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The transverse dynamics of rotating imperfect disks Rosval, Gergory William 1981

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THE TRANSVERSE DYNAMICS OF ROTATING IMPERFECT DISKS  by GREGORY WILLIAM ROSVAL B . A . S c , University of B r i t i s h Columbia, 1976  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES (Department of Mechanical  Engineering)  We accept this thesis as conforming to the required standard.  THE UNIVERSITY OF BRITISH COLUMBIA January @  1981  Gregory William Rosval,  1981  In presenting this thesis in p a r t i a l f u l f i l l m e n t  of the  require-  ments for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library shall make i t freely available for and study.  reference  I further agree that permission for extensive copying of  this thesis f o r scholarly purposes may be granted by the Head of my Department or by his representatives.  It  i s understood that copying  or publication of this thesis f o r financial gain shall not be allowed without my written permission.  Department of Mechanical Engineering The University of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada  V6T  1W5  December  1,  1980  ABSTRACT  The transverse motions of c i r c u l a r saws have undesirable effects on many aspects of c i r c u l a r sawing.  Due to current high man-  ufacturing c o s t s , substantial savings may be realized i f these transverse motions can be reduced. In this thes.is a c i r c u l a r saw i s modelled as a rotating imperfect disk acted upon by a transverse, non-oscillatory point load stationary in space.  Such a model i s known to accurately predict  certain  relevant aspects of the behaviour of a c i r c u l a r saw in i t s operating environment. I n i t i a l l y the free response of a non-rotating perfect disk is cons'iderced.  This model i s then refined by considering the effects  rotational stresses and small imperfections within the disk.  of  The res-  ponse of such a disk to an o s c i l l a t o r y load i s determined, from which the response to a non-oscillatory load may be determined as a special case of p a r t i c u l a r  interest.  Experimental theory presented.  results are given which quantitatively confirm the  ACKNOWLEDGEMENTS  I wish to thank Professors H. Vaughan and S. Hutton for t h e i r supervision of the t h e s i s .  I am also grateful to Dr. J . Brdicko of  Canadian Car Limited both for his advice and for the loan of some of the experimental equipment that was used.  Mr. P. Hurren and Mr. J . Richards  of the University of B r i t i s h Columbia were very helpful in a s s i s t i n g me in the development of a functioning experimental set-up. The fine job of typing this thesis was done by Mrs. Lynn Main, whom I wish to thank.  F i n a l l y , I would l i k e to acknowledge the patience  and encouragement offered by my wife Sandy during the preparation of this thesis.  iv  TABLE OF CONTENTS  Acknowledgements  ii1  Notation  v1  L i s t of Figures  viii  I Introduction . . . .  1  II  4  The Theory of Rotating Disks 11.1  Free Response of a Perfect Non-rotating Disk  11.2  Free Response of an Imperfect Non-rotating Disk . . . .  11.3  An approximate Method of Determining the Free  11.4  11.5  11.7  11  Vibrations of a Non-rotating Disk  25  The Effects of Rotation  30  Rotating Membrane  30  Rotating Disk  33  Southwell's Theorem  34  Disk Displacement with Respect to Rotating and Non-rotating Coordinates  11.6  6  3  8  Forced Response of a Disk  44  Non-rotating Disk  44  Rotating Disk  49  Comments on the Theory and i t s Applications Current Research Saw Behaviour in i t s Environment  56 57  Operating 60  V  III  Experimental V e r i f i c a t i o n of the Theory of Rotating Disks . . . 111.1  Experimental  111.2  Forced Response of a Non-rotating Disk  62  Equipment  62 65  The Location of the Nodes  66  Damping Coefficient  .  Radial and Angular Deflection P r o f i l e s  . . . .  69 71  i  111.3  Forced Response of a Rotating Disk  75  Observed Frequencies as a Function of the Rotational  Speed  Response to a S t a t i c Load IV  Closing Remarks  References  76 78 85 87  NOTATION  clamp radius disk radius flexural r i g i d i t y ; energy dissipated by damping Young's Modulus observed frequency damping dissipation  coefficient  disk half-thickness generalized force amplitude of load natural frequency natural frequency radial displacement function radial  coordinate  radial location of load temporal response function; kinetic energy time coordinate s t r a i n energy displacement in space-fixed coordinates proportionality  constant for membrane frequency  displacement in d i s k - f i x e d coordinates angular phase angles s t r a i n energy  coefficient  temporal phase angles angular space-fixed coordinate  vii  A  non-dimensional frequency parameter  ©  angular displacement function  e  angular d i s k - f i x e d coordinate  6p  angular location of load in the disk coordinate system  y  kinetic energy c o e f f i c i e n t  a (r)  radial membrane stress  a (e)  tangential membrane (hoop) stress  p  mass density  v  Poisson's r a t i o  fi  disk rotational  to  excitation  T  variable c o e f f i c i e n t in radial  £  damping r a t i o  speed  frequency function  vi i i  LIST OF FIGURES Figure 1  Perfect Disk Dimensions and Coordinate Systems  5  Figure 2  Non-dimensional Frequency Parameter versus Clamping Ratio  28  Figure 3  Radial Function Coefficient versus Clamping Ratio  29  Figure 4  Membrane Stresses in a Rotating Membrane  31  Figure 5  Deformed Element of a Rotating Membrane  32  Figure 6  Observed Resonance Frequencies versus Rotational Speed . .  41  Figure 7  Membrane Stresses in a Non-rotating Tensioned Disk . . . .  58  Figure 8  Saw Response versus Rotational Speed  61  Figure 9  Proximity Sensor Calibration Curve  63  Figure 1 0  Amplitude of the Response versus location of the Load  Figure 1 1  Relative Amplitude versus Excitation Frequency  70  Figure 12  Radial P r o f i l e at 0 Hz. (Disk C) .  72  Figure 1 3  Radial P r o f i l e at 90 Hz. (Disk C ) . \  Figure 1 4  Theoretical Angular P r o f i l e s at 0 Hz  74  Figure 1 5  Angular P r o f i l e at 0 Hz (Disk C)  75  Figure 1 6  Observed Resonance Frequencies versus Rotational Speed . .  77  Figure 1 7  Displacement versus Rotational Speed (Disk C)  79  Figure 1 8  Angular P r o f i l e at 2400 R.P,M.(Disk  80  Figure 1 9  Displacement versus Rotational Speed (Disk D)  81  Figure 20  Forward T r a v e l l i n g Component Observed with Two Sensors . .  83  . . . .  . .  68  7  C)  3  1  I.  Introduction The use of c i r c u l a r saws in the manufacture of lumber i s very  common, and in an attempt to reduce the costs associated with this process much research has been conducted.  Many of the problems r e s u l t -  ing from the use of a c i r c u l a r saw may be attributed saw in the direction normal to i t s plane.  to motions of the  These motions, which are  referred to as transverse v i b r a t i o n s , have an undesirable effect on cutting accuracy, kerf losses, the quality of the cut surfaces, saw l i f e and ambient noise l e v e l s .  One of the primary goals of researchers  in t h i s area i s to reduce the kerf losses by making the saw  thinner.  This must be done without suffering the adverse effects caused by an increase in the transverse vibrations resulting from a reduction in the l a t e r a l s t i f f n e s s of the saw. * In attempting to predict the behaviour of a c i r c u l a r saw in i t s working environment, many d i f f i c u l t i e s are encountered.  Since the  saw and i t s environment are continually changing, no complete solution exists which considers a l l aspects of the problem simultaneously. However, i t  i s possible to model t h i s system in a way such that useful  qualitative  and quantitative  results may be determined from fundamental  principles. Most simply, a c i r c u l a r saw may be modelled as a non-rotating, undamped complete d i s k .  The natural frequencies and shapes of the free  vibrations of such a disk were determined by Kirchoff in 1850 . 1  Later in  that century Rayleigh made a s i g n i f i c a n t contribution to this problem, in p a r t i c u l a r in r e c o n c i l i n g theoretical predictions with experimental results . 2  He introduced the idea that small imperfections within the  2  disk could s i g n i f i c a n t l y a f f e c t i t s behaviour.  Near the end of the  19th  century Zenneck formalized the ideas of Rayleigh regarding the influence of imperfections on the free vibrations of a d i s k . 3  The inclusion of stresses due to the rotation of the disk was the next major step in the development of the model. by Lamb a n d Southwell in 1921 *. 1  Theorem was f i r s t  It  was in this work that Southwell's  This theorem states that  intBoduced.  This was done  under  certain  conditions an approximate method i s available which establishes a lower bound on the fundamental frequency of vibration of a body.  This work,  however, did not consider the influence of a central clamp covering a portion of the d i s k .  Later that year Southwell published a paper in  which the effects of a clamp were considered, and where his theorem was demonstrated with numerical examples T 5  It  wasn't until  1957 that the experimentally observed forced  response of a rotating disk was s a t i s f a c t o r i l y explained.  This inform-  ation was made available in a paper by Tobias and Arnold, where i t was shown that achieving agreement between theoretical  and experimental  results requires considering the effects of minor imperfections within the d i s k . 6  The directions of research since this time have generally f a l l e n into one or more of the following categories; the control of membrane s t r e s s e s , ' ' * ? ' 7  control m e t h o d s . ' 1 5  1 6  8  9  1  1 1  *  1 2  a l t e r i n g saw g e o m e t r i e s ' 1 3  1 4  and external  Although numerical results can be obtained when  the problem can be mathematically modelled, obtaining a suitable model which includes a l l aspects of the problem presents a major  difficulty.  For this reason an approach combining both theory and experimentation very u s e f u l .  is  3  Because experimentation should be an important part of an investigation in this f i e l d , this thesis deals with rotating disks on both a theoretical  and an experimental basis.  It  i s the purpose of this  thesis to present a theory of the forced response of rotating disks and to present experimental results which v e r i f y this theory. ed theoretical  While a d e t a i l -  development of the forced response of idealized disks i s  given, emphasis i s placed on the theory required to account for the departure of the c h a r a c t e r i s t i c s of real disks from those of idealized disks.  4  II  The Theory of Rotating Disks Two types of disks are analyzed in this t h e s i s .  what are referred to as "perfect d i s k s " .  The f i r s t  are  These disks are homogeneous ,  i s o t r o p i c , completely c i r c u l a r and are of constant thickness throughout. The second type, c a l l e d "imperfect d i s k s " , possess material  properties  and geometries which depart s l i g h t l y from those of perfect d i s k s .  All  real disks are to some extent imperfect, and the influences of these imperfections on the d i s k s ' behaviour must be taken into account when experimental  results are  interperted.  Because the disk i s rotating in space while observations are made from points stationary in space, two coordinate systems w i l l be defined. (r,e)  The coordinate system fixed in the disk w i l l be denoted as the  system, whereas that fixed in space w i l l be denoted as the (r,  system.  The radial coordinate r i s the same in each case.  y)  These two  coordinate systems and the physical dimensions of a perfect disk are shown in figure  1.  The disk rotational  speed i s  in the positive y d i r e c t i o n , and  the origins of both angular coordinates are taken to be coincident at t = 0.  Therefore a point located at an angle e in the disk is located in  space at an angle y given by: Y = fit-6 S i m i l a r i l y , the reverse transformation i s : 0 =  nt-y  The transverse displacement as a function of e i s denoted by w, and as a function of y by u.  Figure 1 Perfect Disk Dimensions and Coordinate Systems  I n i t i a l l y the method of solution of the "free vibrations of a p e r f e c t , c e n t r a l l y clamped non-rotating disk i s outlined.  The results  obtained from this analysis form the basis for predicting the disk behaviour when the problem i s compounded with imperfections, stresses and a transverse load.  rotational  6  II.1  Free Response of a Perfect Non-rotating Disk The equilibrium plate bending equation f o r a homogenous,  i s o t r o p i c plate  is:  (II.1)  2 1  v*w = (J/D  Here q i s the transverse load per unit area and D i s the flexural r i g i d i t y given by:  D  =  E(2h) /12(l-v ) 2  3  The biharmonic operator i s to be expressed i n polar coordinates. For the case of a f r e e l y vibrating non-rotating d i s k , the only load i s that due to the i n e r t i a l damping.  forces and internal  and external  I f damping i s neglected, equation ( I I . l ) may be written as:  (II.2)  * • - f r  There are four boundary conditions which the solution of equation ( I I . l ) must s a t i s f y .  They are:  (a)  no deflection at the clamp  (b)  no slope at the clamp  (c)  no internal  (d)  twisting moment/shear stress condition at the free edge  radial  bending moment at the free edge  (Kirchoff boundary condition)  7  A separation of variables form of a solution may be assumed in t h i s case:  (11.3)  w = R(r)T(t)e(e)  When equation (II.3) i s substituted into equation ( I I . 2 ) , a suitable expression f o r 0 i s found to be:  (11.4)  e(e) =  B^os ne + B si.n ne 2  The necessity that w ( r , t , e ) = w ( r , t , e + 2ir) requires that n i s an integer.  