e e l m u s t r o t a t e f a s t e r t h a n t h e o t h e r . I n t h i s c a s e , t h e p l a n e t p i n i o n s n o t o n l y r e v o l v e a r o u n d t h e a x i s o f t h e a x l e s , b u t t h e y a l s o r o t a t e a b o u t t h e i r o w n a x e s . T h e g e a r t e e t h o n t h e p l a n e t p i n -i o n s a n d t h e s i d e g e a r s m e s h a n d t h i s p e r m i t s t h e w h e e l s t o r o t a t e a t d i f f e r i n g s p e e d s . F i g u r e 1 3 s h o w s d e t a i l s o f a t y p i c a l b e v e l g e a r d i f f e r e n -t i a l u n i t a n d F i g u r e 14 s h o w s t h e p r o p o s e d a p p l i c a t i o n o f s u c h a u n i t a s a n i n t e r l o c k i n g m e c h a n i s m . T h e r i n g g e a r i s r e m o v e d a n d t h e d i f f e r e n t i a l c a s e b o l t e d t o o n e e n d o f t h e h a u l b a c k d r u m . T h e a x l e s a r e r e p l a c e d b y t h e m a i n a n d a u x i l i a r y s h a f t s a n d t h e h a u l b a c k d r u m f i l l e d w i t h o i l t o a c t a s a n o i l b a t h f o r t h e d i f -f e r e n t i a l u n i t s ' g e a r s . T h e s h a f t s r o t a t e o n b e a r i n g s i n t h e d r u m e n d c a p s ; t h e d r u m i t s e l f a l s o r o t a t e s o n b e a r i n g s . F i g u r e 1 5 i s a s i m p l i f i e d s c h e m a t i c o f t h e p r o p o s e d i n t e r l o c k e d w i n c h s e t . T h e . d i f f e r e n t i a l i s a s p e c i a l c a s e o f p l a n e t a r y g e a r t r a i n : i t s s p e e d a n d t o r q u e r e l a t i o n s h i p s c a n b e s h o w n t o b e e q u i v a l e n t t o t h o s e o f t h e c o n v e n t i o n a l e p i c y c l i c p l a n e t a r y . T h e u s e o f a n a u t o m o t i v e d i f f e r e n t i a l g e a r u n i t a s a n i n t e r l o c k i n g 3 2 Figure 13. The Esse n t i a l Components of a Typical Automotive-Type Bevel Gear D i f f e r e n t i a l Unit. 33 Figure 14. The Proposed Application of the Automotive-Type Bevel Gear D i f f e r e n t i a l as an Interlock Mechanism. HAULBACK DRUM Figure 15. Schematic of the Proposed Interlocked Winch Set. var. M fix. H a u n. LT P R I M E M O V E R B. J l H A U L B A C K D R U M M A I N L I N E D R U M 35 device should have the following advantages over.the e p i c y c l i c planetary: (a) The shape and size of a t y p i c a l automotive d i f f e r e n t i a l make i t suitable for placement inside the haulback drum. This location has been successfully employed, using a con-ventional planetary, to the Pee Wee interlocked yarder (Mann & M i f f l i n , 1979) and has the main advantage that i t removes the necessity of using an external planetary transmission case. Care must be taken to ensure that the drum i s s t r u c t u r a l l y r i g i d enough, under expected loads, to maintain the d i f f e r e n t i a l input/output shafts within acceptable alignment l i m i t s . The location and concen-t r i c i t y of the unit's bearing and mounting surfaces (Figure 12) should simplify the machining, f a b r i c a t i o n and i n i t i a l alignment of the drum/differential components due to t h e i r concentric arrangement. (b) The number of interlocked winch sets produced by any one manfacturer i s small. The production of gear compon-ents for such winches i s necessarily on a small scale, so the cost of tooling has to be written o f f over a small number of machines re s u l t i n g i n a high unit cost. There may also be problems of r e f i n i n g production tech-niques over a small production run. This problem i s highlighted with any type of planetary gear t r a i n , the p i t c h - l i n e v e l o c i t y of the planet gears i n p a r t i c u l a r i s extremely high, frequently requiring the hardening and subsequent grinding of the gear tooth surfaces to obtain the necessary tolerances. This represents an additional, and i n many cases, substantial cost (Anderson, 1974) . The automotive d i f f e r e n t i a l i s produced i n large quanti-t i e s , production techniques are highly refined and the cost of a unit i s r e l a t i v e l y low. The d i f f e r e n t i a l i s supplied as a completely engineered unit with a l l gear bearings and thrust bearings matched to the torque/speed capacities of the unit; by paying careful attention to the torque rating of the unit, interlock gear l i f e should be long. Spares, or replacement units, should be well dis t r i b u t e d , e a s i l y obtained and r e l a t i v e l y inexpensive. The f a m i l i a r i t y of maintenance personnel with the unit should also be an advantage. Possible disadvantages of the d i f f e r e n t i a l are: (a) The d i f f e r e n t i a l unit w i l l be heavier than an e p i c y c l i c planetary with the same torque ratin g . (b) I t i s possible to l i n k the e p i c y c l i c planetary i n such a way that the main input shaft need carry a considerably lower torque than a sim i l a r shaft transmitting the same 37 power to a d i f f e r e n t i a l u n i t . T h i s i s due to the g r e a t e r r e d u c t i o n p o s s i b l e with the c o n v e n t i o n a l p l a n e t a r y (the d i f f e r e n t i a l always has a 2:1 r e d u c t i o n ) . T h i s means t h a t with the c o n v e n t i o n a l p l a n e t a r y , any c l u t c h e s r e q u i r e d i n the main i n p u t s h a f t power t r a i n w i l l r e q u i r e a lower torque r a t i n g , w i t h i t s attendant r e d u c t i o n i n s i z e , weight and c o s t . 2) The Torque/Speed R e l a t i o n s h i p s W i t h i n the D i f f e r e n t i a l The d i f f e r e n t i a l may be regarded as a s p e c i a l case of a p l a n e t a r y gear t r a i n i n which there i s a maximum p o s s i b l e reduc-t i o n of 2:1. The angular v e l o c i t y r e l a t i o n s h i p s of the gears i n F i g u r e 15 are g i v e n by Cowie (1961, p. 305) as: W = W + W C _A B ( 3 ) 2 where W,. W and W_, are the angular v e l o c i t i e s of A , B C gears A, B and the p l a n e t gear c a r r i e r C r e s p e c t i v e l y . The torque r e l a t i o n s h i p s w i t h i n the d i f f e r e n t i a l can be d e t e r -mined by performing a gear f o r c e a n a l y s i s on the d i f f e r e n t i a l . The gear f o r c e a n a l y s i s which f o l l o w s uses the method des-c r i b e d i n Hall,Hoilowenko and L a u g h l i n (1961). In t h i s a n a l y s i s , i t i s assumed t h a t the i n t e r - g e a r f o r c e s a c t at the mid-points of gear f o r c e s . A c l o c k w i s e r o t a t i o n , as viewed from the r i g h t hand s i d e of the page i s c o n s i d e r e d to be p o s i t i v e . F i g u r e 16 shows the e s s e n t i a l components and dimensions of the b e v e l gear 38 Figure 16. Gear Dimensions for the Gear Force Analysis d i f f e r e n t i a l under analysis. Consider gear c a r r i e r , C, to be held stationary; then application of a clockwise torque, T A , on gear A produces a force, at point M which i s given by: T A (acting into the page) (4) a There i s an equal and opposing force, at P, which i s given by T A (acting out of the page) (5) a This force produces a torque on gear D which i s given by: T A x d (6) a This torque produces a force, at R, which i s given by: T A x d (acting out of the page) (7) a x e This i s an equal and opposing force, at S, which i s given by: T A x d (acting into the page) (8) a x e This force produces a counter-clockwise torque, TB, which i s given by: T A x d x b (9) a x e Since a = b and d = e, and since counter-clockwise i s considered to be negative: TA = -TB (10) Thus, with the gear c a r r i e r C held stationary, the torques on gears A and B are equal i n magnitude but opposite i n d i r e c t i o n . 40 The torque produced on gear c a r r i e r C can be s i m i l a r l y determined by considering gear B to be held stationary. A p p l i -cation of a clockwise torque, TA, on gear A produces a force, at P, which i s given by: T'A (acting up the page) (11) a This force produces a moment about point R, which i s given by: T A x (d + e) (12) a Since d = e, t h i s moment i s also given by: T A . x 2e (13) a This moment produces a force, at Q, which i s given by: 2 x T A (acting up the page) (14) a This force, acting through point Q, produces a torque on gear c a r r i e r C ,Tc, which i s given by: 2 X T A x c ^ 1 5 j a Since a = c, and since both torques are acting i n the same clockwise d i r e c t i o n , the torque produced i n gear c a r r i e r C by a torque applied on gear A i s given by: Tc = 2 x T A (16) Thus the torque relationships within the d i f f e r e n t i a l corres-pond to the inverse of the speed relationships i n Equation 3. 41 3) Power-flow within the D i f f e r e n t i a l With the torque and speed relationships now defined, i t i s possible to derive expressions which describe the flow of power through the d i f f e r e n t i a l . I t i s necessary to adopt a sign convention to aid i n the development of these expressions: power i n i s p o s i t i v e , i . e . , torque and angular v e l o c i t y must be acting i n the same d i r e c t i o n for power to enter the system (Page 5 ) . In addition, with reference to Figure 14: TA = -TB d o ) t c = 2 x T A (16) wc = wA + wB (3) 2 where torque i n lb f t W = angular v e l o c i t y i n rev min and A, B and C are subscripts r e f e r r i n g to gears A and B and planet gear c a r r i e r C, respectively. P = Tx w ( 1 7 ) 5252 where P = power i n Horsepower 5252 = constant /33000 2 x if P A + P B + P C = 0 (18) With any type of planetary i t i s possible to specify only three parameters. In the case of the d i f f e r e n t i a l used as an i n t e r -locking device, the angular v e l o c i t y of the input gear, A, the required angular v e l o c i t y of the gear c a r r i e r , C (in t h i s case the haulback drum also) and-ithe required torque on C w i l l be known. In gear t r a i n analysis, i t i s customary to select the "driver" gear (Cowie 1961, p. 294) and relate a l l other component speeds and torques to t h i s gear. It i s convenient to consider the haulback drum as the dri v i n g component, thus the gear c a r r i e r C becomes the d r i v e r . The speed of the a u x i l i a r y input, gear B, from equation 3, i s given by: WB = 2 WC " WA (19) In t h i s application, both gears A and B w i l l have the same torque: t h i s torque i s dictated by the torque on the gear c a r r i e r C. The torque on the a u x i l i a r y input, gear B, from equation 16, i s given by: T B = Tc 2 (20) The power input/output at gear B i s : P B = (2 Wc - WA) x T c 5252 x 2 (21) 4) Power-flow i n the Interlocked Winch Set These basic equations may now be used to construct a power-flow model for the winch set shown i n Figure 14. The r a t i o across the d i f f e r e n t i a l as i t i s employed here i s 2:1 ( i . e . , turning gear A or B w i l l produce half as many turns of the planet gear carrier/haulback drum, C). It i s assumed that the r a t i o between gear E and F i s s i m i l a r l y 2:1 to compensate f o r t h i s r e d u c t i o n . With the m a i n l i n e v e l o c i t y , the m a i n l i n e and haulback l i n e t e n s i o n s and the dimensions of -the drums a l l known, the power flow r e l a t i o n s h i p s are as f o l l o w s : The m a i n l i n e power i s : P„ = V x T„„ M M M 2xrfx5252 (21) where T = l i n e t e n s i o n (lb) V = l i n e v e l o c i t y ( f t min "*") P = power (horsepower). The r a t i o o f gear E to gear F i s 2:1: the angular v e l o c i t y of gear F i s equal to t h a t o f the m a i n l i n e drum and the angular v e l o c i t y o f gear A i s equal t o t h