THE UNIVERSITY OF MICHIGAN COLLEGE OF ENGINEERING Department of Mechanical Engineering Student Project Report SPRING-DAMPER SUSPENSION SYSTEM ANALYSIS FOR HORIZONTAL AXIS WASHING MACHINES Brian Sowards ORA Project 371550 supported by: WHIRLPOOL CORPORATION BENTON HARBOR, MICHIGAN administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR June 1972

TABLE OF CONTENTS Page ABSTRACT iii INTRODUCTION 1 THE MATHEMATICAL MODEL 1 DERIVATIONS OF EQUATIONS OF MOTION 4 WALKING BEHAVIOR 11 SUPPLEMENTARY EQUATIONS 13 SUSPENSIONS DESIGN 14 BASIC PROCEDURE 18 EXPERIMENT 20 CONCLUSIONS 22 THE BELLEVILLE SPRING SUSPENSION 23 ii

ABSTRACT A major concern of washing machine design is controlling walking and load transmission to the floor. A suspension system must do this without large deflection. Given preliminary design data about a horizontal axis washing machine, a procedure was developed to design a spring-damper suspension system. Parts of the procedure were given limited testing and accuracy within 35 was found. Suggestions for improving the design procedure and suspension system conclude the report. iii

INTRODUCTION The horizontal axis washing machine, while having advantages over vertical axis machines, has some disadvantages. One of the problems encountered in these machines is controlling the dynamics of the spin cycle when the load in the basket is not evenly distributed. Machine movement (walking) and large load transmission to the floor (i.e., causing dishes to shake off kitchen shelves) are the two basic concerns of the spin cycle dynamics problem. There are at least two methods for dealing with these conditions. The first is to balance the basket during the spin cycle to eliminate the imbalance. This has been investigated by Whirlpool, and the last production model contained a balancing mechanism. Unfortunately, the mechanism was very expensive and therefore was discontinued. The second method is to suspend the chassis (i.e., springs) to absorb deflections without walking and transmitting excessive loads to the floor. This method is the subject of this report. A mathematical model of a horizontal axis washing machine is constructed and tested and a design procedure formulated. THE MATHEMATICAL MODEL A mathematical model of a spring suspended chassis was formulated. It describes the physical machine of Figure 1. This is a six-degree-of-freedom system with a rotating imbalance. Ordinarily a ground reference frame would be chosen at A. From A, a. vector Q describes the position of the machine, as represented by coordinate system B imbedded in the machine halfway along the drum's axis of rotation (or at any convenient point). The rotation of B relative to A is described by a transformation matrix (i.e., the Euler transformation). A vector D in frame B locates the imbalance in the basket. A vector describing the imbalance relative to ground, QD, requires three components of vector Q, a transformation matrix and three components of vector D. I decided that the machine could be modeled adequately witha simple set of linear, ordinary differential equations whose solutions could be obtained analytically. Although numerical methods were feasible, it is easier to work with an explicit expression of the form x = FUNCTION (machine parameters), where x is the machine displacement and the parameters are damping coefficients, masses, spring constants, etc. This form of equation is more flexible.

Machine Imbalance Drum r B\ Sori ng (Vertical) -z~..s^~f S i Damper l/11II//}/},/i////1i Figure 1 The mathematical model formulation was attempted using the standard reference frame, and was abandoned because the complexity of expressions of QD appeared to require a very tedious derivation of the governing equations. To simplify QD, the coordinate system A for the ground frame was to consist of three axes fixed to ground located in the center of the drum axis such that at static equilibrium the z axis runs along the axis of rotation of the drum. Coordinate systems A and B are superimposed at static equilibrium and vector Q becomes the deflection of the machine in the dynamic state. Since QI is small compared to IDI, RI, or other vectors of the problem (IQI.5 in., D| 15 in., R| 8 in.), the coordinate systems A and B are assumed superimposed DURING THE DYNAMIC STATE. This does not mean deflections are assumed to be zero; it is the deflections we wish to solve for. It means deflections are assumed not to influence the position vectors locating the imbalance masses. Note also, that the angles of machine rotation Ox, Oy, Oz are also assumed small to justify the above assumption. These assumptions will reduce the former vector QD to only three components (see Figure 2). This unusual approach is taken because the Lagrange equation will be used. 2