The expression f o r T i s also immediately found to be of the  form: (11.5)  Tift) = Ci cos p t + C sin p t n  2  n  The expressions f o r the natural frequencies p are unknown at t h i s stage n  except that they are dependent on the integer n. The solution as given by equation (11.3) may then be written a s :  (II.6)  w  n  = R( ) [Bi cos ne + B s i n ne][Cj cos p t + C sin p t ] ;r  2  n  2  n  The subscript n now appears on w, indicating that there i s a solution f o r each value of n selected. In order t o determine the radial function R (r)and the frequencies Pn> consider one term of equation ( I I . 6 ) , which may be written as:  (II.7)  w  n = A R, (r) sin ne cos p t n  n  8  After substituting equation (11.7) into the equation of motion, the function R(r) may be determined by solving two d i f f e r e n t i a l Bessel's equation and the Modified Bessel's equation ..  equations;  Equation (II.7)  5  i s then:, (11.8)  w  )n  =[a!J  Here Jn» Y , I , n  n  + a Y  n  2  n  + a I 3  n  + a^Kn] sin ne cos p t n  and K are Bessel and Modified Bessel functions of the n  f i r s t and second kinds, whose arguments are dependent on D and p . n  Substitution of equation (II.8) into the boundary conditions y i e l d s a c h a r a c t e r i s t i c equation, from which the natural functions may be determined.  frequencies and radial  For a p a r t i c u l a r value of n there are  an i n f i n i t e number of frequencies and radial functions. radial  Each of these  functions may be i d e n t i f i e d by the number of values of r at which  there i s no transverse motion.  These c i r c l e s , excluding the central one  due to the clamp, are cctlled nodal c i r c l e s , and the number of them occurring i s denoted by s , where s = 0, 1, 2 . . . . It may also be seen from equation (II.8) that there exists n diameters of zero transverse motion, known as nodal diameters.  The s nodal c i r c l e s , n nodal diameters  response of equation (II.8) may be written a s :  (11.9)  w,  n)S  =A , n  s  :.R  n>s  ( r ) sin ne cos p , s t n  For reasons which w i l l become apparent when the forced response of a rotating disk i s discussed", i t i s only the zero nodal c i r c l e s modes which are of concern.  The response of the disk in those modes where  s f o i s therefore neglected and the subscript s may be ommitted. Equation (II.9) i s then:  9  w  n  =  R  n () r  s  i  n  n^cos p t n  This expression, however, was developed by considering only one term of equation ( I I . 6 ) .  When a l l four terms are considered the n nodal diameter  response i s :  w  n  =  R  n ()  C i i  r  B  c  c  o  s  n e  c  o  Pnt + B ^  s  cos ne s i n p t n  + B C ! sin ne cos pnt + B C s s i n ne s i n pnt] 2  It  2  2  i s possible to write this expression in two forms using the trigono-  metric  identity:  a sin x + b cos x = c cos (x - ?) ?  where:  a  2  + b  and  2  5  = c  2  = tan  b/a  By applying t h i s i d e n t i t y to the angular trigonometric terms we obtain: (11.10)  w n  = Rn(r)  [A  cos (ne-e ) s i n p t + A  n l  x  n  n 2  cos (ne - e ) 2  cos p t ] n  The second form i s obtained by applying the identity to the temporal trigonometric terms:  (II. 11)  Wp = R (r) n  [A i n  s i n ne cos (p t-e}))n-A cos n  n2  ne cos (Pnt-e )] 2  Equations ( 1 1 . 1 0 ) and ( 1 1 . 1 1 ) are the two forms of the n nodal free response of a perfect d i s k .  Although they are equivalent, i t  i s the  form given by equation ( 1 1 . 1 1 ) which w i l l be used for the remainder of this thesis for reasons which w i l l become apparent when the response of an imperfect disk i s i c o n s i d e r e d . There are four constants in equation ( 1 1 . 1 1 ) which must be determined from the i n i t i a l  conditions rM the free response.  It  will  now be shown that although t h i s expression i s referred to as the n nodal diameter response, i t does not in general consist of what i s commonly known as nodal diameters. If we set w  n  = 0 in equation ( 1 1 . 1 1 ) , the following expression  for e r e s u l t s :  + a r i t  a  n  „  n  n e  _ A cos (p t " AnT 60S (p t n ?  n  n  - ) -el) E 2  From t h i s expression i t i c a n be seen t h a t , i n general, the angular locations where: v the displacement i s zero i s a function of time.  If,  however,  e i = e this becomes: 2  tan ne =  - An / A i 2  n  =  constant  which i s the familar case of a free response with nodal diameters fixed in the d i s k .  Referring to equation ( 1 1 . 1 1 ) i t can be seen that this motion  results when the disk i s i n i t i a l l y deformed in the shape w, _;| V"ti-al n  [Am sin ne  + A  n 2  cos ne] and released at t =  p . n  n  =  R  n( ) r  n  As a second example, i f the temporal phase angles are such that e  2  = ei+ 3ir/2  and i f A  = A  n 2  , the location of the-nodal l i n e s is  given by: tan ne = tan ( p t n  e =  Bnt -  -  e x  )  .  e1  These nodal l i n e s are t r a v e l l i n g around the disk at a constant speed of e = p /n  .  The i n i t i a l  conditions of this motion may be obtained by  substituting e = £\ + 3 IT /2 into equation (11.11). 2  in creating these i n i t i a l  Due to the  difficulty  conditions, t h i s motion i s not commonly observed  experimentally. Both of the above cases, the vibration fixed in the disk and the t r a v e l l i n g wave, are special cases of the more general r e s u l t .  From  equation (11.11) i t can be seen that in general the n nodal diameter free response consists of two o s c i l l a t i n g n nodal diameter shapes, with the nodes of one located half-way between those of the other.  Although the  frequencies are equal, the time phases are not.  II.2  Free Response of an Imperfect Non-rotating Disk As previously discussed, an imperfect disk i s here defined to  be a disk whose geometry and physical properties d i f f e r s l i g h t l y from those of the idealized perfect d i s k .  While the effects of the imperfect-  ions could be determined quantitatively  i f the imperfections can be  mathematically modelled, this i s not l i k e l y to be the case.  The following  theoretical  developments do not require such a knowledge of the nature  of the imperfections, yet they provide very useful  qualitative  information. The energy method i s used here to investigate the effects of imperfections on the free vibrations of a non-rotating d i s k . The 3  assumed expression f o r the total free response i s given by equation (11.12) where ^(t) and v(t)  (11.12)  are to be determined.  w = | [ $ ( t ) R (r) cos ne n  n  + v ( t ) R ( r ) s i n ne] n  n  Two types of energy w i l l be considered here; the kinetic energy of the transverse motion and the s t r a i n energy of deformation. The energy dissipated by damping i s considered in a l a t e r section. (i) Kinetic Energy:  The kinetic energy of the transverse  vibrations of the disk i s given by: rdrdedz  T vol From equation (11.12):  W  ?=  M i n m Vm R  vv n  m  R  [  c o s  n e  s i n ne s i n me  c o s  m e  + 2h i  n m  +  cos ne s i n me]  Due to the orthogonality of the trigonometric terms in (w) , 2  the only remaining terms after the kinetic energy  integration  would be those containing cos ne cos me and s i n ne s i n me f o r  In t h i s case, the kinetic energy i s of the form:  (11.13)  T  =  jj h u  n  *n  h p  +  *n  n  where: (11.14)  y  n  =  2ppir h j[  R (r)]  r dr  2  n  T h i s , however, assumes that the disk i s perfect. Although the imperfections are considered to be general in nature, as an i l l u s t r a t i v e example consider the case where the imperfection consists of a density variation in the angular direction.  (11.15)  If  the density can be represented as:  p(e) =  i: C|< cos ke + D|< sin ke k=0  the results of the kinetic energy integration w i l l be somewhat d i f f e r e n t than in the perfect disk case. kinetic energy integral  for this density imperfection i s :  p(w) r = n I n m^*n*m k mn 2  The integrand of the  R  R  Dfc sin ke) + Dk sin ke )  c  o  s  n e  c  o  s  m e  ( k cos ke + e  sin nesin me (C|< cos ke + +  2 i * n  m  (C^ cos ke + D|< sin ke  cos ne sin me x )]r  Of the six terms within this summation, i t cannot be said that any one w i l l vanish for a l l values of k, m, and n when the  integration i s performed over the range of e.  If,  however,  the density imperfections are s m a l l , a l l values of C|< and Dj,, in equation (11.15) are also small with the exception of C . 0  The general form of the kinetic energy expression i s then: (11.16) T  =  W  h  where a  n m  a  ™ ®n*  m  +  2  + bnm *n*m  n  In a d d i t i o n , the values of a  are s l i g h t l y d i f f e r e n t  n m  n m  and b  i s small for a l l n m  , f o r n = m,  from what they would be f o r a perfect disk.  Strain Energy:  For a non-rotating d i s k , the s t r a i n energy  i s that due to bending. is:  ™ *n*m  and b m are small for n f m, and c  n and m.  (ii)  c  In p o l a r - c y l i n d r i c a l coordinates this  6  -  f(T  "  IV  0  (n.17)  E  '  z  z  I  2  2(l-v )  ,, u  _  /I  9W  iF "  IF  9?  )  + 2(1-v)  5  1 aw  2  , 9W 2  v  (3 w .  ) I  2  _9_ [1_9W 9r \ r 99  F 9F  +  1  F " ae*)  9 W\2 2  2  9^w\  , 1 +  . +  F"  a? ")  2  2  rdrdedz  If the disk i s considered to be p e r f e c t , the resulting expression for the s t r a i n energy i s :  (11.18)  U= E  h g $ n  2 n  +  h p^n  2  15  Where:  Here R denotes R ( r ) .  As can be seen from equation  n  (II.17),each  term of t h e . s t r a i n energy integrand contains e i t h e r cos me cos ne, sin ne s i n me or cos ne s i n me. Again, due to the orthogonality of the trigonometric terms, no cross-terms appear in the s t r a i n energy expression for a perfect disk.  However, i f the disk i s  imperfect, the general form of the s t r a i n energy i s : (11.20) U  =  h  I E a mn  n m  *n*m  +  2 E  n m  $w n  m  ++  b  Here the r e l a t i v e magnitudes of a m> Cnm and B n  to those of a  n m  , c  n m  and b  n m  n m  n m  vv n  m  are s i m i l a r  .  Lagrange's equations f o r the free undamped vibrations of this system a r e :  6  The results of substituting equations (11.16) and (11.20) into Lagrange's equations are:  (11.21)  za $ + a $ + c Y + C ¥ m nm m nm m nm m nm m  =0  (11.22)  l b Y + 6 T + C $ + c $ m nm m nm m nm m nm m  =0  Here there are two systems of l i n e a r d i f f e r e n t i a l  equations, each system  consisting of an i n f i n i t e number of equations, one equation for each value of n. By the theory of systems of l i n e a r d i f f e r e n t i a l constant c o e f f i c i e n t s , solutions are sought of fche form 2 3  and  <p. = Y. e  conservative.  X t  equations with $. = X. e J J  x t  However, A must be pure imaginary since the system i s  .  The assumed form of the solutions to equations (11.21) and  2  (11.22) are therefore taken as: $. = X. cos (tot - y) J  (11.23)  Yi  J  = Y j COS ( a i t - Y )  These assumed solutions may be substituted into each of the equations of (11.21) and (11.22).  The r e s u l t , f o r example, of substituting into the  n=2 equation of (11.21) i s :  (a i  - io a i)  + (a  2  2  2  - i o a ) X ++ 2  2 2  22  2  —  i +  (c  - i o c i ) Yj+ ( c 2  2 1  2  - o) c ) Y + — 2  2 2  22  2  =0  17  All  such r e s u l t i n g equations may be written in matrix form as:  (11.24) (in-oj&aii)  (C -UJ C )  (a -oj a )  ( c - a ) C=]1 2 )  (c{ -aj c )  (b -a) b )  (c~ -w c )  (B -u bj  (a| -oj a )  (c -io c )  (a -a) a )  (c -w c,  (c -aj c )  (b -a) b )  i(:c -u) c )  (b -u> b  2  1  11  2  21  1  2  2 1  2 1  2  1 1  1 1  2  11  11  2  2 1  2 1  2  2 1  2 1  2  1 2  12  2  2  12  2  x  2  2 2  2 2  2  22  0 0 0  2  1 2  2 2  f cT  h  2  1 2  2  *  Y  2  22  22  2  2  1  1  1 1  1 V.  ^  J  or: [M] (A)  (11.25)  A non-trivial det[M] = 0 .  =  (6}  solution to equation (11.25) exists i f and only  In the case where the imperfections are s m a l l ,  if  the non-  diagonal elements of the matrix [M]aare small in r e l a t i o n to the diagonal elements.  The solutions to the equation  det [1M] = 0  are then approximately  determined by s o l v i n g : (an  -u)c a )(6 2  u  1 1  - to b )(a 2  1 1  1 1  - u> a )(b 2  2 2  22  - w b ) — 2  22  2 2  = 0  The values of u which are the solutions to t h i s equation w i l l be denoted by p and q:  (H.26)  where  Pi = a n / a n  p|  qf. = B n / b u  q  p. = q.. , since  a... =  2  b.. D^.  and  2  -  a  2 2  /a 2 2  —  = b  2 2  / b 22  aj_. = b. n n  /  Each eigenvalue of equations (11.26) , p. or q. , may be s u b s t i t uted into equation (11.24) from which the r e l a t i v e magnitudes of the components of the matrix {A} may then be determined.  For example, the assumed  s o l u t i o n s , equations (11.23), obtained by putting  w = Pi  jn equation  (11.24) may be written as:  $i = P i i c o s t p j t soe{?) t $  Here P  1 1 S  P  2  1  •••> B  2 1  f  P iCos(p t  = 2  1 1 S  B  determined f o r to = p  2 1  .  "ti  •••  a n c  ' i e  u e n  •j = P c o s ( p . t - e.)  (11.27)  f  E  h  c  o  2  1  x  = B c o s ( p ! t - ej) 21  °te X  j i  =  1 1  1 S  X  Y  2  1 S  Y  2  . . . and Y as  For <o= p. and <o= q . , $.aarid f . may be written as J  (11.26)  j  = B cos(p t - e )  ^  ~ i> e  Q . . c o s ( q t - Cj) i  D..cos(q .t - c..) i  The most general forms of the solutions f o r $. and v . are the sums of the solutions given by equations (11.26) and (11.27):  (11.28)  With a change of the dummy indices i and j  , equations (11.28) may be sub-  19  stituted into the i n i t i a l  expression for the free v i b r a t i o n , equation (11.12),  yielding:  w = i(R (r) | m m  m  Rjr) m  It  cos me z [P c o s ( p t - eJ L mn n n  + Q cos(q t - c )] + mn ^n nJ  sin me z  + U cbs(q t - t )] j mn ^n n J j  mn  m n  r  n  n  cos(p t - e J n n  L mn  mn  r  is possible to rearrange this expression to identify the shape of the  response at a frequency p at the frequency p  n  n  (or q ) . n  If we denote by w  (  w n2  )  t  n  e  r e s  Ponse  ( P ) » the result i s : n  w = cos(p t - e ) z [P R cos me + B R sin me] ni ^nt n l mn m mn m J m  (11.29) w„„ = cos(q„t - c„) z Ft) „R cos me + D R sin mel nz ^n n L mn m mn m J m  m  m  Clearly the effects of the imperfections are to not only a l t e r the natural frequencies, but to also a l t e r the shapes of the modes.  