a t of gear E. The angular v e l o c i t y o f gear A i s : W. = 2 WM (22) A 2 x?fx % (23) where R = the drum r a d i u s ( f t ) W = angular v e l o c i t y (rev min ^) The torque on gear A i s : -?A = Tc 2 (24) where T = torque ( l b f t ) The torque on gear c a r r i e r C i s : TC = T H x R H (25) The power at the gear c a r r i e r C i s : Pc = Wc x " f c 5252 (26) The angular v e l o c i t y of gear c a r r i e r C i s Wc = VH 2 x Tf x R„ (27) ri It i s assumed that the required mainline v e l o c i t y i s known. The haulback l i n e v e l o c i t y i s not equal to mainline v e l o c i t y for a l l carriage positions i n the span. The l i n e v e l o c i t y r a t i o may be calculated; i t i s given by Pease (1977) as: L V R = LH (1) " LH (0) • V ( l ) ~ hi (0) ( 2 8 ) LVR = r a t i o of v e l o c i t y of Haulback l i n e to v e l o c i t y of i n l i n e L = l i n e length,in the span (ft.) and subscripts (0) and (1) represent the i n i t i a l and present positions of the carriage i n the span, respectively. The l i n e v e l o c i t y of the haulback l i n e , V„, i s : V„ = V M x LVR (29) H M Thus the power at the gear c a r r i e r C, i s equal to the haulback power and i s : PC •= P R = V H * T H 2 x 5252 (30) The required angular v e l o c i t y of the a u x i l i a r y input gear, gear B, i s : WD = 2 x Wc - W_ (31) The torque on gear B, from equation 20, i s : f B = Tc 2 The power at gear B i s : P n = (2 x Wc - W ) x fc 2 x 5252 (32) The power l e a v i n g the d i f f e r e n t i a l on gear A i s : p _p _p (33) A H B K ' The net power requirement from the prime mover, P^, i s : PY = PM " P H (34) Assuming t h e r e are no l o s s e s i n the system, i s a r e f l e c t i o n of the work done i n changing the l e v e l of p o t e n t i a l energy i n the system, i n p a r t i c u l a r , t h a t of the l o a d : P r e p r e s e n t s the instantaneous power requirement to b r i n g i n the c a r r i a g e and the l o a d . Thus with the r e q u i r e d c a b l e t e n s i o n s and v e l o c i t i e s known, t h i s model permits the c a l c u l a t i o n of both the t o t a l y a r d i n g power requirements and the magnitude and d i r e c t i o n s of the power flows i n the h y d r o s t a t i c ( a u x i l i a r y ) c o n t r o l c i r c u i t . Chapter IV o u t l i n e s a computer program which may be used to model the running s k y l i n e system and the proposed i n t e r l o c k e d winch s e t . 46 CHAPTER IV A COMPUTER MODEL OF THE DIFFERENTIAL-INTERLOCKED WINCH The power-flow model developed i n Chapter I I I was com-bined w i t h a ca b l e mechanics model to permit r e a l i s t i c m o d e l l i n g of c a b l e t e n s i o n s and the de t e r m i n a t i o n o f the gear torques developed w i t h i n the d i f f e r e n t i a l . These torques may be used to s e l e c t a s u i t a b l y r a t e d d i f f e r e n t i a l u n i t , or used to d e t e r -mine the maximum l o a d s i z e and/or span l e n g t h f o r a g i v e n d i f f e r e n t i a l u n i t . The model may a l s o be used t o determine the r e q u i r e d torque/speed c h a r a c t e r i s t i c s of the main and a u x i l i a r y t r a n s -m i s s i o n s . A d d i t i o n a l l y , the e f f e c t of changing v a r i o u s system parameters, e.g., drum and ca b l e dimensions, span l e n g t h and span s l o p e , o r l o a d s i z e , on the magnitude of the power flows w i t h i n the drum s e t , may be examined. The c a b l e mechanics model used i s a p a r a b o l i c model and permits m o d e l l i n g of a running s k y l i n e system. The ca b l e model was developed by Guimier (1977) ; d e t a i l s of the model can be found i n Appendix A. Appendix E c o n t a i n s a l i s t i n g of the program. The running s k y l i n e and power-flow model was implemented u s i n g a program w r i t t e n i n BASIC language f o r the Hewlett-Packard 9 84 5A desktop computer. The program was w r i t t e n so t h a t the parameters of the system are grouped a t the beginning o f 47 t h e p r o g r a m . They a r e g r o u p e d l o g i c a l l y , e a c h g r o u p o f p a r a -m e t e r s d e s c r i b i n g d i f f e r e n t components o f t h e c a b l e and t h e drum s y s t e m . T h e s e p a r a m e t e r s may be q u i c k l y c h a n g e d and t h e e f f e c t on t h e power f l o w s c a n be o b s e r v e d . The p r o g r a m u s e s mnemonic v a r i a b l e s w h i c h a r e s e l f -d e s c r i p t i v e . I n a d d i t i o n , a l l s i g n i f i c a n t p r o g r a m l i n e s o r s t e p s a r e e x p l a i n e d i n t h e i n - p r o g r a m d o c u m e n t a t i o n . The p r o -gram l i n e s h a v e been k e p t d e l i b e r a t e l y s h o r t and e a c h s t e p i n t h e p o w e r - f l o w model, i s r e p r e s e n t e d by a s e p a r a t e p r o g r a m l i n e . The p r o g r a m models t h e i n h a u l p h a s e o n l y ; t h e c a r r i a g e i s l o a d e d and maximum l i n e t e n s i o n s c a n be e x p e c t e d t o o c c u r . The p r o g r a m c o n t a i n s two o p t i o n s . The f i r s t m a i n t a i n s c o n s t a n t t e n s i o n i n t h e r u n n i n g s k y l i n e a t t h e u p p e r l i n e s u p p o r t and c a l c u l a t e s t h e l o a d p a t h t h a t t h i s t e n s i o n p e r m i t s . T h i s o p t i o n assumes t h e l o a d i s f u l l y s u s p e n d e d . The s e c o n d o p t i o n m a i n t a i n s a c o n s t a n t c a r r i a g e h e i g h t o v e r a g r o u n d p r o f i l e t h a t i s r e p r e s e n t e d by a s t r a i g h t l i n e c o n n e c t i n g t h e b a s e s o f t h e h e a d s p a r and t h e t a i l s p a r . The l i n e t e n s i o n s t h a t a r e r e q u i r e d t o m a i n t a i n t h i s l o a d p a t h a r e c a l c u l a t e d . T h i s o p t i o n a l s o m o d e ls t h e e f f e c t o f f r i c t i o n b etween a d r a g g i n g l o g l o a d and t h e g r o u n d , u s i n g a d r a g g i n g l o a d model d e v e l o p e d by C a r s o n ( 1 9 7 5 ) . A p p e n d i x B g i v e s d e t a i l s o f t h i s d r a g g i n g l o g m o d e l . F i g u r e 17 i l l u s t r a t e s t y p i c a l e x p e c t e d l o a d p a t h s f o r e a c h o f t h e s e o p t i o n s , w h i c h c a n be e x p e c t e d t o p r o d u c e minimum and maximum a u x i l i a r y t r a n s m i s s i o n power f l o w s , r e s p e c t i v e l y . 48 F i g u r e 17. T y p i c a l Load Paths Modelled by the Computer Program. Constant S k y l i n e Tension 49 The program produces two kinds of output: graphs showing the v a r i a t i o n i n power flow components with the po s i t i o n of the carriage i n the span, and a summary table l i s t i n g a l l of the s i g n i f i c a n t system parameters and variables for s p e c i f i e d span positions. Figure 18 i s a flow chart for the program (Appendices C and D d e t a i l other models and algorithms used i n the program). 50 Figure 18. Flowchart for the Computer Program. START Specify system parameters] Span p o s i t i o n = t a i l h o l d Increment p o s i t i o n across the span |£ no > Calculate v e r t i c a l p o s i t i o n of the carriage from the ground p r o f i l e Calculate v e r t i c a l p o s i t i o n of the carriage from the allowable l i n e tensions Calculate l i n e tensions * Calculate the l i n e lengths i n the span and on the drums Calculate the l i n e speed r a t i o Calculate the drum wrap r a d i i Calculate torque, angular v e l o c i t y and power f o r each drum END 51 CHAPTER V THE RESULTS OF THE COMPUTER MODEL The r e s u l t s of the combined c a b l e mechanics and i n t e r -l o c k powerflow computer model d e s c r i b e d i n Chapter IV are presented as a s e r i e s of graphs and t a b l e s which show the v a r i a t i o n of s e l e c t e d system parameters with the p o s i t i o n of the loaded c a r r i a g e i n the span. The values o f these parame-t e r s are presented f o r s p e c i f i e d span p o s i t i o n s . The extreme span p o s i t i o n s s e l e c t e d r e p r e s e n t the c l o s e s t p o s i t i o n s to the l i n e supports to which the c a r r i a g e may be taken, the a c t i o n o f the weight of the l i n e s out i n the span prevents the c a r r i a g e from hanging c l o s e r to the l i n e supports. The extreme p o s i t i o n s of the span r e p r e s e n t areas where the c a r r i a g e w i l l e i t h e r be moving s l o w l y , or decking w i l l be t a k i n g p l a c e , so the power flows at these p o s i t i o n s are not c r i t i c a l . The dimensions o f the drums used i n the computer model were taken from those of the drums i n the Lantec I n d u s t r i e s i n t e r l o c k e d winch s e t . Appendix F c o n t a i n s a t a b l e o f drum dimensions and l i n e s p e c i f i c a t i o n s recommended f o r a running s k y l i n e system. The drums are wide and have a l a r g e b a r r e l diameter, a design which g i v e s a hig h per-wrap l i n e c a p a c i t y and which r e s u l t s i n a sm a l l number of changes i n r e l a t i v e m a i n l i n e and haulback drum wrap r a d i i as the c a r r i a g e c r o s s e s the span. 52 F i g u r e s 19 and 20 show the r e s u l t s of m o d e l l i n g a f u l l y suspended l o a d and a running s k y l i n e maintained a t a constant t e n s i o n at the upper l i n e support. Tables 1 and 2 d e t a i l the system parameters and span geometry, and c o n t a i n the s u p p o r t i n g data, f o r F i g u r e s 19 and 20 r e s p e c t i v e l y . In both f i g u r e s the p l o t s of haulback (running s k y l i n e ) and m a i n l i n e power are v i r t u a l l y l e v e l , s t r a i g h t l i n e s . Tables 1 and 2 show t h a t the s k y l i n e t e n s i o n has been maintained constant, the s m a l l changes i n the l i n e v e l o c i t y r a t i o across the span mean t h a t the l i n e v e l o c i t i e s are to a l l i n t e n t s and purposes constant; thus, the product of l i n e t e n s i o n and l i n e v e l o c i t y , l i n e power, i s v i r -t u a l l y c o nstant. The m a i n l i n e t e n s i o n requirement i s h i g h e r i n u p h i l l y a r d i n g , w h i l s t i n d o w n h i l l y a r d i n g , the haulback t e n s i o n r e q u i r e -ment i s h i g h e r . Since the l i n e speeds can be c o n s i d e r e d to be equal, the m a i n l i n e power i s h i g h e r i n u p h i l l y a r d i n g , the h a u l -back l i n e power hig h e r i n d o w n h i l l y a r d i n g . T h i s d i f f e r e n c e i n l i n e power i s r e f l e c t e d i n the curves f o r y a r d i n g power, Hp Y. In F i g u r e 19, u p h i l l y a r d i n g , the y a r d i n g power r e q u i r e -ment i s p o s i t i v e f o r a l l p o s i t i o n s across the span: the prime mover must supply power to r a i s e the p o t e n t i a l energy of the l o a d . F i g u r e 20, d o w n h i l l y a r d i n g , shows t h a t the y a r d i n g power i s ne g a t i v e f o r a l l p o s i t i o n s across the span: as the c a r r i a g e moves across the span, the p o t e n t i a l energy of the lo a d i s reduced and the r e l e a s e d energy must be d i s s i p a t e d , probably i n the haulback drum brakes. 