THE TRANSVERSE DYNAMICS OF ROTATING IMPERFECT DISKS by GREGORY WILLIAM ROSVAL B . A . S c , University of Br i t ish 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 partial ful f i l lment of the require-ments for an advanced degree at the University of Br i t ish Columbia, I agree that the Library shall make i t freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Mechanical Engineering The University of Br i t ish Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 December 1, 1980 ABSTRACT The transverse motions of c i rcular saws have undesirable effects on many aspects of c ircular sawing. Due to current high man-ufacturing costs, substantial savings may be realized i f these transverse motions can be reduced. In this thes.is a c i rcular saw is modelled as a rotating im-perfect disk acted upon by a transverse, non-oscillatory point load stationary in space. Such a model is known to accurately predict certain relevant aspects of the behaviour of a c ircular saw in i ts operating en-vironment. In i t i a l l y the free response of a non-rotating perfect disk is cons'iderced. This model is then refined by considering the effects of rotational stresses and small imperfections within the disk. The res-ponse of such a disk to an osci l latory load is determined, from which the response to a non-oscil latory load may be determined as a special case of part icular interest . Experimental results are given which quantitatively confirm the theory presented. ACKNOWLEDGEMENTS I wish to thank Professors H. Vaughan and S. Hutton for their supervision of the thesis . 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 Br i t ish Columbia were very helpful in assisting 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 ina l ly , I would l ike 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 is t of Figures v i i i I Introduction . . . . 1 II The Theory of Rotating Disks 4 11.1 Free Response of a Perfect Non-rotating Disk 6 11.2 Free Response of an Imperfect Non-rotating Disk . . . . 11 11.3 An approximate Method of Determining the Free Vibrations of a Non-rotating Disk 25 11.4 The Effects of Rotation 30 Rotating Membrane 30 Rotating Disk 33 Southwell's Theorem 34 11.5 Disk Displacement with Respect to Rotating and Non-rotating Coordinates 3 8 11.6 Forced Response of a Disk 44 Non-rotating Disk 44 Rotating Disk 49 11.7 Comments on the Theory and i ts Applications 56 Current Research 57 Saw Behaviour in i ts Operating Environment 60 V III Experimental Veri f icat ion of the Theory of Rotating Disks . . . 62 111.1 Experimental Equipment 62 111.2 Forced Response of a Non-rotating Disk 65 The Location of the Nodes 66 Damping Coefficient . 69 Radial and Angular Deflection Profi les . . . . 71 i 111.3 Forced Response of a Rotating Disk 75 Observed Frequencies as a Function of the Rotational Speed 76 Response to a Static Load 78 IV Closing Remarks 85 References 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 coeff icient 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 strain energy displacement in space-fixed coordinates proportionality constant for membrane frequency displacement in disk-f ixed coordinates angular phase angles strain energy coeff icient temporal phase angles angular space-fixed coordinate v i i A non-dimensional frequency parameter © angular displacement function e angular disk-f ixed coordinate 6p angular location of load in the disk coordinate system y kinetic energy coeff icient a (r) radial membrane stress a ( e ) tangential membrane (hoop) stress p mass density v Poisson's ratio fi disk rotational speed to excitation frequency T variable coeff icient in radial function £ damping ratio 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 . . 68 Figure 11 Relative Amplitude versus Excitation Frequency 70 Figure 12 Radial Prof i le at 0 Hz. (Disk C) . 72 Figure 1 3 Radial Prof i le at 90 Hz. (Disk C ) . \ 7 3 Figure 1 4 Theoretical Angular Profi les at 0 Hz 74 Figure 1 5 Angular Profi le 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 Prof i le at 2400 R.P,M.(Disk C) 80 Figure 1 9 Displacement versus Rotational Speed (Disk D) 81 Figure 20 Forward Travell ing Component Observed with Two Sensors . . 83 1 I. Introduction The use of c i rcular saws in the manufacture of lumber is very common, and in an attempt to reduce the costs associated with this process much research has been conducted. Many of the problems resul t -ing from the use of a c i rcular saw may be attributed to motions of the saw in the direction normal to i ts plane. These motions, which are referred to as transverse vibrations, have an undesirable effect on cutting accuracy, kerf losses, the quality of the cut surfaces, saw l i f e and ambient noise levels . One of the primary goals of researchers in this area is 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 lateral st i f fness of the saw. * In attempting to predict the behaviour of a c ircular saw in i ts working environment, many d i f f i c u l t i e s are encountered. Since the saw and i ts environment are continually changing, no complete solution exists which considers a l l aspects of the problem simultaneously. However, i t is possible to model this system in a way such that useful qualitative and quantitative results may be determined from fundamental pr inciples. Most simply, a c ircular saw may be modelled as a non-rotating, undamped complete disk. The natural frequencies and shapes of the free vibrations of such a disk were determined by Kirchoff in 18501. Later in that century Rayleigh made a signif icant contribution to this problem, in particular in reconcil ing theoretical predictions with experimental r e s u l t s 2 . He introduced the idea that small imperfections within the 2 disk could s igni f icant ly affect i ts 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. This was done by Lamb a n d Southwell in 19211*. It was in this work that Southwell's Theorem was f i r s t i n t B o d u c e d . This theorem states that u n d e r c e r t a i n conditions an approximate method is 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 disk. Later that year Southwell published a paper in which the effects of a clamp were considered, and where his theorem was demonstrated with numerical examples5T It wasn't until 1957 that the experimentally observed forced response of a rotating disk was sat is factor i ly 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 fal len into one or more of the following categories; the control of membrane s t r e s s e s , 7 ' 8 ' 9 * 1 ? ' 1 1 * 1 2 altering saw geomet r i es 1 3 ' 1 4 and external control m e t h o d s . 1 5 ' 1 6 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 d i f f i c u l t y . For this reason an approach combining both theory and experimentation is very useful. 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 is the purpose of this thesis to present a theory of the forced response of rotating disks and to present experimental results which verify this theory. While a de ta i l -ed theoretical development of the forced response of idealized disks is given, emphasis is placed on the theory required to account for the departure of the characteristics of real disks from those of idealized disks. 4 II The Theory of Rotating Disks Two types of disks are analyzed in this thesis. The f i r s t are what are referred to as "perfect disks". These disks are homogeneous , isotropic , completely c i rcular and are of constant thickness throughout. The second type, called "imperfect disks", possess material properties and geometries which depart s l ight ly from those of perfect disks. Al l real disks are to some extent imperfect, and the influences of these imperfections on the disks' behaviour must be taken into account when experimental results are interperted. Because the disk is rotating in space while observations are made from points stationary in space, two coordinate systems wil l be defined. The coordinate system fixed in the disk wil l be denoted as the (r,e) system, whereas that fixed in space wil l be denoted as the (r, y) system. The radial coordinate r is the same in each case. These two coordinate systems and the physical dimensions of a perfect disk are shown in figure 1. The disk rotational speed is in the positive y d i rect ion, 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 Simi lar i ly , the reverse transformation i s : 0 = nt-y The transverse displacement as a function of e is denoted by w, and as a function of y by u. Figure 1 Perfect Disk Dimensions and Coordinate Systems In i t i a l l y the method of solution of the "free vibrations of a perfect, centrally clamped non-rotating disk is outlined. The results obtained from this analysis form the basis for predicting the disk be-haviour when the problem is compounded with imperfections, rotational stresses and a transverse load. 6 II.1 Free Response of a Perfect Non-rotating Disk The equilibrium plate bending equation for a homogenous, isotropic plate i s : 2 1 ( I I .1) v*w = (J/D Here q is the transverse load per unit area and D is the flexural r ig id i ty given by: D = E ( 2 h ) 3 / 1 2 ( l - v 2 ) The biharmonic operator is to be expressed in polar coordinates. For the case of a freely vibrating non-rotating disk, the only load is that due to the inert ia l forces and internal and external damping. If damping is neglected, equation (II.l) may be written as: (II.2) * • - f r There are four boundary conditions which the solution of equation (II.l) must sat is fy . They are: (a) no deflection at the clamp (b) no slope at the clamp (c) no internal radial bending moment at the free edge (d) twisting moment/shear stress condition at the free edge (Kirchoff boundary condition) 7 A separation of variables form of a solution may be assumed in this case: (11.3) w = R(r)T(t)e(e) When equation (II.3) is substituted into equation (II.2), a suitable expression for 0 is found to be: (11.4) e(e) = B^os ne + B2si.n ne The necessity that w(r,t ,e) = w(r, t ,e + 2ir) requires that n is an integer. The expression for T is also immediately found to be of the form: (11.5) Tift) = Ci cos p n t + C 2 sin p n t The expressions for the natural frequencies p n are unknown at this stage except that they are dependent on the integer n. The solution as given by equation (11.3) may then be written as: (II.6) wn = R( ;r) [Bi cos ne + B 2 sin ne][Cj cos p n t + C 2 sin p n t ] The subscript n now appears on w, indicating that there is a solution for each value of n selected. In order to determine the radial function R (r)and the frequencies Pn> consider one term of equation (II.6), which may be written as: (II.7) w n = A n R, (r) sin ne cos p n t 8 After substituting equation (11.7) into the equation of motion, the function R(r) may be determined by solving two dif ferent ial equations; Bessel's equation and the Modified Bessel's equation 5.. Equation (II.7) is then:, (11.8) w)n =[a!J n + a 2 Y n + a 3 I n + a^Kn] sin ne cos p n t Here Jn» Y n , In, and Kn are Bessel and Modified Bessel functions of the 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 yields a characterist ic equation, from which the natural frequencies and radial functions may be determined. For a particular value of n there are an in f in i te number of frequencies and radial functions. Each of these radial functions may be identi f ied by the number of values of r at which there is 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 is 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 as: (11.9) w, n ) S = A n , s : .R n > s (r ) sin ne cos p n , s t For reasons which wil l become apparent when the forced response of a rotating disk is discussed", i t is only the zero nodal c i rc les modes which are of concern. The response of the disk in those modes where s f o is therefore neglected and the subscript s may be ommitted. Equation (II.9) is then: 9 wn = R n ( r) s i n n^cos p n t This expression, however, was developed by considering only one term of equation (II.6). When a l l four terms are considered the n nodal diameter response i s : w n = R n ( r) C B i c i c o s n e c o s Pnt + B ^ cos ne sin p n t + B 2 C! sin ne cos pnt + B 2 C 2 ss in ne sin pnt] It is 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 2 = c 2 and 5 = tan b/a By applying this identity to the angular trigonometric terms we obtain: (11.10) w n = Rn(r) [ A n l cos (ne-e x) s in p n t + A n 2 cos (ne -e 2 ) cos p n t ] The second form is obtained by applying the identity to the temporal trigonometric terms: (II. 11) Wp = R n(r) [ A n i sin ne cos (pnt-e}))n-An 2cos ne cos (Pnt-e 2)] Equations (11 . 