STATISTICAL ANALYSIS OF VEHICLE VIBRATION AND DYNAMIC LOADr AND SELECTION OF SUSPENSION DESIGN PARAMETERS Iby KONG-HUI GUO Visitling Scfolar* Report No. UM-MEAM —82-15 *on visit from the Changchun Motor Vehicle Research Institute, The People's Republic of China.

Foreword This report represents the English rendering of work published by Mr. K.H. Guo -in China in 19-75. Apart from the general interest of this work, it was also useful in providing the necessary modeling background for an optimal design project at the University of' Michigan. This work resul ted to the article entitled "A Design Procedure for Optimization of Vehicle SuspensiLon-s", authored by X.P. Lu, HL Li and myself, to appear in tIChe Inte~rnational Journal of Vehicle Design. Thus the present report was prepared as a bas-ic reference for the model development used in the optimization project.'the report was made possible through the translating efforts of X.P. Lu and H.L. Li, both of whom I would like to thank. Ann Arbor P. Papalambros March 1983 Assistant Professor

Abstract A suspension design method is proposed for obtaining statistical values of vehicle vibration and dynamic load easily. The method is based on the random vibration analysis of a twodegree of freedom suspension system. Concise relationships between the RMS of vibrations, dynamic loads and corresponding level distribution, and the design parameters are given. The rational selection of damping and suspension stroke and the method for estimating speed limits on rough road are studied. Suggestions related to experimental evaluation are included.

INTRODUCTION Roughness of road surface always stimulates the corresponding vibrations and dynamic loads of vehicles. Reducing these vibrations and dynamic loads is always a goal for automobile designers. Because of the random properties of the road surface, vibrations and dynamic loads of a vehicle will be random. For reducing the dynamic load tramsmitted to passengers and cargos, it is necessary to reduce vibrations. For improving the durability of loading parts and reduce vehicle weight, it is necessary to reduce the dynamic loads transmitting to the parts. For improving direction control of the vehicle at high speed, it is necessary to reduce the dynamic load between tires and road surface, because vibration of adhesion will cause a reduction of "dynamic cornering stiffness." When driving on a rough road, the vehicle speed is usually decreased significantly, since the level of vehicle vibration is too high and uncomfortable for the driver, especially when heavy hitting on the suspension bumper occures. For decreasing the probability of bumper hitting, it is necessary to design the travel stroke of suspension appropriately while trying to reduce the relative displacement between sprung and unsprung masses. From the viewpoint of durability of suspension springs, this latter part is also desirable. No procedure and calculation method appears to having been derived to-date which is convenient enough for engineering application. To some extent, suspension design is still performing 1

2 with a rough calculation and experience. A more precise calculation method is proposed in this paper. By using this method, t-he statistical charactICeristic-s (such as root mean square, by which probability levels are estimated directly by a Gaussian hypothesis) of vibrations and dynamic loads will be esti-mated easily by explicit formulas. The optimal damping selection, estimation of speed limit on rough road and experimental evaluation are discussed. 2. STATICAL CALCULATION OF VEHICLE VIBRATION AND DYNAMIC LOAD ON ACTUAL ROAD SURFACE. A large amount of statisti1cal measurement data on road roughness have been acquired in several countries. All these data imply that a certain regularity exists behind the random movement. The distribution of road roughness is essentially similar to the Normal (Gaussian) distribution, and the frequency constitution of different road surfaces are very similar to each other. The power spectral density of different surfaces may be expressed by an unique formula as follows S = A/Q 2(cm2 /c/m)(1 where, s: spacial frequency (circle/rn) n: -frequency index (dimensionless), representing a relative distribution of frequency components. For most of road surface, n 2. A: coefficient, representing the intensity of the road

It has been proved tIChat, for a linear system, if the input is Gaussian random process then the output of the system must be a Gaussian random process too. F1or a Gaussian dist-ributed random variable (which may represent the roughness, vibration acceleration or dynamic'load), we will1 have a clear understanding about the distribution of x if and only if the mean value m~ and the root mean square a are known. The probability density will be 9 P(x) = __e 2 xm)' 2 I127r a 2 x2 The probability N(a) of x exceeding a-certain level(s) may be obtained from integrating equation (2) a N(a) = P(x)dx (3) -a The results of the integration have been made already in a form of table in many mathmatical handbooks or bookS on -probability theory. For instance lxi > a = 3a If N(a) = 0.0027 lxi > a = 3.5a, N(a) = 0.00047 1xl > a = 4a, N(a) = 0.00006 lxI > a= 4.89a, N(a) = 106 If we like the probability of vehicle vibration acceleration > lg to be less than 0.00006, then we should ensure a.. < 0..25g z (the probability of ~ > 4a.. is 0.00006) z