Oy Machine t Imbalance / )/L Drum -I D ) ex Figure 2.6 6(T-V 6(T-v) (whe T = kinetic i (1) where T = kinetic energy of system V = potential energy of system Q. = nonconservative forces associated with system This equation will require three vectors of the type QD to locate two imbalances in the basket and the center of mass of the machine in a six-degreeof-freedom system. These vectors must be used for the kinetic and potential energy of the system and therefore were chosen to be as simple as possible. The Lagrange approach with the first coordinate system would consume a much larger portion of the project time. The angular coordinates are the rotations observed around the individual axes if no other deflections were present in the system. Due to the deflections being small, this is a reasonable assumption which further simplifies the problem. 3

DERIVATION OF EQUATIONS OF MOTION The kinetic energy of the system (see Figure 3) is: 1.2 1.2 1 M2 1 I 22 + 1 I 02 T = -Mp + Mp +10 +-IO +-IO T 2 11 2 2P2 2 5 2 x x 2 yy 2 zz -~ -^ -+ A + A A A where r r3 = unit vectors Similarly P2 = r2+ r r4,r3 are fixed magnitudes on the basket: dr3/dt = 0 = angle of drum rotation relative to machine Figure 3 4

If the spring rates in the x and z directions are known in terms of the spring rate in the y direction, the potential energy of the x, y, and z coordinates is 1 2 1 2 1 2 V = k x +-k y +-kz (3) 2x 2y 2 z The rotational potential energy is 1 2k. 2e 1 2 V = L sin \ - k e (4) ~ 8 ii i 2 i Because deflections are small, it is assumed there is no coordinate coupling in the potential energy terms. Once T and V are known, the tedious job of differentialing T-V is done. If the damping coefficients in the x, y, and z directions are known, the nonconservative forces are: Q = -C x x Q = -C y y y Q, = -C z z z L2C + L2C z y y z 2x x L2C + L2C Q 2 XZ( — 6 ) -C 9'e2 v(y) e y Y Y L2C + L2C Qe = - yx x (Y 0) = -Ce () Th 2, Z The resulting differential equation of motion in the x-direction is: 5

m r + m r + M M + r - 1 xl 2 x 1 z x x z z x y y1 -m 2 +2)r 2 r - r I 2 l 2 z x x z z x y y 72 2 m2 Z x2 x x z z2 2y - M + ex- eez - Y + k = -Cx (6) z x z x y x x where 4x, $y, and $z are the components of the vector $. X R +x, Y R + y, Z =R+z (7) ~x ~y z where RXRy,Rz = components of R (constants) x,y,z = machine deflections Now rxl, ryly rzl, rx2, ry2x rz2 are the components of single vectors from the origin of the ground frame to ml and m2, respectively. This equation is still not simple enough for an analytical solution. The following assumptions are made: M > m1 + m2 0 = W > 0,,9,9,9 z x y x y z r r cos t (8) x The resulting equation is: 2 2 Mx + Cx+kx = mlW r cos Wt + m 2) r cos(ct +) (9) x x 2 This equation is solvable. The expressions in the other coordinate systems are of the same form. 6

m r + m r + MY - m + r - r - r 1 Y 2 x x1 -m 24 + r - r - r 2 z Y2 z2 y x x2 - M e + eYy ez e0 x + k Y = Q (10) z x Y y z z y y m r + m2r + MZ m10- m r rr 1 2 z x z x xxLj -2m 02 + 2)r - r - r 2 Lx z2 Z y y2 z x x2 dt I 9 +M (2T + Y2N - ZX - YX ) dt x + Mx (Z X)0 - z- yj 2 2 2 2 N + m 2r r + 2w2r r + m 2r r -2cw2rr 1 z1 1 3 2 z 2 2 z4 + (L2k + L k = (12) 4 \y z z2 x dt (fl + M V(Z + X2)3 - YZ6 - YX6 ) + 1 YL-k + L2k = Q (14) 4 y y xy y z 7 md~ + m2 +1 25 + y2r 2 2 1tzx z x y 2 z x zx 1 + - 2 2 2 +-xL +LkI =X ( + \LkLk' (14 +M +x - yz - Yx