10 ) and ( 1 1 . 1 1 ) are the two forms of the n nodal free response of a perfect disk. Although they are equivalent, i t is the form given by equation (11 . 11 ) which wil l be used for the remainder of this thesis for reasons which wil l become apparent when the response of an imperfect disk isiconsidered. There are four constants in equation (11 . 11 ) which must be determined from the i n i t i a l conditions rM the free response. It wil l now be shown that although this expression is referred to as the n nodal diameter response, i t does not in general consist of what is commonly known as nodal diameters. If we set w n = 0 in equation ( 1 1 . 1 1 ) , the following expression for e results: + a r i „ n _ A n ? cos (p nt - E 2 ) t a n n e " AnT 60S (p nt -el) From this expression i t ican be seen that, in general, the angular locations where: v the displacement is zero is a function of time. If, however, ei = e 2 this becomes: tan ne = - A n 2 / A n i = constant which is the familar case of a free response with nodal diameters fixed in the disk. Referring to equation ( 1 1 . 1 1 ) i t can be seen that this motion results when the disk is i n i t i a l l y deformed in the shape w,n_;|nV"ti-al = R n( r ) [Am sin ne + A n 2 cos ne] and released at t = p n . 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 ines is given by: tan ne = tan (p n t - e x ) . e = Bnt - e 1 These nodal l ines are travel l ing 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 2 = £\ + 3 IT /2 into equation (11.11). Due to the d i f f i cu l ty in creating these i n i t i a l conditions, this motion is not commonly observed experimentally. Both of the above cases, the vibration fixed in the disk and the travel l ing wave, are special cases of the more general result . From equation (11.11) i t can be seen that in general the n nodal diameter free response consists of two osci l la t ing 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 is here defined to be a disk whose geometry and physical properties d i f fer s l ight ly from those of the idealized perfect disk. While the effects of the imperfect-ions could be determined quantitatively i f the imperfections can be mathematically modelled, this is not l ike ly 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 is used here to investigate the effects of imperfections on the free vibrations of a non-rotating d i s k . 3 The assumed expression for the total free response is given by equation (11.12) where ^(t) and v(t) are to be determined. (11.12) w = | [$ n ( t ) R n(r) cos ne + v n ( t ) R n(r) sin ne] Two types of energy wi l l be considered here; the kinetic energy of the transverse motion and the strain energy of deformation. The energy dissipated by damping is considered in a later section. (i) Kinetic Energy: The kinetic energy of the transverse vibrations of the disk is given by: T vol rdrdedz From equation (11.12): W ? = M i RnRm [Vm c o s n e c o s m e + vnvm sin ne sin me + 2hnim cos ne sin 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 sin ne sin me for In this case, the kinetic energy is of the form: (11.13) T = jj h u n *n + h p n *n where: (11.14) y n = 2ppir h j [ R n ( r ) ] 2 r dr Th is , however, assumes that the disk is perfect. Although the imperfections are considered to be general in nature, as an i l lus t ra t ive example consider the case where the imperfection consists of a density variation in the angular direct ion. If the density can be represented as: (11.15) p(e) = i: C|< cos ke + D|< sin ke k=0 the results of the kinetic energy integration wil l be somewhat different than in the perfect disk case. The integrand of the kinetic energy integral for this density imperfection i s : p(w) 2r = n I RnRm^*n*m c o s n e c o s m e ( ek cos ke + k m n Dfc sin ke) + sin nesin me (C|< cos ke + Dk sin ke ) + 2 i n * m cos ne sin me x (C^ cos ke + D|< sin ke )]r Of the six terms within this summation, i t cannot be said that any one wil l vanish for a l l values of k, m, and n when the integration is performed over the range of e. If, however, the density imperfections are small, 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 is then: (11.16) T = h W a ™ ®n* m + 2 c ™ *n*m + bnm *n*m where a n m and bnm are small for n f m, and c n m is small for a l l n and m. In addit ion, the values of a n m and b n m , for n = m, are s l ight ly different from what they would be for a perfect disk. ( i i ) Strain Energy: For a non-rotating disk, the strain energy is that due to bending. In polar-cyl indrical coordinates this i s : 6 - f(T E z 2 I (3 2w . 1 aw . 1 92W\2 " IV 2 ( l - v 2 ) ) I + F 9F + F2" ae*) 0 , , , 92W /I 9W , 1 9^w\ ( n . 1 7 ) ' z u _ v ) iF 5 " IF 9? + F2" a?2") + 2(1 - v ) _9_ [1_9W 9r \r 99 rdrdedz If the disk is considered to be perfect, the resulting expression for the strain energy i s : (11.18) U = E h g n $ n 2 + h p ^ n 2 15 Where: Here R denotes R n ( r ) . As can be seen from equation (II.17),each term of the.strain energy integrand contains either cos me cos ne, sin ne sin me or cos ne sin me. Again, due to the orthogonality of the trigonometric terms, no cross-terms appear in the strain energy expression for a perfect disk. However, i f the disk is imperfect, the general form of the strain energy i s : (11.20) U = h I E a n m *n*m + 2 E n m $nwm ++ b n m vnvm m n Here the relative magnitudes of anm> Cnm and B n m are similar to those of a n m , c n m and b n m . Lagrange's equations for 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: z a $ + a $ + c Y + C ¥ =0 m nm m nm m nm m nm m l b Y + 6 T + C $ + c $ =0 m nm m nm m nm m nm m Here there are two systems of l inear dif ferential equations, each system consisting of an in f in i te number of equations, one equation for each value of n. By the theory of systems of l inear dif ferential equations with constant c o e f f i c i e n t s 2 3 , solutions are sought of fche form $. = X. e x t J J and <p. = Y. e X t . However, A must be pure imaginary since the system is conservative. 2 The assumed form of the solutions to equations (11.21) and (11.22) are therefore taken as: $. = X. cos (tot - y) J J (11.23) Y i = Yj COS (ait - Y) These assumed solutions may be substituted into each of the equations of (11.21) and (11.22). The resul t , for example, of substituting into the n=2 equation of (11.21) i s : (a 2 i - io 2 a 2 i ) + ( a 2 2 - io 2 a 2 2 ) X2++ — i + ( c 2 1 - io 2 c 2 i ) Yj+ ( c 2 2 - o) 2 c 2 2 ) Y 2 + — =0 (11.21) (11.22) 17 All such resulting equations may be written in matrix form as: (11.24) ( in-oj&aii) ( C 1 1 - U J 2 C 1 1 ) (a 1 2 -o j 2 a 1 2 ) ( c 1 2 - a) 2C] (c{ 1 -aj 2 c 1 1 ) (b 1 1 -a) 2 b 1 1 ) ( c~ 2 -w 2 c 1 2 ) (B12-u2bj ( a| 1-oj 2 a 2 1 ) ( c 2 1 - i o 2 c 2 1 ) (a 2 2 -a) 2 a 2 2 ) ( c 2 2 - w 2 c , ( c 2 1 - a j 2 c 2 1 ) (b 2 1 -a ) 2 b 2 1 ) i ( :c 2 2 -u) 2 c 2 2 ) (b22-u>2b2 = 1 2 ) h f cT 0 * x2 0 Y 2 1 0 1 1 1 V. J 1 ^ / or: (11.25) [M] (A) = (6} A non-tr iv ia l solution to equation (11.25) exists i f and only i f det[M] = 0 . In the case where the imperfections are small , the non-diagonal elements of the matrix [M]aare small in relation to the diagonal elements. The solutions to the equation det [1M] = 0 are then approximately determined by solving: ( a n -u)c u 2 a 1 1 ) ( 6 1 1 - t o 2 b 1 1 ) ( a 2 2 - u> 2 a 2 2 ) (b 2 2 - w 2 b 2 2 ) — = 0 The values of u which are the solutions to this equation wil l be denoted by p and q: (H.26) P i = a n / a n qf. = B n / b u p| - a 2 2 / a 2 2 — q 2 = b 2 2 / b 2 22 where p. = q.. , since a... = D^. b . . and aj_. = b. n n Each eigenvalue of equations (11.26) , p. or q. , may be substit -uted into equation (11.24) from which the relative magnitudes of the comp-onents of the matrix {A} may then be determined. For example, the assumed solutions, equations (11.23), obtained by putting w = Pi jn equation (11.24) may be written as: $i = Pi icostpj t soe{?) t - "ti = B 1 1 cos (p 1 t - e x ) Here P 1 1 S P 2 1 •••> B 1 1 S B 2 1 • • • a n c ' e i u e n ° t e X 1 S X 2 Y 1 S Y 2 . . . and Y as determined for to = p . For <o= p. and <o= q. , $.aarid f . may be written as $ 2 = P 2 iCos(p 1 t f 2 = B 2 1 cos(p! t - ej) J (11.26) •j = P j i c o s ( p . t - e.) Q. .cos(q i t - Cj) (11.27) f j = E h c o ^ ~ ei> D..cos(q i.t - c..) The most general forms of the solutions for $. 