4 the RMS of suspension tCravel u less than one third of the V. maximum stroke (the probability of Y > 3u is 0.27%). If we like the probability of dynamic load P'larger then static load G to be less than 4.7 ten thousandths, then we should ensure the RIMS of dynamic load a less 1/_30.5) timesG (the probability of JP1 > 3.5G is 0.00047) and so forth. Thus for a random process with Gaussian distribution, a calculation problem of level distribution of' vibrations and dynamic loads will become a calculation problem of RMS value of vibrations and dynamic loads on the actual road. The RMS of a random process x can be obtained by integrating over its power spectral density-S x(o) as following = *;LS (Co) dw1 (4) - cc where the dimension of frequency w is rad/sec. if a dimension ofE- c/sec is used instead of rad/sec, let 27Tf -(o then, co 00 =IS (f)df = 4 S (f)df (5) x -~co 0 where the last equality holds due to SX (f) being an even function. I f S (w) represent a power spectral density (PSD) of the x output vibration or dynamic load, then it will relate with PSD of road surface as follows: Sx (M = W(jw)12 Sq (os) (6)

5 S =jc,(=11) Since the transfer function W(s) is a fraction in which the order of numerator- is alIways- less than the order o-f the denominator, and power spectrum densities are always an even function, i.e. S (ct) =Vs (jWk) v s (-jW) q q q with -regard to equation (6), e-quation (4) can be expressed as co a 2 ~~~Gj) dw (7) x _ TOF ( j ()*F(j (j3 Lettina S jW, we get 2 _ 1 r' G s) x 2iTj F(s)F(s)d(8 2n-2 2-n-4 ~..+ sawy nee where, G(s) =b 0s +1b1 +bn-1isawyanen function since all of power index of s are even. n n-i F(s)= a s + as ~. + a 01 0 Equation (8) represents an integration of a complex variable function. The int egration path is along the whole imaginary axis. According lto Jordan's t h — o -Y_-er e, w e c.-an c- losee the intec.-qrati-_ng path by adding an infinite half circle without causing any difference of the results, and the value along the closed path will equal to the sum of all residues in the left half plane. After some maniolations, the result of equation (8) is obtained as

6 where -a1 a 0 0 0 0 0.. 0 0~~~~~ A I-a3 a~ -a1 a 0a 0 0... = i- a4 -3 a2 -1 a0 0.. 0 n -a21C1a 2n-1 ~~~~~~~~~~n lb a 0 0 0 0 0 0... 0 b a -a1 0 0 0 0.. 0 A - b a -a a -a a 0.. 0 1 2 4 3 2 1 0 For a Gaussian variable, oncea is obtaineed, all probability distributions will be known. 3. TYPICAL ACTUAL ROAD SURFACE AND INPUT POWER SPECTRUM DENSITY As mentioned above, random roughness of road surfaces is approximately Gaussian. The spacial PSD of road surfaces can be expressed as follows S (Q) = -(cm/c/in) (10) If the vehicle speed is v, the temporal PSD will be [5] 1 2 2 S'(f) =- S (Q) =Av/f (cm /c/sec), (11) Letig =2'rfrd/) ndnoin ta

.7 the temporal PSD of road surface can be expressed as 1 2 S (X) = 2rAv - (cm /rad/sec) (12) q,2 Noting further that the coefficient of right hand side of equation (12) only contains A and v and both are of first order, the input PSD (function S (M)) of different road surfaces are q only dependent on the product Av. Increasing road roughness constant A is equivalent to increasing vehicle speed v at the same rate. Once the vibration and dynamic load of any A and v is calculated, the vibration and dynamic load for other A and v will be directly deduced by modifying a simple constant. It is known from equations (4) and (6) that if the value (Av) increases n times, the output (vibration and dynamic load) will increase n times also. In order to relate this calculation to actual road surfaces, some typical values of A are as follows: cross country rough road A = 1 (cm -c/m) low quality road surface A = 0.1 ( " ) high quality road surface A = 0.01 ( ) excellent high speed road A = 0.001 ( " ) 4. STATISTICAL CALCULATION OF VIBRATION ACCELERATION OF VEHICLE BODY For most modern motor vehicles, the dynamic index of mass distribution are approximately equal to 1. So the front and rear suspension can be considered as separate systems as shown in Fig. 1. The differential equations of motion for each system are as follows