These equations are suitable to numerical or modal solutions. However, assumptions similar to the x-direction were used to reduce the equations further. 2 2 My - Q + KY = mmwtr sin wut + m r sin(ot +A) (15) y y 1 2 Mz -Q + K Z = 0 (i6) z z 1 /2 2 2 I Q+( L k + L k)e -m D r2r sin cwt -m 2w rqr sin(wt+A) (17) 3r &eY 4 +kL W 4 ie -Q +I(~k~kL"~)o-m~w~rqr sinwcta (8 1/2 2 2 -m 2r r sin(wt +A) (18) 1-Q 2 2. (19) IZZ + 1 ky + L k = -MR (a) zz 9 4 xy y z y z z \ Because the force generated by the rotating imbalances acts directly through the center of the coordinate system, it is impossible for these forces to gena torque in the 9- direction. The forcing function in the ez direction is, therefore, implicit in -MR0-9. To show this, a free body diagram of the force on the machine (assumed only to include the effects of the rotating unbalance, i.e., spring force is small compared to this force) should be drawn. This force will act at a distance R = IRJ from the center of mass. By writing Newton's motion equations, one obtains F = M, T = -FR = Io (20) These equations yield 0 = -MRX/I. The O9 equation becomes 22 l 2 2 MR I -Q + f(Ly + kyL)9 = yx (21) z z 4 \xy y z I z The second derivative of the solution of the x equation is proportional to the forcing function of the 9^ direction. I is taken about the center of mass. 8

Furthermore, an important assumption must be included to get the last set of differential equations. That assumption is: R =R = 0 (22) x z i.e., the center of mass of the machine is on the vertical centerline of the machine (the y axis). These uncoupled equations can allow an unlimited number of imbalances to be accounted for. This is accomplished by a geometric exercise that reduces the effect of two imbalances into a single effective imbalance. In the uncoupled x-y plane (see Figure 4): a b cL 2 2 2 mw r sin ct + m2 r sin(ot +A) = mc r sin(ct +A) (23) -1/ 1 A = tan a b sin A tan C -b sin A (24) L sin[ALl y m2 r2/ pM Imbalance resulting from &A ~AL combination of mi and m2 mi Figure 4 In the same manner, the torsional effect of two imbalances can be resolved into a single effect. 9

2 2 2 mlC rlr5 sin ct + m2C r2r4 sin(cot + A) = r r r sin(ct +A) a b c9 (25) The above formulas apply again for this A9 and c. Plugging this result into the differential equations results in a set of six uncoupled equations with a single forcing function for each. The solutions of these equations are x = X sin(ct +a) (26) y Y sin(t t+ 2) (27) z = 0 (28) 0 = X sin(ot + C) (29) O = SS sin(wt +c ) (50) y y 0 = ( z sin(cut+C5) (31) where ai = phase angles, see equations (39)-(43) 2 mr c a X = - (52) J (4c ()2 + (4k - 2)2 2 mr wc Y a (33) j (4c c)2 + (4k - M)22 where mr C = the result of resolving the two imbalances a r = r = r, so m is an equivalent imbalance a 1 2 -mr r c02 x J (2COx )2 + (2kx - Icu2)2 -mr r cu2 ~ = ab (55) (2cO)2 + (2ke - iy2)2 10

mrb is the equivalent torsional effect Z = O (36) -mr R3? M /I ~ ay z(37) z j(2ez)2 ) + (2kz - I ) J(4cX )2 + kX M2)2 WALKING BEHAVIOR The mathematical model should also contain a routine for predicting if a washing machine will walk. On a smooth floor, an experimental chassis with a spring suspension (see Figure 5) was observed to begin walking with an oscillating sliding motion (all four feet moved the same distance in the same direction simultaneously). The machine did not pivot about one foot. It was assumed the total loads on the individual feet of the machine produce the same effect as the average load applied to all four feet simultaneously (because the feet are connected by a rigid structure). The problem then becomes: -u(vertical forces on the feet) > (horizontal forces on the feet) u frictional coefficient Lx \ fX \CMC,Foot of Machine Floor Figure 5 11