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 vibrat ion, equation (11.12), y ie ld ing: w = i ( R m ( r ) cos me z [Pm n c o s ( p t - e J + Q m n cos(q t - c )] + m | m n L mn r n n mn ^n n J R j r ) sin me z cos(p t - e J + Umn cbs(q t - t )] j m n L mn r n n mn ^n n J j It is possible to rearrange this expression to identify the shape of the response at a frequency p n (or q n ) . If we denote by w ( w n 2 ) t n e r e s Ponse at the frequency p n (P n )» the result i s : w = cos(p t - e ) z [P R cos me + B R sin me] ni ^nt n m l mn m mn m J (11.29) w„„ = cos(q„t - c„) z Ft) „Rm cos me + Dm R sin mel nz ^n n m L mn m mn m J Clearly the effects of the imperfections are to not only al ter the natural frequencies, but to also alter the shapes of the modes. If the coefficients a , b . . . are known, i t is possible to determine the relative 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", § 90) 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* n are the natural freq-uencies of a perfect disk. If i t is assumed that th«is contribution is negligable, tjjat i s , that the imperfection does not alter the actual shapes of the modes of vibrat ion, equations (11.29) are: (11.30) w = R cos (p t -e ) [ P cos ne + B sin ne] ni n V f n n' nn nn w „ = R cos (q t - r ) [Q cos ne + D sin ne] n2 n X M n wnn nn 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 n i = Rn <f>n cos (ne - cxn) V = R n *n S I N ( N E " P N } 1 2 2 ' where * = VP - + B - cos (p t - e ) n v nn nn r n nv it = Jo1 + d1. cos (q r ) v n v w n n nn V H n - ? n ; and a n = tar f i B n n / P n n 8 = - tan" 1 0 I D„ n n ^nn nn If i t is assumed that the imperfections do not alter the shape of the response, the coordinates <j> and are the disk normal coordinates, n n since a free vibration is possible which is characterized by the vanishing 2 3 of a l l $ and ^ n except one . - ' When the kinetic and strain energies of the imperfect disk are expressed in terms of these normal coordinates, 21 the coeff icients of the cross-terms must be zero. By comparing the expressions for * , ¥ , <j> and 4 the relationships between the two r n n MI n sets of coordinates is seen to be: *n = *n c o s °n " *n s 1 n 3 n v = sin a + t cos & n n n n 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 = R + B n ^ + C n v * n The coefficients A , B etc. are functions of the angles a and e as n n 3 n n well as the coeff icients a„„ , b„„ etc. Setting the coeff icients of the nn nn 3 cross-terms, that is C n and C^, to zero results in the following expressions for t * n and $ n : (be - cB) t a n 2a n + (ba - aB) tan a n + (ca - ac) = 0 (11.33) (ca - ac) tan 2 6 n - (ba - ab) tan g n + (be - cb) = 0 The sub-scripts "nn" have been ommitted. In addition to the above two expressions for a and 6 , their ratio is obtained as: r n n tan a n caa ac (d i .34 ) = : r tan B n cb - be 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 2 «n + 2c tan a (H-35) a + b t a n 2 c n + 2c tana „ _ 5 + a tan 2 3„ - 2c tan s % - E! D_ b + a tan 2 B n - 2c tan Bn Equations (11.33), (11.34) and (2.35) require, for their use, an exact knowledge of the imperfections. This is not l ike ly to be known except in the simplest of cases. Howewer, several signif icant results are available from these equations. From equations (11.35), i t is seen that p n (q n) is a function of &n ( B n ) , a n d also a , b etc. which are constants dependent solely on the disk physical character ist ics. If stationary values of p or q are r n n sought, the resulting expressions for ctn and *3n are those given by equations (11.33). That i s , the nodal l ines , determined by a n a n d 3 n , are located such that the frequencies p and q are stationary in value. However, for a perfect disk, where a n n = t>nn, a n n = t>nn and c = c = 0, equations (11.35) show that p„ = q„ regardless of the nn nn ^ r n ^n values of a OB £ . n n A second result regarding a and is available from equations (11.33) and (11.34). Clear ly , for a perfect disk, both of these express-ions for a and 3 are indeterminant, and hence the location of the nodes n n cannot be determined solely from the disk physical character ist ics. In the case where the disk is imperfect but the nature of the imperfections is such that their influence on the kinetic energy is much greater than on the strain energy, equations (11.33) may be written approximately as: -c t a n 2 a n + (b - a) tan « n + c =0 -c tan 2 6 n + (a - b) tan B n + c =0 When their influence on the strain energy predominates: - c " t a n 2 a + (5 - a) tan a + c = 0 a n ' n -c tan 2 3 n + (a - b) tan e n + c = 0 In either case i t can be seen that the extent to which the anglesca^ and 3 n d i f fer is greater for greater imperfections within the disk. The free vibrations of a disk with small imperfections may be taken as: w = I [ A hi R n c o s ( n 0 " a n ) c o s ( p n t " (11.36) -, + A p 2 R n ; s i n (ne - B ) cos ( % t t - 6 ) ] where the differences between « n>3 n and p n , qJn in general depend on the nature and extent of the imperfections, and where a and 3 are such n n that p and q are stationary in value, n n The two terms of equation (11.36), that i s : w ni A R cos (ne - a- ) m n v n' cos (pt -V f n w, IT2 = A R sin (ne - ft ) r\z n n cos (p t are referred to as the two configurations of the n - nodal diameter vibrat ion. Each is independent of the other, that i s , each is a mode of vibrat ion, and the amplitudes, A n l and A n 2 , and the time phases, e n and c n , are dependent entirely on the i n i t i a l conditions. It should be pointed out that there are very special cases where imperfections may be present and either the phase angles a n and e n are equal, or the frequencies p n and q n are equal. As an example of the f i r s t case, consider a mass imperfection symmetric about some angular location on the disk. Any free vibration must also be symmetric about that point. 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 n = e^. 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 n and q n be stationary in value results in p n = p^ . when 2n is not an integer multiple of m. This second case is known as the Zenneck r u l e 1 3 Although either the difference between a n and 3 n , or between P n and q n could be considered measures of the imperfection of a disk, i t is the lat ter that is the accepted practice. While i t is possible to experimentally determine the coefficients a n , b n e t c . , this would be extremely d i f f i c u l t to do. Since the desired information is the natural frequencies p and q and the location of the ^ M n M n nodes, these can be determined directly and with much greater ease by experimental methods which are described la ter . It i s , however, desirable to obtain numerical results so that theoretical predictions can be compared with experimental results. It is found that theoretical 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 is consistent with experimental evidence. The following section gives an approximate method whereby num-erical 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 con-ditions yields the natural frequencies and the relative magnitudes of the coefficients of the Bessel and Modified Bessel functions. Another possible approach is that known as the Rayleigh-Ritz method. This is the approach which wil l be taken here. In order to use the Rayleigh-Ritz method, the shape of the vibration must be re-presented as a kinematically - admissible orthogonal sequence with un-known coef f ic ients . Using this shape an expression for the natural frequencies may be obtained. This expression is a function of the unknown coef f ic ients , and minimizing the frequency with respect to these coefficients yields an upper bound on the fundamental frequency pf t ie free vibrat ion. It is not necessary to assume an angular shape, however, since i t is known to consist of nodal diameters which must be symmetrically distributed around the disk. It is the radial functions R n(r) which are unknown. A kinematically •admissible representation of the radial functions i s : 5 Here the x '^ are the unknown coeff ic ients. The value of this function and i ts slope are both zero at the clamp as required. If only the f i r s t two terms are taken, the approx-imation to the radial function i s : R n(r) - ( b'-a r-a R n ^ = T G ) x 2 + T i x 3 r-a b-a The range of x is then 0 <x<1• Since Rn(a) may be multiplied by a constant without affecting the result in any way, we can write: (11.37) Rn (x) = x 2 + T X 3 Because the angular function is known exactly, i f the frequency is de-termined by use of the radial function given by equation (11.37) then minimized with respect to j , the resulting minimum is an upper bound on the natural frequency of that n nodal diameters, zero nodal c i rc les mode. The expressions for the kinetic and strain energies as previously given by equations(II.13) and (11.18) are: 2T = u $$2 + y $2 n n p n n 2U = 6 $ 2 + 6 f 2 •u p n n p n n where the values of u and e are determined from the integrals of n n equations (11.14) and (11.19). If damping is neglected and a harmonic time function is assumed, equating the maximum kinetic and strain energies results in an express-ion of the form: "Eb 2 " *2 Ki f + K2 x + Ka The coefficients $a, S 2 , and S3 result from the strain energy ca lcu l -ation whereas K i , K2 and K3 result from the kinetic energy calculat ion. Al l coefficients are functions of the disk dimensions a and b, Poisson's ra t io , and the integer n. The value of t which minimizes p is ava i l -n able in closed form in this case. It is possible to express the minimum 0.1 0.2 a/b 0.3 Figure 2 Non-dimensional Frequency (parameter versus Glamping Ratio the Rayleigh-Ritz method (dashed l ines ) . The Rayleigh-Ritz values of X are generally more accurate for low values of the clamping ratio a/b, with the exception of the n = 0 mode whose error is essential ly independent of the clamping rat io . For values of the clamping ratio greater than approximately 0.2, the use of only two terms for the"approximating function is obviously insuf f ic ient . The values of the coeff icient t which minimizes the frequencies P n are given in figure 3 . Using these values of T i t is 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 resul t . The f i r s t is the previously discussed effect of imperfections which are treated experimentally in a later chap-ter . The second effect is that of the rotational stresses, which wil 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 tensi le membrane stresses which exist when the disk is rotating. In i t ia l ly the influence of these stresses on the free vibrations of a perfect disk are investigated, then an approximate method is described by which this influence on the free vibrations of an imperfect disk may be included. Rotating Membrane In order to introduce the effects of rotational stresses, i t is convenient to consider a rotating disk with no flexural r ig id i t y ; that i s , a membrane. This approach wil l also be useful in a later section, where an empirical relationship for the effects of rotation wi l l be presented. The equilibrium equation for a non-vibrating, rotating circular membrane is obtained from the free-body diagram shown in figure 4. Due to symmetry there are no shear stresses, and the hoop stresses a 0 are constant around the disk. The body force B is of magnitude n2rdm where dm is the mass of the element. The boundary conditions on the stress distr ibution 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 partial clamp. This type of clamp (or col lar) is the type commonly found in practice. It allows a radial displacement of a l l points on the disk, and serves only to prevent a transverse displacement at the co l la 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 : 1 8 r 2 + / k i b 2k (b 2 - r 2 ) pto 2 2/-J '3 r1 pur where k i , k 2 and k 3 are functions of the disk and col lar radii and Poisson's rat io . It is now assumed that the membrane is deformed arb i t rar i ly in the transverse direct ion, 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 direct ion, due in the radial case to both the variation in the radial stress and to the curvature in the radial direct ion, 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, result-ing in the equation of motion: 1 a / • 3w\ . 