If  the c o e f f i c i e n t s  a , b . . . are known, i t is possible to determine the r e l a t i v e magnitudes mn mn ° of P . B „ and Q , D and hence evaluate the natural frequencies and mn mn mn mn ^ mode shapes ("The Theory of Sound", § 9 0 ) . 2  It  can also be shown that the  contribution of the cos me shape to that of the n nodal diameter mode is proportional to / -  1  | w, where here p ^ and p *  uencies of a perfect d i s k .  If  it  n  are the natural  freq-  is assumed that th«is contribution is  negligable, tjjat i s , that the imperfection does not a l t e r the actual shapes of the modes of v i b r a t i o n , equations (11.29) are:  (11.30)  w = R cos (p t -e ) [ P cos ne + B sin ne] ni n n n' nn nn V f  w „ = R cos (q t n2 n n  ) [Q nn  - r  X M  cos ne + D sin ne] nn  w  J  The two independent expressions for the n nodal diameter free vibrations of the disk are those given by equations (11.29), or approximately by equations (11.30).  Equations (11.30) may be written  as:  (11.31)  w  ni  V  = R  = R  n  n  <f> cos (ne - cx ) n  *n  n  S I N  (  N  "  E  where  P  *  n  it v  and  a  n  n  8 n  If  it  } N  1 2 2 ' = VP - + B cos (p t - e ) nn  v  = Jo1 v w  nn  nn  r  + d.  cos (q  1  nn  =  tarfi B  =  -tan"  1  n n  n  V H  /  P  n  nv  -  ?  r ) n  ;  n n  0 I D„ ^nn nn n  i s assumed that the imperfections do not a l t e r the shape of the  response, the coordinates <> j  and  n  are the disk normal coordinates, n  since a free vibration i s possible which i s characterized by the vanishing 2 3  of a l l $  and ^  n  except o n e . - '  When the kinetic and s t r a i n energies of  the imperfect disk are expressed in terms of these normal coordinates,  21  the c o e f f i c i e n t s of the cross-terms must be zero.  By comparing the  expressions for * , ¥ , <> j and 4 the relationships between the two n n MI n r  sets of coordinates i s seen to be:  *n  v  n  *n  =  =  c  o  ° n " *n  s  sin a  n  n  +  t  s  n  1  n  3  n  cos &  n  Upon substitution of these equations into the kinetic and strain energy equations, (11.16) and (11.20), there results the following:  (n.32)  2  U  =  B  +  R  ^  n  +  C  n  v  *  n  The c o e f f i c i e n t s A , B e t c . are functions of the angles a and e as n n n n well as the c o e f f i c i e n t s a „ „ , b„„ e t c . Setting the c o e f f i c i e n t s of the nn nn 3  3  cross-terms, that i s C expressions for t *  n  and C^, to zero results in the  n  following  and $ : n  (be - cB) t a n a  n  +  (ba - aB) tan a  n  + (ca - ac) = 0  (ca - ac) t a n 6  n  -  (ba - ab) tan g  n  + (be - cb) = 0  2  (11.33) 2  The sub-scripts "nn" have been ommitted.  In addition to the above two  expressions for a and 6 , t h e i r r a t i o i s obtained as: n n r  tan a  caa  ac  : cb -  r be  n  (di.34)  =  tan B  n  Lagrange's equation may be used with the expressions for T and U, equations (11.32), to determine the natural frequencies.  This results  in: 2  _  a + b tan  (H-35)  «n + 2c tan a  2  a + b tan c 2  „ %  _ -  n  + 2c t a n a  5 + a tan 3„ E!  - 2c tan s  b + a tan B  - 2c tan B  2  2  n  D_ n  Equations (11.33), (11.34) and (2.35) require, for their use, an exact knowledge of the imperfections.  This i s not l i k e l y to be known except  in the simplest of cases. Howewer, several s i g n i f i c a n t results are available from these equations. of &  ( ), B  n  n  From equations (11.35), i t a n  i s seen that p  n  (q ) n  i s a function  d also a , b e t c . which are constants dependent s o l e l y on  the disk physical c h a r a c t e r i s t i c s .  If  stationary values of p or q are n n r  sought, the r e s u l t i n g expressions for ct and *3 are those given by n  equations (11.33).  n  That i s , the nodal l i n e s , determined by a  are located such that the frequencies p  and q  n  and3 , n  are stationary in value.  However, for a perfect d i s k , where a c  nn  = t> , a  n n  nn  n n  = t>  nn  and  = c = 0, equations (11.35) show that p„ = q„ regardless of the nn ^ n ^n r  values of  a  n  OB £ . n  A second result regarding a (11.33) and (11.34).  and  i s available from equations  C l e a r l y , f o r a perfect d i s k , both of these express-  ions for a and 3 are indeterminant, and hence the location of the nodes n n cannot be determined s o l e l y from the disk physical c h a r a c t e r i s t i c s .  In  the case where the disk i s imperfect but the nature of the imperfections i s such that t h e i r influence on the kinetic energy is much greater than on the s t r a i n energy, equations (11.33) may be written approximately a s : -c t a n a  n  +  (b - a) tan «  -c t a n 6  n  +  (a - b) tan B  2  2  n  + c  =0  + c  =0  n  When t h e i r influence on the s t r a i n energy predominates:  -c"tan  2 aa  n  -c t a n 3 2  n  +  (5 - a) tan a ' n  + c  =  0  +  (a - b) tan e  + c  =  0  n  In either case i t can be seen that the extent to which the anglesca^ and 3  n  d i f f e r i s greater for greater imperfections within the d i s k . The free vibrations of a disk with small imperfections may  be taken as:  w  =  I [ hi A  R  n  c  o  s  (  n  0  " n  c  o  s  ( p  nt  a  )  B  ) cos ( t t  "  (11.36) + A  p 2  R s i n (ne n ;  %  -  6  -, ) ]  where the differences between  « >3 and p , qJ in general depend on the n  n  n  n  nature and extent of the imperfections, and where a  and 3 n  are such n  that p and q are stationary in value, n n The two terms of equation (11.36), that i s :  w  ni  =  w,  IT2  A R m n  cos (ne - a- ) cos ( p t n' n  A R r\z n  sin (ne - ft ) n  v  -  V f  cos (p t  are referred to as the two configurations of the n - nodal diameter vibration.  Each is independent of the other, that i s , each i s a mode  of v i b r a t i o n , and the amplitudes, A a n d A nl  , and the time phases, e  n 2  and c , are dependent e n t i r e l y on the i n i t i a l n  It  n  conditions.  should be pointed out that there are very special cases where  imperfections may be present and either the phase angles a equal, or the frequencies p  n  and q  n  are equal.  n  and e  n  are  As an example of the  f i r s t case, consider a mass imperfection symmetric about some angular location on the disk. that point.  Any free vibration must also be symmetric about  This requires that a node of one configuration and an  anti-node of the other pass through the point.  Since the shapes of the  configurations are assumed to be unaltered, i t must be that a  = e^.  n  As an example of the second case, i f the imperfections are symmetric about m equally spaced locations on the disk, the requirement that p and q  n  be stationary in value results in p  multiple of  m.  n  n  = p^. when 2n i s not an integer  This second case is known as the Zenneck r u l e  1 3  Although e i t h e r the difference between a P  n  it  and q  n  n  and 3 , or between n  could be considered measures of the imperfection of a d i s k ,  is the l a t t e r that i s the accepted p r a c t i c e . While i t i s possible to experimentally determine the c o e f f i c i e n t s  a , b n  n  e t c . , this would be extremely d i f f i c u l t to do.  information i s the natural  frequencies p and q ^ n n M  Since the desired  and the location of the  M  nodes, these can be determined d i r e c t l y and with much greater ease by experimental methods which are described l a t e r . It  i s , however, desirable to obtain numerical results so that  theoretical predictions can be compared with experimental is found that theoretical  results.  It  results based on a perfect disk assumption  and adjusted according to information obtained experimentally from the free response of an imperfect disk predicts a forced response that i s consistent with experimental evidence. The following section gives an approximate method whereby nume r i c a l results for the free vibrations of a perfect non-rotating disk may be obtained.  II.3  An Approximate Method of Determining the Free Vibrations 6if a Non-rotating Disk The natural frequencies of a non-rotating disk may be determined  by several different approaches.  One, previously mentioned, requires  solving the equation of motion for the radial functions R ( r ) .  The  result of requiring the radial functions to conform to the boundary conditions y i e l d s the natural frequencies and the r e l a t i v e magnitudes of the c o e f f i c i e n t s of the Bessel and Modified Bessel functions.  Another possible approach i s that known as the method.  Rayleigh-Ritz  This i s the approach which w i l l be taken here.  In order to  use the Rayleigh-Ritz method, the shape of the vibration must be represented as a kinematically known c o e f f i c i e n t s .  - admissible orthogonal sequence with un-  Using this shape an expression for the natural  frequencies may be obtained.  This expression is a function of the  unknown c o e f f i c i e n t s , and minimizing the frequency with respect to these c o e f f i c i e n t s y i e l d s an upper bound on the fundamental  frequency  pf t i e free v i b r a t i o n . It it  i s not necessary to assume an angular shape, however, since  i s known to consist of nodal diameters which must be symmetrically  distributed around the disk. are unknown. functions i s :  It  i s the radial functions R (r) n  A kinematically •admissible representation of the  which radial  5  R (r) n  -  (  r-a  b'-a  Here the x^' are the unknown c o e f f i c i e n t s . The value of this function and i t s slope are both zero at the clamp as required.  If  imation to the radial  R  only the f i r s t two terms are taken, the approxfunction i s :  n ^  =  T G  )  x 2  +  T  i  x  3  r-a b-a  The range of x i s then 0 <x<1•  Since R (a) may be multiplied by a n  constant without affecting the result in any way, we can write: (11.37)  R  (x)  n  = x  + T X  2  3  Because the angular function i s known e x a c t l y , i f the frequency i s determined by use of the radial function given by equation (11.37) then minimized with respect to j , the resulting minimum i s an upper bound on the natural  frequency of that n nodal diameters, zero nodal c i r c l e s mode.  The expressions for the k i n e t i c and s t r a i n energies as previously given by equations(II.13) and (11.18) are: 2T = u $$ n n  +  yp  2U = 6 • n  +  p  2  u  p  $ n 2  n  $ n  6 n  f n  2  2  where the values of u and e are determined from the integrals of n n equations (11.14) and (11.19). If  damping i s neglected and a harmonic time function is assumed,  equating the maximum k i n e t i c and s t r a i n energies results in an expression of the form: "Eb " 2  *2  Ki f  +  K x 2  + K  a  The c o e f f i c i e n t s $a, S , and S3 result from the s t r a i n energy c a l c u l 2  ation whereas K i , K2 and K3 result from the k i n e t i c energy c a l c u l a t i o n . A l l c o e f f i c i e n t s are functions of the disk dimensions a and b, Poisson's r a t i o , and the integer n.  The value of t which minimizes p i s a v a i l n able in closed form in this case. It i s possible to express the minimum  0.1  0.2  0.3  a/b Figure 2 Non-dimensional  Frequency (parameter versus Glamping Ratio  the Rayleigh-Ritz method (dashed l i n e s ) . The Rayleigh-Ritz values of X are generally more accurate for low values of the clamping r a t i o a / b , with the exception of the n = 0 mode whose error i s e s s e n t i a l l y independent of the clamping r a t i o .  For  values of the clamping r a t i o greater than approximately 0.2, the use of only two terms for the"approximating function is obviously i n s u f f i c i e n t . The values of the c o e f f i c i e n t t which minimizes the frequencies P  n  are given in figure 3 .  Using these values of T i t i s possible to  approximate the shape of the free vibrations of a perfect disk.  However,  this approximation neglects two effects which must be included in order to arrive at a useful r e s u l t .  The f i r s t i s the previously discussed  effect of imperfections which are treated experimentally in a l a t e r chapter  .  The second e f f e c t i s that of the rotational  s t r e s s e s , which w i l l  now be discussed.  1.0  .  :  r—  .  Figure 3 Radial Funtion Coefficient versus Clamping Ratio  II.4  The Effects of Rotation The effects of rotation considered here are those  attributable  to the t e n s i l e membrane stresses which e x i s t when the disk i s r o t a t i n g . I n i t i a l l y the influence of these stresses on the free vibrations of a perfect disk are investigated, then an approximate method i s described by which t h i s influence on the free vibrations of an imperfect disk may be included. Rotating Membrane In order to introduce the effects of rotational convenient to consider a rotating disk with no flexural i s , a membrane.  stresses, i t  is  r i g i d i t y ; that  This approach w i l l also be useful in a l a t e r s e c t i o n ,  where an empirical relationship for the effects of rotation w i l l be presented. The equilibrium equation for a non-vibrating, rotating c i r c u l a r membrane is obtained from the free-body diagram shown in figure 4. Due to symmetry there are no shear s t r e s s e s , and the hoop stresses a are constant around the disk.  0  The body force B is of magnitude n rdm  where dm is the mass of the element.  2  The boundary conditions on the  stress d i s t r i b u t i o n are: 1.  zero radial stress at the clamp.  2.  zero radial stress at the free edge.  The f i r s t boundary condition is for what is referred to as a p a r t i a l clamp.  This type of clamp (or c o l l a r ) is the type commonly found in  practice.  