If this equation holds, the machine will not walk. Since the forces on the feet of the machine are functions only of the weight of the machine and deflections of the springs in the suspension, (F = kx), the walking equation becomes: -u"y Y - + > y+ + 2+ ezR y 4zx c y2y zC xc yy 2(38) Rc is defined in Figure 5. This is a time dependent equation. The quantities in it are defined / 4c c x = X cos(t +AL- a) a = tan - (39) 4k - M / \ x / dx x = - A defined previously dt L 4c y = Y sin(wt +ALT ) tan 4 4k - Mw y dy dt -l 2cQ XC 0) = ( sin(wot+A - ) 7 = tan - (41) 2ke - I o x x de a = X x dt 1 2cey C 0 = Q) cos(wt +A- ) 6 = tan y 2 (42) ty ty 2ke - I c y y dO y dt -1 2CQz2c 9 ~= z cos(ct +AL - r) = tan (43) 2kz -I c1W z z AL and A0 defined from equations (24) and (25). 12

SUPPLEMENTARY EQUATIONS In addition to the derived equations, some additional equations were needed to relate the mathematical model to the physical machine. In the book Mechanical Springs by A. M. Wahl, springs are treated as elastic columns in two cases of interest. The first case deals with the lateral spring rate (p. 70) (see Figure 6). k = lateral spring rate x 6 4 k = 10 d((44) c nD (204 h2 +.265 D L s!y is the vertical spring rate; the spring was designed for (Fy = kyY). ky h Y = 1.44 L.204 - +.26 (45) kx k D2 These equations are for round wire springs where E = 30 x 106 psi and G = 11.5 x 106 psi or E/G = 2.6. 8 H- Lateral Deflection \\ I — ^ Kx, Lateral / Ky gv Rate Elastic,//../>. Column Figure 6 Observation of the experimental chassis on springs showed still another problem area. Certain springs are incapable of supposing the weight of the machine without buckling. This instability occurred if the spring was long or the spring rate, k, was small. The second case is where Wahl provides two equations to predict if a spring will buckle under a given load (pp. 279-82). 15

6cr =.812 [1 ~+1 - 6.87(D/o)2] ~0 &cr G d 4o Per = 6 G (46) 64 r n where G = modulus of rigidity (steel - 11.5 x 106 psi) ~o = free length of spring n = number of active coils d = wire diameter r = mean radius of coils, D = 2r h = compressed length of spring s C = given by graph in Mechanical Springs, p. 74 L Pcr =vertical load on springs that produces buckling 6cr = lateral deflection which produces bucling with load Pcr Because coulomb damping is used more on washing machines than viscous damping, one needs an equivalent viscous damping coefficient developed from the coulomb damping factor. This exercise is carried out in Mechanical Vibrations by Tse, Morse, and Hinkle on p. 177. The result is: Ceq = 4Fk(l- r2) (47) w i Fo 1- (4F/J Fo)2 where r = frequency ratio = w/un; F is the frictional resisting force of the damper, and Fo = meo2 for a rotating imbalance. Also e = eccentricity. SUSPENSION DESIGN With the mathematical model of the suspension complete, how does one use it? The pupose of this project originally was: Given the machine parameters, design a suspension that does not walk, does not transmit vibration to the floor, and deflects within allowable limits. 14

So a design procedure was formulated to try and do this. The first step was to try to show that for the given machine parameters there is a single function of the following form: function value = function (displacement, walking behavior) (48) This function value should have a maximum or minimum value when the best compromise is found between walking and deflection behavior. The damping coefficient, c, and spring constants, k, would be the variables. Optimization routines generally take a complicated function and find the maximum or minimum points of the function. It was thought that by combining the walking equation and displacement equations directly, the result would be something of the form function value = A +B(deflection) + C(walking behavior) A,B,C = constants m,n = powers (49) and this could be optimized. The problem was how to find AB,C,n, and m. This is when the problem of spring buckling was first noted. This would have to be included too. Formulating this single function seemed difficult, the task was given up and another method sought. The next step was to consider optimizing the deflection alone and using walking behavior and stability equations as constraints. This would mean writing and optimization computer program from scratch, as no optimization computer programs I found could handle constraints. This appears reasonable, until you realize that the deflection equations can be optimized by inspection. The larger the values of c and k (the operating point is above the natural frequency), the smaller the deflections. The result of optimizing deflection would be k and c values so large, that if they would be any larger, the machine would walk (see Figure 7). Washing machines are not to be designed on such "borderlines." You need a factor of safety. Optimization was discarded as a design procedure. The only alternative was to admit defeat to the original objectives of the research. The following compromise can be made. For any given machine parameters, determine the values of the spring constants and damping coefficients which will make the machine stable and will not let the machine walk. This amounts to describing the region labeled "solutions" in Figure 8. 15