1 3 / 3w\ , 3§W _ n 7 - 3 ? \ r a r 3?j + FT 36 [°Q 3?j " p W ~ ° The boundary conditions on the transverse displacement are: 1. zero transverse displacement at the co l la r . 2. f in i te transverse displacement at the free edge. The method of solving this differential equation is very lengthy, but numerical results are a v a i l a b l e . 1 8 The resulting expression for the transverse vibrations are of the same form as given in equation (I1.9), but the radial functions are not the same. A signif icant result is that for a given disk the natural frequencies are directly proportional to the disk rotational speed. That i s : '(11.38) Pn 2 = V n ^ where V n is a function of n and the disk physical properties and dim-ensions. Rotating Disk It is possible to formulate the differential equation for a rot-ating disk considering both rotational and bending stresses using information previously presented. Equation (II .1), the plate bending equation, is the equilibrium equation for a non-rotating disk subjected to a transverse load, which was taken to be the inert ial 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. There-fore i t is possible to use the plate bending equation for a rotating disk, where the transverse load is due to both the inert ia l forces and the rotational stresses. The resulting equation of motion i s : 34 0 V 4 W = 1 - 1 r 9r + 1 9W R2 39 \ 99 This equation was solved numerically by Eversman and Dodson and the results published in 1969. 1 9 The shape of the vibration is again found to consist of nodal c i rc les 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 ratio a/b, the disk thickness and the rotat-ional speed. It is found, however, that the influence of these stresses on the disk natural frequencies is not very large for disks and clamping ratios of the dimensions corresponding to those typical of c ircular saws. As an example, using results from the paper by Eversman and Dodson, an 18" diameter disk, 0.150" thick with an 8" diameter col lar 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 is in fact an upper l imit since typical rotational speeds of c ircular 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 is general in nature, not pertaining specif-i ca l l y to rotating disks. It is derived direct ly from Raycleigh's Theorem, 35 which states that the natural frequency of the fundamental mode of vibrat ion, as calculated from an assumed shape of def lect ion, is an upper bound for the exact value. In order to develop Southwell's Theorem as i t applies to rotating disks, 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 r ig id i ty) Assuming simple harmonic motion, the maximum strain and kinetic energies of the transverse vibration may be written as: Strain Energy = U (S) Kinetic Energy = p 2 T (S) If damping is neglected, the frequency is given by: For small deflections, the work done by the bending stresses is independ-ent of that done by the membrane stresses. Therefore: U (S T) = UB (S T) + UM (S T) Where Ug and Aare the bending and membrane potential energies. The frequency is then: . p | _ UB (S T) + UM (S T) T (S T) = n 2 + D 2 * PB PM Here p B and are what the natural frequencies would be i f the disk was assumed to vibrate in the shape Sj under the action of either the bending or the membrane stresses. That i s : P B * ' T (S T) - 2 U M <ST> P M ^ " T ($T) However, i f only bending stresses or only membrane stresses were present, the shape would be S B or respectively. Therefore the exact values of p B and p M .are.given by: P B ! = U B ( V T ( S B ) P M 2 = "M ( S M ' T ( S M ) By Rayleigh's Theorem, for the fundamental mode: P B 2 * P B 2 P M 2 > - P M 2 Therefore the expression for the frequency p T becomes: (".39) p T 2 z P 2 + p 2 Equation (11.39) is Southwell's Theorem. It can ben seen that i f the potential energy of a freely vibrating body is 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 relat ion-ship i f the equality is taken to hold: P T 2 = P B 2 + P M 2 However, equation (11.38) indicated that the relationship between p 2 and was: P M 2 = V " 2 Therefore: (11.40) P T 2 = P B 2+ Vn 2 In the previous example of the 0.150" disk, the increase in the n = 1 mode natural frequency in going from 0 to 5300 r.p.m. was 19.5%. If the exact value of V is u s e d , 1 8 the increase is found to be 17% (232 Hz.) by equation (11.40). The actual difference in the 5300 r.p.m. natural frequency by the two methods is only 2%. Evidently, for disks of the physical dimensions similar to those typical of c i rcular saws, equation (11.40) provides a satisfactory approximation for the natural frequencies as a function of rotational speed. While the coeff icient V n is available from the l i terature , there are several factors in practice which can signi f icant ly al ter i ts value from that calculated theoret ical ly. However, i t may be approximated quite easi ly experimentally. Since for a real disk there wil l l ike ly be two natural frequencies associated with the n nodal diameter free vibrat-ion, there would be two relationships of the form of equation (11.40): pf • »i * ve 2 i - i * v 2 Experimentally determining the coefficients V n necessitates observing the disk natural frequencies at various rotational speeds. However, observations wi l l generally be made from points stationary in 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 co-ordinate systems may be i l lustrated by considering the free vibration given by: wn = A n Rn (r) cos (ne -e g ) cos (p n t - t 0 ) This vibration is given with respect to the (r,e) coordinate system fixed in the disk. For s impl ic i ty , let e0= 0, t 0 = 0 for the mode under consideration. Then: (11.41) wn = A n R p(r) cos ne cos p p t Using a trigonometric ident i ty , equation (11.41) can be written as: (11.42) wn = ^ A n R n(r) [cos (p n t - ng ) + cos (p p t + ne ) ] Consider the f i r s t term of equation (11.42), which wi l l be denoted as w (1). That i s : n n ( D = J * A n R n(r) cos (p n t - ne ) w. This term, at any part icular instant, is identical in form to the original mode shape given by equation (11.41). The same is true of the second term of equation (11.42). These two terms each contain "nodes", the locations of which, in general, are given by: 8(1) = "V + ( 2 K + 1 ] l n 2n 6(2) = -V + ( 2 K + 1 ) I R -n 2n k = 0,1,2 . . . In addition, 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 con-sidered 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 ravel -l ing in opposite directions around the disk. The above results are stated with respect to the (r,e) corodinate system. If the disk is rotating, 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 will be denoted u n- Substitution of the transformation into equat-ion (11.42) y ie lds : u n = h A n R n ^ L ^ P p * + n Y ~ nnt) + cos(p nt - n Y + nnt)] The response wi 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 Rn (r) [ cos( p n - nfl)t + cos( pn+ nfl) 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 ie ld a simple method of determining the part icular mode shape associated with a resonance peak. The method is to excite the disk randomly and record the observed natural frequencies at several rotational speeds. The observed frequencies f as a function of the rotational speed n would appear-as shown on figure 6 (neglecting imperfections). 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 physically by considering the two components of a mode as given by equation (11.42). When the disk is not rotating these two components, which are travel l ing in opposite direct-ions attifehessame speed in the disk, are also travel l ing at equal but opposite speeds in space. However?9when the disk is rotating, the com-ponent travel l ing in the direction of rotation, the "forward-travelling component", is moving faster in space than when the disk is stationary. Just the opposite is true of the "backward-travelling component". A signif icant phenomenon may be noticed in figure 6. At some rotational speed one observed frequency of each mode, except for n = 0, becomes zero. From equation (II.45-a) for f = 0, we have: p = nn *n If this value of P n is substituted into the expression for the speed of the backward-travelling component, equation (II.43-a), the result i s : 6(1) = a Since the positive e direction is that opposite to the rotation, i t can be seen that the backward-travelling component is stationary in space. S imi lar i ly , the forward-travelling component is travel l ing at twice the djitskdspeed. As the rotational speed is increased even further, the back-ward-travelling component actually begins to move forwards in space. The above description does not consider the effects of either imperfections or rotational stresses. The existence of imperfections wi l l double the number of observed frequencies, while the rotational stresses wil l increase their 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: Apparently when V n > n 2 there is no poss ib i l i ty of the observed frequency becoming zero. Physically this occurs when as the rotational speed increases, the speed in the disk of the backward travel l ing component is increasing at a faster rate. This phenomenon does in fact occur in practice as wi l l be shown experimentally in a later chapter. If an observed frequency does become zero, the rotational speed at which this occurs is very signi f icant when the forced response of rotating disks is considered. This is investigated in the following section. The lower branches of these observed frequencies become zero when: 44 II.6 Forced Response of a Disk 6 The transverse loading of a c i rcular saw arises from the interaction with the work piece. Since this transverse load is dependent on many factors, primarily the wood i t s e l f , i ts spectral density function is unknown. What is known, however, is 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 is referred to as the "cr i t ica l speed theory". This theory has been veri f ied with circular saws in their working environment?4 The iload used in the following development is taken to be P cosoot, since a stat ic load may then be taken as a speci f ic case of a more general result . The energy method is used here to determine the forced response of imperfect disks. When the effects of imperfections are neglected the response of a perfect disk is obtained. While i t is 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 igni f icant ly dif ferent. NNdn-rotatiiig Disk For an imperfect disk, the free vibration in the n nodal diameter modes is taken to be: (11.46) = $ n Rn c o s ( n e _ aj w n 2 = \ R n s 1 n ( n e " Pn> where $ and Y are the normal coordinates previously denoted by <j> and n n n r n The kinetic and strain energies of the non-rotating disk, from equations (11.32) are (11.47) 2T = A i 2 + B i 2 n n on n n 2Un = A„ :#2 + B„ i 2 n n n n n The damping of this system is known to be very small. Experi-mental resul ts , given in a later chapter, verify that the damping may be neglected when considering the forced response except when the disk is osci l lated at very close to one of i ts natural frequencies. In order to provide a theoretical basis for this experimental resul t , damping wil l be assumed to be viscous, in which case the energy dissipated may be ex-pressed as: (11.48) 2Dn = G n i * * + G n o <52 v ' n ni n m n In using Lagrange's equation i t is necessary to determine the generalized force associated with the load P cosieaYt. The generalized force is the quantity selected such that the pwoduct of this quantity and a virtual change in the generalized coordinate is equivalent to the virtual work done. Since each configuration from equation (11.46) behaves independ-ently there wi l l be two generalized forces associated with the n nodal dia-meter response. The load P cos cot is located at ( r p , e p ) . The displacement of the load due to a motion in the f i r s t configuration i s : w , (r . e j = « R (r)) cos (ne„ - a ) ni v p p n n v p' v p n' The work done <5iw"ni during a virtual change in $ n is given by: 6¥„ = P cos cot R (r ) cos (ne„ - a ) 6 $ ni n p' pp n' n and s imi lar i ly for sV 2 . J n2 -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 p ) sin (ne p - PR) cos cot Lagrange's equations are: d /3T I 3D A 3U M I J T j Wn + 3T n = M i » n/ n n (11.50) dt I 3¥ h-/ + . 