It  allows a radial displacement of a l l points on the d i s k ,  and serves only to prevent a transverse displacement at the c o l l a r .  Figure 4 Membrane Stresses in a Rotating Membrane  With these boundary conditions the membrane stresses at any point may be determined.  They a r e :  r  2  + /k  1 8  i  (b  2  -  r ) 2  pto  2  '3  r  1  pur  b k 2/-J 2  where k i , k Poisson's  2  and k  ratio.  3  are functions of the disk and c o l l a r r a d i i and  It  is now assumed that the membrane i s deformed a r b i t r a r i l y  in the  transverse d i r e c t i o n , but that the stresses do not change, as shown in figure 5.  Figure 5 Deformed Element of a Rotating Membrane  As can be seen, on each element there is a net restoring force in the transverse d i r e c t i o n , due in the radial  case to both the variation  in  the radial stress and to the curvature in the radial d i r e c t i o n , and in the angular case due to the curvature only.  This net restoring force  may be equated to the rate of change of momentum of the element, ing in the equation of motion:  1 a / • 3w\ . 1 3 / 3w\ , 3§W _ 7 - 3 ? \ r 3 ? j FT 36 [°Q 3?j " W ~ ° n  r a  +  p  result-  The boundary conditions on the transverse displacement are: 1.  zero transverse displacement at the c o l l a r .  2.  f i n i t e transverse displacement at the free edge.  The method of solving this d i f f e r e n t i a l numerical results are a v a i l a b l e .  equation is very lengthy, but  The resulting expression for the  1 8  transverse vibrations are of the same form as given in equation (I1.9), but the radial  functions are not the same.  f o r a given disk the natural the disk rotational  '(11.38)  speed.  Pn  where V  2  =  V  A s i g n i f i c a n t result is that  frequencies are d i r e c t l y proportional  to  That i s :  n  ^  i s a function of n and the disk physical properties and dim-  n  ensions. Rotating Disk It  is possible to formulate the d i f f e r e n t i a l  ating disk considering both rotational information previously presented.  equation for a rot-  and bending stresses using  Equation ( I I . 1 ) , the plate bending  equation, i s the equilibrium equation for a non-rotating disk subjected to a transverse l o a d , which was taken to be the i n e r t i a l forces in the free vibrations case.  However, the stresses due to rotation were seen  to cause a net transverse load when the membrane was deformed. fore i t  There-  is possible to use the plate bending equation f o r a rotating d i s k ,  where the transverse load i s due to both the i n e r t i a l forces and the rotational  stresses.  The resulting equation of motion i s :  34  0  V  4  W  =  r  1 - 1  +  9r  1 2 R  39 \  9W 99  This equation was solved numerically by Eversman and Dodson and the results published in 1 9 6 9 .  19  The shape of the vibration i s again found  to consist of nodal c i r c l e s and diameters, but the radial functions are different from those in either the non-rotating disk or rotating membrane cases. As would be expected, the effects of the rotational  stresses are  dependent upon the clamping r a t i o a / b , the disk thickness and the r o t a t ional speed.  It  is found, however, that the influence of these stresses  on the disk natural  frequencies i s not very large for disks and clamping  ratios of the dimensions corresponding to those typical of c i r c u l a r saws. As an example, using results from the paper by Eversman and Dodson, an 18" diameter d i s k , 0.150" thick with an 8" diameter c o l l a r has a natural frequency of 198 Hz.for n = 1 when the disk is stationary, as compared to 237 Hz. at a rotational  speed of 5300 r.p.m.  This is an increase of only  19.5% which i s in fact an upper l i m i t since typical rotational  speeds of  c i r c u l a r saws are somewhat lower than 5300 r.p.m. Under these conditions, reasonably accurate results may be obtained by an approximate method based on Southwell's Theorem. Southwell's Theorem^ Southwell's Theorem i s general in nature, not pertaining s p e c i f i c a l l y to rotating d i s k s .  It  is derived d i r e c t l y from Raycleigh's Theorem,  35  which states that the natural  frequency of the fundamental mode of  v i b r a t i o n , as calculated from an assumed shape of d e f l e c t i o n , i s an upper bound for the exact value.  In order to develop Southwell's Theorem  as i t applies to rotating d i s k s , three deflection shapes must be defined: Sj . . . resulting from membrane and bending stresses SB . . . resulting from bending stresses only (no rotation) SM . . . resulting from membrane stresses only (no flexural  rigidity)  Assuming simple harmonic motion, the maximum s t r a i n and kinetic energies of the transverse vibration may be written as: Strain Energy  =  U (S)  Kinetic Energy = If  p  T (S)  2  damping i s neglected, the frequency i s given by:  For small d e f l e c t i o n s , the work done by the bending stresses is independent of that done by the membrane s t r e s s e s .  U (S ) T  Where Ug and  =  U  B  (S )  +  T  U  Therefore:  (S )  M  T  Aare the bending and membrane potential  frequency i s then: . p | _ UB (S ) + UM (S ) T  T  T  =  *  n  2  PB  +  P  (S ) T  D  2  M  energies.  The  Here p  B  and  are what the natural  assumed to vibrate  frequencies would be i f the disk was  i n the shape S j under the action of either the bending  or the membrane s t r e s s e s . That i s : P  B* '  -  T (S ) T  U 2  "  PM^  M < T> T ($ ) S  T  However, i f only bending stresses or only membrane stresses were present, the shape would be S or  respectively.  B  p  B  Therefore the exact values of  and p . a r e . g i v e n by: M  P !=  T  P  =  2 M  (S ) B  "M T  V  (  B  U  B  (  S  M '  (S ) M  By Rayleigh's Theorem, f o r the fundamental mode:  P  P  *  2 B  2 M  >  -  PB  P  M  2  2  Therefore the expression f o r the frequency p becomes: T  (".39)  p  2 T  z  P  2  +  p  2  Equation (11.39) i s Southwell's Theorem.  It  can ben seen that i f  the  potential energy of a freely vibrating body i s due to the action of two (or more) systems of stresses which act independently, then a lower bound on the natural  frequency of the fundamental mode may be determined by con-  sidering the effect of each system separately. Equation (11.39) may serve as the basis for an empirical  relation-  ship i f the equality i s taken to hold:  PT  2  P B  =  2  PM  +  2  However, equation (11.38) indicated that the relationship between p  2  and  was: P  =  2 M  V  "  2  Therefore: (11.40)  P  2 T  =  P + 2  B  Vn  2  In the previous example of the 0.150" d i s k , the increase in the n = 1 mode natural If  frequency in going from 0 to 5300 r.p.m. was 19.5%.  the exact value of V is u s e d ,  (232 Hz.) by equation (11.40). natural  1 8  the increase is found to be 17%  The actual difference in the 5300 r.p.m.  frequency by the two methods is only 2%. Evidently, for disks of the physical dimensions s i m i l a r to those  typical of c i r c u l a r saws, equation (11.40) provides a satisfactory approximation f o r the natural speed.  frequencies as a function of  rotational  While the c o e f f i c i e n t V  n  is available from the l i t e r a t u r e ,  there  are several factors in practice which can s i g n i f i c a n t l y a l t e r i t s value from that calculated t h e o r e t i c a l l y . quite e a s i l y experimentally.  However, i t may be approximated  Since f o r a real disk there w i l l l i k e l y be  two natural frequencies associated with the n nodal diameter free  vibrat-  i o n , there would be two relationships of the form of equation (11.40):  pf • »i * ve i  -  i  *  v  2  2  Experimentally determining the c o e f f i c i e n t s V observing the disk natural  n  necessitates  frequencies at various rotational  speeds.  However, observations w i l l generally be made from points stationary i n space, whereas the equations describing the transverse motion have been given with respect to a coordinate system fixed in the disk.  The  following section therefore describes the results of the transformation to a coordinate system fixed in space.  II.5  Disk Displacement with Respect to Rotating and Non-rotating Coordinates  The significance of the difference between the observed disk displacement as expressed in terms of the rotating or non-rotating coordinate systems may be i l l u s t r a t e d by considering the free vibration given by: w  n  =  A  n  R  n  (r)  cos (ne - e ) g  cos ( p t - t ) n  0  This vibration i s given with respect to the (r,e)  coordinate system  fixed in the disk.  = 0  consideration.  For s i m p l i c i t y , l e t e = 0, t 0  for the mode under  Then:  (11.41)  w  n  = A  n  R (r)  w  n  cos ne cos p t  p  Using a trigonometric i d e n t i t y ,  (11.42)  0  = ^ A n R (r) n  p  equation (11.41) can be written as:  [cos ( p t n  - ng ) + cos ( p t + ne ) ] p  Consider the f i r s t term of equation (11.42), which w i l l be denoted as w (1). n  That i s :  w. ( D n  = J*A  n  R (r) n  cos ( p t n  - ne )  This term, at any p a r t i c u l a r i n s t a n t , is identical mode shape given by equation (11.41). of equation (11.42).  in form to the  original  The same is true of the second term  These two terms each contain "nodes", the locations  of which, in general, are given by:  8(1)  = "V  (2K +  n  6(2)  =  -V -n  1 ]  +  l  2n  +  (2K +  1)IR  2n  k = 0,1,2  In a d d i t i o n , these "nodes" are moving in the disk at speeds of:  ...  n (11.43)  e (2)  =  Ifn n  We conclude therefore that the response of equation (11.41) may be considered equivalently as consisting of two shapes, each identical to the shape of the mode i t s e l f ,  but of half the amplitude, and which are t r a v e l -  l i n g in opposite directions around the disk. The above results are stated with respect to the (r,e) system.  If  corodinate  the disk i s r o t a t i n g , the response with respect to the non-  rotating coordinate system (r,y)  may be obtained with the use of the  previously given transformation:  e =  fit  - y  To avoid confusion, the response as observed from the stationary coordinate system w i l l be denoted u n  Substitution of the transformation into equat-  ion (11.42) y i e l d s : u  n  =  h  A  n  R  n ^  L^Pp*  + Y ~ nnt) + n  cos(p t - n n  Y  + nnt)]  The response w i l l be observed at some point in space, sayY= 0, in which case i t can be written as:  (11.44)  u  n  =  h A  n  R  n  (r)  [ cos( p - nfl)t + n  cos( p + nfl) n  t]  The frequencies seen by a space stationary observer, denoted by f , are therefore:  (11.45)  f  n  =  (  p  p  - nn  Equations (11.45) are very useful since they y i e l d a simple method of determining the p a r t i c u l a r mode shape associated with a resonance peak. The method i s to excite the disk randomly and record the observed natural frequencies at several rotational as a function of the rotational (neglecting imperfections).  speeds.  The observed frequencies f  speed n would appear-as shown on figure 6  It can be seen that at low rotational  speeds  the slopes of the lines are approximately +n.  Rotational Speed  n  Figure 6 Observed Resonance Frequencies versus Rotational Speed  This result may be interperted p h y s i c a l l y by considering the two components of a mode as given by equation (11.42).  When the disk i s not  rotating these two components, which are t r a v e l l i n g i n opposite d i r e c t ions attifehessame speed in the d i s k , are also t r a v e l l i n g at equal but opposite speeds i n space.  However? when the disk i s r o t a t i n g , the com9  ponent t r a v e l l i n g in the direction of r o t a t i o n , the "forward-travelling component", i s moving faster in space than when the disk i s stationary. Just the opposite i s true of the "backward-travelling component". A s i g n i f i c a n t phenomenon may be noticed i n figure 6. rotational  At some  speed one observed frequency of each mode, except f o r n = 0,  becomes zero.  From equation (II.45-a) f o r f  p *n If this value o f P  n  = 0, we have:  = nn  i s substituted into the expression f o r the speed of  the backward-travelling  component, equation ( I I . 4 3 - a ) , the result i s : 6(1)  =a  Since the positive e direction i s that opposite to the r o t a t i o n , i t can be seen that the backward-travelling S i m i l a r i l y , the forward-travelling djitskdspeed.  As the rotational  ward-travelling  component i s stationary in space.  component i s t r a v e l l i n g at twice the  speed i s increased even f u r t h e r , the back-  component actually begins to move forwards i n space.  The above description does not consider the effects of e i t h e r imperfections or rotational  stresses.  The existence of imperfections w i l l  double the number of observed frequencies, while the rotational stresses  w i l l increase t h e i r values in accordance with the relationship given by equation (11.40).  The observed frequencies of the n nodal diameter  vibration considering these two effects is then:  The lower branches of these observed frequencies become zero when:  Apparently when V  n  > n  2  there is no p o s s i b i l i t y of the observed frequency  becoming zero. Physically t h i s occurs when as the rotational  speed increases,  the speed in the disk of the backward t r a v e l l i n g component is increasing at a faster rate.  This phenomenon does in fact occur in practice as w i l l be  shown experimentally If  in a l a t e r  chapter.  an observed frequency does become zero, the rotational  which t h i s occurs i s very s i g n i f i c a n t when the forced response of disks is considered.  speed at rotating  This i s investigated in the following s e c t i o n .  44  II.6  Forced Response of a Disk  6  The transverse loading of a c i r c u l a r saw arises from the interaction with the work piece.  Since this transverse load i s dependent  on many f a c t o r s , primarily the wood i t s e l f , i s unknown.  