53 gure 19. Interlock Power Flow Components v. Carriage Posi t i o n : 1200 f t . Span, U p h i l l Yarding and F u l l y Suspended Load. 388 258 288 158 188 58 8 -58 -188 -158 H -288 \ -258 ': -388 -POWER COMPONENTS vc. CflRRIRGE POSITION Hp H I I I I I I I I I | I I I I I I I I I 11 I I I I |V| > I I |^l I I I I I I I I | I II M I II I | M I I I I II I | 288 488 688 ^-—8,88 1888 1288 Hp - " ^ "P. CRRRIRGE POSITION < Dletanoe from the tmllhold ) i f t . B Hp Hp Hp HP, M H B Mainline Power Haulback Power A u x i l l i a r y Transmission Power Yarding Power 54 TABLE 1 : Yarding conditions and summarized supporting data for the system modelled i n Figure 19 SUMMARY TfiBLE SPAN LENGTH : 1206.00 ft ( 365.76 m ) ELEVATION DIFFERENCE BETWEEN SUPPORTS : 600.00 f t ( 182.68 m > ( Uph i 1 1 yard i ng ) LOAD: <Car-ri age+Logs> : 2500.00 lb (. 113-5.99 kg ) MID-SPAN DEFLECTION <•/.*/ : 3.94 MAINLINE VELOCITY:-500. 00 f t -'min (-2.54 m/s) MODEL OPTION: Constant running s k y l i n e t e n s i o n < 13OOO.O0 lb ; 5896.76 kg F u l l y suspended load SPAN :•: 10. 00 20. 00 40. 00 60 . 00 95. 00 (from t a i 1 ho 1d) TENSION Haulback : lb 130 0 0.0 0 13000.00 13000.00 13000.00 13000.00 Haulback : kg 5898.37 5898.37 5898.37 5898.37 5898.37 Mai n 1 i ne : 1 b 14338.25 14347.20 14353.04 14343.14 14289.53 Mai n 1 i ne : kg 6505.56 6509.62 6512.27 6507.78 6483.45 DRUM RADIUS RATIO < hau l b . : mai n ) . 76 . 85 . 95 1.11 1 . 38 TORQUE Hau1bac k drum:1b f t 14218.75 1354 1.67 12864.58 12187.50 10833.33 Haulback drum:N m 19280.62 18362.50 17444.37 16526.25 14 690.00 M a i n 1 i n e d r u rn: 1 b f t 11948.54 12703.25 13455.97 14940.77 16373.42 Mai n1i ne drum:N m 16202.22 17225.61 18246.30 20259.69 22202.36 Gear B(aux.):lb f t 7109.37 6770.83 6432.29 6093.75 5416.67 Gear B(aux. ): N m 9640.31 9181.25 8722.19 8263.12 7345.00 LINE VELOCITY RATIO (Haulb.:Mai n.) -1.01 -1.01 -1.01 -1 . 00 -1 . 00 REVS.per MIN. Hau1back drum 73. 61 77. 13 80. 90 85. 13 95. 34 Mai n1i ne drum -95.48 -89.86 -84.87 -76.38 -69.44 Gear B (aux.input) -43.75 -25.46 -7. 94 17.48 51 . 80 POWER C,< hau1bac k) : hp 199.27 198.88 198.16 197.54 196.66 C,(haul back) : kW 267.03 266.50 265.54 264.70 263.52 A,(mainline) : hp -217.22 -217.36 -217.45 -217.30 -216.48 A,( m a i n l i n e ) : k W -291.08 -291.26 -291.38 -291.18 -290.C 9 B,(au x.trans) : hp 59. 22 32. 82 9. 73 -20.28 -53.42 B , (au x.t r an s ) : k W 79. 36 43. 99 13. 04 -27.18 -71.58 Y,(y a r d i ng) : hp 17. 95 18. 48 19. 29 19. 76 19. 83 Y,(yardi ng) : kW 24. 05 24. 76 25.84 26. 48 26. 57 55 F i g u r e 20. I n t e r l o c k Power Flow Components v. C a r r i a g e P o s i t i o n : 1200 f t . Span, Downhill Y a r d i n g and F u l l y Suspended Load. 388 258 288 158 188 t. a g.58 t 2 8 £ - 5 8 -188 -158 -288 -2581-POWER COMPONENTS vs. CRRRIRGE POSITION Hp H II I I l t l I I | l I I I I II II | M V l l l l \} | l I I I I I l i I \ i i M l M l l | M I I I I l l I | ?02 488 088 a vS88 1288 Hp M Hp, -302L CRRRIRGE POSITION < Distance from the tallhold ) t f t . B 56 TABLE 2 : Yarding Conditions and Summarized Supporting Data for the System Modelled i n Figure 20 SUMMARY TABLE SPAN LENGTH : 1206.06 f t ( 365.76 m > ELEVATION DIFFERENCE BETWEEN SUPPORTS : -606.00 f t (-182.88 m ) ( Ii o w n h i l l y a r d i n g > LOAD: (Carr i age+Logs) : 2506.60 lb ( 1 133.99 kg > MID-SPAN DEFLECTION W> : 4.12 MAINLINE VELOCITY:-560. 00 f t m i n (-2.54 n i s- > MODEL OPTION: Constant running i k y l i n e t e n s i o n < 13666.00 lb ; 5896.76 kg > F u l l y suspended load SPAN X (from tai1 ho 1d> 10. 06 20. 66 40. 06 60. 66 95. 06 TENSION Haulback : lb Hau1bac k : k g Ma i n l i n e : 1b Mai n 1 i ne : kg 12580.16 5767.88 16939.11 4 96 3.3 6 12586.11 5767.85 11029.41 5664.27 12580.04 5767.82 11200.39 5681.85 12586.61 5767.61 11359.45 5154.01 12586.00 5707.86 11611.09 5268. 19 DRUM RADIUS RATIO < hau 1 b. : mai n) .81 . 85 1 . 06 1.11 1 . 38 TORQUE H au 1 b ac k d r u rn: 1 b f t Haulback drum: N rn Mai n l i n e drum :1b f t Mai n 1 i ne drum: N m Gear B(aux.):lb f t Gear B<aux.):N m 13759.55 18657.95 9685.67 13133.77 6879.77 9328.97 13104.2S 17769.46 9765.63 13242. 19 6552. 14 8884.70 12449.06 16880.85 1 10S3.72 15029.53 6224.50 8440.42 11793.76 15992.34 11832.76 16645.22 5896.88 • 7996.17 10483.33 14215.40 13364.38 18640.73 5241.67 7167.76 LINE VELOCITY RATIO (Haulb.:Mai n.) -1.61 -1.01 -1.01 -1 . 66 -1 . 00 REVS.per MIN. Hau1bac k drum Mai n1 i ne drum Gear B (aux.input) 73. 47 -89.86 -32.78 77. 04 -69.86 -25.65 86.86 -80.40 .91 85. 1 1 -76.38 17. 45 95. 36 -69.44 51. 72 POWER C,(haulback) : hp C,(haulback) : kW A,< mai n1i ne) : hp R,(mainline) : k W E,(aux.t rans) : hp l , ( a u x . t r a n s ) : kW Y, (y ar d i ng) : hp Y,(yardi ng) : kW 192.49 257.94 -165.73 -222.07 42. 94 57. 54 -26.77 -35.87 192.21 257.57 -167.69 -223.91 32. 61 42.89 -25,12 -33.66 191.66 256.83 -169.68 -227.38 -1. 08 -1.44 -21.98 -29.45 191.12 256.10 -172.69 -230.61 -19.59 -26.26 -19.03 -25.50 196.23 254.91 -175.91 -235.71 -51.62 -69.17 -14.32 -19.19 57 In F i g u r e s 19 and 20 the curve f o r the power f l o w i n g i n the a u x i l i a r y ( h y d r o s t a t i c ) t r a n s m i s s i o n , Hp , shows the way i n which t h i s power flow r e v e r s e s d i r e c t i o n as the c a r r i a g e c r o s s e s the span. The p o i n t at which t h i s curve c r o s s e s the x - a x i s ( i . e . , where there i s zero power flow) corresponds to the p o i n t i n the span a t which both drums are r o t a t i n g a t the same abso-l u t e speed, and the a u x i l i a r y i n p u t gear, gear B, i s s t a t i o n a r y . T h i s i s the " l o c k - p o i n t " . F i g u r e s 21 and 22 show the power flow i n the i n t e r l o c k f o r the two extremes of span p o s i t i o n ; the val u e s of the power flow components are taken from Table 1. F i g u r e : 21 shows the d i r e c t i o n o f power flow when the c a r r i a g e i s at the t a i l s p a r . Power from both the haulback drum and the prime mover e n t e r s the m a i n l i n e drum v i a gear F and leaves on the m a i n l i n e . T h i s power, s u b j e c t t o l o s s e s due to the change i n p o t e n t i a l energy of the lo a d , enters the haulback drum on the haulback l i n e , e n t e r i n g the d i f f e r e n t i a l on gear c a r r i e r C. In t h i s case, the haulback drum i s not r o t a t i n g f a s t enough due to i t s l a r g e wrap r a d i u s , so angular v e l o c i t y , hence power, must be added to the d i f f e r e n t i a l t o maintain the r e q u i r e d speed on gear A. The power added a t gear B i s s u p p l i e d v i a gear D, the v a r i a b l e displacement h y d r a u l i c pump ( a c t i n g as a pump), the f i x e d d i s -placement h y d r a u l i c motor ( a c t i n g as a motor) and gears H and I. Since gears F, E and D have a f i x e d r e l a t i v e speed r a t i o , the power l e a v i n g the d i f f e r e n t i a l , v i a gear A, s p l i t s a t gear 5 8 Figure 21. Schematic of the Power Flows i n the Winch Set when the Carriage i s Near the T a i l h o l d . Hp R : 59-22 Figure 2 2 . Schematic of the Power Flows i n the Winch Set when the Carriage i s Near the Headspar. 60 E a c c o r d i n g t o t h e t o r q u e r e q u i r e m e n t s o f g e a r s F a n d D . A n y a d d i t i o n a l t o r q u e r e q u i r e m e n t o n g e a r F i s m a d e u p b y t h e p r i m e m o v e r , v i a g e a r G . F i g u r e 22 s h o w s t h e d i r e c t i o n o f p o w e r f l o w w h e n t h e c a r r i a g e i s n e a r t o t h e h e a d s p a r . I n t h i s c a s e , t h e s m a l l w r a p r a d i u s o f t h e h a u l b a c k d r u m c a u s e s i t t o r o t a t e f a s t e r t h a n t h e a n g u l a r v e l o c i t y n e e d e d t o m a i n t a i n t h e r e q u i r e d a n g u l a r v e l o c i t y o n g e a r A . A n g u l a r v e l o c i t y m u s t b e r e m o v e d a t g e a r B ; t h u s , t h e p o w e r f l o w s v i a g e a r s I a n d H , t h r o u g h t h e f i x e d d i s p l a c e m e n t m o t o r ( a c t i n g a s a p u m p ) , t o t h e v a r i a b l e d i s p l a c e -m e n t p ump ( a c t i n g a s a m o t o r ) a n d t h r o u g h t o g e a r s D a n d E . T h i s p o w e r i s r e c o m b i n e d w i t h t h e p o w e r l e a v i n g o n g e a r A , a t g e a r E , a n d i s r e c i r c u l a t e d t o t h e m a i n l i n e d r u m v i a g e a r F . A g a i n , a n y a d d i t i o n a l t o r q u e r e q u i r e m e n t a t g e a r F i s m a d e u p b y t h e p r i m e m o v e r . A n e x a m i n a t i o n o f T a b l e s 1 a n d 2 s h o w s t h a t t h e r e q u i r e d a n g u l a r v e l o c i t y o f t h e a u x i l i a r y i n p u t g e a r , g e a r B , d e t e r m i n e s t h e d i r e c t i o n o f t h e p o w e r f l o w i n t h e a u x i l i a r y t r a n s m i s s i o n b e c a u s e t h i s g e a r m u s t t u r n a g a i n s t t h e t o r q u e o n t h e h a u l b a c k d r u m , w h i c h i s a l w a y s p o s i t i v e . F i g u r e 23 a n d T a b l e 3 i l l u s t r a t e t h e e f f e c t o f s h o r t e n i n g t h e l e n g t h o f t h e s p a n t o 6 0 0 f e e t w h i l s t m a i n t a i n i n g t h e s p a n s l o p e , l o a d w e i g h t , e t c . t h e s a m e a s t h a t u s e d i n T a b l e 1 . T h e y a r d i n g p o w e r r e q u i r e m e n t r e m a i n s p o s i t i v e f o r a l l p o s i t i o n s a c r o s s t h e s p a n , b u t e x a m i n a t i o n o f t h e p o w e r f l o w c u r v e f o r t h e a u x i l i a r y t r a n s m i s s i o n r e v e a l s t h a t f o r m o s t o f t h e s p a n , Figure 23. Interlock Power Flow Components v. the Carriage Posit i o n : 600 f t . Span, U p h i l l Yarding and F u l l y Suspended Load. 388 r 258 288 158 188 3 S.58 c. 2 8 e-58 i -188 H -158 '--288 \ -258 \ -3881 POWER COMPONENTS vs. CflRRIRGE POSITION Hp H — l — l — l — l — l — l — I — h H — i — i — i — i — i — i i i i | i i i i i > i o i — | 288 488 688 Hp M — »P. CflRRIRGE POSITION ( Distance from the tailhold ) (ft. B 62 TABLE 3 : Yarding Conditions and Summarized Supporting Data for the System Modelled i n Figure 23 SUMMARY TABLE SPAN LENGTH : 600.66 f t ( 182.88 tn ) ELEVATION DIFFERENCE BETWEEN SUPPORTS : 366.06 f t ( 91.44 m ) .(. U p h i l l yard i ng > LOAD: (Cart-i age + Logs ) : 2560.66 lb ( 1 133.99 kg ) MID-SPAN DEFLECT I ON ( V. ) : 3.28 MAINLINE VELOCITY:-560. 66 f t .-min (-2.54 »/•*> MODEL OPTION: Constant running s k y l i n e t e n s i o n ( 13666.66 lb ; 5896.76 k F u l l y suspended load SPAN X (from t a i 1 h o 1 d ) 16. 66 26 . 66 40 . 00 60. 00 95. 86 TENSION Haulback : lb Haulback : kg Mai n l i n e : lb Mainline : kg 13 6 66.6 6 5898.37 14189.42 64 38.63 13600.00 5898.37 14262.36 6443.96 13006.66 5898.37 14223.66 6453.29 13000.00 5898.37 14236.91 6459.58 13660.6 6 5898.37 14245.62 64 63.26 DRUM RADIUS RATIO (hau l b . : tnai n) . 79 . 87 . 87 . 95 1 . 65 TORQUE Hau 1 bac 1: drum:1b ft Hau 1 bac k drurn: N m Ma i n l i n e d r u m:1b f t Mainline d r u m:N m Gear BCaux.):1b f t Gear B(au x. ): N m 16256.6 6 22635.66 14641.62 19646.43 8125.06 11617.56 15572.92 21116.87 14794.12 26066.83 7786.46 1055.8.44 15572.92 21 1 16.87 14815.69 2 0 6 96.67 7786.46 10558.44 14895.83 26198.75 15571.62 21115.12 7447.92 16099.37 14218.75 19286.62 16322.42 22133.2 6 7169.37 9646.31 LINE VELOCITY RATIO (Haulb.:Main.) -1.61 -1.01 -1.01 -1 . 60 -1.68 REVS.per MIH. Hau1bac k drum Mai n1ine drum Gear B (aux.input) 64. 23 -86.46 -32.35 66. 94 -76.38 -18.90 66. 77 -76.38 -19.23 69. 64 -72.75 -6.21 72. 76 -69.44 6.51 POWER C,(haul back) : hp C,(haul back) : kW A,(mai n l i ne) : hp A, (mai n l i ne) : kW E,(aux.trans) : hp B, (aux.trans) : k W Y,(yard i ng) : hp Y,(yarding) : kW 198.74 266.31 -214.97 -288.66 56. 64 67. 05 16. 23 21 . 75 198.47 265.95 -215.16 -288.32 28. 02 37.54 16. 69 22. 37 197.97 265.29 -215.48 -288.74 28.52 38. 21 17. 50 23. 45 197.52 264.67 -215.69 -289.62 8.81 11.81 13.17 24. 35 196.81 263.73 -215.81 -289.19 -8.81 -11.81 19. 66 25. 