8 Mi + kz-) + c (z -)=0 (13) Mi + m + C ( C- ) 0 k wn~ere, Vert ac, -J icmeto pun as ehic< body) ~:vertical displacement of unsprung mass (wheels) a: elevation of road surface M: sprung mass m: unsprung mass c: spring constant of suspension c,: ve-22rtical stiffness of the tire k: coefficient of damping The transfer function of -body acceleration z vs road input a is as follows: _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _(1 4 ) g S ~28:(1~1)S3~(O~ W~ )S +2CWSwk 2 c k 2 c k M where, - > co - - 5: the Laplace variable. From equation (12) the PSD of road surface may be-written as S q(c) = 2'rvAv jW- j and letting S= jaw, yields S ()=2irAv11 _ (15) Referring to equation (4), (6), (8), the RMS of body accl -eleration is

9 = ~ f 1~-(io)2 S ()dw 2 = 1 Ai~Vf 1(s). z (-s) ds qT c q 2 j wS q -s c 00 ~ ~ ~ ~ ~ ]o G (s) r~~~~~~.6 whnere C (s) =b S + -b S 4+ bS ~+b o 2 3 1~~~~~ 4 3 9 F (s) = a0S ~ aS ~ a,) + a S a 0 ~ I3 4 2 2 2 2 2 2 a0 a, a 2 c(l~I) a 2 = 0 a3 =CCk r a Ok 0 0 b1 4 k b2,WF, = 2 b, 0 Substituting these values into equation (9), yields Z 2 (-1W A1(17) Z = 2aA 2iv 0 k 2(~i 1 0 0 k 2 ~~~~2 2?2 22 0cwk Pk 0 Ok 0 0 0 0 0 1 0 0 2 4 2 2 22 4 4 2~~O 222 2 2 0 ok c0k2 ok ~ 0 k 0 0 0 0

10 Expanding the determinants A and A1, many terms are canceled and the following simple result is obtained..2 k 0 k -1+ U z -2Av- 2 2'i Av(~ + 4~) (18) where, ~ = is damng conis obtained, where, is ingdcontant. Once z i 0 a.. and the Drobabilit1 —v distribution are known. z From equation (18), the following observations can be made: (1) When M and m are constant, reduction of either suspension spring rate or tire stiffness will cause a decrease of (2) Differentiating z vs ~i, we get - ~~~4 4&2which is always less than zero. So we know t~hat increasing ii will cause a decrease of z, i.e. either increasing M or reducing m will cause a decrease of ~ (3) If we increase the damping constant'C he first term of equation (18) will increase and the second term will decrease monotonicallyv. Only a particular value of 4~ relates to the minimum of'Z 2 Differentiating equation (18) and letting dz = 0 the particular damping constant 1~ ~Min is obtained as follows ~f - -1 l+ji c k_ 19k'~min 2 ii ck - f (9 where, -- - (M+m)g is the static deflection of tire, - ck, - Mg is the static deflection of suspension spring. s c

When we take ip= LVjn he RMS value of boyacleains equal to _2 3 1+11 0k - 2r~v0 C - -2TriAvw ~ s (20) 0 f~k -2TrAv~- / - k +, M v From equation (20) further observations are: (4) where M and m are constant,, z is proportional to cV ck. and reducing spring constant c is more effective than reducing 112 tire stiffness c k for reducing z (5) The damping constant 1~m corresponding to minimum z is proportional to, i.e.., the softer the susuension 5 spring, the smaller tIChe mn But the -softer t1-he tire the larger the ~ imin. Example 1: A medium truck (loaded) has the following characteristics: P = 4= ck 3.52, f = 6 cm, f= 2.13 cm, m c s k =O 127rds ~2= 13(rad)2 (02 =2300 (rad)2* From 127 a/, 0 13 s I k s equation (19) we calculate the damping constant for minimizing z as'p.. = =k 0.298 zmin 2i/ f 5 From equation (20) the corresponding z2 is 2 2 3 lT' V 27300Av mmn - T~0 Ui c

12 36km/h, then Av 10 cm C/S and.2 ~~~~~2 cy. 522 cm/s = 0.533 g z From the Gaus,_-sian di tr'ibution table the p-robabilt of+ji (when the cargo begins to lift up from the bed) is equal to 6% Example 2: A large-size car, (with normal load) has the following characteristics: ii = 6.12, k= 8.35 suspension static deflection f= 18 cm, tire static deflection S f ~~~2 2 kf 2.51 cm,, = 54.6,, wk 2790. The dampina constant corresponding to the minimum of vertical acceleration of spring mass is nowf 1 /`l.87 The corresponding RMS value of vertical acceleration of sprung mass is 3 __ _ k_ = TrAvI / l(a k 7920Av If the car is driven on the same road with same speed as the truck in example 1, (A=l, v=10), then *&.. =0.288g, which is z only at level of 56% of the truck in example 1. From the Gaussian distribution table, we found that probability of f~1>lg is equal to 0.05% which is a value of a hundredth compared with the truck mentioned in exampled 1. As explained later, for a high speed car, from the viewpoint of direction control, it is preferred to choose a value ~ larger