i t s spectral density function  What i s known, however, i s that there is generally a s i g n i f i -  cant load at a very low frequency, usually taken to be zero.  The theory  of a rotating disk responding to a s t a t i c , space stationary point load i s referred to as the " c r i t i c a l speed theory".  This theory has been v e r i f i e d  with c i r c u l a r saws in t h e i r working environment?  4  The iload used in the following development i s taken to be P cosoot, since a s t a t i c load may then be taken as a s p e c i f i c case of a more general result. The energy method is used here to determine the forced response of imperfect d i s k s .  When the effects of imperfections are neglected the  response of a perfect disk i s obtained.  While i t  i s possible to determine  the forced response of a rotating disk and consider the non-rotating response as a special case, this is not the approach taken here.  The non-rotating  and rotating cases are treated separately, since one aspect of the problem, the damping, is s i g n i f i c a n t l y  different.  NNdn-rotatiiig Disk For an imperfect d i s k , the free vibration in the n nodal diameter modes i s taken to be:  (11.46)  =  w  n2  =  $  n  \  R  R  n  n  c  s  o  1  s  n  (  (  ne  n  e  j  _  a  " P> n  where $ and Y are the normal coordinates previously denoted by < > j and n n n r  n The kinetic and s t r a i n energies of the non-rotating d i s k ,  from equations (11.32) are 2T  n  =  A i + n on  B i n n  =  A„ :#2 + n n  B„ i n n  2  2  (11.47) 2U n n  2  The damping of this system i s known to be very small. Experimental r e s u l t s , given in a l a t e r chapter, verify that the damping may be neglected when considering the forced response except when the disk i s o s c i l l a t e d at very close to one of i t s natural provide a theoretical  frequencies.  basis for this experimental  In order to  r e s u l t , damping w i l l  be assumed to be viscous, in which case the energy dissipated may be expressed as:  (11.48) '  2D n  v  n  =  G ** ni n n i  +  G  <5 n 2  m  n o  In using Lagrange's equation i t i s necessary to determine the generalized force associated with the load P cosieaYt.  The generalized  force i s the quantity selected such that the pwoduct of this quantity and a virtual work done.  change in the generalized coordinate i s equivalent to the v i r t u a l Since each configuration from equation (11.46) behaves independ-  ently there w i l l be two generalized forces associated with the n nodal d i a meter response.  The load P cos cot i s located at ( r , e ) . p  p  The displacement  of the load due to a motion in the f i r s t configuration i s :  w , (r . e j ni p p  =  v  « R (r)) n n p' v  cos (ne„ - a p n' v  )  The work done <5iw" during a v i r t u a l change in $ ni  6¥„  ni  =  P  cos  cot  R  (r ) cos (ne„ - a ) n p' pp n'  6$  n  i s given by:  n  and s i m i l a r i l y f o r s V . 2  n2  J  -The generalized forces are therefore:  (11.49)  Mi = P R (r ) cos (ne - an) cos <ot n p'i p n  M  2  = P R  n  (r )  s i n (ne  p  -  p  PR)  cos  cot  Lagrange's equations are:  d /3T I I J T j » n/  3D W n  3U 3T n  A +  n  M =  M  n  i  (11.50) .  . 3 D dt I 3 ¥ - / h  +  n.  3U  u  n  which i s the standard form f o r a viscously damped forced system. for $  n  and ?  n  and substituting into equations (11.46) results i n :  Solving  47  P R  (11.51)  n  (r ) R (r) cos (ne - a ) cos (ne - a ) cos (cot-n ) n  R  n  n  w, ni 2  P R n  n  (r )  .a,  2  R (r) s i n ( n e - B ) s i n (ne - p ) cos (<ot-c )  p  n  p  n  n  n  2  B  (q  2  nN^n  -a, ) 2  2  G ^ + Y"_n la,2  where  pg  * A ^ j  and  n  = tan-HG.- 7 (P  C n  = tan-!(G  qJ {  2  n?  (q|  =  B /B n  n  Aj  - a, ) 2  /UB ) n -0,2)  These equations are the most general form of the n nodal diameter response of a non-rotating imperfect disk to a point load of magnitude P cos o,t. In general there i s a response in both configurations, 1 ft i If, however, the load i s located at e = f- a + £-)- or e = g / n , there p n n 2' p n w i l l be no response in configuration one or two, respectively.  These are  the locations of the free vibrations nodes of these configurations.  Re-  c a l l i n g that the nodes of one configuration are located approximately at the anti-nodes of the other (a ~ B ), the maximum amplitude of the resn  ponse of one configuration i s obtained when the amplitude of the other i s a minimum.  It* i s also apparent from equation (11.51) that i f the excitation frequency i s P (q )> the response w i l l be almost e n t i r e l y in n  n  configuration one (two), unless the load is applied at or near a node of that configuration. Both of the above observations are useful experimentally when the v a l i d i t y of equations (11.51) i s examined, and when the extent to which a disk i s imperfect i s to be determined. One check of equations (11.51) that does not require a physical experiment i s to observe the result when the disk i s considered to be perfect.  The vibration must be symmetric about the load irregardless of i t s  l o c a t i o n , with an anti-node located at the point of application of the load.  If the disk i s perfected  = 3 , A * B , p = q and G = G . The n n n n n n ni nz n nodal diameter response may then be written as: n  w, n  +  s i n (ne  - a ) s i n (ne n  As can be seen, the response i s as expected.  It  i s also interesting to  note that the response of a perfect disk i s e s s e n t i a l l y the same as that of an imperfect disk when the load i s applied at the node of one of the  configurations.  There would, however, be a small difference due to  changes in the values of A  n >  f e c t i o n s , and also because a  p p  n  and G caused by the presence of impern  i s only approximately equal to ^ f o r an  imperfect disk. Rotating Disk To f a c i l i t a t e  the description of the forced response of a  rotating d i s k , several terms must be defined: 1. fixed v i b r a t i o n :  This i s the usual form of the free vibration of  a d i s k , given, f o r example, by: w = A R s i n ne cos at n n n where a i s the frequency.  The nodes, located at e = 0,  are rotating with the disk. (11.52)  w  =  p  ^p.. n This vibration may be written as:  h A R [ s i n (ne - at) + s i n (ne + at)] n  p  where the forward and backward t r a v e l l i n g components are apparent. In non-rotating coordinates, this appears a s : (11.53)  u  =  n  h A  2. t r a v e l l i n g waves:  n  R [ s i n (nnt - ny - at) + s i n (nn t - ny + at)] n  These waves, backward and forward t r a v e l l i n g ,  are identical in form to the backward and forward t r a v e l l i n g components described above.  However, the two components of the fixed  vibration are of equal amplitude, whereas a single wave may e x i s t by i t s e l f .  (11.54) ' v  A backward t r a v e l l i n g wave i s given by:  w n  =  A R s i n (ne - at) n n  u„ n  =  A„ R„ s i n ( nnt - ny - at) n n  and a forward t r a v e l l i n g wave by:  (11.55)  w  u  n  n  =  =  A  A  n  R s i n (ne + at) n  n n R  3. steady d e f l e c t i o n :  s  i  n  ^  n f i t  "  n  y  +  a t  ^  If the rotational  speed  and the frequency  a  a are such that no. = a , the backward t r a v e l l i n g component or backward t r a v e l l i n g wave becomes stationary in space.  For example,  substituting a = no, into equation (11.54) f o r u y i e l d s : p  (11.56)  u n  =  -A R s i n n n n  Y  This i s not a function of time.  If the steady deflection i s a  result of a backward t r a v e l l i n g component becoming stationary i n space, the forward t r a v e l l i n g component w i l l be t r a v e l l i n g at twice the disk speed, y i e l d i n g an observed frequency of f  = 2na .  The method of determining the forced response of a rotating disk i s very similar to that f o r a non-rotating disk.  The k i n e t i c energy  of the transverse vibrations i s the same in both cases.  The potential  energy U i s of the same form, except the c o e f f i c i e n t s A and B are now n n n functions of ft due to the membrane stresses. The damping i s a p a r t i c u l a r i l y d i f f i c u l t problem since the vibration w i l l suffer s i g n i f i c a n t windage at high rotational though viscous damping has been assumed in the past,  6  '  1  5  speeds. A l -  i t i s unjust-  i f i e d s i n c e , f o r example, the windage suffered by a backward t r a v e l l i n g wave i s s i g n i f i c a n t l y less than that of a forward t r a v e l l i n g wave.  As  well as being a function of the transverse v e l o c i t y , the damping w i l l also be a function of the rotational  speed and the instantaneous amplitude  of the transverse displacement.  Except at near resonance, the  effect  of damping i s quantitative only; the nature of the response is the same as in the undamped case.  In the absence of a reasonable  theoretical  means of including the effects of damping i t w i l l be neglected in the theory that follows. The generalized forces may be determined in the same way as for a non-rotating disk except that the location e of the load v a r i e s . p  It  i s assumed that a = £ and this angle i s taken to be zero in the disk n n coordinate system.  Since both origins are coincident at t = 0 and the  load i s taken to be located at y = 0, i t s disk coordinates are ( r , p  The generalized forces are then:  which may be written as: MT. = h P R  N  (r )  M2 = h P R  N  (r ) [sin (10 + n f i ) t ^ s i n  nfi)t]  [cos («•;-+ nfi)t-t+cos ( «  p  (  nfi)t]  u  Applying Lagrange's equations, the response i s found to be:  (II.§7)  cos (a) + nfi)t w ni  =  h P [ R (r )/A ] n p" L  +  v  cos (to, [ p  2  n nJ  R (r)cdsne n ' v  nn)t  - (to- nfi) ] 2  ' [p  2  - (oi+ nfi) ] 2  fit).  V * ^ p B  P C  -  ( r  ]  R  n  (T) S i n n8  sin  (co + nO)t  Cq  - (to + nfi  2  ) ] 2  s i n (to'-'rifl)t [q  The natural  n  ) / B  - (ai- nn) ]  2  2  frequencies p  n  and q  n  here are functions of the rotational  speed n , since they are obtained from the r a t i o s :  Pn  =  q n  =  2  M  The c o e f f i c i e n t s A  n  R  A  /  S  ,n  B / B" n n  and B are not functions of the rotational p  i t can be seen from equations (11.57) that i f the natural  speed, so  frequencies are  known either from experimentation, or approximately from equation (11.40), that the only unknown e f f e c t of rotation When  is on the radial  functions R ( r ) . n  = 0, equations (11.57) reduce to the previously developed  expression f o r the response of a non-rotating disk subjected to a load P cos fflt located at an anti-node of configuration 1.  The resonance con-  dition is p = u . n However, when the disk i s rotating there are four possible resonance conditions, given by:  w  =  Pn" ^ n  (0  These resonance frequencies coincide with what have previously been referred to as the observed natural  frequencies of the disk.  Although  the response as predicted by equations (11.57) becomes i n f i n i t e at any of these resonance frequencies because damping has been neglected, i t can be seen that the response near resonance,when u ~ " P * nfi-or n  oo - q  n  ± nn, i s the shape and approximate frequency of the free response  of the d i s k . Since c i r c u l a r saw i n s t a b i l i t y i s known to be caused by a s t a t i c l o a d , the response to such a load w i l l now be discussed in d e t a i l . Under the action of a constant load P, the response, as given by equation (11.57) i s : (11.58)  w  n i  w„ n  2  = P F = P F„ n  cos ne cos nnt sin ne s i n nnt 2  R (r ) R (r) F„ = -"—P— "i A [ p - (nn) ] n n ' n  where  n  2  2  L f  and  v  R (r ) R (r) _ = P , -_ n B [q - (nn) ] n  F  n  2  2  2  n  Resonance occurs when p = nn or q = nn. n n  If a non-rotating disk i s  r  3  subjected to a s t a t i c l o a d , and the rotational  speed i s then increased,  the f i r s t speed at which resonance occurs i s known as the c r i t i c a l speed. In general, i t can be seen that this type of resonance occurs at the same rotational  speed at which the backward t r a v e l l i n g component of a  free vibration would be stationary in space; that i s , when f If the disk i s perfect, p  n  = q  n  and A = B . n  n  = 0.  In this case  the response i s :  w n  = w ni  +w  nz  = P F [cos ne cos nnt + sin nesin nnt] n = P F  n  [cos (ne -  nnt)]  This i s the expression for a backward t r a v e l l i n g wave (see equation 11.54). Substituting e = ntv--Y>  (11.59)  u  p  t  n  e  = P F  response in space-stationary coordinates i s :  n  oos ny  This backward t r a v e l l i n g wave, being fixed in space, is what has been defined as a steady d e f l e c t i o n . speed at any rotational  It  i s t r a v e l l i n g in the disk but i t s  speed i s such that i t appears fiixed in space.  Resonance occurs when the speed of this wave in the disk becomes equal to the wave speed of the backward t r a v e l l i n g component of the free v i b ration in this mode.  This i s the usual condition for resonance; the  system i s forced to respond at the rate at which i t  does f r e e l y .  When the effects of imperfections are included, the response to a s t a t i c load is somewhat more complicated.  In this case, the total  n nodal diameter response from equation (11.58) i s :  (11.60)  w = P (F - F ) cos ne cos nnt + P F cos (ne - nnt) n nj n n 2  2  The f i r s t term here is a fixed vibration of frequency nn and the second term is a backward t r a v e l l i n g wave stationary in space.  