46 6 3 p o w e r e n t e r s t h e d i f f e r e n t i a l i n t h e m a n n e r d e p i c t e d i n F i g u r e 2 1 . I n F i g u r e s 19 a n d 20 ( 1 , 2 0 0 f o o t s p a n ) t h e e x c u r s i o n s o f t h e a u x i l i a r y t r a n s m i s s i o n p o w e r c u r v e a b o u t t h e x - a x i s a r e r e l a t i v e l y s y m m e t r i c a l ; t h i s m e a n s t h a t t h e p o w e r f l o w s , i n b o t h d i r e c t i o n s , i n t h i s h y d r a u l i c p o w e r l o o p a r e a p p r o x i m a t e l y e q u a l a n d t h a t t h e m a x i m u m s i z e o f t h e h y d r o s t a t i c t r a n s m i s s i o n r e q u i r e d w i l l b e a p p r o x i m a t e l y 60 H p . I n F i g u r e 23 t h e a u x i l i a r y t r a n s m i s s i o n p o w e r f l o w c u r v e , Hp , i s n o t s y m m e t r i c a l a n d t h e B m a x i m u m s i z e o f h y d r o s t a t i c t r a n s m i s s i o n r e q u i r e d w i l l b e 5 0 H p . a s d i c t a t e d b y t h e p o w e r r e q u i r e m e n t a t t h e t a i l s p a r , d e s p i t e t h e f a c t t h a t n e a r t h e h e a d s p a r , o n l y 9 H p . ( a p p r o x i m a t e l y ) w i l l b e c i r c u l a t i n g . T h e s y m m e t r i c a l Hp_ c u r v e i n F i g u r e s 19 a n d 20 s h o w t h a t B t h e 1 , 2 0 0 f o o t s p a n i s o p t i m u m f o r t h e d r u m d i m e n s i o n s , l i n e d i a -m e t e r s a n d l i n e l e n g t h s m o d e l l e d . I t i s p o s s i b l e t o p r o d u c e a s i m i l a r s y m m e t r i c a l c u r v e f o r t h e 6 0 0 f o o t s p a n . T o d o t h i s , t h e H p n c u r v e m u s t b e s h i f t e d d o w n w a r d s s o t h a t t h e e x c u r s i o n s o f t h e c u r v e a b o v e a n d b e l o w t h e x - a x i s a r e a p p r o x i m a t e l y e q u a l . T h i s m a y b e a c h i e v e d b y r e d u c i n g t h e l e n g t h a n d / o r d i a m e t e r o f t h e h a u l b a c k c a b l e o r b y a l t e r i n g t h e d i m e n s i o n s o f t h e h a u l b a c k d r u m , o n a " c u t a n d t r y " b a s i s . B y s h i f t i n g t h e a u x i l i a r y t r a n s -m i s s i o n p o w e r c u r v e d o w n w a r d , t h e r e q u i r e d s i z e o f t h e h y d r o -s t a t i c t r a n s m i s s i o n i s r e d u c e d ; t h i s w i l l a l s o r e d u c e t h e f i r s t c o s t o f t h e t r a n s m i s s i o n . A d r u m s e t o p t i m i s e d f o r a s p a n o f 6 0 0 f e e t w i l l n o t o p e r a t e e f f i c i e n t l y b e y o n d t h i s s p a n l e n g t h . 64 F i g u r e s 24 and 25 (Tables 4 and 5 r e s p e c t i v e l y ) show the e f f e c t of m a i n t a i n i n g the c a r r i a g e a t a constant (5-foot) h e i g h t above a s t r a i g h t l i n e ground p r o f i l e . The e f f e c t of f r i c t i o n between a dragging l o a d and the ground i s a l s o modelled. Because the lo a d path and the ground p r o f i l e are p a r a l l e l s t r a i g h t l i n e s , t h i s f r i c t i o n w i l l be constant across the span. The curves f o r both m a i n l i n e and haulback l i n e power e x h i b i t marked minimae and maximae due p r i m a r i l y t o the g r e a t l y i n c r e a s e d t e n s i o n requirements i n the l i n e s when the c a r r i a g e i s at midspan. F i g u r e s 24 and 25 correspond t o the same y a r d i n g c o n d i t i o n s as those modelled i n F i g u r e s 19 and 20 r e s p e c t i v e l y . F i g u r e 24 shows the power requirement i n u p h i l l y a r d i n g . The l a r g e i n c r e a s e i n y a r d i n g power requirement when compared to F i g u r e 19 i s due to the f a c t t h a t a d d i t i o n a l power i s r e q u i r e d to overcome the f r i c t i o n between the l o g and the ground. In F i g u r e 25 the amount of power t h a t must be d i s s i p a t e d i n d o w n h i l l y a r d i n g i s reduced by the power r e q u i r e d to over-come the f r i c t i o n between the l o g and the ground. An examination of the a u x i l i a r y power flow curves i n F i g u r e s 24 and 25, and o f the numerical values i n the support-i n g t a b l e s (Tables 4 and 5 respectively) shows t h a t when com-pared t o the constant s k y l i n e tension/suspended l o a d model, aux-i l i a r y power c o n s t i t u t e s a g r e a t e r percentage of m a i n l i n e and haulback power. Th i s i n c r e a s e d power flow i s caused by the much g r e a t e r v a r i a t i o n i n l i n e speed r a t i o , across the span, .required i n the 65 Figure 24. Interlock Power Flow Components v. the Carriage P o s i t i o n : 1200 f t . Span, U p h i l l Yarding and Dragging Log Load. 66 LE 4 : Yarding Conditions and Summarized Supporting Data for the System Modelled i n Figure 24 SUMMARY TABLE SPAN LENGTH : 1200.00 ft < 365.76 m ) ELEVATION DIFFERENCE BETWEEN SUPPORTS : 600.00 f t ( 182.88 l» ) ( U ph i 1) yarding ) LOAD: (Carriage+Logs) : 2500. 00 lb < 1133.9'? kg ) MID-SPAN DEFLECTION <:•;:> : 2.92 MAINLINE VELOCITY:-5O0. 00 f t /min C-2.54 m/s) MODEL OPTION: Constant c a r r i a g e height ( 5.00 f t ; 1.52 m ) Dragging load SPAN :•: (f rom t ai 1 h o 1 d) 10.00 20. 00 40. 00 60. 00 95. 00 TENSION Haulback : lb Haulback : kg Mainline : lb M a i n 1 i n e : k g 5777.61 2621.42 7309.61 3316.5 2 16159.15 4609.41 11833.10 5368.92 14946.02 6781.32 16631.31 7545.97 14612.94 6630.19 16267.92 7381.09 2967.18 1346.27 5012.67 2274.35 DRUM RADIUS RATIO ( hau l b . : ma.i n ) .76 . 85 . 95 1.11 1 . 38 TORQUE Haul back drum: 1 b f t Haulback d r u rn: N m Mai n1 i ne drum:1b f t Mai n 1 i ne drum: N tn Ge ar B<*ux.>:lb f t Gear B <•'. aux. > : N m 6319.26 8568.91 6091.34 8259.86 3159.63 42S4.46 10582.44 14349.79 10477.22 14207.11 5291.22 7174.90 14790.33 20055.69 15591.86 21142.56 7395.17 10027.84 13699.63 18576.70 16945.75 22978.44 6849.82 9288.35 2472.65 3352.91 5743.69 7788.44 1236.32 1676.46 LINE VELOCITY RATIO (Haulb.:Main.) -.84 -.93 -.97 -.98 -1.01 REVS.per MIN. Hau1bac k drum Mai n1i ne drum Gear E (aux.input) 61.18 -95.48 -68.59 71.00 -89.86 —37•73 77. 84 -84.87 -14.07 83. 20 -76.38 13. 62 96. 22 -69.44 53.57 POWER C,<hau1bac k) : hp C,(haul back) : kW A,(mainline) : h p A, (mai n 1 i ne ) : k W . B, (aux.trans) : hp B,(aux.trans) : kW Y,(yard i ng) : hp Y,(yarding) : k W 73. 62 98.65 -110.74 -148.39 41 . 26 55. 29 37. 12 49. 74 143.66 191.70 -179.27 -240.22 38. O 1 50. 94 36. 21 48. 52 219.19 2 9 3.72 -251.96 -337.63 19. 82 26. 55 32. 77 43. 91 217.61 290.79 -246.46 -330.25 -17.77 -23.81 29.45 39.46 45. 30 60. 71 -75.94 -101.76 -12.61 -16.90 30. 64 41 . 06 67 Figure 25. Interlock Power Flow Components v. the Carriage P o s i t i o n : 1200 f t . Span, Downhill Yarding and Dragging Log Load. 68 TABLE 5 : Yarding Conditions and Summarized Supporting Data for the System Modelled i n Figure 25 SUMMARY TABLE SPAN LENGTH : 1200.00 f t ( 365.76 n, ) ELEVATION DIFFERENCE BETWEEN SUPPORTS : -600.00 f t (-182.88 m ) ( Dounh i 1 1 y a r d i n g ) LOAD: ( C a r r i a g e + L o g s ) : 2500.00 l b ( 1133.99 kg > MID-SPAN DEFLECTION ( :•: ) : 2.92 MAINLINE V E L O C I T Y : - 5 0 0 . OO f t /'min (-2.54 m-'s) MODEL OPTION: C o n s t a n t c a r r i a g e h e i g h t ( 5.OS f t ; 1.52 m ) D r a g g i ng 1oad SPAN ( f r o rn t a i 1 h o 1 d ) 10. 08 20 . 80 40 . 00 60 . 00 95. OO TENSION H a u l b a c k : l b Hau1bac k : kg Mai n l i ne : 1 b M a i n l i n e : kg 7056.90 3201.86 5444.52 2470.29 11828.18' 5366.69 10395.46 4716.64 16937.34 7684.82 15678.47 7113.64 16362.31 7423.92 -15241 .49 6915.38 2944.13 1335.81 2678.54 1215.31 DRUM RADIUS RATIO (h au 1 b. : rn a i n) .81 . 85 1 . 08 1.11 1 . 33 TORQUE Hau1bac k drum :1b f t H au1b ac k d r u m:N m Mai n1i ne drum:1b f t Mai n l i ne drum:N m Gear B ( a u x. ): 1 b f t Gear B(aux.>:N m 7718.48 10466.26 4820.67 6536.83 3859.24 5233.13 12321.02 16707.31 9204.32 12481.05 6160.51 8353.65 16768.91 22727.79 15515.15 21038.54 83S0.45 1 1363.89 15339.67 28888.59 15876.55 21528.60 7669.83 10408.29 2453.44 3 3 2 6.86 3069.16 4161.78 1226.72 1663.4 3 LI N E VELOCITY PATIO ( H a u l b . : M a i n . ) -.85 -.94 -. 97 -.98 -1.01 R E V S . p e r MIN. H a u l b a c k drurn Mai n 1 i ne drurn Gear B ( a u x . i n p u t ) 61 . 80 -89.86 -56.12 71 . 47 -89.86 -36.78 78. 09 -80.40 -4.64 83. 37 -76.38 13. 97 96. 67 -69.44 54. 46 POWER C , ( h a u l back ) : hp. C , ( h a u l b a c k ) : kW A , ( m a i n l i n e ) : hp A, (mai n l i n e ) : kW B, ( a u x . t r a n s ) : hp E , ( a u x . t r a n s ) : kW Y , ( y a r d i ng) : hp Y , ( y a r d i ng) : kW 90.83 121.71 -82.48 -110.53 41 . 24 55.26 -8. 34 -11.18 167.67 224.68 -157.49 -211.04 43. 14 57. 81 -10.18 -13.65 249.28 333.92 -237.53 -318.29 7.4 0 9. 92 -11. 67 -15.64 243.58 326.30 -230.91 -309.41 -20.41 -27.34 -12.60 -16.88 45. 16 60. 51 -48.58 -54.38 -12.72 -17.05 -4. 58 -6.14 69 constant carriage height model; t h i s i n turn means that larger amounts of angular v e l o c i t y must be added or removed at the d i f f e r e n t i a l . The upswing of the a u x i l i a r y transmission and yarding power curves i n Figures 24 and 25 i s due to the r e l a t i v e l y large increase i n tension requirements i n the mainline r e l a t i v e to the haulback l i n e , as the carriage approaches the headspar. Figure 26 i l l u s t r a t e s the e f f e c t of changing the slope of the span from 100% (downhill yarding) to -100% (u p h i l l yarding) whilst maintaining a l l other system variable values the same as those detailed i n Table 1. The yarding power requirement i s plotted against span p o s i t i o n . The curve for 0% span slope shows the amount of power required to overcome the f r i c t i o n between the log and the ground only, as the pote n t i a l energy of the log remains the same. Figures 27, 28 and 29 i l l u s t r a t e the e f f e c t of varying the load between 1,000 and 5,000 pounds for span slopes of 50% (downhill yarding), 0% and -50% (u p h i l l yarding) respectively. When the span slope i s 50%, increasing the load increases the amount of power to be dissipated i n the haulback drum brakes; the r e l a t i v e l y close spacing of the curves i s due to increasing f r i c t i o n with increasing log weight, which absorbs some of the power which would otherwise require d i s s i p a t i o n i n the brakes. With 0% span slope, the increasing yarding power requirement simply r e f l e c t s the increasing f r i c t i o n with log weight. With 70 F i g u r e 26. Yardin g Power Requirement v. C a r r i a g e P o s i t i o n f o r Span Slopes Ranging from -100% to 100% : 2000 l b . Log Load. 68 58 48 38 f 28 YRRDING POWER vs. CflRRIRGE POSITION £18 m c 2 8 SPAN SLOPE -100%] . . . - 6 6 % l u P h l l t _3 3 %jyarding 0% 1 1 1 1 1 m 11111111 H 1 1 1 1 1 1 1 1 1 1 1 1 n 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 n 11111 o/i 111 3 3 % U i - i e z £ -28 -38 -48 r 1 0 0 % downhill yardi ng -58 -68 CflRRIRGE POSITION (distance from the tallhold)tft. 71 Figure 27. Yarding Power Requirement v. Carriage P o s i t i o n for Load Weights Ranging from 1000 l b . to 5000 l b . ; 50% Span Slope (Downhill Yarding). 88 78 68 58 48 38 «-2B e 3 O a 18 « c ? 8 or-18 ut z -28 -38 -48 -58 V -68 L YRROING POWER v«. CflRRIRGE POSITION ' LOAD WEIGHT (lb) 1288 CflRRIRGE POSITION (distance from the ta11hold)ift. 72 Figure 28. Yarding Power Requirement v. Carriage Position for Load Weights Ranging from 1000 l b . to 5000 l b . ; 0% Span Slope. 88 78 68 58 48 38 t-28 § o a 10 i 2 8 z £-28 -38 -48 h -50 -68 L YRRDING POWER v«. CflRRIRGE POSITION 111 1 1 1 1 1 l 1 1 m n 1 1 1 | H 1 1 » M11111111 i l l 11111111111 11 l I I II I 111 208 488 688 880 1808 1288 I- CflRRIRGE POSITION (dletanoe from the tat 1 ho Id) if t. 73 Figure 29. Yarding Power Requirement v. Carriage Pos i t i o n for Load Weights Ranging from 1000 l b . to 5000 l b ; -50% Span Slope (Uphill Yarding). 