1 3 2.THE STATISTICAL CA.LCULATION OF SUSPENSION TRAVEL. From equation (13), letting z- Y, the differential equations becomes MY ~ (M+m),~ + = q(21) k ~k From this, the transfer function of suspension travel Y vs road input q is found as 2 Y (s) k~~~~1 (22) qr 4 2+1 2 2 2 22 Ref erring to equations (4) and (6), the RMS of suspension travel will be 00 co 12~~~~~~~~~~~~0 2 1;~~~ 2' rd K 2 TrAv Q q~~~~~0 Letting 2 -yiev[1l____d 27T-4 (s)F(-s) 6 4 2 where, G(s) b S b S ~ bS ~ b o 1 2 3 4 3 2 F(s)= a S + aS + aS + a S+a o 1 2 3 4 2 2 2 2 2 a 1~. a 2c 4l+ij) = 1~lW a3 2= 2co a = c o 0 b2 0 kk0,b1=k0, 4b = 0 4A 2 (-) 1 Y = ~2TF Av2 - -(23a)

14 where, ~~-28(1+1A) 1 ~0 0 2 2 2 K (l+'~) W0~)-2s (l~1jL) I 0 W w2O -2ew (l+ii)a-tW+W 0 k ~~k 0 k I 0 0 0 2CO2 O k 0 ~~~10 0 0 (l~I3)o~~ k2(~i A1 ~~4 2 2 2 2 2 I k O k k O k o 0 0 22o Expanding A and Al, many terms alre canceled and a simple result is obtained as follows: 2' 2A Tll~ (24) It is seen that, under a certain operating condition (A and v kept constant), the value Y only depends on c and IJ.. When M k il= is constant, YL will be inversel proortonamt Increasing damping e will cause an effective approach for reduc-ing the RMS of suspension travel. Once a i Y is obtained, the probability distribution of y suspension travel will be obtained also, by referring to a Gaussian distribution table. Exam-ole 3: The same truck as in example 1, ji 4, c two

1 5 on the same road (A=1, v=l0m/s), the bumping stroke [Y]=6 cm. Calculate the probability of bumper hitting and 7>4cm as follows: From ec~uation (24). 2 AV __ = _ A __ 0.516Av Y I/ = 0.7-19 /A-v= 2.27 cm [71 2.64 V Referring to GaussIan distribution table, the probability of 1Y1>2.64 a is 0.008. y 6. THE SPEED LIMIT ON ROUGH ROAD When a vehicle is driven on a rough roadi usually has to s-:low down to avoid unacceptable vibrations. Some oarticular vehicles, e.g. ambulances or trucks carrying some fragile cargo, even driven on a smoother road, require a low level of vibration, and can only be driven under a certain low speed. Thus, the speed limit on rough road is usually a significant index of vehicle ride quality. There are two cases which the vehicle speed is restricted with no reqard to the engine power, (1) Bumping stroke [7] is large enough, but the acceleration of body vibration cs.. exceeds a certain level which is z unacceptable for driver or passengers. In other words, in this case, the probail -ity4 of - ben lre ta cran au, is-_ I —- -I — L -'

1 6 (2) Bumping stroke LY] is not'large enough, while a exceeds an allowable value so that the probability of bumper hitting reaches an endurance limit, because intense bumper hitting causes an unacceptable shock. S'uppose the endurance limit of RMS of vehicle vibration iE2 ~~2 is [a.] From equation (18), letting 2=[.., the f irstC kind of speed limit (related to case (I)) is obtained as follows or a 1 = A ~~~~~[ + (25 rV ~ 0.573 [2; [Va A 3 4 _ c 4ivt Deteminig [ I acoring o bmpin stoke nd ccepabl La 2A l3ii adet[amini= [Y] Thevalue tof bumand stimply the probabilteo y~~ bume (carigo throwaing) an Y>[b](bumper hitoeuting) For4an orinary trusckord aind off-road vehicleeittis chocsenta (2) =a2 =o 3.w