The frequency  of the fixed vibration i s such that i t s backward t r a v e l l i n g component is also stationary in space, thus contributing to the steady d e f l e c t i o n . Equation (11.60) may therefore be written as:  w„ = h P ( F - F ) cos (ne + nftt) n ni ri2 n  + h P (F„  + F  ) cos (ne - nnt)  Toca space-stationary observer this appears as:  u = h P(F - F n n^  (11.61)  n  n  + h P(F  ua  ) cos (2nnt -  + F  ) cos  n) Y  ny  The observed frequency of the forward t r a v e l l i n g component i s 2no,, as previously discussed. If  p  i s considered to be the lower of the two natural  n  fre-  quencies of the n nodal diameter configurations, when no i s less than P , which i s the range of 'rotational n  speeds of interest here, equation  (11.61) may be written as:  u = h P n 2  1 -  \ /Bn " B \q* - (no)'  F  cos (2nftt -  n  + %PP B n  v^n  (nn)  :  F^ cos  ny  ny)  As can be seen, the amplitude of the forward t r a v e l l i n g component r e l a t i v e to that of the steady deflection increases as the c r i t i c a l speed i s approached.  Although the amplitude of the forward t r a v e l l i n g  component never exceeds that of the steady v i b r a t i o n , this theory, which neglects damping, predicts that they are of very nearly the same amplitude close to the c r i t i c a l speed.  Recalling that a vibration fixed in  the disk i s composed of forward and backward t r a v e l l i n g components of equal amplitude, at close to the c r i t i c a l speed the response can be seen to consist almost e n t i r e l y of a -fixed v i b r a t i o n .  This contrasts with the  response of a perfect d i s k , where only a steady deflection is present. It  must be remembered, however, that a fixed Vibration is  moving in space at the disk rotational ion is stationary in space.  It  speed, whereas the steady d e f l e c t -  i s expected therefore that, due to windage  the amplitude of the fixed vibration r e l a t i v e to that of the steady d e f l e c t ion w i l l be s i g n i f i c a n t l y less than that predicted by the above theory. This i s confirmed in a l a t e r chapter which offers experimental  verification  of the theory which has been presented here. II.7  Comments on the Theory and i t s Applications  The responses of non-rotating and rotating imperfect disks to a transverse point l o a d , given by equations (11.51) and (11.57), were determined in order to predict the behaviour of a c i r c u l a r saw in i t s operating environment. theoretical  Although such a model neglects many f a c t o r s , the  results obtained have been v e r i f i e d in sawing operations as  being f a i r l y accurate in predicting certain aspects of a saw's behaviour, most s i g n i f i c a n t l y the c r i t i c a l s p e e d .  24  One of the major advantages of  of this model i s that those factors which are important in determining the saw's behaviour are e a s i l y i d e n t i f i e d .  Several of the areas of  current research on saw vibrations are based on concepts which were d i s cussed in the rotating disk theory. Current Research One of the most common methods of reducing the transverse motions of c i r c u l a r saws i s known as tensioning, having been used in some form ffior approximately.100 years.  Tensioning i s the process of  the saw membrane stresses to increase the c r i t i c a l The membrane stresses for an i n i t i a l l y due to rotation only.  The maximum rotational  altering  speed of the saw. stress free saw are  speed of the saw must be  less than that speed at which a s t a t i c load causes resonance.  When rotat-  ional ly induced membrane stresses only are present, this may be approximated by:  Q  2  This speed is dependent on the c o e f f i c i e n t V which i s a measure of the n  influence of the rotational  stresses.  It  can be seen that i f this i n f l u -  ence could be increased, the result would be a higher c r i t i c a l speed. One method of tensioning involves p l a s t i c a l l y deforming the saw in compression in a narrow annul us, indicated by the dashed line figure  in  7. When the saw i s not r o t a t i n g , the radial stresses w i l l be com-  pressive throughout, while the hoop stresses w i l l be compressive on the inner portion of the saw, and t e n s i l e on the outer portion as shown. These  i n i t i a l stresses are such that when the saw is rotating at i t s  rated  speed a l l membrane stresses are t e n s i l e , although not of the same value as when the saw is not tensioned.  Since the magnitude of the  stress is reduced by this process, those modes whose membrane energies of deformation are due primarily to the radial reduced natural  frequencies.  number of nodal diameters.  radial potential  stress w i l l have  These modes are those consisting of a low However, due to the increase in the  tensile  hoop stress in the outer portion of the saw, those modes consisting of a larger number of nodal diameters w i l l have an increase in t h e i r frequencies.  natural  With the correct amount of tension i t i s possible to i n -  crease the natural  frequency of that mode which determines the  speed of the saw.  Figure 7  Membrane Stresses in a Non-rotating Tensioned Disk  critical  The same result may be obtained by another method which i s less common.  If  the saw i s heated at the c o l l a r a thermal gradient w i l l  e x i s t where the rim is at a lower temperature than the inner portion. This gradient may be selected such that the membrane stresses are s i m i l a r to those described above,  also resulting in an increase in the c r i t i c a l  speed. The effects of tensioning may be determined numerically for an idealized saw.  1  0  ,  1  1  In p r a c t i c e , however, there are unknown factors  which influence the optimum amount of induced tension, such as the heat generated by c u t t i n g .  It  i s necessary that any method by which the des-  ired amount of tension i s determined be f l e x i b l e to allow for the changing conditions of the operating environment. A second major d i r e c t i o n of research, a l t e r i n g saw geometries, has recently become of interest t h e o r e t i c a l l y due to the a v a i l a b i l i t y such techniques as the f i n i t e element method.  of  One of the most common  departures of saw geometries from those of perfect disks i s the presence of holes and radial s l o t s .  Slots are p a r t i c u l a r l y common for two reasons.  F i r s t l y , they i n h i b i t the motion of waves t r a v e l l i n g around the saw, and secondly, the compressive hoop stresses at the rim resulting from the heat generated during cutting are reduced because the saw can expand into the slots. Although holes and slots cannot be considered small imperfecti o n s , they have the same effect as small imperfections on the amplitude of the steady d e f l e c t i o n .  It  is p o s s i b l e , with the proper selection of holes  and s l o t s , to reduce the amplitude of this d e f l e c t i o n .  An approach cur-  rently being persued is the selection of the number, size and locations of  holes and slots to optimize this amplitude  reduction. ' 1 3  1 4  As i s the case with tensioning, the effects of holes and slots on the behaviour of an idealized saw can be determined numerically. Although this predicted response can be v e r i f i e d under controlled experimental conditions, v e r i f i c a t i o n in the f i e l d , which i s the ultimate t e s t , has in the past proven d i f f i c u l t to achieve. Saw Behaviour i h i t s Operatirig Eiivirdiiment It  is d i f f i c u l t  to evaluate the performance of a c i r c u l a r saw  in i t s operating environment by monitoring i t s transverse displacement. A more direct and simpler method i s to observe the quality of the wood cut by the saw. critical  This is in fact the method i n i t i a l l y  speed t h e o r y .  used to verify  the  2 4  It may be d e s i r a b l e , however, to monitor the saw's displacement.  This displacement may, f o r example, be used as an input for an  automatic control s c h e m e . ' 1 5  1 6  The d i f f i c u l t i e s encountered can be seen  by considering the response predicted by the imperfect disk theory.  As  previously stated, the load is known to consist predominantly of a large s t a t i c component, which results in a space-stationary backward t r a v e l l i n g wave and a vibration fixed in the saw.  In addition to this response, due  to the very low damping of the system there is the p o s s i b i l i t y of an observable response at the natural  frequencies due to random e x c i t a t i o n .  Considering these two types of responses, the observed frequencies as a function of rotational  speed would appear as shown in figure 8. Here the  responses in the n •? 1 and n = 2 modes only are shown for c l a r i t y . cluding the zero frequency responses, at any rotational  In-  speed the total  response consists of responses at eleven different frequences.  Since there  61  w i l l be responses in other modes as w e l l , i t can be seen that evaluating the behaviour of a saw at a p a r t i c u l a r rotational  speed would be very  difficult. Figure 8  Rotational Speed  n  Saw Response versus Rotational Speed  ;Under laboratory conditions however, the situation is somewhat different.  In this case the load can be c a r e f u l l y controlled and measured,  and the rotational  speed set at any desired l e v e l .  It  is then possible to  investigate the v a l i d i t y of a theory under conditions more closely resembl i n g those on which the theory was based. The next chapter describes experiments which were conducted to verify the theory which was developed in this chapter.  Ill  Experimental V e r i f i c a t i o n of the Theory of Rotating Disks Experiments were devised to v e r i f y the theory presented in the  previous chapter.  The outcomes of this theory were the forced responses  of non-rotating and rotating d i s k s , given by equations (11.51) and (11.57). Before these results could be v e r i f i e d , i t was necessary to conduct several preliminary experiments.  For each r e s u l t presented both the  experimental  and the theoretical means used to obtain the r e s u l t are described in d e t a i l . III.l  Experimental Equipment The disks used to obtain the experimental results were prepared  from steel blanks from which c i r c u l a r saws are manufactured.  Their dimen-  sions were: thickness  diameter  disk  A  0.050"  18.4"  disk  B  0.085"  18.2"  disk  C  0.085"  18.2"  disk  D  0.050"  18.3"  Although tests were conducted with a l l d i s k s , only the results f o r disks C and D are given here since the other disks simply supported the results obtained with these two.  A l l disks possessed no large imperfections and  were complete except for a central hole approximately one inch in diameter. The c o l l a r s used were three inches in diameter. The c o l l a r assembly was p r e s s - f i t t e d onto the rotor shaft of a one-half horsepower Reliance D.C, motor.  The rotational  speed was  infinitely  variable over a range of 0 to 2500 r . p . m . , with the speed being monitored by  a Dynamics Research Corp. incremental shaft encoder and displayed digitally. The transverse motion of the disks was measured with spacestationary Bentley eddy current proximity sensors, and t h e i r accompanying power supply and d r i v e r s . most tests used only one. in figure 9.  Two sensors were c a l i b r a t e d , although  The c a l i b r a t i o n curve of this sensor is shown  This curve was obtained with the use of a dial gauge.  •2.'  4  6  Output Voltage  8 (volts)  Figure 9 Sensor Calibration Curve  10  The proximity sensors were positioned  approximately  0.10"  from the surface of the d i s k , giving a l i n e a r response range of nearly ± 0.050".  It was possible to position the sensors at any desired l o c -  ation in space. A Spectral Dynamics Spectrascope II  spectral analyzer was used  to measure the response spectrum, and to ensure an accurate load frequency when the disk was excited s i n u s o i d a l l y .  This is a single channel analyzer.  It was used in close conjunction with a Telequipment model DM64 dual channel o s c i l l o s c o p e .  The two channels of the oscilloscope allowed a d i r e c t  observation of the phase differences between either the load and the r e s ponse, or between the responses at two different points. Disk excitation was provided by an electromagnet fixed in space at a distance of approximately one-quarter inch from the surface of the d i s k , at a radius of seven inches.  Power to the electromagnet was supplied  by a power a m p l i f i e r , a D.C. power supply or both.  The input to the power  amplifier was an A . C . or random signal provided by a Bruel and Kjaer Type 1024 Sine-Random Generator. The electromagnet was secured to a shaft which was inserted in a ball bushing allowing rotational  and axial freedom.  However, axial mot-  ion was prevented by securing the end of the electromagnet shaft to a Bruel and Kjaer P i e z o e l e c t r i c Force Transducer Type 8200, which was then connected to a heavy support.  The output from this transducer was directed  through a charge amplifier from which the magnitude of the force was continously available as a voltage.  Calibration was obtained with the use of  weights which were accurately weighed.  The free responses of the disks were not investigated experimenta l l y because the presence of damping, even though very s m a l l , created difficulties  in obtaining accurate measurements.  A l l results given here  are f o r the forced responses o f the d i s k s . It was desirable to apply three types o f loads to the d i s k s ; a s t a t i c l o a d , a sinusoidal load and a random load.  The s t a t i c load was  achieved simply by applying a D.C. voltage to the electromagnet. A s i n u soidal l o a d , however, cannot be obtained with the use of only one e l e c t r o magnet.  When a load such as B cosoot was desired, this input was added to  a D.C. voltage of magnitude P, y i e l d i n g a resultant load o f P(l+  cosut).  Since the applied loads were o f a magnitude such that the response was within the linear range, i t was p o s s i b l e , i n some cases, to obtain the response to a sinusoidal load by neglecting the D.C. component of the proximity sensor output.  