88 78 68 58 48 38 «-28 • 3 O g-18 e 5 8 K-18 UJ X -28 -38 -48 -58 I--68 L YRRDING POWER vs. CflRRIRGE POSITION LOAD W E I G H T ( l b ) 5000 I 1 I I I II I I | I H I II I I I | I I I I I IHI | I I I I I I I I I | I I I I I I I I I | I II I I I H I | 288 488 688 888 1888 1288 CflRRIRGE POSITION (distance from the talIhold)ift. 7 4 a -50% span slope, the amount of yarding power again increases with increasing load, the wider spacing of the curves r e f l e c -t i n g the increased power required to change the potential energy of the load over a greater v e r t i c a l distance and the increasing f r i c t i o n force between the load and the ground. Although the ef f e c t s of f r i c t i o n and the hydraulic losses i n the a u x i l i a r y transmission have not been included i n the power flow model, the summary tables produced by the computer program contain a l l of the esse n t i a l information required for the design of an interlocked drum set using t h i s type of d i f f e r e n t i a l unit. The program i s also e a s i l y altered so that changes i n the drum dimensions, l i n e lengths and sizes, loads and span geometry may be made, and the ef f e c t s of these changes observed. Optimiza-ti o n of the drum set design for d i f f e r e n t span lengths and maxi-mum loads i s also f a c i l i t a t e d . In examining Tables 1 and 2 i t i s apparent that the maxi-mum torque loading on the d i f f e r e n t i a l unit, which i s the same as the torque on the haulback drum, i s 14,200 lb f t (obtained from Table 1 i n the 10% span p o s i t i o n ) . This figure i s the max-imum torque loading produced on the d i f f e r e n t i a l when a f u l l y suspended 2,500 lb load i s flown over a 1,200 f t span with a -50% (u p h i l l yarding) span slope. The load comprises a 500 lb carriage and a 2,000 lb log; a size of log i s t y p i c a l for t h i n -ning and smallwood yarding. The value for the maximum torque loading for the given span and load conditions may be used as a guideline i n the the selection of a suitable d i f f e r e n t i a l unit; s i m i l a r l y , any system parameters may be changed and the maximum res u l t i n g torque loading determined and used to select a suitable gear unit for the new conditions. It was hoped to obtain information about the torque and power ratings of several commercially manufactured d i f f e r -e n t i a l gear units and use t h i s information to recommend suitable units for d i f f e r e n t classes of winch set (eg. "smallwood" or "Coastal"). None of the manufacturers contacted was w i l l i n g to supply the required information, although two of them, "off the record", indicated that t h e i r larger d i f f e r e n t i a l units had the torque capacity suitable for application i n a smallwood yarder. 76 CHAPTER VI CONCLUSIONS AND RECOMMENDATIONS From the development of the power flow model and i t s testing through the use of the computer program, the nature of, and the factors a f f e c t i n g the power flows i n the interlock have been determined. The bevel-gear d i f f e r e n t i a l unit, from a th e o r e t i c a l standpoint, would seem to be a viable alternative to the conventional e p i c y c l i c planetary gear t r a i n as an inter l o c k i n g mechanism. The location of the d i f f e r e n t i a l unit inside the haulback drum i t s e l f removes the need for an ex-ternal gear box and permits the use of the d i f f e r e n t i a l as a f i n a l reduction gear as well as an inte r l o c k i n g mechanism. This application should be economical of space and reduce the torque loading on the.intermediate gear t r a i n , a d e f i n i t e advantage where the application of clutches i n t h i s gear t r a i n i s considered. It i s unfortunate that the torque and power ratings for an automotive d i f f e r e n t i a l unit are considered to be " c l a s s i -f i e d " information by equipment manufacturers. Should such a unit be capable of transmitting the power and torque required i n th i s application, the advantages i n terms of ease of construction, low unit cost and a v a i l a b i l i t y of spare parts, would be consider-able. It seems l i k e l y that such a unit would be capable of hand-l i n g the loading imposed on i t by, at lea s t , smallwood yarding. As for further recommendations, the modelling of the powerflows and losses i n the hydrostatic transmission would appear to be of prime importance. With the speed and torque outputs required of t h i s transmission already determined, the choice of po t e n t i a l l y suitable transmission components i s sim-p l i f i e d . With t h i s choice made, the intermediate gearing i n the pump and motor drives can also be selected and an i t e r a t i v e solution, which accounts for the hydraulic losses i n the trans-mission, can be employed to determine the required pump and motor displacements for a l l positions i n the span. Pease (1973) outlined a possible method of determining the required solution to the problem of ca l c u l a t i n g flows and displacements. Other important areas requiring further analysis are the f r i c t i o n a l torque (and hence, power) losses at the gear faces, i n p a r t i c u l a r those i n the d i f f e r e n t i a l unit. Within t h i s unit are several gear interfaces at which power losses,and possibly high p i t c h - l i n e v e l o c i t i e s , w i l l occur. A l l power losses res u l t i n heat and although i t i s proposed to run the d i f f e r e n -t i a l unit i n an o i l bath (the haulback drum i t s e l f ) , a check should be made to ensure that the drum has s u f f i c i e n t surface area to dissipate the heat produced at a sa t i s f a c t o r y rate to prevent overheating of the l u b r i c a t i n g o i l and loss of i t s l u b r i c a t i n g properties. Consideration must also be given to the location of the variable displacement pump drive, and, depending upon the location chosen, to the selection of suitable intermediate gearing to prevent reverse power flows (with t h e i r attendant heat production) 78 i n the primary transmission. F i n a l l y , i n r e l a t i o n to the computer modelling, the i n c l u -sion of a dynamic cable mechanics model would be useful i n determining appropriate safety factors for the rigging yarder and interlock design, and i n determining hydraulic c i r c u i t overload-protection device settings. The ultimate test of the proposed design would be to b u i l d either a scale model or an actual working interlocked drum set. Unless the reluctance of the automotive gear manu-facturers to reveal torque and power ratings of these gear units can be overcome, t h i s would be the only way of determining the s u i t a b i l i t y to, and the longevity of such a unit in,the proposed application. 79 BIBLIOGRAPHY Anon. 1905. Logging by steam: Lidgerwood Logging.Systems. Ligerwood Manufacturing Company, New York, N.Y. 127 pp. Anon. 1959. Wire rope handbook. U.S. Steel Corporation, San Francisco, C a l i f o r n i a . 182 pp. Anon. 1969. Model M-3 Interlock Slackline Yarder. (Sales l i t e r a t u r e ) Tyee Manufacturing Company Limited, Vancouver, B r i t i s h Colubmia. 2 pp. Anon. 1979. Specifications of the Lantec Industires Interlocked Winch Set. (Sales l i t e r a t u r e ) Lantec Industries, Richmond, B r i t i s h Columbia. 2 pp. Anderson, CH. 1974. Machine choices and l i m i t a t i o n s . Paper presented to conference on Factors A f f e c t i n g Yarding Systems and Road Spacing, Faculty of Forestry and Centre for Continuing Education, University of B r i t i s h Columbia, Vancouver, B r i t i s h Columbia. pp. 8-17. Beer, F.P. and Johnston, E.R. 1977. Vector Mechanics for Engineers, McGraw-Hill, New York, N.Y. 957 pp. Carson, W.W. 1972. The influence of a u x i l i a r y power unit e f f i c i e n c i e s on the design of a planetary/hydraulic i n t e r -lock yarder. Unpublished technical report, U.S.D.A. P a c i f i c North West Forest and Range Experiment Station, Seattle, Washington. 6 pp. 1975. Analysis of running skyline with drag. U.S.D.A. Forest Service Research Paper PNW-193, Seattle, Washington. 8 pp. and Jorgensen, J.E. 1974. Understanding interlock yarders. U.S.D.A. Forest Service Research Paper PNW-221, P a c i f i c North West Forest and Range Experiment Station, Portland, Oregon. 13 pp. Cowie, A. 1961. Kinematics and design of mechanisms. Inter-national Textbook Company, Scranton, Pennsylvania. 257 pp. Guimier, D.Y. 1977. Experimental study of cable logging systems. M.A. Sc. Thesis, Faculty of Forestry, University of B r i t i s h Colubmia, Vancouver, B r i t i s h Columbia. 210 pp. H a l l , A.S., Hollowenko, A.R. and Laughlin, H.G. 1961. Machine design. Schaums outline series, McGraw-Hill, Toronto, Ontario. 344 pp. 80 Holzbok, W. 1966. Which hydraulic drive? Paper i n Hydraulic and Pneumatic Power and Control (Ed. F.D. Yeaple), McGraw-H i l l , New York, N.Y. pp. 241-254. Jorgensen, J.E. 1971. Steady state analysis of in t e r l o c k i n g yarders. Preliminary report to P a c i f i c North West Forest Engineering Research Station, (Unpublished). 52 pp. 1974. Understanding running skyline yarders. Paper i n Proceedings of Skyline Logging Symposium (Ed. J.E. Jorgensen), University of Washington, Seattle, Washington, pp. 26-34. Mann, C.N. 1977. Running skyline systems for harvesting timber on steep t e r r a i n . Paper 770519, Society of Automotive Engineers, Warrendale, Pennsylvania. 8 pp. and M i f f l i n , R.W. 1979. Operational tests of the. prototype Pee Wee yarder. General technical report PNW-92, U.S.D.A. P a c i f i c North West Forest and Range Experiment Station, Seattle, Washington. 7 pp. Pease, G.E. ( J r . ) . 1973. A steady state model for interlocking yarders. M.Sc. Thesis, University of Washington, Seattle, Washington. 87 pp. Plummer, W.T. and Koury, T. 1974. Rosedale Machine Shop Ecologgers. Paper i n Proceedings of Skyline Logging Symposium (Ed. J.E. Jorgensen), University of Washington, Seattle, Washington, pp. 76-79. Raven, J. 1974. What innovations do we need i n running skylines? Paper i n Proceedings of Skyline Logging Symposium (Ed. J.E. Jorgensen), University of Washington, Seattle, Washington, pp. 69-75. Ryan, M.S. 1977. Computer simulation of logging skylines. M.Sc. Thesis, University of Washington, Seattle, Washington. 182 pp. U.S.D.A. Forest Service. 1969. Glossary of cable logging terms. P a c i f i c North West Forest Experiment Station, Portland, Oregon. 7 pp. Wilkinson, W.H. 1960. Four ways to calculate planetary gear t r a i n s . Machine Design, 32 (1): 155-159. Penton Publica-tions, Cleveland, Ohio. 81 APPENDIX A THE CABLE MECHANICS MODEL 82 THE CABLE MECHANICS MODEL The running skyline system i s modelled using a paraboli cable mechanics model developed by Guimier ( 1 9 7 7 ). This para-b o l i c model was chosen because, when applied to t i g h t c a b l e s — reasonable assumption for logging c a b l e s - - i t provides a s a t i s -factory balance between accuracy and s i m p l i c i t y of cal c u l a t i o n when compared to a catenary model. The model assumes: 1 . that the tension at any point on the cable acts along the tangent to that point; 2. that the cable weight i s uniformly d i s t r i b u t e d along the subchords; 3. that the cable i s i n f i n i t e l y f l e x i b l e . Figure A l shows the running skyline system as i t i s considered i n the model. 