Deflection Solution given _ _Equation / by Optimizing.t \s Y/ Deflection ~a) |~ -Walking Constraint K,C values Figure 7,r Values above this line ~~K P a are no good because the C K s _ - machine will walk SCnsa Solutions' Constan'~~f~ *.- "Values below this line are. /<?^ no good due to instability or C, Damping Factor buckling of the springs Figure 8 A close examination of the stability equations reveals that very soft springs, in general, will be unstable under the weight of the machine and, therefore, unsuitable. Examination of the walking equations (or force transmission curves of rotating imbalances-found in many dynamics textbooks) will show that when spring constant and damping coefficient combinations become large, the machine will walk. Using physical reasoning, if k = o and c = oo, i.e., no suspension at all, when a 4-lb rotating imbalance reaches 500 rpm, the centripetal load generated (approximately 500 lbf) could lift a 200-lb machine completely off the floor. This is the extreme case of walking. If a designer of a washing machine were given the region labeled "solutions," he could pick the values he feels is the best comprmise between walking and instability and then see if the deflections with these k and c values are acceptable. The procedure designed to produce the "solutions" region picks values of k and c in an orderly fashion, and tests the combination for walking behavior or instability with the machine parameters given. It can be changed into a computer program where the acceptable values are remembered and printed.as output. 16

FLOW CHART Read input parameters KmaxI m = mmaximum value of parameters you Increment values__ ofwish to investigate j Inrcrement values of C End Increment values of K,_ompute — unde-fle-te.;Print-out ~Compute und-~ef~l~ected. |~ "solutions" spring length Compute diameter of spring necessary for stability N o{ Length & diameter1 -i ^^s acceptable? -Physicallyj / Yes Compute lateral deflection for Yes given parameters l No / K ~ Kmax [Kmax is input] / Lateral \ No / deflection less \ than maximum stable \ deflection / INo C > Cmax? Yes Compute walking [Cmax is input] behavior Yes Machine No Remember K walks & C values 17

BASIC PROCEDURE An explanation of the following routine follows it. (1) Read (M,g,6c,bk,6A,d,n,G,D,m, e,)o,C,K,D,Lo max max max max max )C = -6C O m = ()C =C + C O m=m +1 CK K = -SK N = 0 C3 K =K + K 9 N = N +1 6 = Mg/K (1 LoMN = (5 +A) + d n c bcr/oo = Mg/K LoMN (i DMN = LoM ((l- ((5cr/~o)/.812- 1)2)/6.87)1/2 ~( If (DmaxDMN) 8 15n 15 ~ If (Lo ax -Lm) 8a 16, 16 ( ) If DMN is complex, go to 8 (17 Choose value of CL () h = LoMN - 6 9)K = 106d /(LnD (204 h +.265 D2)) 3 X = me 2/((cc)2 + (K-Mw02)2)1/ 2 ( @ cr = cr/o LoN 18

(O If (Scr-X) 8, 8, 25 (2) If (-u(K y + C y - Mg/4) - {[K (x+e Lz/2+e R) y y x y z c + C (x+6 Lz/2+e R )]2 + [K (0 R + e Lx/2) x y z c z x c y + C (6 R + 0 Lx/2)] 2)1/) 4, 24, 24 (EG C = CMN @ K=MN If (C - Cax) 27, 100, 100 @ If (Kmax -K) 4, 4, 8 Print LoMN, DMN CMN, KMN for all values which are solutions MN MN MN MN End where g = acceleration of gravity 6c = arbitrary increment of the damping coefficient ok = arbitrary increment of the spring constant 6A = expected maximum deflection; 6A may require a separate equation to evaluate it if real accuracy is to be obtained e = radius of rotation of imbalance C = maximum range of C you want to evaluate max K = maximum range of K you want to evaluate max D = maximum acceptable spring diameter max Lo = maximum acceptable free spring length max Lines ) through ( read the data, increment C and K, and increment subscript indices M and N. Lines (1 and () define the free length of the spring. Lines 2) and ( define the stable diameter of the spring. 19