Applying a random load presents the same problem  as does a siinusoidal l o a d , but since a random load was used only for identi f y i n g resonance frequencies.,this signal was not o f f - s e t by a D.C. voltage.  III.2  Forced Response o f a Non-rotating Disk The response o f a non*rotating disk i n i t s n nodal diameter modes  to a load P costot i s given by equation (11.51).  (III.l)  W  =  s  w n  ni  +  The total response i s :  V  In this section i t w i l l be shown that the observed amplitudes and shapes of the response of a disk agree very c l o s e l y with those determined ically.  theoret-  The f i r s t step in obtaining this v e r i f i c a t i o n was to accurately determine the disk natural  frequencies since the amplitude of the response  i s extremely sensitive to the differences between the excitation and the natural  frequencies.  In order to determine the natural  frequency frequencies,  the disk was excited randomly while i t was not rotating and the resonance peaks displayed on the spectral analyzer were recorded. By rotating the disk slowly so that the influence of the rotational  stresses was minimal,  the  values of n were determined by noting the rate of change of the observed frequencies with respect to the rotational  speed, as previously discussed.  The results are shown below for disk C.  Nodal Diameters  Disk Natural Experimental  0 1  54.0 34.8,  35.2  Frequencies  (Hz)  Theoretical 48.8 40.1  2  61.4  62.2  3  125.0  124.7  The n = 1 configurations of this disk were the only two that displayed an observable difference in their natural  frequencies.  The knowledge  of these two frequencies may be used to determine the location of the nodes of these configurations. The Location of the Nodes If  the disk i s excited at a frequency u> = p^, and the sensor is  located at the point of application of the l o a d , the responses in the n = 1 configurations are:  67  P fR w  1  = n  i  v  (r  ) 1  2  cos (ee  -  sin (e  -  2  p'J  p  « )  1'  cos  (wt  -  n  )  1'  G p 11 i H  P  w 12  lX(r )]i  2  p  ^7  (  q  i  -  p  i  )  2  +  p  ( ^  2  cos (at  p  2  -  ^)  '  The response in these two configurations w i l l be much larger than in any others, and the total response of the disk to this load may be approximated as w = w  u  + w  .  j 2  In addition, i t  can be seen t h a t , except when the load  is applied at or near the node of configuration one, the response in t h i s configuration w i l l be much larger than in the other. The natural and 35.2 Hz.  frequencies, p ' - a n d ^ , of disk C were seen to be 34.8  The frequency of excitation was set i n i t i a l l y  at 34.8 H z . , and  the load and proximity sensor were located at some arbitrary angle,wwith the amplitude of the response being noted. around the d i s k .  The excitation  the procedure was repeated.  This was done at 15°  intervals  frequency was then changed to 35.2 Hz. and  The results are shown in figure  10.  The o r i g i n of the coordinate system has been selected such that the anti-node of configuration two i s located at e = 0.  As expected, the  node of configuration one is located approximately halfway between that of configuration two. nearly  cos (e-a ) 2  n  In a d d i t i o n , the shapes of the two curves are very and  sin (e-e ). 2  n  The deviation from these shapes is due  to several f a c t o r s ; the load covered a f i n i t e area rather than being a point l o a d , there were responses in modes other than the one being resonated,  and any change in the shape of the response due to imperfections was neglected  theoretically.  oo = q  co  = p.  180 Location of Load and Sensor Figure 10 Amplitude of the Response versus Location of the Load  In addition to the location of the nodes, in order to the theoretical  response of the disk i t  c o e f f i c i e n t s G cmd G ni  n 2  .  i s necessary to know the d i s s i p a t i o n  These could be determined approximately  equations (III.2) and the information presented in practical problems a r i s e . radial  calculate  from  figure 10, but several  Most s i g n i f i c a n t l y , there is a dependence on the  functions and these functions have yet to be v e r i f i e d .  There is also  a dependence on the magnitude of the load P, and i t s value at resonance must be so small to prevent a non-linear response that i t measured.  cannot be accurately  Both of these problems are avoided by the method described below.  Damping Coefficient If  it  i s assumed that  a  =s  2  l s  and i f both the load and the  sensor are located at a node of configuration two, the one nodal diameter response may be written as:  Wi  P [Ri(r )]  =  p  AiP:  Here  cos(o)t-ni)  /re  22  +  Pr  5 i s the damping f a c t o r ; the r a t i o of the actual damping  c o e f f i c i e n t of the disk to i t s c r i t i c a l  damping c o e f f i c i e n t .  2 2  The response w i l l be predominantly in configuration one i f  the  excitation frequency i s close to p j , and the load i s applied at an a n t i node of this configuration.  It  is possible to non-dimensionalize this  response and the excitation frequency.  If  the amplitude of the response  is Wj-* when to = UJ^, the amplitude as a function of the excitation frequency can be written as:  (III.3)  The experimental  results are shown in figure 11 (dashed l i n e s ) .  parison purposes the theoretical also plotted ( s o l i d l i n e s ) .  results for  k= 0.00,  0.01  For com-  and 0.02  are  0.95  1.00  Excitation Frequency  1.05 w/p  Figure 11 Relative Amplitude versus Excitation Frequency  It critical.  can be seen that the damping c o e f f i c i e n t is less than 1% of The response at frequencies above the resonance frequency is  larger than expected due to responses in modes other than the one considered t h e o r e t i c a l l y . This experiment was also performed in- the n = 2 mode. assumed that the disk was perfect since the natural were indistinguishable.  Here i t was  frequencies p  2  and q  2  The damping c o e f f i c i e n t was again found to be less  than 1% of c r i t i c a l . There are two effects of damping with which we are concerned.  The  f i r s t i s i t s influence on the modal admittance.  It  can be shown that the  difference in the admittance when calculated for  £ = 0.00 as compared to  71  5 = 0 . 0 1 . i s less than 4 % unless the excitation of the natural  frequency i s within 5%  frequency.  The second effect of damping i s the "smoothing out" of the phase change as the excitation frequency i s varied from lower to higher than the natural  frequency.  The phase angle between the load and the response i s :  Unless the excitation frequency i s within 5 % of the natural difference  frequency, the  in phase for the two cases g = 0 . 0 0 and 5 = 0 . 0 1 i s less than  11°.  Since neither an amplitude error of 4% nor a phase angle error of 1 1 ° are s i g n i f i c a n t , the system damping may be neglected t h e o r e t i c a l l y when the excitation frequency is not within 5 % of a natural When damping i s neglected, i t expressions for the radial  frequency of the disk.  is possible to formulate  approximate  and angular p r o f i l e s of the disk using the rad-  i a l functions obtained by the Rayleigh-Ritz method. Radial and Angular Deflection  Profiles  When a disk i s excited by a load B cos ut, the total response i s the sum of the responses of each mode.  This requires that the location of  the nodes of each configuration be known. that <x = ^approximately, n  any difference between p  n  It  has been shown, however,  and i f the excitation and q  n  frequency 10 i s such that  is negligible the theoretical  disk response is equivalent to that of a perfect disk.  imperfect  This was the ap -  proach taken when the radial disk C.  and angular p r o f i l e s were determined for  The excitation frequencies were 0 Hz and 90 Hz, which are s u i t -  able for neglecting both the differences in the natural  frequencies of  the configurations and the effects of damping. In order to obtain a radial p r o f i l e both the load and the sensor were located at 6 = 0.  The sensor was positioned at various values of r  where the amplitude of the response was noted. The maximum amplitude of the deflection as a function of the radius i s , from equation (11.51): Rp (r_) R (r) PA ( P -co ) 0  w(r)  =  -2  2  0  2  Ri (r ) R (r) PA (pf -co ) x  +  +  —  2  x  For to = 0 Hz, the predominant response was t h e o r e t i c a l l y  found  to be in the n = 1 mode, whereas at to = 90 Hz the n = 2 mode predominated. In both cases the contribution of the n = 4 mode was i n s i g n i f i c a n t . These p r o f i l e s are shown in figures 12 and 13, where the s'olid lines are the theoretical  p r o f i l e s and the dashed lines are those measured.  Figure 12 Radial P r o f i l e at 0 Hz.  (Disk C)  • 002 V  P = 0.51 l b . = 90 Hz.  io  .001"j  2  4  6  Radius  8  (inches)  Figure 13 Radial P r o f i l e at 90 Hz.  It  ((Disk C)  can be seen that although the load for the  u = 90 Hz case  is approximately one half that of the u = 0 Hz case, the response duced by roughly 80%.  i s re-  This i s due to the much greater s t r a i n energy per  unit deflection for the higher modes. The angular p r o f i l e was measured for a s t a t i c load (0 Hz).  The  sensor was located at r = 8.5" and measurements were taken at 30° intervals around the disk.  The theoretical  angular p r o f i l e is obtained from equation  (11.51) which, neglecting damping i s :  w(e)  =  PRo ( r ) p  A  To i l l u s t r a t e  (Pi| -  Ai (p\ - w )  *> ) 2  2  the relative magnitudes of the responses in the various  modes, the theoretical figure 14.  0  (8.5)  RQ  responses for w  0  , w  2  , w  2  and w  3  are shown in  The predominance of the n = 1 mode can be observed.  .0050" .0025" i  180  240  300  60  .0025"  T  .0025"  T  Figure 14 Theoretical Angular P r o f i l e s at 0 Hz.  120  180  The total theoretical p r o f i l e w i s plotted in figure 15 ( s o l i d lines) along with that measured (dashed 1ine).  .010"  T  Figure 15 Angular P r o f i l e at 0 Hz.  Although the experimental  (Disk C)  results for one disk only have been  presented here, these experiments were performed on the other disks as w e l l , y i e l d i n g results which also agreed very closely with the theoretical predictions.  III.3  Forced Response of a Rotating Disk The n nodal diameter forced response of a rotating disk is given  by equation (11.57). interest, i t  Because the response to a s t a t i c load is of p a r t i c u l a r  is this type of load for which the experimental results are pre-  sented. With thas load the theoretical  response i s given by equation (11.58).  As in the case of the non-rotating d i s k , i t  i s essential to have accurate  values of the disk natural  frequencies, which are now a function of the  rotational speed. Observed Frequencies as a Function of Rotational Speed There are two methods which may be used to determine the disk natural  frequencies as a function of the rotational  speed.  The f i r s t  method i s by random e x c i t a t i o n , as was done for the non-rotating disk. This ws the method that was used to obtain the results presented here. The second method i s somewhat more time consuming but i s necessary under certain conditions.  If,  for example, the disk either contained s l o t s ,  was not completely f l a t or was not running t o t a l l y in a plane perpendicular to the shaft a s i g n i f i c a n t response would be observed at integer multiples of the disk rotational  speed.  This would make the resonance peaks caused  by random excitation d i f f i c u l t to i d e n t i f y . rotational  However, i f at a p a r t i c u l a r  speed a sinusoidal load was applied and the frequency swept  over the range of i n t e r e s t ,  from equation (11.57) i t can be seen that a  large response w i l l occur when the excitation frequency corresponds to an observed natural  frequency at that rotational speed.  By random excitation the graph of figure 16 was obtained for 9isk C.  The difference between the natural  frequencies of the two n = 1  configurations i s c l e a r l y observable at a l l  rotational speeds.  An approximate relationship for the natural  frequencies was  previously given as: P ni 2  =  For this d i s k , the values of V pectively.  1 8  Pn nB  +  2  H  l 9  y  V n  ft  V > and V 2  2  3  are 1.34,  2.91  and 5.90 res-  At 2500 r.p.m. this approximation y i e l d s frequencies which  are s i g n i f i c a n t l y higher than those measured, being in error by .6.1%, 7.6% and 6.5% for the one, two and three nodal diameter modes. of these errors i s attributable at 2500 r.p.m. the potential  The size  to the fact that for this disk and c o l l a r  energy of the membrane stresses i s not small  in relation to that of the bending s t r e s s e s , which i s a requirement for an accurate approximation.  —i  500  i  i  1500  2000  1  1000  Rotational Speed n  (rpm)  Figure 16 Observed Resonance Frequencies versus Rotational Speed  '.  i  2500  It was stated in Section II.4  that i f V > n , there would be 2  no p o s s i b i l i t y of an observed frequency of that mode becoming zero. this case,  1 and i t  In  can be seen from figure 16 that the lower branch  of the observed frequencies of this mode w i l l not intersect the horizontal  axis.  For this disk the c r i t i c a l  mode, since i t  speed is determined by the n = 2  is this mode whose lower observed frequency f i r s t becomes  zero. Response to a Stati c Load In the space-stationary coordinate system the n nodal response to a s t a t i c load located aty=  diameter  0 i s given by equation (11.61).  For a perfect disk the total deflection at the load i s :  u(n  ,  - z  E n (r n  P  R  A  It  2  p  n  [p  2  -  (nn)2]  i s not immediately clear from this expression whether the rate of  change of the amplitude with respect to the rotational or negative at any p a r t i c u l a r speed.  