3 D . 3U u n. n which is the standard form for a viscously damped forced system. Solving for $ n and ? n and substituting into equations (11.46) results in : 47 (11.51) w, P Rn (r ) R n(r) cos (ne - aR) cos (ne - a n) cos (cot-nn) ni 2 . a , 2 n 2 P Rn (r p ) R n(r) sin (ne p- Bn) sin (ne - pn) cos (<ot-cn) B n N ^ n G ^ (q 2 - a , 2 ) 2 + Y"_n la, 2 where pg * A ^ j q {J = B n / B n and n = tan-HG. - 7 A j ( P 2 - a,2) C = tan-!(G /UB ) n n? n ( q | -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 is a response in both configurations, 1 ft i If, however, the load is located at e = f- a + £-)- or e = g / n , there p n n 2' p n wi l l be no response in configuration one or two, respectively. These are the locations of the free vibrations nodes of these configurations. Re-cal l ing that the nodes of one configuration are located approximately at the anti-nodes of the other (a n~ B ), the maximum amplitude of the res-ponse of one configuration is obtained when the amplitude of the other is a minimum. It* is also apparent from equation (11.51) that i f the excitation frequency is P n(q n)> the response wil l be almost entirely in 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 val id i ty of equations (11.51) is examined, and when the extent to which a disk is imperfect is to be determined. One check of equations (11.51) that does not require a physical experiment is to observe the result when the disk is considered to be per-fect . The vibration must be symmetric about the load irregardless of i ts locat ion, with an anti-node located at the point of application of the load. If the disk is perfected = 3 , A * B , p =q and G = G . The n n n n n n n ni nz n nodal diameter response may then be written as: w, n + sin (ne - a n) sin (ne As can be seen, the response is as expected. It is also interesting to note that the response of a perfect disk is essential ly the same as that of an imperfect disk when the load is 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 > p n and G n caused by the presence of imper-fect ions, and also because a p is only approximately equal to ^ for an imperfect disk. Rotating Disk To fac i l i t a te the description of the forced response of a rotating disk, several terms must be defined: 1. fixed vibration: This is the usual form of the free vibration of a disk, given, for example, by: w = A R sin ne cos at n n n where a is the frequency. The nodes, located at e = 0, ^ p . . n are rotating with the disk. This vibration may be written as: (11.52) wp = h A n Rp [sin (ne - at) + sin (ne + at)] where the forward and backward travel l ing components are apparent. In non-rotating coordinates, this appears as: (11.53) u n = h A n Rn [sin (nnt - ny - at) + sin (nn t - ny + at)] 2. t ravel l ing waves: These waves, backward and forward t rave l l ing , are identical in form to the backward and forward travel l ing com-ponents described above. However, the two components of the fixed vibration are of equal amplitude, whereas a single wave may exist by i t s e l f . A backward travel l ing wave is given by: (11.54) w = A R sin (ne - at) v ' n n n u„ = A„ R„ sin ( nnt - ny - at) n n n and a forward travel l ing wave by: (11.55) wn = A n Rn sin (ne + at) u n = A n R n s i n ^ n f i t " n y + a t ^ 3. steady deflection: If the rotational speed a and the frequency a are such that no. = a, the backward travel l ing component or back-ward travel l ing wave becomes stationary in space. For example, substituting a = no, into equation (11.54) for u p y ie lds : (11.56) u = -A R sin n Y n n n This is not a function of time. If the steady deflection is a result of a backward travel l ing component becoming stationary in space, the forward travel l ing component wi l l be travel l ing at twice the disk speed, yielding an observed frequency of f = 2na . The method of determining the forced response of a rotating disk is very similar to that for a non-rotating disk. The kinetic energy of the transverse vibrations is the same in both cases. The potential energy U is of the same form, except the coefficients A and B are now n n n functions of ft due to the membrane stresses. The damping is a par t icu lar i ly d i f f i c u l t problem since the vibration wi l l suffer signif icant windage at high rotational speeds. A l -though viscous damping has been assumed in the past, 6 ' 1 5 i t is unjust-i f i ed since, for example, the windage suffered by a backward travel l ing wave is s igni f icant ly less than that of a forward travel l ing wave. As well as being a function of the transverse veloci ty , the damping wil 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 is 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 wi l l be neglected in the theory that follows. for a non-rotating disk except that the location e p of the load varies. It is assumed that a = £ and this angle is taken to be zero in the disk n n coordinate system. Since both origins are coincident at t = 0 and the load is taken to be located at y = 0, i ts disk coordinates are ( r p , fit). The generalized forces are then: The generalized forces may be determined in the same way as which may be written as: MT. = h P R N (r p ) [cos («•;-+ nfi)t-t+cos (« M2 = h P R N (r ) [sin (10 + nf i ) t^sin ( u nfi)t] nfi)t] Applying Lagrange's equations, the response is found to be: ( I I . § 7 ) cos (a) + nfi)t w = h P [ R (r )/A n] R (r)cdsne ni L n v p " n J n v ' ' [p 2 - (oi+ nfi) 2 ] + cos (to- nn)t , [ p 2 - (to- nfi) 2 ] V B * P C ^ ( r p ) / B n ] R n ( T ) S i n n 8 sin (co + nO)t Cq 2 - (to + nfi ) 2 ] - sin (to'-'rifl)t [ q 2 - (ai- nn) 2 ] The natural frequencies p n and q n here are functions of the rotational speed n, since they are obtained from the rat ios: The coefficients A n and B p are not functions of the rotational speed, so i t can be seen from equations (11.57) that i f the natural frequencies are known either from experimentation, or approximately from equation (11.40), that the only unknown effect of rotation is on the radial functions R n ( r ) . When = 0, equations (11.57) reduce to the previously developed expression for 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-dit ion is p = u . n However, when the disk is rotating there are four possible resonance conditions, given by: Pn = A R / S , n q 2 = B / B" M n n n w = P n " 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 in f in i te at any of these resonance frequencies because damping has been neglected, i t can be seen that the response near resonance,when u ~"P n * nfi-or oo - q n ± nn, is the shape and approximate frequency of the free response of the disk. Since circular saw instab i l i ty is known to be caused by a stat ic load, the response to such a load wil l now be discussed in deta 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 = P F cos ne cos nnt w„ = P F„ sin ne sin nnt n 2 n 2 R n(r ) R n(r) where F„ = -"—P- — "i A [p 2 - (nn) 2 ] n L f n v ' R (r ) R (r) and F _ = n P , n -_ n 2 B n [ q 2 - (nn) 2 ] Resonance occurs when p = nn or q = nn. If a non-rotating disk is r n n 3 subjected to a stat ic load, and the rotational speed is then increased, the f i r s t speed at which resonance occurs is 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 travel l ing component of a free vibration would be stationary in space; that i s , when f = 0. If the disk is perfect, p n = q n and A n = B n . In this case the response i s : w = w + w n ni nz = P F [cos ne cos nnt + sin nesin nnt] n = P F n [cos (ne - nnt)] This is the expression for a backward travel l ing wave (see equation 11.54). Substituting e = ntv--Y> t n e response in space-stationary coordinates i s : (11.59) u p = P F n oos ny This backward travel l ing wave, being fixed in space, is what has been defined as a steady deflection. It is travel l ing in the disk but i ts speed at any rotational speed is 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 travel l ing component of the free vib-ration in this mode. This is the usual condition for resonance; the system is forced to respond at the rate at which i t does freely. When the effects of imperfections are included, the response to a stat ic 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 2 n 2 The f i r s t term here is a fixed vibration of frequency nn and the second term is a backward travel l ing wave stationary in space. The frequency of the fixed vibration is such that i ts backward travel l ing component is also stationary in space, thus contributing to the steady deflect ion. Equation (11.60) may therefore be written as: w„ = h P (F n - F ) cos (ne + nftt) n n i ri2 + h P (F„ + F ) cos (ne - nnt) Toca space-stationary observer this appears as: (11.61) u n = h P(F n - F ) cos (2nnt - n Y) n n^ ua + h P(F + F ) cos ny The observed frequency of the forward travel l ing component is 2no,, as previously discussed. If p n is considered to be the lower of the two natural fre-quencies of the n nodal diameter configurations, when no is less than P n , which is the range of ' (11.61) may be written as: rotational speeds of interest here, equation u = h P n 2 1 -\ /Bn " B n \q* - (no)' F cos (2nftt - ny) + %PP B n v^ n (nn) : F^ cos ny As can be seen, the amplitude of the forward travel l ing component relative to that of the steady deflection increases as the c r i t i ca l speed is approached. Although the amplitude of the forward travel l ing component never exceeds that of the steady vibrat ion, this theory, which neglects damping, predicts that they are of very nearly the same ampli-tude close to the c r i t i c a l speed. Recalling that a vibration fixed in the disk is composed of forward and backward travel l ing 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 entirely of a -fixed vibrat ion. This contrasts with the response of a perfect disk, where only a steady deflection is present. It must be remembered, however, that a fixed Vibration is moving in space at the disk rotational speed, whereas the steady deflect-ion is stationary in space. It is expected therefore that, due to windage the amplitude of the fixed vibration relative to that of the steady deflect-ion wil l be s igni f icant ly less than that predicted by the above theory. This is confirmed in a later chapter which offers experimental ver i f icat ion of the theory which has been presented here. II.7 Comments on the Theory and i ts Applications The responses of non-rotating and rotating imperfect disks to a transverse point load, given by equations (11.51) and (11.57), were determined in order to predict the behaviour of a c i rcular saw in i ts op-erating environment. Although such a model neglects many factors, the theoretical results obtained have been veri f ied in sawing operations as being fa i r l y accurate in predicting certain aspects of a saw's behaviour, most s igni f icant ly the c r i t i ca l speed . 