Example 4: The same truck as in example 1 and 3 driven on a rough road A=l(cm-c/m). To calculate the first speed limit: we substitute a.. into equation (25) and get: z n 2 (82 [V ] _ [a.7.] 0.5 73 -2 3 31. 45 k~m/h a1 A 2 4382300 +2600 5 4 4-384 1U 46 ~i The second kind of speed limit is found by substituting [a i [Y]- 2(cm) into equation (26) yielding, y 3 [V ] ~2.2 9 ~u 2 3.8-4 2 [V I ~1-p[a]= 2.29- [2]1 27. 8 kM/h 2 It is seen tIChat the speed limit'of this truck de~pends on bumper hitting and its value is equal to 27.8 km/h. It is noticable that in order to increase vehicle speed on rough road, it is important LCo reduce the suspension spriLng constant c and the tire stiffness ck and to select the damping constant propenly. In the case that the bumping stroke is large enough, it is 1 -Fk acceptable to choose Y, - (see equation (19)). However, it is seen from equation (26), that in the case when the bumping stroke is not large enough, changing c and c k will not affect t-he [Va] 2. in this case, the only effective way is to increase damping. As a result of increased dam-ping, the [a will increase, Va2 but [Va may decrease, i mn So the optimum value of damping will be obtained according to the condition [Va] = [Va]2 when the condition for bumper hitting is dominant. This reason

18 - 2oo / 2 4 (2 7) 0 k~~~~~~ decreasing the spring constant c and choosing damping according to equation (27) (if 4) > 4)..i ) will cause an increase of speed limit. For example, in the case of example 4 244)min- 0.298, [Y] 6cm, [a I 2 cm), according to equation (27):4= 0. 334, ~ = ~4) =4.26(l/s) and substituting into equation (26) yields [Va] [Va] 31.2 km/h. Comparing with the speed limit under 4 ). 0.298, it is increased by 12%. However, when we decide t, —o reduce the spring constant, besides properly choosing damping and bumping stroke, some appropriate arrangements should be made to avoid having rolling pitching and changing off_ body height become unacceotable. 7. DESIGN OF BUMPING STROKE. According to the above discussion, to guarantee the desirable high speed limit, it is needed to choose different bumping stroke [Y] for different types of vehicles. of course, choosing the bumping stroke as large as possible may have the advantage of enhancing the second speed limit. But if the [Vi'is chosen so large that the [Va] 2 becomes larger than [Va] I' then increased [Y] will be no longer helpful, but rather lead to very high center of gravity height and very large spring stress. Thus, it is reasonable tICo choose [VI such that [Val1 [Vali2'

1 9 [a (28) 2 4 c I~TA 0 Substit,-ut-ing [a.. 2i and [a]= into equation (2 8), z ~~~y p gives [Y] = (2 9) _ _ + 2 4 1 l~ia 0 fk As mentioned before, in most cases it is chosen = 2, p = 3, therefore l.5f5 [Y] =(30) /f k Irf the damping is chosen such that the condition of mini-mum.. 2 is satisfied. i.e. 1, substituting into equation (29) we have [Y] = ~_ (31) E / Taking = 2, pj = 3, we find [Y] = 1.06 f (32) 5 This means that we should design the bumping stroke as large as 1.06 times the static deflection under -~=IJ 2/ f condition. For the suspension with relatively small static deflection (e.g. f s< 6 cm), such a design is feasible. But for softer suspensions, because of other considerations (reduc

20 it is necessary to choose smaller [Y] and try to reduce the,-: bumper hitting probability by means of increasing damping ~ in this casete fI is given and dam-ping i~ will be calc~ulated by equation (217), while accepting a certain increase o-f-'Z'. However, if damping 11 is deter-mined in advance for some reasons, then [Y] will be calculated by equation (30) or equation (29). Example 5: Same truck as in example 1, suspension static def lec-tion, f5 6 cm, damping is chosen to meet tChe minimum z2 condition i.e. =4*m = = 0.298. Originally choosing [Y]= 6 cm, the second speed limit condition becomes dominant. In this case, the speed limit (in km A=l od s V> 27. 8 (see example). If we redesign A.- road) is [Val" the bumping stroke according to equation (32), then the bumping stroke will be [Y]= 1.06 x 6= 6.36 cm.- From equation (25) or equation (26) we have [Val [Vail= 31.415 km/h. Comparing ai ~~~2 with original value, the speed limit is increased by 13%. 8. THE PROBABILITY DISTRIBUTION4 OF DYNAMIC LOAD BETWEEN TIRE AND ROAD SURFACE The dynamic load between tire and road surface is P = M2~mr. After solving the transfer function.. ~(s) and ~-(s), the transfer q a function of the relative dynamic load between tire and roadsurface (G is the static load) vs road input q, is obtained G as follows. 2 P/Go - ~ l~~i + 2cS +, 2)S (33 q 43,2 2 22 +2s~~l+~i)S + ~ A + k'! k 0! k.,.