83 Figure A l . The Running Skyline System Dimensions and Components used i n the Parabolic Cable Mechanics Model. With reference to Figure A l : 1 = length of the span x = the pos i t i o n of the carriage i n the span, r e l a -t i v e to the t a i l h o l d S = sag, the position of the carriage r e l a t i v e to the t a i l h o l d l i n e support: sign convention = "down i s p o s i t i v e " ( i . e . i n Figure A l , S i s positive) E = difference i n elevation between the headspar and t a i l h o l d l i n e supports: sign convention = "up i s p o s i t i v e " ( i . e . , i n Figure A l , E i s positive) oc = angle between cable segment and the horizontal, where A, B and C are subcripts representing the t a i l h o l d , headspar and carriage respectively <9 = angle of the subchords ( i . e . l i n e s CA and CB) and the horizontal, measured at the carriage 1, l',2, 3, 3' = l i n e segments W = weight per unit length of the cable segments R = weight of the load and the carriage In t h i s analysis, W^ , W^ , and are a l l weights/unit length of the segments of the same (haulback) cable and are equal. In addition, i t i s assumed that l i n e s 3 and 3' (mainline and slack-p u l l e r respectively) share the load equally and since the power flow model i n Chapter III requires only one l i n e tension, the weight per unit length of each of 3 and 3' may be combined, e f f e c t i v e l y creating a cable with the tension and weight per unit length equivalent to that of the two separate cables. This cable i s designated 3*. 85 Thus: W l = W* + W, w. w. + w The tangents o f the angles of the subchords are: Tan = S x ( 3 5 ) and Tan Q2 = S + E L - x ( 3 6 ) 1) The V e r t i c a l P o s i t i o n of the C a r r i a g e i s Known I f the p o s i t i o n o f the c a r r i a g e , S, i s known, then the h o r i z o n t a l component of the l i n e t e n s i o n i n segment 1 i s : R + x 2 Cos Q1 + W2 (L-x) + W* (L-x) 2 Cos Q, 2 Cos © 2 ~ — The l i n e t e n s i o n at the c a r r i a g e , i n l i n e segment 1, i s x (L-x) SL + xE ( 3 7 ) 1 + (Tan,* ) C l = H 1 1 / + S -( 3 8 ) ( 3 9 ) X 2H 1 Cos Q The l i n e t e n s i o n a t the c a r r i a g e , i n l i n e segment 2 i s equal to the t e n s i o n a t the c a r r i a g e i n l i n e segment 1 and i s : = H, = H, + (Tana' ) c 2 S + E L - x W2 (L-x) 2 H 1 Cos$ 2 (40) (41) 8 6 Squaring equation 41, equating i t t o equation 39 and r e a r r a n g i n g g i v e s : H, 1 +/S + E \L - x - H 2 (S + E) W2 + Cos <90 W2 (L - x) 2 Cos 0n 2 2 n -a = 0 (42) The h o r i z o n t a l component of l i n e t e n s i o n i n l i n e segment 2 i s give n by the s o l u t i o n o f t h i s q u a d r a t i c equation. 2) The Maximum Al l o w a b l e Tension i n the Running S k y l i n e i s Known a) U p h i l l Yarding In u p h i l l y a r d i n g , the maximum t e n s i o n i n the running s k y l i n e w i l l occur a t the upper l i n e support; the yarder tower sheave; t h i s w i l l occur i n l i n e segment 2. From catenary theory, the t e n s i o n i n l i n e segment 2, at the c a r r i a g e i s : T = T (S + E) W, (43) where T = maximum a l l o w a b l e t e n s i o n i n the running s k y l i n e . Due to the c o n t i n u i t y of the running s k y l i n e through the c a r r i a g e , the t e n s i o n a t the c a r r i a g e i n l i n e segment 1 i s equal to the t e n s i o n , a t the c a r r i a g e , i n l i n e segment 2. Equating equations 39 and 43, squaring both s i d e s and r e a r r a n g i n g g i v e s : H, 1 + Cos Q 1 VI1 x 2 Cos Q. - T 0 (44) The h o r i z o n t a l component of t e n s i o n i n l i n e segment 1 i s giv e n by the s o l u t i o n o f the q u a d r a t i c equation. b) Downhill Yarding In d o w n h i l l y a r d i n g , the maximum t e n s i o n i n the running s k y l i n e w i l l occur at the t a i l h o l d . Again from catenary theory the t e n s i o n i n l i n e segment 1, a t the c a r r i a g e , i s : T = T - S fl. c l 1 (45) The t e n s i o n a t the c a r r i a g e , i n l i n e segment 2, i s equal to the t e n s i o n a t the c a r r i a g e i n l i n e segment 1. Equating equations 41 and 45, squaring and r e a r r a n g i n g g i v e s : H, 1 + /S + E L - x - H 2 (S + E) W2 + C o s t 9 0 W2 (L - x) 2 Cos 9 , 2 - T 2 = 0 C l (46) The h o r i z o n t a l component of t e n s i o n i n l i n e segment 2 i s gi v e n by the s o l u t i o n o f t h i s q u a d r a t i c equation. The h o r i z o n t a l component of t e n s i o n i n l i n e segment 3* i s : H 3 * ~ 2 H l H, (47) I f the dragging l o g l o a d model i s used, then the l o a d on the system, R, used i n equations 37 and 50, i s : R = C a r r i a g e weight + Wv (48) and the h o r i z o n t a l component of t e n s i o n i n l i n e segment 3* i s given by: H. 2H + W H " H 2 (49) where, from the Dragging Load Model i n Appendix : Wv = v e r t i c a l component of choker c a b l e t e n s i o n W„ = h o r i z o n t a l component of choker c a b l e t e n s i o n t i 88 The position of the carriage r e l a t i v e to the t a i l s p a r i s : x (L-x) 2 L R + 2 WjX + W2 (L-x) + W*(L-x) 2 Cosft 2 Cos <92 2 Cos 0 2 - x E (50) Since H 1, Cos 6^ and Cos $ 2 axe. functions of S, an i t e r a t i v e solution i s required. 3) The Line Tensions at the Headspar The tension, at the tower, i n the running skyline i s : T B 2 = H2 (51) Cos<sK_. a and since Cos = 1 y 1 + (Tan) 2 (52) T B •= H 2 / 1 + ^ T a n ^ B " l 2 { 5 3 ) E2 / 1 + I S + E + W2 ( L ' ~ X ) ,2 V L " x 2 Cos 0 2 / The tension, at the tower, i n the mainline/slackpuller i s s i m i l a r l y : (54) T B 3 * " H3* / 1 + ( T a n ^ B ) 2 (55) K31t / 1 + fS + E + W* (L - x)\2 (56) L - x 2 Cos Q 2 89 A P P E N D I X B T H E D R A G G I N G L O A D M O D E L 90 Figure B l . The Dragging Load Geometry. ( From Carson (1975)) N, normal force With reference to the above figure, the horizontal and v e r t i c a l forces are: W = T C o s ^ = W - N Cos 9 : + N Sin Q v ^ WR = T Sine* = N Sin G + N^ / Cos G where: T = tension i n the choker cable oC = angle of the choker cable with respect to the v e r t i c a l Q = the l o c a l slope of the ground N = normal force t = log length N assumed location of the centre of gravity of the log, with respect to the choker attachment point the c o e f f i c i e n t of f r i c t i o n between the log and the ground W £(Cosc3 -jusin9+ Sin<9 Tan (Q+/3) + ^Cos 0 Tan «9+/3) ) The v e r t i c a l force due to the dragging log i s : W v W 1 - Cos 0 - Sin 6 Tanyd (CosQ - Sin@) £(1 +/<Tan/3 ) and the horizontal force due to the dragging log i s : W. H W Cos Q - Sinc9Tan^/3 £(1 + /«Tan / 6) (Sin 6 + Cos 9) 92 APPENDIX C THE LINE LENGTH MODEL F i g u r e C l . The Geometry of a Suspended Cable. *• a. 2 a Consider a l i n e suspended between p o i n t s A and B where: d = mid-span d e f l e c t i o n H = h o r i z o n t a l component of t e n s i o n i n the l i n e W = weight per u n i t l e n g t h of the l i n e a = l e n g t h o f the span Q = angle of the chord / = l e n g t h of the l i n e i n the span The mid-span d e f l e c t i o n i s g i v e n by Guimier (1977) as: d = W a 2 2 Cos Q H The l e n g t h of the l i n e i n the span i s g i v e n by Anon. (1959) as: With r e f e r e n c e to F i g u r e A l i n Appendix A, the l e n g t h of each L Cos 9 a 1 + 94 l i n e may be f o u n d by t r e a t i n g e a c h l i n e segment ( i . e . 1, 1', 2, 3 and 3') i n t h e above manner. The t o t a l l e n g t h o f a l i n e i s t h e n g i v e n by a d d i n g t o g e t h e r t h e l e n g t h s o f t h e a p p r o p r i a t e segments. 95 APPENDIX D THE DRUM WRAP-RADIUS MODEL F i g u r e D l . T h e G e o m e t r y o f a L i n e - W o u n d D r u m . Q PI f^ oe^ pooooooo OOOOOOOO0OOQ1 Wrap radius Q = n o m i n a l d i a m e t e r o f t h e r o p e R = b a r e d r u m r a d i u s ( b a r r e l r a d i u s ) P = d r u m w i d t h D = d e p t h o f d r u m f l a n g e n = n u m b e r o f c o m p l e t e l i n e w r a p s I = i n t e r l e a v e f a c t o r : a m e a s u r e o f t h e m e s h i n g o f t h e l i n e s i n s u c c e s s i v e w r a p s A s s u m i n g t h e l i n e w r a p s p e r p e n d i c u l a r t o t h e c e n t r e l i n e o f t h e d r u m : T h e m a x i m u m n u m b e r o f c o i l s a c r o s s t h e d r u m i s : p * Q T h e m a x i m u m n u m b e r o f w r a p s i s : I Q * T h e r o u n d e d - d o w n i n t e g e r v a l u e s o f t h e s e e x p r e s s i o n s a r e u s e d i n t h e w r a p - r a d i u s c a l c u l a t i o n . 9 7 The l e n g t h o f l i n e on the f i r s t wrap 2 r r p R Q th The l e n g t h o f l i n e on the n — wrap i s 2 T 1 P (R + ( n - 1 ) Q I) Q th The wrap r a d i u s of the n-^ wrap i s : R + (n-1) Q I The maximum l i n e c a p a c i t y i s : n - D I Q 2 r f P *V (R + ( n - 1 ) Q I Q n = 1 98 Figure D2. Flowchart for the Drum Wrap-Radius Algorithm. 1 n= 0 * n- n+1 ik L = 2wP(R + (n-1)Q I) Q W 7 = [(n-1)QI]-R radius 99 APPENDIX E LISTING OF THE COMPUTER PROGRAM 100 BEVEL-GEAR DIFFERENTIAL INTERLOCK ANALYSIS PROGRAM IAN PENDLEBLIRY 8 September 1980 * 6 10 28 30 80 4 0 1 _b < 50 68 70 88 90 1 00 1 10 120 1 30 148 150 160 1 70 1 SO 190 2 O 0 210 OFT ION BASE 0 DEG D IM H_p_m a i n C1 8 0), H_p_h au 1 b (18 0 ), H_p_au x ( 1 O O ), D r u m_r at i o(188),To w e r _ t e n_h ( 1 , Toue r _ t e n_m ( 1 0 6), D_r ad_h ( 1 0 O >, D_r ad_m < 1 6 O > , T o r q u e_h < 1 0 O >, H_p_y ar ding < 1 0 0 > DIM H_p_eng i ne <1OO),Line_ue t _ r at i o <108),fi*C 25 3,I * I SO 3,Ang_vel_a<l00 >,Ang_ue ( 1 88 ) , fing_we 1_c(188),M*[213, Torque_b ( 1 08 ) DIM T o r q u e_m(10 8 >, R_P_m_h au 1 b < 1 8 8 J , F:_p_m_m a i n 1 (10 O) ! ! ************************************************************************; Internal=188 ! THE NUMBER OF POSITIONS ACROSS THE SPAN REQUIRED TO BE ANALYSED Xl = 18 !X1 TO X5 ARE THE POSITIONS IH THE SPAN <X span) -MEASURED FROM THE TAILSF'AR- FOR WHICH THE SUMMARY TABLE SHOWS THE ASSOCIATED X2 = 28 !PARAMETERS OF THE YARDING SYSTEM . X3 = 40 !THE- POSITIONS XI AND X5 SHOULD BE CHOSEN TO ENSURE THAT THEY REP--RESENT FEASIBLE CARRIAGE POSITIONS X4 = 68 X5 = 95 L i ne_spe ed = 500 t h i s is not t he ! THE REQUIRED MAINLINE SPEED ( f t . / m i n . ) c a r r i age speed! L i n e _s p e e d = L i n e• spe e d * - 1 Tower ht=48 ! THE HEIGHT OF THE TOWER FA I RLE AD FROM THE GROUND (.ft.) T a i l s p a r _ h t = 40 i C a r r i age_ht=5 ! THE HEIGHT OF THE TAIL-BLOCK' FROM THE GROUND C f t . ) ! THE HEIGHT OF THE CARRIAGE ABOVE THE GROUND FOR ALL POSITIONS IN THE SPAN ( f t . ) 220 Span_length=1288 ! THE (HORIZONTAL) LENGTH OF THE SPAN ( f t . ) 238 E l e v _ d i f f = - 6 8 8 ! THE DIFFERENCE IN ELEVATION OF THE CABLE SUPPORTS ( f t . ) : +we. i f the 'headspar' i s higher than the ' t a i l s p a r ' 240 ! 250 ! ************************************************************************* 260 2 7 O 2 8 0 290 -OUND ! Carriage_weight=588 ! THE WEIGHT OF THE CARRIAGE ( l b s . ) Log_weight=20OO ! THE WEIGHT OF THE' LOG LOAD ( l b s . ) F r i c t i o n coeff=.6 ! THE FRICTION COEFFICIENT BETWEEN THE LOG AND THE GR-300 316 320 * 330 348 358 368 378 371 372 373 400 410 420 Log_ang1e=30 ! THE ANGLE BETWEEN THE LOG AND THE GROUND LINE (degrees) Tensionl=1300O ! SAFE WORKING TENSION OF THE RUNNING SKYLINE : only for constant running s k y l i n e t e n s i o n ) i Omegal=0 Orne ga3 = 2 1 b. (used mega2 = . ~< •Omegal ! THE WEIGHT PER UNIT LENGTH OF THE HAULBACK LINE ( l b s . / f t . ) ! THE WEIGHT. PER UNIT LENGTH OF THE MAINLINE/SLACK-PULLER ! The program assumes that the weight per uni t length of of the mainline and the s 1 a c k _ p u 1 1 i n g l i n e are the ! same and that both of these l i n e s share e q u a l l y the loads imposed upon them ! There i s no p r o v i s i o n f o r c a l c u l a t i n g the i n d i v i d u a l t e n s i o n s i n these l i n e s Rope_diam_m=.625 ! THE DIAMETER OF THE HAULBACK LINE ( i n s . ) Rope_diam_h=.625 ! THE DIAMETER OF THE MAINLINE ( i n s . ) Mainl_length=1400 ! THE LENGTH OF THE MAINLINE ( f t . ) 101 436 H a u l b _ l e n g t h = 2 6 0 O ! THE LENGTH OF THE HRULBRCK L I N E ( f t . ) 446 ! 