2 4 One of the major advantages of of this model is that those factors which are important in determining the saw's behaviour are easi ly ident i f ied. Several of the areas of current research on saw vibrations are based on concepts which were d is -cussed in the rotating disk theory. Current Research One of the most common methods of reducing the transverse motions of c i rcular saws is known as tensioning, having been used in some form ffior approximately.100 years. Tensioning is the process of altering the saw membrane stresses to increase the c r i t i ca l speed of the saw. The membrane stresses for an i n i t i a l l y stress free saw are due to rotation only. The maximum rotational speed of the saw must be less than that speed at which a stat ic load causes resonance. When rotat-ional ly induced membrane stresses only are present, this may be approxi-mated by: Q 2 This speed is dependent on the coeff icient Vn which is a measure of the 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 ca l speed. One method of tensioning involves p last ica l ly deforming the saw in compression in a narrow annul us, indicated by the dashed line in figure 7. When the saw is not rotat ing, the radial stresses wi l l be com-pressive throughout, while the hoop stresses wil l be compressive on the inner portion of the saw, and tensile 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 ts rated speed a l l membrane stresses are tens i le , although not of the same value as when the saw is not tensioned. Since the magnitude of the radial stress is reduced by this process, those modes whose membrane potential energies of deformation are due primarily to the radial stress wil l have reduced natural frequencies. These modes are those consisting of a low number of nodal diameters. 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 wi l l have an increase in their natural frequencies. With the correct amount of tension i t is possible to i n -crease the natural frequency of that mode which determines the c r i t i ca l speed of the saw. Figure 7 Membrane Stresses in a Non-rotating Tensioned Disk The same result may be obtained by another method which is less common. If the saw is heated at the co l lar a thermal gradient wi l l exist where the rim is at a lower temperature than the inner portion. This gradient may be selected such that the membrane stresses are similar 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 pract ice, however, there are unknown factors which influence the optimum amount of induced tension, such as the heat generated by cutt ing. It is necessary that any method by which the des-ired amount of tension is determined be f lex ib le to allow for the chang-ing conditions of the operating environment. A second major direction of research, altering saw geometries, has recently become of interest theoretically due to the ava i lab i l i ty of such techniques as the f in i te element method. One of the most common departures of saw geometries from those of perfect disks is the presence of holes and radial s l o t s . Slots are part icular ly common for two reasons. F i r s t l y , they inhibit the motion of waves travel l ing 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 s lo ts . Although holes and slots cannot be considered small imperfect-ions, they have the same effect as small imperfections on the amplitude of the steady deflection. It is possible, with the proper selection of holes and s l o t s , to reduce the amplitude of this deflection. An approach cur-rently being persued is the selection of the number, size and locations of holes and slots to optimize this amplitude r e d u c t i o n . 1 3 ' 1 4 As is 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 ver i f ied under controlled experi-mental conditions, ver i f icat ion in the f i e l d , which is the ultimate test , has in the past proven d i f f i c u l t to achieve. Saw Behaviour i h i ts Operatirig Eiivirdiiment It is d i f f i c u l t to evaluate the performance of a c ircular saw in i ts operating environment by monitoring i ts transverse displacement. A more direct and simpler method is to observe the quality of the wood cut by the saw. This is in fact the method i n i t i a l l y used to verify the c r i t i c a l speed theory . 2 4 It may be desirable, however, to monitor the saw's displace-ment. This displacement may, for 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 cu l t i es 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 stat ic component, which results in a space-stationary backward travel l ing 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 poss ib i l i ty of an observable response at the natural frequencies due to random excitat ion. 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 la r i t y . In-cluding the zero frequency responses, at any rotational speed the total response consists of responses at eleven different frequences. Since there 61 wil l be responses in other modes as wel l , i t can be seen that evaluating the behaviour of a saw at a particular rotational speed would be very d i f f i c u l t . 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 carefully controlled and measured, and the rotational speed set at any desired leve l . It is then possible to investigate the val id i ty of a theory under conditions more closely resemb-l ing those on which the theory was based. The next chapter describes exper-iments which were conducted to verify the theory which was developed in this chapter. I l l Experimental Veri f icat ion of the Theory of Rotating Disks Experiments were devised to verify the theory presented in the previous chapter. The outcomes of this theory were the forced responses of non-rotating and rotating disks, given by equations (11.51) and (11.57). Before these results could be ver i f i ed , i t was necessary to conduct several preliminary experiments. For each result presented both the experimental and the theoretical means used to obtain the result are described in de ta i l . III.l Experimental Equipment The disks used to obtain the experimental results were prepared from steel blanks from which c i rcular 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 disks, only the results for disks C and D are given here since the other disks simply supported the results obtained with these two. Al l disks possessed no large imperfections and were complete except for a central hole approximately one inch in diameter. The collars used were three inches in diameter. The col lar assembly was press-f i t ted onto the rotor shaft of a one-half horsepower Reliance D.C, motor. The rotational speed was in f in i te ly 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 d ig i ta l l y . The transverse motion of the disks was measured with space-stationary Bentley eddy current proximity sensors, and their accompany-ing power supply and drivers. Two sensors were cal ibrated, although most tests used only one. The calibration curve of this sensor is shown in figure 9. This curve was obtained with the use of a dial gauge. •2.' 4 6 8 10 Output Voltage (volts) Figure 9 Sensor Calibration Curve The proximity sensors were positioned approximately 0.10" from the surface of the disk, giving a l inear response range of nearly ± 0.050". It was possible to position the sensors at any desired loc-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 sinusoidal ly. This is a single channel analyzer. It was used in close conjunction with a Telequipment model DM64 dual chan-nel osci l loscope. The two channels of the oscilloscope allowed a direct observation of the phase differences between either the load and the res-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 disk, at a radius of seven inches. Power to the electromagnet was supplied by a power amplif ier, 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 Piezoelectric 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 con-tinously 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 experiment-a l ly because the presence of damping, even though very small , created d i f f i cu l t i es in obtaining accurate measurements. Al l results given here are for the forced responses of the disks. It was desirable to apply three types of loads to the disks; a stat ic load, a sinusoidal load and a random load. The stat ic load was achieved simply by applying a D.C. voltage to the electromagnet. A sinu-soidal load, however, cannot be obtained with the use of only one electro-magnet. When a load such as B cosoot was desired, this input was added to a D.C. voltage of magnitude P, yielding a resultant load of P(l+ cosut). Since the applied loads were of a magnitude such that the response was within the l inear range, i t was possible, in 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 load, but since a random load was used only for ident-i fying resonance frequencies.,this signal was not of f -set by a D.C. voltage. III.2 Forced Response of a Non-rotating Disk The response of a non*rotating disk in i ts n nodal diameter modes to a load P costot is given by equation (11.51). The total response i s : (III.l) W = s n w n i + V In this section i t wi l l be shown that the observed amplitudes and shapes of the response of a disk agree very closely with those determined theoret-i c a l l y . The f i r s t step in obtaining this ver i f icat ion was to accurately determine the disk natural frequencies since the amplitude of the response is extremely sensitive to the differences between the excitation frequency and the natural frequencies. In order to determine the natural 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 Frequencies (Hz) Experimental Theoretical 0 54.0 48.8 1 34.8, 35.2 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 is excited at a frequency u> = p^, and the sensor is located at the point of application of the load, the responses in the n = 1 configurations are: 67 P fR (r ) 1 2 c o s 2 ( e e - « ) cos (wt - n ) wn = 1 i v p'J p 1' 1' G p 11 H i P lX (r p ) ] i s i n 2 ( e p - cos (at - ^) ^7 ( q i - p i ) 2 + ( ^ 2 p 2 ' The response in these two configurations wi 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 that, except when the load is applied at or near the node of configuration one, the response in this configuration wi l l be much larger than in the other. The natural frequencies, p ' - a n d ^ , of disk C were seen to be 34.8 and 35.2 Hz. The frequency of excitation was set i n i t i a l l y at 34.8 Hz. , and the load and proximity sensor were located at some arbitrary angle,wwith the amplitude of the response being noted. This was done at 15° intervals around the disk. The excitation frequency was then changed to 35.2 Hz. and the procedure was repeated. The results are shown in figure 10. The origin of the coordinate system has been selected such that the anti-node of configuration two is located at e = 0. As expected, the node of configuration one is located approximately halfway between that of configuration two. In addition, the shapes of the two curves are very nearly c o s 2 ( e - a n ) and s i n 2 ( e - e n ) . The deviation from these shapes is due to several factors; the load covered a f in i te area rather than being a point load, there were responses in modes other than the one being resonated, w 12 and any change in the shape of the response due to imperfections was neglected theoret ical ly . 