Lines ( through ) check to see if diameters and lengths of spring are acceptable. Lines ( through ( compute expected lateral deflection. Lines ( and ( check to see that lateral deflection does not exceed the critical buckling deflection of the spring. Line ) insures machine does not walk. Lines ) and ) index C and K to be remembered. Lines ( and ) check whether C or K have been exceeded. max max EXPERIMENT Due to the many assumptions made to arrive at the final form of the deflection equations, it seemed appropriate to seek correlation with the actual deflection of an experimental machine. A chassis with motor and drum was obtained and mounted on springs. Two sets of springs were obtained; one set having a large spring constant (86 lb/in. for each of four springs) and the other set having a small spring constant (8 lb/in. for each spring). Deflections were to be measured at angular drum velocities lower than and exceeding the natural frequency of the washing machine mass on the springs. The 8 lb/in. springs, however, were unstable (which prompted the search for a stability routine) and had poor fasteners at each end due to the large diameter of the spring. Tests were conducted only with the 86 lb/in. springs below the natural frequency. The simple -case of one imbalance in the axially centered plane of the drum was tested (m2 = 0, r3 = 0). Then the x and y displacements were measured. Displacement was measured with a load cell; an amplifier bridge circuit boosted the signal; and the displacement was drawn on a strip chart recorder. Because the deflection readings are so small, it is difficult to obtain accurate data. Barring some basic misunderstanding of the experiment, it is estimated the recorded data should not be more than 10% in error. The following were the machine parameters: 20

Drum radius = 12.5 in. m = 2.1 Ibm M = 92 Ibm k = 86 lb/in. per spring (4 springs), (lateral spring rate, K, was calculated) c = 0 The experimental data: X 1.7 cycles/sec 2.1 cycles/sec y.0346 in..062 in. x.0825 in..135 in. Calculated values for these conditions: lo 1.7 cycles/sec 2.1 cycles/sec y.0234 in..04 in. x.0445 in..079 in. The calculated values are about 61% of the measured values on the average. I discovered that the weight distribution on each side of the drum axis measured at the springs was 44 lb on one side and 48 lb on the other. This means that the center of mass of the machine is not on the vertical centerline of the machine, a basic assumption of the derived equations. The machine was balanced and the experiment repeated. M = 100 Ibm. The experimental data: X 1.57 cycles/sec 1.5 cycles/sec 2.12 cycles/sec y.0299 in. ---.0543 in. x ---.0469 in..115 in. 21

The calculated values: XC 1.57 cycles/sec 1.5 cycles/sec 2.12 cycles/sec y.0208 in. ---.0406 in. x ---.0356 in..0816 in. The calculated values are now 73% of the measured values on the average. The small center of mass offset cost about 12% of the accuracy. Better agreement would be welcome, but it does suggest that the displacement equations of the type derived should be checked at drum rotation velocities above the natural frequencies. The equations derived may be suitable, however, for rough estimates of spring constants and dampers. The lateral spring constant and stability equations were checked. The lateral spring rate was computed to be 287 lb/in. (for all four springs together). It was difficult to measure the lateral spring rate on the test machine and recorded values fluctuated between 200 and 280 lb/in. For the weight of the machine, the stability equations predicted that springs of.188 in. diameter round wire (G = 20x 106 psi) should be 10.9 in. in free length. The 8 lb/in. springs were 12 in. long and unstable. These springs were cut down to 7-1/2 in. and were very stable. The lateral spring constant equations and stability equations are, therefore, considered to be fairly accurate. Obtaining data for evaluating the walking equation's accuracy was difficult and time would not permit this experiment. The mathematical model appears good enough to get a "ball park" value of springs and dampers. It is suggested that additional testing be carried out, especially for drum rotation velocities above the natural frequency. CONCLUSIONS The problems associated with spring suspensions applied to horizontal axis washing machines are: deflection control; load transmission (walking); and stability under the machine weight. Finding a spring-damper combination which will satisfy all three of these conditions simultaneously may be possible, but difficult. An alternative is to satisfy the walking and stability conditions, and see if the resulting deflection of the machine can be 22