speed is positive  This is because the natural  fre-  quencies and to a lesser extent, the radial functions are dependent on the rotational speed. This response was measured for disk C f o r rotational speeds varying from 0 to 2500 r.p.m. assuming the radial  A theoretical  functions were independent of the rotational speed,  and by using the natural  frequencies given in figure 16. The experimental  results (dashed l i n e ) and theoretical figure  17.  response was calculated by  results ( s o l i d line) are shown in  F i g u r e 17 Displacement v e r s u s R o t a t i o n a l Speed  It  ( D i s k C)  can be seen t h a t up t o a p p r o x i m a t e l y 1500 r . p . m .  d e f l e c t i o n remains f a i r l y  constant.  the maximum  A l t h o u g h n o t shown on f i g u r e 1 7 ,  this  r e s u l t s from the f a c t t h a t the d e f l e c t i o n s i n the n = 0 and n = 1 modes are d e c r e a s i n g w h i l e those i n the n = 2 and n = 3 modes are i n c r e a s i n g . The a n g u l a r p r o f i l e may a l s o be determined from e q u a t i o n  (11.61).  For a p e r f e c t d i s k w i t h the l o a d l o c a t e d a t y = 0 and the s e n s o r l o c a t e d a t r = 8.5", it  i s given by:  u{Q,y)  =  I  P  A  R  n  n V  R  (  [p  2  -  n ( 8 . S J cos ny (no) ] 2  T h i s p r o f i l e was measured f o r d i s k C a t 2400 r . p . m . shown i n f i g u r e  18.  and the r e s u l t s are  Figure 18 Angular P r o f i l e at 2400 RPM  (Disk C)  The predominance of the n = 2 mode is observable.  This p r o f i l e  may be compared with figure 15, which is f o r the same disk and l o a d , except at zero rotation speed.  In the non-rotating case, the " t i l t i n g mode", that  i s , the n = 1 mode, predominates. The theoretical fect.  calculations done assumed that this disk was per-  While this is not s t r i c t l y t r u e , at rotational  2500 r.p.m. the small differences between the natural configurations i s inconsequential.  speeds of less than frequencies of the  One of the effects of imperfections  the existence of a 2nfi. observed frequency. ected for this disk at these rotational  is  No such response could be det-  speeds.  These same experiments were conducted with disk D which i s much thinner than disk C.  The steady deflection as a function of the  speed is shown in figure  19, where  rotational  the dashed l i n e is the experimental  result and the s o l i d l i n e is that determined t h e o r e t i c a l l y .  CU  .010  4->  c  CD E CU o  .005  h  <a  P = 0.30  r—  lb.  Q -  500  1000  1500  Rotational Speed n  2000  2500  (rpm)  Figure 19 Displacement versus Rotational Speed  It  (Disk D)  can be seen that up to approximately 1200 r.p.m. the total  deflection decreases with increasing rotational  speed.  Since this disk  i s quite thin (0.050"), the increase in the n = 0 and n = 1 natural  fre-  quencies due to an increase in the wotaitnonal stresses reduces the admittances of these modes to a greater extent than those of the n = 2 and n = 3 modes are increasing, resulting in a net decrease in the amplitude of the steady d e f l e c t i o n .  Above 1200 r.p.m. the response in the n = 2 mode begins  to predominate, and near the c r i t i c a l speed of 2020 rpp.m., the angular  p r o f i l e i s very s i m i l a r to that shown in figure 18, since the c r i t i c a l speed of this disk i s also due to resonance of the n = 2 mode. The angular p r o f i l e was measured at 1955 r.p.m. and the d e f l e c t ion under the load was found to be 0.014".  However, at t h i s speed a  s i g n i f i c a n t vibration at approximately 130 H was observed, with an ampz  litude of 0.0035". rotational  The frequency of this vibration i s four times the disk  speed.  That t h i s vibration i s due to the forward t r a v e l l i n g component of a fixed vibration i s readily v e r i f i e d by the use of two proximity sensors, located at  y^ and  Yg in the space-fixed coordinate system.  From equation  (11.61) the forward t r a v e l l i n g component as measured by each proximity sensor i s given by:  unA  unB  These two signals may be observed simultaneously on a dual-beam o s c i l l o s c o p e , appearing as shown in figure 20. Here the sensor B i s assumed to be in the positive y direction with respect to sensor A. difference between u is  2I  [  &B "  Y  A ' ]  n f l  Since the time t  and u  n R  d  is ^ - E Y  B  - y^l,  the phase  expressed as a fraction of the wavelength  83  time  Figure 20 Forward T r a v e l l i n g Component Observed With Two Sensors  At 1955 r.p.m. the two natural H  z  and 66.9 Hz-  frequencies p  are  65.9  The two nodal diameter response at this rotational  speed  2  and q  2  may therefore be written as:  +  y>  1 + 0.42  In order to evaluate this expression a knowledge of the nature of the imperfections is necessary so that the ratio A / B 2  2  can be determined.  In  f  B)  general, the imperfections w i l l effect both the k i n e t i c energy (A and the potential energy (A  f B" ). n  p  n  However, by assuming f i r s t one type  of imperfection and then the other, the amplitudes of neither the t r a v e l l i n g component nor the steady deflection are s i g n i f i c a n t l y  forward altered.  For small imperfections, i t may be assumed in this expression that A  2  = B . 2  It  can be seen that due to the imperfections, the amplitude of  the steady deflection i s substantially decreased.  At 1955 r.p.m.  this  decrease i s roughly 25% that of the perfect disk steady deflection in the n = 2 mode.  This aspect of the e f f e c t of the imperfections cannot  be d i r e c t l y v e r i f i e d since the steady deflection due to a single mode cannot be accurately i s o l a t e d from the t o t a l . The amplitude of the forward t r a v e l l i n g component, though, was measured (0.0035") and this compares quite well with the theoretical of 0.005".  It  value  must be remembered that a vibration fixed in the disk i s  moving r e l a t i v e to the a i r , in this case in excess of 150 feet per second at the rim, and i t s amplitude is expected to be s i g n i f i c a n t l y less than that calculated neglecting windage. Considering the assumptions made in order to obtain numerical values for the forced response of rotating disks the experimental  results  presented here are in f a i r l y close agreement with the theoretical predictions.  Although the quantitative  agreement is not as good as in the  non-rotating case, equation (11.57) has been shown to reasonably accurately represent the response of an imperfect rotating disk to a s t a t i c load.  IV  Closing Remarks A theory has been presented here for the free and forced responses  of nonSrotating and rotating centrally clamped imperfect d i s k s .  Initially  the free response of a non-rotating c e n t r a l l y clamped perfect disk was considered.  In order to achieve a theoretical  response which more closely  agrees with experimental observations, the influence of small imperfections within the disk were then considered.  The primary effects of these imper-  fections on the free response were shown to be that nodal lines e x i s t at d e f i n i t e locations in the d i s k , and the natural frequencies of the two configurations of a nodal shape are d i f f e r e n t ,  the extent to which depends on  the nature and magnitude of the imperfections. To obtain a reasonable model of a c i r c u l a r saw i t was necessary to consider the e f f e c t s of rotational  stresses on the d i s k ' s behaviour.  It  was shown that these effects can be determined by an approximation based on Southwell's Theorem.  For disks of the physical dimensions and at  rot-  ational speeds typical of c i r c u l a r saws, t h i s approximation y i e l d s acceptable r e s u l t s . In determining the forced response of a d i s k , the load was taken as a transverse point load stationary in space.  The forced responses of non-  rotating and rotating imperfect disks were determined separately, and the perfect disk responses in both cases were achieved by simplifying a more general r e s u l t .  The primary use of the theoretical non-rotating disk res-  ponse was for interpreting-experimental to which the disks were imperfect.  results to determine the extent  86  The rotating disk response to a transverse load was investigated in detail.  For a perfect disk at any rotational  speed, resonance of a mode  consisting of nodal diameters was shown to be possible at two excitation frequencies.  For an imperfect disk the response was s i m i l a r except there  were four resonance frequencies due to the frequency difference of the two configurations of each shape. Of p a r t i c u l a r interest was the response to a s t a t i c load since this is the basis of the c i r c u l a r saw c r i t i c a l speed theory.  At rotational  speeds well away from the resonance speed, the responses of perfect and imperfect disks were shown to be s i m i l a r .  However, as the resonance  speed was approached these responses became very d i f f e r e n t .  The perfect  disk modal response to a s t a t i c load was shown to consist e n t i r e l y of a backward t r a v e l l i n g wave stationary in space, whereas the imperfect disk response consisted of this wave plus a vibration fixed in the disk.  The  frequency of this fixed vibration was such that i t s backward t r a v e l l i n g component was also stationary in space thus contributing to the steady deflection.  The magnitude of the fixed vibration r e l a t i v e to that of the  backward t r a v e l l i n g wave increased as the resonance speed was approached. Near the resonance speed, this theory, which neglected damping, indicated that the response consisted almost e n t i r e l y of a fixed wibpation, contrasting s i g n i f i c a n t l y with the perfect disk resonance response. The major aspects of the theory presented were v e r i f i e d experimentally. The agreement between theoretical and experimental  results was good con-  sidering the theoretical approximations and experimental accuracy.  87  REFERENCES 1.  K i r c h o f f , G.,"Ueber die Schwingungen einer elastischen Scheibe", C r e l l e ' s J o u r n a l , v o l . 40,  2.  1850.  Rayleigh, J . W . S . , The Theory of Sound, New York, Dover Publications, 1945  3.  Zenneck, J . , "Ueber die freien Schwingungen nur annahernd volkommener kreisformiger P l a t t e n " ,  4.  1899 Anrialen der Physik, v o l . 67, p.164.  Lamb, H. and Southwell, R.V., "The Vibrations of a Spinning Disk", Proc. Royal Soc. of London, v o l . 9 9 , 1921, p.  5.  272.  Southwell, R.V., "On the Free Transverse Vibrations of a Uniform C i r c u l a r Disk Clamped at i t s Center; and on the Effects of Rotation,"  6.  Proc. Royal Soc. of London, v o l . «1)01 , 1922, p.  133.  Tobias, S. A. and Arnold, R. N., "The Influence of Dynamical Imperfection on the Vibration of Rotating Disks", Inst. Mech. Engr. P r o c , v o l . 171 , 1957, p. 669  7.  Mote, C. D., "Effect of In-plane Stresses on the Vibration  Character-  i s t i c s of Clamped-free D i s c s " , PhD T h e s i s , Department of Mechanical Engineering, University of C a l i f o r n i a , Berkeley, 8.  1963.  Dugdale, D. S . , "Theory of C i r c u l a r Saw Tensioning", The International Journal of Production Research, v o l . 4, No. 3, 1966, p.  9.  Szymani, R., "Evaluation of Tensioning Stresses in C i r c u l a r Saws", University of C a l i f o r n i a Forest Products Laboratory,  10.  237.  1973.  Mote, C. D. and Nieh, L. T . , "Control of C i r c u l a r - d i s k S t a b i l i t y with Membrane S t r e s s e s " , Experimental Mechanics,'vol. 11, No. 11, p. 490,  1971  88  11.  Mote, C. D., "Free Vibration of I n i t i a l l y Stressed C i r c u l a r Disks" Trans. ASME, vol.' 87B, p. 258,  12.  1965  Nieh, L. T . , "Rotating Disk S t a b i l i t y ,  Spectral Analysis and Thermal  E f f e c t s " , PhD T h e s i s , Department of Mechanical Engineering, University of C a l i f o r n i a , Berkeley, 13.  1972.  Dugdale, D. S . , "Effect of Holes and Slots on Vibration of C i r c u l a r Saws," Departmentoof Mechanical Engineering, University  of  S h e f f i e l d , England. 14.  Mote, C. D., " S t a b i l i t y  Control Analysis of Rotating Plates by  Finite Element: Emphasis on Slots and Holes", Trans. ASME, v o l . 94G, p. 64, 15.  1972  E l l i s , W. E . , "Active Electromagnetic Vibration Control in  Rotating  D i s c s " , PhD T h e s i s , Department of Mechanical Engineering, University of C a l i f o r n i a , Berkeley, 16.  1976.  R a d c l i f f e , C. J . and Mote, C. D., "On Line Control of Saw Vibration: Active Guides", Department of Mechanical Engineering, of C a l i f o r n i a , Berkeley,  17.  University  1979.  Vogel, S. M. and Skinner, D. W., "Natural Frequencies of Transversely Vibrating Uniform Annular P l a t e s " , Trans. ASME, p. 926,  December,  1965. 18.  Eversman, W., "Transverse Vibrations of a Clamped Spinning Membrane", AIAA Journal, v o l . 6, No. 7, p. 1395,  19.  1968  Eversman, W. and Dodson, R.O., "Free Vibration of a Centrally Clamped Spinning Circular Disk", AIAA J o u r n a l , v o l . 7, No. 10, p. 1969  2010,  89  20.  Benson, R.C. and Bogy, D . B . , "Deflection of a Very Flexible Spinning Disk Due to a Stationary Transverse Load", Trails ASME, v o l . 45, p. 636,  21.  1978  Timoshenko, S. and Woinowsky-Krieger, S . , Theory of Plates and S h e l l s , New York, McGraw-Hill, 1959  22.  Thomson, W. T . , "Vibration Theory arid A p p l i c a t i o n s " , Englewdod C l i f f s , N. J . , P r e n t i c e - H a l l , I n c . ,  23.  1965  Ritger, P. and Rose, N . , " D i f f e r e n t i a l Equations with Applications," New York, McGraw-Hill Book Company, 1968  24.  Mote, C. D., " V e r i f i c a t i o n of Saw S t a b i l i t y Theories", Department of Mechanical Engineering, University of C a l i f o r n i a , Berkeley.  

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