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 calculate the theoretical response of the disk i t is necessary to know the dissipation coefficients Gn icmd G n 2 . These could be determined approximately from equations (III.2) and the information presented in figure 10, but several practical problems ar ise . Most s ign i f icant ly , there is a dependence on the radial functions and these functions have yet to be ver i f ied . There is also a dependence on the magnitude of the load P, and i ts value at resonance must be so small to prevent a non-linear response that i t cannot be accurately measured. Both of these problems are avoided by the method described below. Damping Coefficient If i t is assumed that a 2 = s 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 [R i ( r p ) ] cos(o)t-ni) A i P : / r e 22 + P r Here 5 is the damping factor; the ratio of the actual damping coeff icient of the disk to i ts c r i t i c a l damping c o e f f i c i e n t . 2 2 The response wil l be predominantly in configuration one i f the excitation frequency is close to p j , and the load is applied at an ant 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 = U J^, 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 ines ) . For com-parison purposes the theoretical results for k= 0.00, 0.01 and 0.02 are also plotted (sol id l ines ) . 0.95 1.00 1.05 Excitation Frequency w/p Figure 11 Relative Amplitude versus Excitation Frequency It can be seen that the damping coeff icient is less than 1% of c r i t i c a l . The response at frequencies above the resonance frequency is larger than expected due to responses in modes other than the one consid-ered theoret ical ly. This experiment was also performed in- the n = 2 mode. Here i t was assumed that the disk was perfect since the natural frequencies p 2 and q 2 were indistinguishable. The damping coeff icient 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 is i ts 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 frequency is within 5% of the natural frequency. The second effect of damping is the "smoothing out" of the phase change as the excitation frequency is varied from lower to higher than the natural frequency. The phase angle between the load and the response i s : Unless the excitation frequency is within 5% of the natural frequency, the difference in phase for the two cases g = 0 . 0 0 and 5 = 0 . 0 1 is less than 1 1 ° . Since neither an amplitude error of 4% nor a phase angle error of 1 1 ° are s igni f icant , the system damping may be neglected theoretical ly when the excitation frequency is not within 5% of a natural frequency of the disk. When damping is neglected, i t is possible to formulate approximate expressions for the radial and angular prof i les of the disk using the rad-ia l functions obtained by the Rayleigh-Ritz method. Radial and Angular Deflection Profi les When a disk is excited by a load B cos ut, the total response is the sum of the responses of each mode. This requires that the location of the nodes of each configuration be known. It has been shown, however, that <xn = ^approximately, and i f the excitation frequency 10 is such that any difference between p n and q n is negligible the theoretical imperfect disk response is equivalent to that of a perfect disk. This was the ap -proach taken when the radial and angular prof i les were determined for disk C. The excitation frequencies were 0 Hz and 90 Hz, which are sui 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 prof i le 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 0(r) Ri (r ) Rx (r) w(r) = - 2 P- + P- + — A 0 (P 2 -co2) Ax (pf -co2) For to = 0 Hz, the predominant response was theoretical ly 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 ins igni f icant . These profi les are shown in figures 12 and 13, where the s'olid lines are the theoretical prof i les and the dashed lines are those measured. Figure 12 Radial Prof i le at 0 Hz. (Disk C) • 002 V P = 0.51 lb . io = 90 Hz. .001"j 2 4 Radius (inches) 6 8 Figure 13 Radial Profi le at 90 Hz. ((Disk C) It 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 is re-duced by roughly 80%. This is due to the much greater strain energy per unit deflection for the higher modes. The angular prof i le was measured for a s tat ic load (0 Hz). The sensor was located at r = 8.5" and measurements were taken at 30° intervals around the disk. The theoretical angular prof i le is obtained from equation (11.51) which, neglecting damping i s : To i l lust ra te the relative magnitudes of the responses in the various modes, the theoretical responses for w0 , w2 , w2 and w3 are shown in figure 14. The predominance of the n = 1 mode can be observed. w(e) = PRo (r p ) RQ (8.5) A 0 (Pi| - *>2) Ai (p\ - w2) .0050" .0025" i 180 240 300 60 120 180 .0025" T .0025" T Figure 14 Theoretical Angular Profi les at 0 Hz. The total theoretical prof i le w is plotted in figure 15 (sol id lines) along with that measured (dashed 1ine). .010" T Figure 15 Angular Profi le at 0 Hz. (Disk C) Although the experimental results for one disk only have been presented here, these experiments were performed on the other disks as wel l , yielding results which also agreed very closely with the theoret-ical 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). Because the response to a stat ic load is of particular interest, i t is this type of load for which the experimental results are pre-sented. With thas load the theoretical response is given by equation (11.58). As in the case of the non-rotating disk, i t is 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 is by random excitat ion, as was done for the non-rotating disk. This ws the method that was used to obtain the results presented here. The second method is somewhat more time consuming but is necessary under certain conditions. If, for example, the disk either contained s lo ts , was not completely f la t or was not running total ly in a plane perpendicular to the shaft a signif icant 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 ident i fy . However, i f at a part icular rotational speed a sinusoidal load was applied and the frequency swept over the range of interest, from equation (11.57) i t can be seen that a large response wil 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 is clearly observable at a l l rotational speeds. An approximate relationship for the natural frequencies was previously given as: P 2 = P 2n + V ft2 ni HnB yn For this disk, the values of V l 9 V2> and V 3 are 1.34, 2.91 and 5.90 res-p e c t i v e l y . 1 8 At 2500 r.p.m. this approximation yields frequencies which are s igni f icant ly higher than those measured, being in error by .6.1%, 7.6% and 6.5% for the one, two and three nodal diameter modes. The size of these errors is attributable to the fact that for this disk and col lar at 2500 r.p.m. the potential energy of the membrane stresses is not small in relation to that of the bending stresses, which is a requirement for an accurate approximation. —i 1 i i '. i 500 1000 1500 2000 2500 Rotational Speed n (rpm) Figure 16 Observed Resonance Frequencies versus Rotational Speed It was stated in Section II.4 that i f V > n 2 , there would be no poss ib i l i ty of an observed frequency of that mode becoming zero. In this case, 1 and i t can be seen from figure 16 that the lower branch of the observed frequencies of this mode wil l not intersect the horizon-tal axis. For this disk the c r i t i ca l speed is determined by the n = 2 mode, since i t 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 diameter response to a stat ic load located aty= 0 is given by equation (11.61). For a perfect disk the total deflection at the load i s : u(n, - z P ERn ( r p n 2 A n [p 2 - (nn)2] It is not immediately clear from this expression whether the rate of change of the amplitude with respect to the rotational speed is positive or negative at any part icular speed. 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 for rotational speeds varying from 0 to 2500 r.p.m. A theoretical response was calculated by assuming the radial functions were independent of the rotational speed, and by using the natural frequencies given in figure 16. The experimental results (dashed line) and theoretical results (sol id line) are shown in figure 17. Figure 17 Displacement versus Ro ta t i ona l Speed (Disk C) I t can be seen tha t up to approx imate ly 1500 r .p .m. the maximum d e f l e c t i o n remains f a i r l y cons tan t . A l though not shown on f i g u r e 17, t h i s r e s u l t s from the f a c t tha t the d e f l e c t i o n s i n the n = 0 and n = 1 modes are decreas ing wh i le those i n the n = 2 and n = 3 modes are i n c r e a s i n g . The angu lar p r o f i l e may a l s o be determined from equat ion (11 .61 ) . For a p e r f e c t d i sk w i th the l oad l oca ted a t y = 0 and the sensor l o c a t e d a t r = 8 . 5 " , i t i s g iven by: u{Q,y) = I P R n (V R n (8 .SJ cos ny A n [ p 2 - ( n o ) 2 ] Th 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 . and the r e s u l t s are shown i n f i g u r e 18. Figure 18 Angular Profi le at 2400 RPM (Disk C) The predominance of the n = 2 mode is observable. This prof i le may be compared with figure 15, which is for the same disk and load, except at zero rotation speed. In the non-rotating case, the " t i l t ing mode", that i s , the n = 1 mode, predominates. The theoretical calculations done assumed that this disk was per-fect . While this is not s t r i c t l y true, at rotational speeds of less than 2500 r.p.m. the small differences between the natural frequencies of the configurations is inconsequential. One of the effects of imperfections is the existence of a 2nfi. observed frequency. No such response could be det-ected for this disk at these rotational speeds. These same experiments were conducted with disk D which is much thinner than disk C. The steady deflection as a function of the rotational speed is shown in figure 19, where the dashed l ine is the experimental result and the sol id l ine is that determined theoret ical ly . CU 4-> c CD E CU o <a r— Q -.010 .005 h P = 0.30 lb. 500 1000 1500 Rotational Speed n (rpm) 2000 2500 Figure 19 Displacement versus Rotational Speed (Disk D) It can be seen that up to approximately 1200 r.p.m. the total deflection decreases with increasing rotational speed. Since this disk is quite thin (0.050"), the increase in the n = 0 and n = 1 natural f re-quencies due to an increase in the wotaitnonal stresses reduces the admit-tances 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 deflection. 