Referring to equation (4) and equation (6) the RMS of the relative dynamic load will be 2 P ~ IP - 2_____ S Recalling S (M T~ and letting co = 4 yelds, q2 ID k ~~~1 G(s) (L) = 2m.Av-d[ j ds] (34) g2 2 -1j F(s)F(-s') -r0O where, G(S) = b S + ~bS + bS ~ b o 1 23 4 3 2 F(s) = a S + aS + aS ~a S +a o 1 2 3 4 0 (l~}i)~~~ 2(,0 2 2 2 2 2 L 1 a = 2c (l+p), a2 (l~11)~ co +k 3 =a 2c,, CL = Subsit.-uting these into equation (8) and equation (9) yields P2 __ 1 2 2TvAv KU 4 AW where A, A1 are the determinants defined by equation (9). After expanding there determinants, many terms are cancelled and finally the following result is obtained: P2 TrW 2 22 42 G 2g2~~~~ ~ (%) ~ + 4 8 Wk (35) 4.g = l~ la Notice t' —hat f f1Euto s 2' k 2 g,0=,Euain(5 0 k~~~~~~~

2 2 p 2 Tr yf (Z) = _r__v s 2 (l+ s (6 G 2 1 _ p2 After dete_,'_,!rmini~ng ( by e qu atI on (5 or (3 6),the IRMNS value of the dynamic load will be obtained as =G (-) (37) P G From this and the probability tables, the probability distribution of the relative dynamic load will be obtained. -Example 6: To find the probability distribution of tIChe dynamic load between the tire and road surface for the same car as in example 2, vi 6.12, f = 18 cm,f= 2.51 cm,, sk = 7.39 rad/s,' 0.187 (the minimum z condition). From equation (36) we have 2 7rAv ~ ~ l)2 2 (-) - 2 ~~~~Ii.8-) 6.12+4(0.187) *7.12-7.181 G ~2 l. 8 7.39 -6.12 0.187 0. 02 95AV = 0.172 i/Av If the car is driven at a speed v 40m/s (144km/h) on a low 2 U quality road surface of A=0.l cm.c/m, then = 0.172 /Av=-0.334. The probability distribution can be found in the available Gaussian distribution table. For example, the probability of the wheel leaving the rload surface i. s eQual t,o them probab 1 iity of a P <-G o 0.344 which is equal to 0.00185. The damping value which relates to the minimum dynamic load

23 equation (6) with respect to.Lettin k = 0, we have 2 /(6f-l) 2+14 1 ________ f Obviousl, in general, this ip is only relating to minimum P but it is not necessar7 to be the same as reana casein, If. " Pm r minimum z" T n most~ cases, <mm slagrthan'in the larger the f compared with fk the larger the difference between and m will be. Some compromise may be possible from global considerations. Substituting equation (38) into equation (36), at the minimum point the RMS of tire dynamic load ~~~~~~will be 2 2irAv /2, [(. -l) +I ~~~~(39) fs0 E ind the opti oad between t ue and proba dynamic load. Substituting e - 7.18 i 6.12 into z equation (38) yields /(e f _1)2~i However, 0.187 (see examole 2) which is smaller than'. From the view-point of dynamic load, the dampingr

24 should be much larger than that from the viewpoint of body vibration. When designing, it is preferable to plot out the z - and 4 ~ curves, then decide a best value of damping which may be a compromise of both requirements. Generally a value close to -min is more preferable when zmin ride comfort is being emphasized, and the value close to ~Pmin is more preferable, when the road holding consideration is dominant. From equation (39) the RMS value of dynamic load between tire and road surface may be obtained for' = bPmi as follows p- 2w~ G 1 1 2Av /[(6.183)2+6.12]1 7.12 7.18 = 0.0204Av 18 -7.39'6.12 2 If (Av) = 4 cm c/s, then =p 0.286G. Compared with example 6, the dynamic load is decreased by 17%. 9. THE PROBABILITY DISTRIBUTION OF DYNAMIC LOAD OF OTHER LOADING PARTS The dynamic load transmitted by a connecting element 3 (Fig. 2) to a cargo or an equipment (M,) which is a part of the sprung mass,may be counted as an inertial force by Newton's Second Law: Pl = Mi So when a.. and the probability distibution of' is obtained, z these are just the RMS and probability distribution of the P1 relative dynamic load (-) transmitted to M1, e.g. in example 1, an 0. 533g, thereby, p = 0.533 Mfg.