456 ! * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 460 ! 476 Drum_i,.ii dth_m = 22. 5 ! THE WIDTH OF THE MAINLINE DRUM MEASURED INSIDE THE FLANGES ( i n s . ) 480 Drum_width_h=22.5 ! THE WIDTH OF' THE HAULBACK DRUM MEASURED INSIDE THE FLANGES ( i n s . ) 490 FIange_m=7.5 ! THE DEPTH OF THE FLANGE OF THE MAINLINE DRUM MEASURED FROM THE BARREL OF THE DRUM TO THE RIM OF THE FLANGE ( i n s . ) 500 F l a n g e _ h = ? . 5 ! THE DEPTH OF THE FLANGE OF THE HAULBACK DRUM MEASURED FROM THE BARREL OF THE DRUM TO THE RIM OF THE FLANGE ( i n s . ) 510 Drum_rad_m=10 ! THE RADIUS OF THE BARREL OF THE MAINLINE DRUM ( i n s . ) 526 Drum_rad_h=10 ! THE RADIUS OF THE BARREL OF THE HAULBACK DRUM ( i n s . ) 536 ! 546 ! * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 550 ! 560 I n t e r l e a v e d ! A MEASURE OF THE OVERLAP OF SUCCESSIVE LINE WRAPS ON THE WINCH DRUMS ( d i m e n s i o n 1 e s s ) : e x p r e s s e d as a d e c i m a l 570 ! f r a c t i o n ( n o t p r e s e n t l y u s e d ; s h o u l d be d e t e r m i n e d e m p i r i c a l l y ) 580 ! 590 ! * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 606 ! 616 P r o f i l e d ! FLAG : = 1 .i f c o n s t a n t c a r r i a g e h e i g h t i s r e q u i r e d 626 ! = 0 i f c o n s t a n t r u n n i n g s k y l i n e t e n s i o n i s t o be mai nt a i n t d 630 ! 64 6 ! * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 650 IF NOT P r o f i l e THEN 760 660 G r o u n d _ a n g l e = ATN( ( E l e v _ d i f T - T o w e r _ h t + T a i 1 s p a r _ h t ).-"Span_l e n g t h ) ! THE SLOPE OF THE CHORD BETWEEN THE HEAD- AND TAILSPARS ( d e g r e e s ) : u s e d i n t h e a b s c e n -670 ! -ce o f a p r o f i l e f o r t h e d e t e r m i n a t i o n o f t h e f o r c e s r e q u i r e d t o d r a g t h e l o g 6 8 O I n t _ c o m p = C 0 S (G r o u n d_an g 1 e.) - S I N ( G r o u n d_an g 1 e•) * T A N ( L o g_an g 1 e) 690 V _ l o ad_c onip = L o g _ i , i e i g ht * (1 -1 n t _ c cm p.* ( 2 * (1 + T AH ( L o g _ a n g 1 e ) * F r i c t i on_c o e f f > ) * ( S I N ( G r o u n d _ a n g l e ) + F r i c t . i o n _ c o e f f * C 0 S ( G r o u n d _ a n g 1 e ) ) ) ! THE VERTICAL COMPONENT 700 ! OF THE WEIGHT OF THE DRAGGING LOG ( l b s . ) 716 H_load_comp=Log_wei g h t * ( I n t _ c o m p / ( 2 * ( 1 + T A N ( L o g _ a n g 1 e ) * F r i c t. i on_c o e f f ) ) * ( S I N ( G r o u n d _ a n g 1 e ) + F r i c t i o n _ c o e f f * C 0 S ( G r o u n d _ a n g 1 e ) ) ) ! THE HORIZONTAL COMPONENT 730 ! OF THE WEIGHT OF THE DRAGGING LOG ( l b s . ) 740 T o t a l _ l o a d = C a r r i a g e _ u e i g h t + V _ l o a d _ c o m p ! THE TOTAL ( V E R T I C A L ) LORD ON THE SYSTEM ( l b s . ) 750 GOTO 770 760 T o t a l _ 1 o a d = C a r r i age_we i g ht + Log_we i g h t 776 FOR 1=1 TO I n t e r v a l ! LOOP To"ANALYSE THE SYSTEM: MOVES THE CARRIAGE FROM T HE T A I LSF'AR TO THE HEADSPAR IH S P E C I F I E D INCREMENTS 780 P o s i t i o n = S p a n _ l e n g t h / I n t e r v a l * I - . 6 0 1 ! INCREMENTS THE POSITION OF THE CARRIAGE ACROSS THE SPAN , STARTING AT THE TAILSPAR . 796 IF P r o f i l e THEN 870 800 CALL S a g ( P o s i t i o n , E 1 e v _ d i f f , S a g , S p a n _ l e n g t h , T o t al_1oad,Omega1,0mega2,Ome ga3 , H o r t en 1,Hort e n 2 , H o r t e n 3 , T e n s i on 1,Speo1mx,S_p1us_e,L_m i n u s _ x , A n g l e i , A n g 1 e 2) 810 ! IF E l e v _ d i f f < 6 THEN 712 820 ! T o u ) e r _ t e n _ h ( I ) = T e n s i o n l 830 ! GOTO 880 840 T a n _ a l p h a _ b 2 = F N T a n _ b ( S p e o l m x , 0 m e g a 2 , L _ m i n u s _ x , H o r t e n 2 , A n g l e 2 ) 850 T o u e r _ t e n _ h ( I ) = H o r t e n 2 * ( 1 + T a n _ a l p h a _ b 2 " 2 ) A . 5 860 GOTO 1000 870 Sag = P o s i t i o n * ( ( E l e v _ d i f f - T o u e r _ h t + T a i 1 s p a r _ h t ) / S p a n _ l e n g t h ) - T a i 1 s p a r _ h t + Car r i a g e _ h t ! THE HEIGHT OF THE CARRIAGE R E L A T I V E TO THE TAILSPAR L I N E SUPPORT 880 S a g = S a g * - l 890 CALL G e o m ( S a g , P o s i t i o n , S p a n _ l e n g t h , E l e v _ d i f f , S p e o l m x , S _ p l u s _ e , L _ m i n u s _ x , A n g l e l , A n g l e 2 ) 906 I n t _ h l = 2 * 0 m e g a l * P o s i t i o n / ( 2 * A n g l e l ) + 0mega2*L_mi n u s _ x / ( 2 * A n g l e 2 ) + 0mega3*L_rn i n u s _ x / ( 2 * A n g l e 2 ) + T o t a l _ l o a d 102 910 H o r t e n l = I n t _ h l * P o s i t i on*L_iii i nus_x-' ( 2* ( Sag*Span_l engt h + Pos i t i on*E1 e v d i f f )) ! HORIZONTAL COMPONENT OF TENSION IN LINE SEGMENT #1 920 Tan_al pha c 1 =FNTan_c ( S ag/F'os i t. i on, Pos i t i on, Omegal , Hort en 1, Ang 1 e l ) 930 Quad_a=l+SpeolKix'2 940 Quad_b=-(0mega2*S_pl us_e ).-'Angl e2 950 Quad_c = < OmEga2*L_m inus_x •' (2*Ang1e2 >)-2-Horten 1 - 2* <1 + Tan_alpha c1*2> 960 Horten2=FNQuad_so1n(Quad_a,Quad_b,Quad_c) (HORIZONTAL COMPONENT OF TENSION IN LINE SEGMENT #2 978 Tan_alpha_b2=FHTan_b<Speo1 mx,0mega2,L_mi nus_x,HortenS,Angle2) 980 Tower_ten_h'C I )=Horten2* a+Tan_al pha_b2 "2)'-. 5 ! TENSION IN THE HAULBACK LINE AT THE TOWER 990 Horten3=2*Hortenl+H_load_comp-Horten2 ! HORIZONTAL COMPONENT OF TENSION IN THE MAINLINE/SLACK-PULLING LINE 1000 Tan_alpha_b3 = FHTan_b(Speo1mx,0mega3,L minus x,Horte n3,Ang1e-2) 1018 Tower_ten_m( I )=Hot-ten3*C 1+Tan_al pha_b3 •'2)'-. 5 (TENSION IN MAINLINE / S L -ACK-PULLING LINE AT THE HEADSPAR LINE SUPPORT. 1028 Hau1b_segl=FNSub1engthCPos ition,Omegal,flnglel,Hortenl> 1030 Hau1b_seg2 = FNSub1engt h(L_m i nus_x,Ome ga2,Ang1e2,Hor t en2) 1040 Haulb_1en_span=Haulb_seg1*2+Hau1b_seg2 ! LENGTH OF HAULBACK OUT IN THE SPAN 1050 Mai n l _ l en_span = FNSub 1 engt h(L_m i nus_x, 0t(,ega3 , Angl e2, Hort en3) ! LENGTH OF MAINLINE OUT IN SPAN 1051 IF 1=50 THEN M i d_sag= (Sag + E 1 ev_d i f f v'2) -'Span_l engt h ( MID-SPAN DEFLECTION 1060 IF I >X1 -1 THEN 1100 1070 Line_len_hl=Haulb_len_£pan ! INITIAL HAULBACK LINE LENGTH USED IN THE LINE VELOCITY RATIO CALCULATIONS 108O Line_len_ml=Mainl_len_span ! INITIAL MAINLINE LENGTH USED IN THE LINE VEL--OCITY CALCULATIONS 1090 GOTO 1300 1108 Li n e _ u e l _ r a t i o (I ) = (Hau1b_len_span-L i ne_len_h1)/(Mai n1_1en_span-L i ne_len_m1) ( THE LINE VELOCITY RATIO ( Haulback : M a i n l i n e ) 1118 Hau1b_len_drum=Haulb_length-Hau1b_len_span ( LENGTH OF LINE ON HAULBACK DRUM 1120 Mai nl_len_druriv=Mainl_l ength-Mai n 1_1 en_span ! LENGTH OF LINE ON MAINLINE * DRUM 1138 Wrap_no_h = FNDrurn_wraps(F1ange_h,Rope_di am_h,Int er1eawe,Drum_ui dt h_h,Drum_ra d_h,Haulb_len_drum), ! NUMBER OF WRAPS ON HAULBACK DRUM 1148 Wrap_no_m = FNDrum_v.irap£ (F 1 ange_rn, R o p e d i am_m, I nt er 1 eawe , Drum_w i dt h_m, Drum_ra d_m,Mainl_len_drum) ! NUMBER OF WRAPS ON MA INLINE^SLACK-PULLING DRUM. 1150 D_rad_h( I ) = Drurn_rad_ti + Wrap_no_h*Rope_di am_h*Interl eaue ! EFFECTIVE RADIUS OF HAULBACK DRUM 1160 D_rad_m( I) = Dt-um_rad_m + Wr ap_no_tii*Rope_d i am_m* Inter 1 eave ! EFFECTIVE RADIUS OF MAINLINE/SLACK-PULLING DRUM 1170 Drum_ratio(I)=D_rad_m(I).-D_rad_h(I) ! RATIO OF THE RADIUS OF THE MAINLINE /SLACK-PULLING LINE DRUM TO THE RADIUS OF THE HAULBACK DRUM 1180 Torque_h(I ) = D_rad_h(I ) 12*Touier_t en_h(I ) ! TORQUE ON HAULBACK DRUM 1190 Torque_m(I) = D_rad_m(I)/12*Touer_ten_m(I ) ! TORQUE ON MAINLINE DRUM 1200 Torque_b(I)=Torque_h(I)>'2 ( TORQUE ON GEAR B ( i . e . aux. input) 1210 R_p_m_hau1b(I)=Li ne_£peed*Li n e _ u e l _ r a t i o ( I ) ( 2 * 3 . 1 4 2 * ( D _ r a d _ h ( I ) / 1 2 ) ) ! RPM OF HAULBACK DRUM 1220 R_p_m_mai nl (I )=Li ne_speed-'(2*3. 1 42*(D_t-ad_m(I )/12)) ! RPM OF MAINLINE DRUM 1230 Ang_uel_a(I)=2*R_p_m_mainl(I> !ANGULAR VELOCITY OF GEAR A 1248 Ang_vel_c(I)=R_p_m_hau1b(I) (ANGULAR VELOCITY OF GEAR C 1258 Ang_vel_b(I)=2*Ang_uel_c(I)+Ang_Mel_a(I) !ANGULAR VELOCITY OF GEAR B 1266 H_p_haulb(I) = Tower_ten_h(I)*Li ne_£peed*Li n e _ M e l _ r a t i o(I)x(2*3.142*5252) ! HORSEPOWER REGENERATED AT THE HAULBACK DRUM. 1270 H_p_mai n(I)=Tower_ten_m(I)*Li ne_speedx(2*3.14 2*5252) ! HORSEPOWER REQUIRED AT THE MAINLINE DRUM. 1280 H_p_aux(I)=Ang_vel_b(I)*-Torque_h(I)x(2*5252) ! HORSEPOWER REQUIRED IN THE AUXILLIARY (HYDRAULIC) TRANSMISSION 1290 H_p_uarding(I)=-H_p_main(I)-H_p_haulb(I) ! NET HORSE-POWER REQUIREMENTS i . e . POWER TO YARD 1300 NEXT I 1310 CALL Plot(Span_ length,Interual,H_p_mai n(*),H_p_hau1b(*),H_p_aux(*),H_p_ya r d i n g ( * ) , X l , X 5 ) (CALLS THE SUB-ROUTINE WHICH PLOTS THE MA INLI HE,HAULBACK, 1320 ! AUXILLIARY TRANSMISSION , AND NET HORSEPOWERS vs. THE POSITION IN THE SPAN. 1330 ! 1 0 3 1346 ! * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 1350 ! 1360 ! THIS SECTION PRODUCES THE SUMMARY TABLE OF LINE-TENSIONS,HORSEPOWER COMP--ONENTS ETC. 1376 ! 1371 PRINTER IS 0 1380 FIXED 2 1390 IMAGE "I I I I I 1400 IMAGE 14 10 PRINT " SUMMARY TABLE " , L I N < 2 ) 1420 PRINT " SPAN LENGTH : ";Span_length; " f t . "; " < "; Sp.ar._l engt h/3. 2808; " m . >". L I N < 1 ) 1430 PRINT " ELEVATION DIFFERENCE BETWEEN SUPPORTS J ";E1e v _ d i f f ; " f t . " ; " C" ; E 1 e v d i f f / 3.2808; " ft.. )" 1446 It="< U p h i l l y a r d i n g > " 1450 IF E l e v _ d i f f < 0 THEN It="< Downhill y a r d i n g V 1460 PRINT " " ; It,LINC1) 1470 T 1oad=Carr i age_we i ght+Log_we i ght 1480 PRINT " LOAD: (Carriage+Logs> : " ; T _ l o a d ; " 1 b T _ l o a d / 2 . 2 0 4 6 ; " kg. >",L IN c 1) 1481 PRINT " MID-SPAN DEFLECTION <:•:) : " ; M i d_sag* 1 60 , L I N 1 ) 1-490 PRINT " MAINLINE VELOCI T Y: "; L i ne_spe ed; " f t . / t n i n . <"; L i ne_speed* . 66568; "i«/s. )" , L I N < 1 ) 1500 IF NOT P r o f i l e THEN 1540 1510 PRINT " MODEL OPTION: Constant c a r r i a g e height < " ; C a r r i a g e _ h t ; " f t . ;";Carr iage_ht/3. 2808;" m.)" 1520 PRINT " Dragging 1 oad", LINC2> 1530 GOTO 1560 1540 PR I NT " MODEL OPT I ON: Const ant runn i ng sk y 1 i ne t ens i on <"; Tens i on 1;"1b. ; " J Tens i or. 1 /2. 2046; " kg.)" 1550 PRINT " F u l l y suspended 1oad",LINC2) 1560 PRINT USING 1400 1570 At="SPAN X " 1580 A l t = " " 1590 CALL Table1<At,Alt,XI,X2,X3,X4,X5) 1591 A*="<from t a i l h o l d ) " 1592 CALL Table2<fi«> 1600 PRINT USING 1390 1610 A*="TENSION " 1620 CALL Table2(At> 1630 B*="Haulback" 1640 Bl*=" : l b . " 1650 CALL Tab 1e1< Bt,B11,Tow«r_ten_h<X1>,Tower_ten_h<X2),Tower_ten_h<X3>,Tower_te n_h<X4),Tower_ten_h<X5)) 1660 Cl*=" : k g . " 1670 CALL TableSCBt,Cit,Tower_ten_h<X1),Tower_ter._h<X2),Tower_ten_h<X3),Tower_te n_h < X4 ), Tower_t er._h < X5 ) , 1/2. 204 ) 1680 Ct="Mainl ir.e" 1696 CALL Tabl e 1 <Ct, Bl t, Tou.er_teri_n.CXl ), Tower_t er._rn<X2), Touer_ten_m<X3), Tower_t e n_m <X4),Tower ten_m< X5)) 1766 CALL Tab 1e3<C*,C11, Tower_t en_rn CX1 ), Tower_t en_m < X2), Tower_t en_m < X3), Tower_t e n_m(X4),Touer_t en_m(X5) , 1/2.264) 1716 PRINT USING 1390 1720 A*="DRUM RADIUS RATIO" 1736 CALL Table2CAt) 1746 R*=" (haul b. : ir.ai n)" 1756 Rl*="" 1760 CALL Tab 1 e 1 (Rt, R 11, Drur.._rat i o<Xl ), Drum_rat i o<X2 ), Drur.._rat i oCX3 ), Drut . i_rat i o< X4),Drum_rat i o<X5)) 1770 PRINT USING 1390 1786 At="TORQUE " 1790 CALL Table2CAt) 1800 B2*=" drum:lb.ft" 1810 CALL Tablel<B*,B2*,Torque_h<XI).Torque_h<X2),Torque_h(X3),torque_h<X4),Torq ue_h<X5)) 1820 C2*=" drum:N.m" 1830 CALL Table3(Bt,C2t,Torque_h<Xl),Torque_h<X2),Torque_h(X3),Torque_hO<4),Torq ue_hCX5),1.356) 104 1846 CALL T a b lel(C*,B2t,Torque_m <X1),Torque_m(X2),Torque_K<X3> ,Torque_m<X4>,Torq ue_m(X5) ) 1 850 CALL Tab 1 e3(C*, C2*, Torque m'•' X1 ) , Torque n<X2 >, Torque m<X3> , Torque m<X4 >, Torq ue_i»(X5), 1. 