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 prof i le is very similar to that shown in figure 18, since the c r i t i ca l speed of this disk is also due to resonance of the n = 2 mode. The angular prof i le was measured at 1955 r.p.m. and the deflect-ion under the load was found to be 0.014". However, at this speed a s igni f icant vibration at approximately 130 H z was observed, with an amp-litude of 0.0035". The frequency of this vibration is four times the disk rotational speed. That this vibration is due to the forward travel l ing component of a fixed vibration is readily ver i f ied by the use of two proximity sensors, located at y^ and Yg in the space-fixed coordinate system. From equation (11.61) the forward travel l ing component as measured by each proximity sensor is given by: These two signals may be observed simultaneously on a dual-beam osci l loscope, appearing as shown in figure 20. Here the sensor B is assumed to be in the positive y direction with respect to sensor A. Since the time t d is ^ - E Y B - y^l, the phase difference between u n f l and u n R expressed as a fraction of the wavelength u nA u nB is 2I [&B " Y A ] ' 83 time Figure 20 Forward Travell ing Component Observed With Two Sensors At 1955 r.p.m. the two natural frequencies p 2 and q 2 are 65.9 H z and 66.9 Hz- The two nodal diameter response at this rotational speed may therefore be written as: In order to evaluate this expression a knowledge of the nature of the im-perfections is necessary so that the ratio A 2 / B 2 can be determined. In general, the imperfections wil l effect both the kinetic energy (Ap f B n) and the potential energy (A f B"n). However, by assuming f i r s t one type of imperfection and then the other, the amplitudes of neither the forward travel l ing component nor the steady deflection are s igni f icant ly altered. For small imperfections, i t may be assumed in this expression that A 2 = B 2 . + y> 1 + 0.42 It can be seen that due to the imperfections, the amplitude of the steady deflection is substantially decreased. At 1955 r.p.m. this decrease is roughly 25% that of the perfect disk steady deflection in the n = 2 mode. This aspect of the effect of the imperfections cannot be direct ly ver i f ied since the steady deflection due to a single mode cannot be accurately isolated from the t o t a l . The amplitude of the forward travel l ing component, though, was measured (0.0035") and this compares quite well with the theoretical value of 0.005". It must be remembered that a vibration fixed in the disk is moving relative to the a i r , in this case in excess of 150 feet per second at the rim, and i ts amplitude is expected to be s igni f icant ly 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 fa i r l y close agreement with the theoretical pred-ic t ions . 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 stat ic load. IV Closing Remarks A theory has been presented here for the free and forced responses of nonSrotating and rotating centrally clamped imperfect disks. In i t ia l ly the free response of a non-rotating centrally 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 l ines exist at definite locations in the disk, and the natural frequencies of the two con-figurations of a nodal shape are dif ferent, the extent to which depends on the nature and magnitude of the imperfections. To obtain a reasonable model of a c ircular saw i t was necessary to consider the effects of rotational stresses on the disk'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 rcular saws, this approximation yields accept-able results. In determining the forced response of a disk, 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 result . The primary use of the theoretical non-rotating disk res-ponse was for interpreting-experimental results to determine the extent to which the disks were imperfect. 8 6 The rotating disk response to a transverse load was investigated in de ta i l . 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 similar except there were four resonance frequencies due to the frequency difference of the two configurations of each shape. Of particular interest was the response to a stat ic load since this is the basis of the c i rcular 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 simi lar . However, as the resonance speed was approached these responses became very dif ferent. The perfect disk modal response to a s tat ic load was shown to consist entirely of a backward travel l ing 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 ts backward travel l ing component was also stationary in space thus contributing to the steady deflect ion. The magnitude of the fixed vibration relative to that of the backward travel l ing wave increased as the resonance speed was approached. Near the resonance speed, this theory, which neglected damping, indicated that the response consisted almost entirely of a fixed wibpation, contrast-ing s igni f icant ly with the perfect disk resonance response. The major aspects of the theory presented were ver i f ied experimentally. The agreement between theoretical and experimental results was good con-sidering the theoretical approximations and experimental accuracy. 87 REFERENCES 1. Kirchoff, G.,"Ueber die Schwingungen einer elastischen Scheibe", Cre l le 's Journal, vo l . 40, 1850. 2. Rayleigh, J .W.S . , The Theory of Sound, New York, Dover Publications, 1945 3. Zenneck, J . , "Ueber die freien Schwingungen nur annahernd volkommener kreisformiger Platten", 1899 Anrialen der Physik, vo l . 67, p.164. 4. Lamb, H. and Southwell, R.V., "The Vibrations of a Spinning Disk", Proc. Royal Soc. of London, vol .99, 1921, p. 272. 5. Southwell, R.V., "On the Free Transverse Vibrations of a Uniform Circular Disk Clamped at i ts Center; and on the Effects of Rotation," Proc. Royal Soc. of London, vo l . «1)01 , 1922, p. 133. 6. 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 , vo 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 Discs", PhD Thesis, Department of Mechanical Engineering, University of Ca l i forn ia , Berkeley, 1963. 8. Dugdale, D. S . , "Theory of Circular Saw Tensioning", The International Journal of Production Research, vo l . 4, No. 3, 1966, p. 237. 9. Szymani, R., "Evaluation of Tensioning Stresses in Circular Saws", University of Cal i fornia Forest Products Laboratory, 1973. 10. Mote, C. D. and Nieh, L. T . , "Control of Circular-disk Stabi l i ty with Membrane Stresses", Experimental Mechanics,'vol. 11, No. 11, p. 490, 1971 88 11. Mote, C. D., "Free Vibration of In i t ia l ly Stressed Circular Disks" Trans. ASME, vol. ' 87B, p. 258, 1965 12. Nieh, L. T . , "Rotating Disk S tab i l i t y , Spectral Analysis and Thermal Ef fects" , PhD Thesis, Department of Mechanical Engineering, University of Ca l i forn ia , Berkeley, 1972. 13. Dugdale, D. S . , "Effect of Holes and Slots on Vibration of Circular Saws," Departmentoof Mechanical Engineering, University of Sheff ie ld, England. 14. Mote, C. D., "Stabi l i ty Control Analysis of Rotating Plates by Finite Element: Emphasis on Slots and Holes", Trans. ASME, vo l . 94G, p. 64, 1972 15. E l l i s , W. E . , "Active Electromagnetic Vibration Control in Rotating Discs" , PhD Thesis, Department of Mechanical Engineering, University of Cal i forn ia , Berkeley, 1976. 16. Radcl i f fe , C. J . and Mote, C. D., "On Line Control of Saw Vibration: Active Guides", Department of Mechanical Engineering, University of Cal i forn ia , Berkeley, 1979. 17. Vogel, S. M. and Skinner, D. W., "Natural Frequencies of Transversely Vibrating Uniform Annular Plates", Trans. ASME, p. 926, December, 1965. 18. Eversman, W., "Transverse Vibrations of a Clamped Spinning Membrane", AIAA Journal, vol . 6, No. 7, p. 1395, 1968 19. Eversman, W. and Dodson, R.O., "Free Vibration of a Centrally Clamped Spinning Circular Disk", AIAA Journal, vo l . 7, No. 10, p. 2010, 1969 89 20. Benson, R.C. and Bogy, D.B., "Deflection of a Very Flexible Spinning Disk Due to a Stationary Transverse Load", Trails ASME, vo l . 45, p. 636, 1978 21. Timoshenko, S. and Woinowsky-Krieger, S . , Theory of Plates and Shel ls , New York, McGraw-Hill, 1959 22. Thomson, W. T . , "Vibration Theory arid Applications", Englewdod C l i f f s , N. J . , Prent ice-Hal l , Inc. , 1965 23. Ritger, P. and Rose, N., "Differential Equations with Applications," New York, McGraw-Hill Book Company, 1968 24. Mote, C. D., "Verif ication of Saw Stabi l i ty Theories", Department of Mechanical Engineering, University of Cal i forn ia , Berkeley.
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The transverse dynamics of rotating imperfect disks Rosval, Gergory William 1981
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Title | The transverse dynamics of rotating imperfect disks |
Creator |
Rosval, Gergory William |
Publisher | University of British Columbia |
Date Issued | 1981 |
Description | The transverse motions of circular saws have undesirable effects on many aspects of circular sawing. Due to current high manufacturing costs, substantial savings may be realized if these transverse motions can be reduced. In this thesis a circular saw is modelled as a rotating imperfect disk acted upon by a transverse, non-oscillatory point load stationary in space. Such a model is known to accurately predict certain relevant aspects of the behaviour of a circular saw in its operating environment. Initially the free response of a non-rotating perfect disk is considered. This model is then refined by considering the effects of rotational stresses and small imperfections within the disk. The response of such a disk to an oscillatory load is determined, from which the response to a non-oscillatory load may be determined as a special case of particular interest. Experimental results are given which quantitatively confirm the theory presented. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-03-26 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0095520 |
URI | http://hdl.handle.net/2429/22703 |
Degree |
Master of Applied Science - MASc |
Program |
Mechanical Engineering |
Affiliation |
Applied Science, Faculty of Mechanical Engineering, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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