12 5 For the dynamic load- of some unsprung parts, an extra analysis is needed. Now imag-ining thatI — t,-he unsprung mass is — di.vi'ded int-o two parts m1 and M. (Fig. 3), the force transmitted by connecting e~lement 3 will be considered as the dvnamic load betwee-n par-,t 1 and part 2. xz where, P0 (m)g 22 _ - S ~2cS+W3 2 c-sw22 ~SL0.q2 2 p P ~-2 cS +w x ( 2 liii 0 -0 4 3k22 2S2 2 2 0 ~ ~ ~ RFerom nt equation (4)4nd quaionl(6 XD _ 12~j X/0 Hp 0 q q 42 s FrecallingioS(4) 2an/d equaind (6=-, hsbeoe g 0

2 6 ID 00 joP p P P ('x) _ 1X/ 0~ (- X/ O.S S)s P0 2TV jo q q 2 c g2 2T ~(S) F-s where, 6 4 2 G(s) =b S + bS ~ bS + b o 1 23 4.3 F(s) =a0S ~ aS ~aS2 ~a3S +a4 1 ~~~~2 2__C 4 b,.=-, b, =4- b2 = -l b, 0 2 2 2 2 2 a0 1, a1 2c (1+ii), a2 (l+vI) ocok a3 2ewk a4 = 0 Comparing these with equation (33) and (34), for dynamic -Load b~etwee-n tiea.7d ro.a-d, h n' dfeec that iIn M G (s), 11 substituted by - - From equation (8) and (9) we get P W4 ~~~~~4 A ~~ 2 Wk (-1) A,1 = 2TAV g2 2a, A Expanding A and A1, we have. P 2 (K)2 2 (X) TrAv k 2 2 22 P 9 -~~~~~ + ) ~ + 4c w 2g8v 0 0 k 7rAy li1t __ - 2 42 (42 2 1+vi~~~ ~~k +11I+ 4-l(lvJ4(42I f kW1 CompDaring it with equation (35) and (36), tChe only difference is thatthe fist t~rm___ 2 2 -f i st at I — hef 1 ttr is substitL-uted by -— (o is subsitutd byr- l~) Wen - (-) Is cacuaedb Px P

Example 8: To find the probability distribution of dynamic load transmitted to rearL axle for tChe same car in example o. The value m, can be de-termined by subtracting t"he wheel weight from unsprung mass m. It is givenIm 1.2 1 cm m1.3 =1.'l=s.1 Vi=6.12, f5 18 cm, "k =.1c, =.9ls =1 Ylmin =0.1187. From equation (42) x _ TrAy l~ i 2 2 K (-) - [( 1) + 1i + 4~ (l+p)1 -: ] = 0.01253Av 0 2f~w1-1P fkf S 0~~ Suppose the car is driven on an intensified proving track of A=l CMZ c/m, at a speed v = 16 in/s (558km/h), a = 0.112 p /16= 0.448P I x 0 0 From this the whole probability distribution can be found by referring to a Gaussian distribution table, e.g. the, probability of Px> P0 is 0.025. 10. INDOOR ENDURANCE TEST AND DURABILITY PREDICTION OF LOADING PARTS. Recently, the technique of quickening the indoor endurance test is being developed. Besides the random loading endurance test by means of an expansive electro-hydraulic loading system, an effective method is the mixed circulation loading method. A stable durability index can be obtained by this loading method k such that Z7No const., and k 6.8 (N - amount of loading a [8]'

28 Usually, an eight-step loading program is adopted which is drawn up according to the RMS value p of the load measured by road test. The load levels and cycles of each step for a big cycle is shown in Table 1. For quickening the test proP cess, sometimes, the first three steps ( < 1.75), which have only little influence on the durability are neglected. Thus, a big cycle will consist of only 497 cycles instead of 4000 cycles. In this way, the testing process will be much quicker. To date, the simplest constant amplitude endurance test is still adopted in many cases. Obviously for estimating durability, it is reasonable to determine the loading amplitude according to RMS value of actual load in operation, e.g. 3.5oa or 3u may be selected as a loading amplitude. Hereby, the dynamic effects to durability may be included. Using the method to calculate the PRMS of dynamic loads, it is possible to estimate the durability of the loading parts. For example, after redesigning suspension system of a vehicle, the RMS value of dynamic load of the axle decreases by 10%, and 6.8 since the durability index k = 6.8, (1+0.1) z 2, we may expect that the durability, under new suspension system will be as long as twice than that before. From above analysis, it is noted that the RMS of dynamic loads are always proportional to v'Av. Therefore the influence of road roughness and operating speed to the durability of loading parts may be estimated, e.g. if operating speed keeps constant,