356) 1854 M*="Gear B ( a u x . ) : 1 b . f t " 1855 Ml*="Gear B( aux. >: N. m. " 1856 CALL T a b l e l ( M*, A1 X , Tor que_b < X 1 > , Torque_b ( X2 ). Torque_b ( X3), Torque_b<X4) . Torq ue_b(X5)> 1857 CALL Tabl e3(MU, A1 *•, Tor que_b(X1 ;•, Torque_b (X2), Torque_bCX3), Torque_b<X4 ) , Tor que_b(X5),1.356) I860 PRINT USING 1390 1870 A*="LINE VELOCITY" 1886 CALL Table2(A*) 1890 A*="RATIO (Haulb.:Main.>" 1900 CALL Tab 1 e l ( Al•, Al *, L i n e _ v e l _ r a t i o < X 1 •', L i ne_ve1_rat l oCX2 ), Li ne_ve l _ r a t . i ©<X3> ,L i ne we 1 r a t i o(X4),Line ve1 r a t i o<X5>> 1910 PRINT USING 1390 1926 A*="REVS.per MIN. " 1930 CALL Table2(A») 1940 Ml*=" drum" 1950 CALL T ab1e1(B *,M1 $,R_p_m_h a u 1 b <X1),R_p_m_h a u 1 b <X 2>,R_p_m_h a u 1 b (X 3 ) , R _ p _ m _ h a u1b(X4 ) ,R_p_m_hau1b (X5)) 1960 CALL T a b l e l ( C * , M 1 1 , R _ p _ m _ m a i n1<X1> , R_p_ m _ m a i n1<X2>,P_p_m_mai n1<X3>,R_p_n._ma i n1(X4),R_p_m_mai n1<X5)) 1970 M*="Gear B <aux.input)" 1980 CALL Tablel<Mt,A1 *,Ang_ve1_b<X1>,flng_vel_b<X2),Ang_ve1_b(X3),fing_vel_b<X4>. Ang_uel_bCX5)) 1990 PRINT USING 1390 2000 D$="C,<haul back)" 2010 A*="P0WER " 2020 CALL Table2CA») 2036 Bl*=" : hp." 2046 D2*=" : kU." 2050 CALL Tab 1e1C B*,H11,H_p_hau1b <X1),H_p_hau1b C X2),H_p_hau1b <X3), H_p_hau 1 b ( X4), H_p_haulbCX5)) i 2066 CALL T ab 1 e 3 < D * , L 2 *, H_p_h a u 1 b < X1 ) , H_p_h a u 1 b X 2), H_p_h au 1 b i X 3 ) , H_p_h a u 1 b C X 4 ), H_p_haulbCX5>,1.34) 2670 D*="A,<mainline)" 2080 CALL Tabl el ( l i t , D l * , H_p_ir.ai n<Xl j, H_p_ m a i n(X2), H_p_ m a i n(X3), H_p_ m a i n(X4), H_p_ main(X5)) 2690 CALL T ab1e 3(D $•,B 2 *,H_p_m a i n <X1),H_p_ma i n(X 2),H_p_m a i n(X 3),H_p_m a i n(X 4 >,H_p_ mai n(X5),1.34) 2100 TJt="B, ( aux. t r a n s ) " 2110 CALL Tablel(Bt,HI*,H_p_au>:(Xl),H_p_aux(X2),H_p_aux(X3),H_p_aux(X4),H_p_aux(-X5)) 2120 CALL Table3(D*,H2t,H_p_aux(Xl),H_p_aux(X2),H_p_aux(X3),H_p_aux(X4),H_p_au>:( X5),1.34 ) 2130 D* ="Y,(yarding)" 2140 CALL T a b l e l CBS,bit,H_p_yardi ng(X1),H_p_yardi ng(X2),H_p_yardi ng(X3),H_p_yard ing(X4),H_p_yarding(X5)) 2150 CALL T ab1e 3(D*,D 2 *,H_p_y ar d i n g(X1),H_p_y ar d i n g(X 2),H_p_y ard i n g(X 3),H_p_y ar d i ng(X4),H_p_yardi ng(X5),1.34) 2160 PRINT USING 1390 2170 ! 2180 END ! END OF MAIN PROGRAM 2190 ! 2260 ! ***************************************************** 2216 ! SUERTNE. TO DETERMINE POSITIVE ROOT OF QUADRATIC EGUATION 2220 DEF FNQuad_sol n((3uad_a, 0uad_b, C)uad_c ) 2236 RETURN (-Quad_b + SGlR(Gluad_b'-2-4*Quad_a*Quad_c ) )/• (2*Quad_a) 2240 FNEND 2250 ! SUERTNE. TO DEFINE TANGENT OF LINE ANGLE AT THE CARRIAGE 2260 DEF FNTan_c(Tan_c1,Tan_c2,Tan_c3,Tan_c4,Tan_c5) 2270 RETURN Tan_c1-Tan_c2*Tan_c3/(2*Tan_c4*Tan_c5) 2286 FNEND 2296 ! 2300 ! SUBRTNE. TO DEFINE TANGENT OF LINE ANGLE AT A LINE SUPPORT 2316 DEF FNTan_b(Tan_bl,Tan_b2,Tan_b3,Tar 1_b4,Tar,_b5) 2320 RETURN Tan_b 1+Tan_b2*Tan_b3''(2*Tari_b4*Tan_b5) 2 3 3 6 FNEND 1.05 2340 ! 2350 ! SUBRTNE. TO DETERMINE THE LENGTH OF fi LINE SEGMENT 2360 DEF FNSub 1 engt h < Sub_l , Sub_orne • ga, Sub_ang I € , Sub_hor t en > 2 3 ? 0 Dm i d = Su b o rn e g a* S u b 1 " 2 < 8 * S u b an g 1 e * S u b h o r t e- n ) 2380 RETURN Sub_l/Sub_ang1e*i1+8/3*CD_mid/Sub_l>*2> 2390 FNEND 2400 ! 24 10 ! SUBRTNE. TO DETERMINE NUMBER OF COMPLETE LINE-WRAPS ON fi DRUM 2420 DEF FNBrum_uir aps (F1 ange, Rope_di am, Inter 1 save, Drurn_wi dth, Drurn_rad, Drurn_l i ne_ ten!' 2430 FOR Hc_wrap= = l TO I NT < F 1 ange/•-. Rope_d i am* I nt er 1 eace •'>> 2440 IF Nc_wraps>l THEN 2470 24 50 Urap_1 engt. h = 2*3. 142*1 NT < Dr urn_i,.' i dt h/Rope_d i am >* < Dr urn_r ad + C No_uraps-1 ) *Rope_d i am*Int er1eaue >/12 2466 GOTO 2486 2476 Wrap_l e ngt h=Wrap_l engt h + 2*3 . 142*1 HT < Brum_u i dt h/Rope_d i am > * < Drurn_r ad + (No_ura ps - 1) *Rope_di am*Int er1eave)/12 2486 IF Wr ap_l ength<=Drum_l i ne_l en THEN 2560 2496 GOTO 2520 2506 NEXT No_uraps 2510 RETURN 6 2526 RETURN No_uiraps-l 2530 FNEND 2546 ! 2556 ! 2566 ! SUBRTNE TO CfiLCULfiTE INTERMEDIATE VARIABLES DESCRIBING THE SYSTEM GEO METRY. 2576 SUB G e o m < S ag,P o s i t i o n,S p an_1e n g t h,E1e u_d i f f,S p e o1 rn :>;, S_p 1 u s_e, L_m i n u s_x, A n g 1 e l , Angle2> 2586 L_m i nus_x=Span_lengt h-Pos i t i on 2596 S_pl us_e = Sag + El e','_di f f 2666 Speo 1 tnx = S_p 1 us_e/L_mi nus_x 2616 Angle1=C0S<ATNiSag/Posi t i on>) 2620 Angle2 = C0S(ATNCSpeo1mx>> 2630 SUBEND , 2640 ! 264 1 ! SUBRTNE. TO CALCULATE VERTICAL CARRIAGE POSITION RELATIVE TO TAILHOLD LINE SUPPORT 2650 ! 2666 SUB Sag iPos i t i on,E1ev_di ff,Sag,Span_lengt h,Tot al_1oad,Omegal,Ome ga2,Ome ga3, Hort en 1,Hort en2,Hort en3,Ten;i on 1,Speo1mx,S_p1 us e,L_m i nut x,Ang1e1,Ang1e2 J 2676 Sag=0 2680 Sag_loop_count=6 2690 Sag_loop_c ount =Sag_loop_count + 1 27 60 CALL Geora<Sag,Position,Span_lengt h,E1ew_diff,Speolmx,S_p1us_e,L_m i nus_x,Ang lei,Ang1e2> 2710 IF E l e y _ d i f f < 6 THEN 2746 2726 CALL Horten_2_l< Sag,Speo1mx,S_p1us_e,L_rni nus_x,Posi t i on,Ang1e1,Ang1e2,Omega 1,Ome ga2,Hort en 1,Hort en2,Tens i on 1) 2736 GOTO 2750 2746 CALL Hort en_l_2(Sag,Spe o1mx,S_p1us_e,L_m i nus_x,Pos i t i on,fing1e1,fing1aZ,Omega 1, 0rnega2, H o r t e n l , Horten2, Tensi onl) 2756 Horten3 = 2*Horten 1-Horten2 2766 Int_sag = 2*0megal*Positiori/<2*Anglel > + 0rnega2*L_m i nus_x/ (2* fing 1 e2> +0rne ga3*L_m i nus_x/< 2*fing1e2) 2776 New_sag = Pos i t i on*L_rn i nus_x./ < 2*Span_l engt h*Hort en 1 j * < Tot a l _ 1 oad+ I nt_sag > -E1 e v_d i f f * P o s i t i o n/S p an_1e ng t h 2788 IF fiBS<Neui_sag-Sag.X. 61 THEN 2836 2796 IF Sag_loop_count>56 THEN 2826 2SO0 Sag=New_sag 2816 GOTO 2690 2826 PRINT ."ITERATION PROBLEM FOR POSITION " . P o s i t i o n 2830 Sag=New_sag 2840 SUEEND 2850 ! SUE-ROUTINE TO CALCULfiTE THE HORIZONTAL COMPONENTS OF LINE TENSIONS 2860 SUB Horten_2_l<Sag,Speolmx,S_plus_e,L_mi nus_x,Posi t i on,Anglel,Angle2,Omegal ,0mega2,Hortenl,Horten2,Tensi onl) 2876 Quad_a=l+Speolmx'2 2886 Quad_b=S_plus_e*0mega2/Angle2 2890 Quad_c = C0mega2*L_m i nus_x/<2*fingle2>) A2-Tensionl'-2 2900 Horten2=FNQuad_soln(Quad_a,Quad_b,Quad_c) , 106 c 291 0 Tan_al pha_c 2=FNTan_c < Speo 1 mx, Omegal , L_rn i nus_x, Hort en2 , Ang 1 e 2) 2920 Quad_a=l+(Sag'Posi t i on)'2 2930 Guad_b«-<S*g*0<«egal)/ftngl e 1 294 0 Quad c = (0»€ gal -Pos i t i on < 2* Angl e l ) ) "'2- (Ten-, I on 1 -S_p 1 us_e*0mega2 ) "2 2950 Hort en 1=FNGuad_so1n< Quad_a,Quad_b,Quad_c) 2960 SUBEND 2970 ! 2980 I SUB-ROUTINE TO CALCULATE THE HORIZONTAL COMPONENTS OF LINE TENSIONS 2990 SUE Hort en_l_2 ', Sag, Speo 1 mx, S_p l u : e , 1. __m i nufc_x, Pos i t i on, Ang 1 e 1, Ang 1 e2, Omega 1 , Oniega2, Hort en 1 , Hort en2 , Tens i on 1 J 3000 Quad_a=l+(Sag'Posi t i on)'"2 3010 Quad_b=Sag*Omegal/Ang1e1 _^ 3020 Quad_c = (. Omega 1 *Pos i t i on' i 2* Ang 1 e 1 )) "-2-Ts n= i on 1 "2 3036 Hort en 1 =FNGuad_so 1 n< Quad_a, Quad_b, Quad_c ) 3040 Tan_al pha e 1 =FNTan_c < Sag -'Pos i t i on , Pos. i t i on, Ome gal , Hort en 1 , Angl e 1 ) 3050 Quad_a=l+Speolmx-2 3060 Quad_b=-<0»ega2*S_p 1 us_e) -'Ang 1 e2 3070 Quad_c = (0mega2*L_mi nus_x •' i2*Ang 1 e2 )) '"2- < Tens i on 1 -Sag*0mega2) '"2 308O Hort er.2 = FNQuad_so 1 n<Guad_a, Suad_b, Quad_c ) 3090 SUBEND 3100 ! 3110 ! SUB-ROUTINE TO PLOT THE MAINLINE , HAULBACK , A U X IL LIA R Y TRANSMISSION , AND NET HORSEPOWERS vs. THE POSITION OF THE CARRIAGE. 3120 SUB Pl o t CSpan_length,Interval,H_p_mai n<*>,H_p_haulb<*>,H_p_aux<*>,H_p_yardi ngC*),XI,X5) 3130 ! PLOTTER IS 13, "GRAPH ICS" 3140 ! GRAPHICS 3141 PLOTTER IS 7,5,"9872A" 3150 DEG 3160 LORG 5 3170 CSIZE 2.5 3180 LINE TYPE 1,1 3190 LOCATE 10,105,5,95 320O SCALE O,Span_length,-300,300 3210 AXES 20,10,0,0,10,5,2 " 3220 FOR Q=-366 TO 300 STEP 50 3230 MOVE -Span_length/25,Q 3240 LABEL USING "K.";Q 3250 NEXT C! 3260 FOR (3 = 200 TO Span_l ength STEP 200 3270 MOVE Q,-20 3280 LABEL USING "K";G 3290 NEXT Q 3300 MOVE Span_l engt h-'2,-295 3316 LABEL USING "K";"CARRI AGE POSITION < Distance from the t a i l h o l d ) : f t . " 3320 LDIR 90 3336 MOVE -Span_l engt h-'l 3, 0 3340 LABEL USING "K";"POWER : horsepower" 3350 LDIR 0 3366 MOVE Span_length/2,295 s 3370 LABEL USING "K";"POWER COMPONENTS vs. CARRIAGE POSITION" 3386 LINE TYPE 4 3390 PLOT Span_lengt h.' Int erval*X1,H_p_mai n<Xl>,-2 3400 FOR Q=X1 TO X5 3410 DRAW Span_l ength-'Interval *Q, H_p_mai nCQ) 3420 NEXT Q 3430 LINE TYPE 1,1 3440 IPLOT 30,0,-2 3450 LABEL USING "K";"Hp" 3460 RPLOT 15,4,-2 3470 LABEL USING "K";"M" 3486 LINE TYPE 8 3490 PLOT Span_length/1nterval*X1,H_p_haulb(Xl) , -2 3506 FOR G=X1 TO X5 3510 DRAW Span_length/Interva1*G,H_p_hau1b<G) 3520 NEXT Q 3530 LINE TYPE 1,1 3540 I PLOT 36,6,-2 3550 LABEL USING "K";"Hp" 3560 RPLOT 15,4,-2 1 0 7 3570 LflBEL USING "K";"H" 35S0 LINE TYPE 1 3590 PLOT Spari_l 6 -ngt n I n t e r v a 1 #KJ, H_p_aux(Xl 1 , -2 3660 FOP G = X1 TO X5 3610 DRAW Span_length'Inter^al*n,H_p_aux(Q) 3620 NEXT GJ 3630 MOVE Span_length*.9,-270 3640 IPLOT 70,0,-1 3650 IPLOT 30,0,-2 3660 LINE TYPE 1,1 3670 LflBEL USING "K";"Hp" 36S0 IPLOT 25,4,-2 3690 LABEL USING "K";"B" 3691 LINE TYPE 3 3700 PLOT Span_l engt, h- I nt, e r ya 1 * X l , H_p_yardi ng(Xl >, -2 3710 FOR Q=X1 TO X5 3720 DRAW Span_l e n g t h <•' I n t. e r v a! *Q. H_p_yar di ng(G) 3730 NEXT G 3731 LINE TYPE 1 3740 MOVE Span_length*.9,-256 3750 IPLOT 70,0,-1 3760 IPLOT 30,0,-2 3770 LflBEL USING "K"J"Hp" 3780 IPLOT 25,4,-2 3790 LflBEL USING "K";"Y" 3800 ! DUMP GRAPHICS 3816 PAUSE 3820 ! GCLEAR 3830 ! EXIT GRAPHICS 3840 SUBEND 3856 ! 3860 ! * * * * * * * * * * * * * * * * * * * * * J » « » H « « H « H * ; B i H H H H - « H H H H H i . * « - - 5 3876 ! SUB-ROUTINES TO PRODUCE THE SUMMARY TABLE 3880 ! 3896 SUB Tab 1e1(A*,Alt,A,B.C,D,E> 390O DIM N*[46] 3916 N* = A*8<A1* » 3920 PRINT " | " ; Nt; TAB ( 22 ) ; " | " ; fl ; TAB ( 34 > ; " j " ; E ; T AE C 46 ); " | " ; C; T AB •: 58 ':> ; " | " ; D ; T AB •' 69 ); " | "; E; TAB (SO); " | " 3930 SUBEND 3940 SUB Tab1e2CV*) 3956 PRINT "|";Vt;TfiB<22>;"|";TAB(34);"|»;TAB(46);"|";TAB(5S);"|";TAB(6 9 ); " |";TA B(80) J " | " 3960 SUBEND 3976 SUB Table3(fl*,Al*,A,B,C,D,E,F) 3980 DIM L*C461 3996 L* = A*S<A1* 4666 PRINT " | ";L*;TABC22); " I";A*F;TAB(34);"| ••;B*F;TAB(46);" |";C*F;TAB(58);"|";D* F;TAB(69);"|";E*F;TAB(86);" |" 4016 SUBEND 1 - 0 8 APPENDIX F SPECIFICATIONS OF THE WINCH DRUMS AND LINES USED IN THE COMPUTER MODEL 109 Table F I . Table of Winch Drum and L i n e S p e c i f i c a t i o n s Used i n the Computer Model.^ M a i n l i n e Drum Haulback Drum Drum Width Flange Depth B a r r e l Radius 22.5 i n (52.1 cm) 7.5 i n (19.05 cm) 10.0 i n (25.4 cm) 22.5 i n (52.1 cm) 7.5 i n (19.05 cm) 10.0 i n (25.4 cm) .Mainline Haulback L i n e Nominal Diameter T o t a l Length Safe Working Tension 0.625 i n (15.87 mm) 1,400 f t (426.7 m) 13,700 l b (6214.3 kg) 0 .625 i n (15 .87 mm) 2,600 f t (792.5 m) 13,700 l b (6214.3 kg) 2) These s p e c i f i c a t i o n s , l i n e s i z e s and l i n e l e n g t h s are taken from the drum s p e c i f i c a t i o n s and recommended l i n e s p e c i f i c a t i o n s of the Lantec I n d u s t r i e s I n t e r l o c k e d Winch Set (Anon. 1979) 110 APPENDIX G IMPERIAL AND METRIC UNITS, AND THE CONVERSION FACTORS USED IN THE COMPUTER MODEL 111 Table Gl. Table of Imperial Conversion Factor 1 foot 1 inch (in) 1 pound (lb) 1 pound force (lb) 1 pound foot (lb ft) 1 horsepower (Hp) 1 foot per second (ft s ^) 1 foot per minute (ft min ^) and Metric Units, and the Used i n the Computer Model. = 0.3048 metres (m) = 2.540 centimetres (cm) = 0.4536 kilograms (kg) = 4.44 8 Newtons (N) = 1.356 Newton metres (Nm) = 0.7457 kilowatt (kW) = 0.3048 metres per second Cm s" 1) = 18.288 metres per minute (m min~l) 112 

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