accommodated. If the deflection is too large, some of the machine parameters will have to be changed. Changing the parameters to reduce deflection is an intuitive operation, and the following examples should help illustrate how it is done (W > wn). 1. From the walking equation (38), as Rc becomes smaller it is possible to use stiffer springs without the machine walking. This means the center of gravity should be as low as possible. 2. From the deflection equation (37), if Ry is made small, the angular Oz deflections hould be reduced. In other words, the drum axis should be close to the center of mass. 3. From the deflection equations in general (32)-(37), increasing the sprung mass of the machine will reduce deflections once beyond the natural frequency. Choosing spring rates and damping coefficients can be done using the routine given in this report. The values received should be rough approximations due to the difficulty of getting an exact mathematical model of the machine. A more accurate mathematical model may be obtained using the coupled differential equations given by equations (9)-(14) or one may start from scratch and derive his own equations. These equations should require more detailed development and solution by computer. It is felt that further testing of the deflection equations should be done, especially for drum rotation velocities above the natural frequencies of the system. The importance of having the center of mass on the vertical centerline of the machine, equations (22), should be emphasized also. Anticipating problems, the start-up phase, where drum rotation velocity goes from 0 to 50 rad/sec, will be difficult. Some informal testing was done with the experimental machine. It was found that excessive deflections resulted when trying to go through the natural frequency at a moderate pace. When the machine was rushed through the natural frequency, the d'Alembert reaction torque (J') on the drum caused the machine to "dance" around the floor. A separate device, such as a dual spring rate or damping coefficient, may be needed to cope with this problem. THE BELLEVILLE SPRING SUSPENSION Belleville springs offer an unusual load-deflection relationship which might be exploited in this situation. The curve in Figure 9 is the one of interest in washing machine suspensions. 23

C b c -oa 0 I b I I I a Deflection b c Figure 9 A brief study was done into Belleville spring design, and it is considered possible to design a spring with the following properties. The weight of the machine will load the spring to the static deflection located at b in Figure 9. With the rotating unbalance spinning at frequencies above the natural frequency of the system, the machine can deflect from b to a when it is lifting up on the suspension. This is the moment when the machine will walk with present suspensions. The Belleville spring, however, has a very "soft" spring rate from b to a; there is a lot of deflection allowed without much changing of the normal force on the feet of the machine. This will help keep the machine with Belleville springs from walking by maintaining a high frictional force on the floor while the machine is deflecting the suspension. When the machine is traveling downward on the suspension, the Belleville spring deflects frombto c. The spring rate from b to c is much "stiffer" than a to b. This will help keep deflections small, but also will help keep the machine stable. From the stability equations already presented, it can be seen, in general, that larger spring rates are more stable than softer ones for a given diameter and free length of the spring. The shape of the Belleville spring also enhances the stability. It is a slightly conical circular disk with a hole in the center (see Figure 10). The load is applied on the circumference of the hole on the inside. This shape cannot buckle as coil springs do. It is also a very compact spring. A rough guess for the size spring needed to support a washing machine might be 5 in. in diameter and 1/2 in. tall. 24

F F Cutaway Side View Belleville Spring Figure 10 There is a small amount of damping action associated with deflection of a Belleville spring, especially when they are stacked on top of each other. It is also suspected that since the spring rate changes so drastically as the spring deflects, a Belleville spring will not have a true natural frequency. The instantaneous natural frequency will fluctuate over large ranges as the spring constant changes with deflection. This should aid in keeping deflections reasonable when running up through the natural frequency to the operating speed of the machine. In summary, a properly designed Belleville spring offers many advantages over coil springs. Although no actual designing of the spring was done, the design parameters are considered flexible enough to allow design of the spring to produce these davantages. 25