-'J the coefficient A decreases to 50%, then a will decrease to or original value. Since ()6 10, then the durability 1v2 of loading parts will decrease to 0 of original value. Thus the importance of improving road surface is seen. The influence of operating speed is just the same as the coefficient A. Improperly increasing operating speed for 20% will cause a reduction of durability of loading part to a half. 11. EVALUATION BY STEP RESPONSE TEST When a vehicle is driven on a certain road surface, the probability distribution of vertical acceleration z, suspension travel Y, and dynamic load on road P or on any point P x can be calculated according to thle method deduced in this paper. Usually, it is necessary to make a final check by the actual load on a proving track and to fLigure out the RSMS or probability distribution. Certainly, this is reliable but expansive. A simpler experimental evaluation method is suggested as follows. Suppose that step from road input function (with an altitude of H) is simulated at the wheel contact point, (by driving over a step, or given a step function by a hydraulic actuator), and output x (may be z, Y, P, P etc.) is recorded X as shown in Fig. 4. There is no difficulty to calculate an integral of 2 E = [x(t)] dt (43) 0 H On the other hand, the value E may also be calculated by input q(t) and transfer function W(s). The Laplace transform of

30 H step input q(t) is so that Laplace transform of output x(t) will be H x(s) = W(s)S (44) S~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [7] According to the integral theorem of the original function 7 co ~~c 0+JCO 2 cI x (t)dt = 2j x(s)x(-s)ds (45) ~~~~~~0 c-J Since the system is stable, there is no pole on right side of the S plane. So, we may consider the integrating path such that c0o 0, thus -i 00 00 2 ] 2 H W(s) W(-s) f x' (t)dt = 2j (-s) ds (46) 2-F. j s -s 0 Comparing equation (46) with equation (16), (23), (34), (41) yields 2 00 E 1 2 X't)dt H 20 2TAv 00 - = (t)dt = 2AvE (47) Hx 2 H 0 The equation (47) showing that, because the PSD is proportional I to L, which is the same as the energy spectral density with a (D0 proportionality constant. Therefore, the RM1S of any output (z, Y, P, P etc.) may be obtained from the square integral of x the step input response. For example, from a step input test, the transient curve of z is recorded and the related square integral is figured out as

31 E. j []dt 0. 00254 -g sec/ cm2.Suppose a certL-ain road surface of A=i cmc/manaveil speed of 1Im/s —ec are considered,. -the IRMS value of 2 will1 be..2 ~~~~~~~2 z =2TrAvE.. 0.16g,. 0.4g zz And if, meanwhile, the stress at a certain point TL(t), is recorded the related sqruare integral is calculated as 00 [Tt)dt 390 2 2 E- =t 398 kg - m sec. 0 H 2 Under the operating condition (Av)=l0 cm c/ sec,, the RMS of the stress will be 2 ~ ~ ~ ~ ~~2 4 T = 2nTAvE~ T 250000 kg /cm = 51-00 kg/cm2 By refer-ri..ng to G~aussian dist-ribution table, t'-he probability distribution ofl 2 and T will be obtained. e.g. the probabilit~y of 2 > lg equal to 0.01124 and the probability of T > 1500 kg/cm 2 is equal to 0.00277. ItI — is necessary to discuss the problem of magnitude and direction of step input. Because of the different damping coerf-icient. in different direction of the damper, it is o)referable to do both upward and downward step input test and to take the averaae va-lue. For a l-inear,- system,. the different value of H will be no influence on the related square integral

References -1~~~~~~~~! Ir' [11 J PDRbn Sn introduct-n to RaDndom V1-r-aionl, I964. [2] S. H Crandal l and W. D. Mark, "Iandom Vibration in Mechanical System"! 1964. [r3] V. Bohn,, he Transeform Analvsis or Linear System" The People's Education Publishing Agency, 1963 (Chinese Trans. ). [4] Solodofnikof, "Theory of Automatic Control" the Industrial Publishing Agency of China (Chinese Trans.), 1961. [5] Kong-hui Guo, "A method for Vibration and Dynamic Load Analysis of Motor Vehicles", Automobile Technology, No. and No. 2, 1975 (in Chinese). [6] ichiroh Kaneshige, "Application of Probability Theory to the Design Procedure of an Endurance Test Track Surface", SAE Paper 690111, 1969. [71 A abkof i "Theory of Linear C ircui-t" the Peo p: Education Publishing Agency, 1963 (Chinese Trans.). [8] "The Simulation of Operating Load in Endurance Test" Foreign Automobile, No. 2, 1970 (In Chinese). [Q] "Quickening Indoor Endurance T'est of Spindles', Foreign Automobile, No. 4, 1974 (In Chinese). 33