P. T. Thawani Climate Control Operations, Ford Motor Company, S. Commerce Drive, Dearborn, Michigan 48120

Size: px

Start display at page:

Download "P. T. Thawani Climate Control Operations, Ford Motor Company, S. Commerce Drive, Dearborn, Michigan 48120"

Beverly Melton
5 years ago
Views:

1 Prediction of the vibro-acoustic transmission oss of panar hose-pipe systems M. L. Munja Department of Mechanica Engineering, Indian Institute of Science, Bangaore , India P. T. Thawani Cimate Contro Operations, Ford Motor Company, S. Commerce Drive, Dearborn, Michigan Vibro-acoustic energy traves through hose was as ongitudina waves and fexura waves, apart from the sound waves through the fuid medium inside. Longitudina waves in the hose wa are couped to the sound waves inside by means of the hose-wa Poisson s ratio. Both in turn get couped to bending or fexura waves because of the energy transfer or interaction at the bends. For any of these three types of waves incident on one end of a hose, waves of a the three types may be transmitted on the other end because of their dynamica couping with one another. Therefore, in the present paper, expressions have been derived for the 33 transmission oss matrix for a two-dimensiona or panar piping system in terms of eements of the overa 88 transfer matrix of the system. These expressions have then been used in a comprehensive computer program to evauate the vibro-acoustic performance of hoses, with particuar appication to the automotive cimate contro systems with gaseous as we as iquid media. Finay, parametric studies have been made that have ed to some genera design guideines Acoustica Society of America. S PACS numbers: Mv, Jr, Gf JEG INTRODUCTION Rubber hoses have been used for severa decades for transport of fuids so as to isoate vibrating machinery from accessories ike exhaust and intake muffers or receiver bottes/penums. Hoses are capabe of executing fexura vibrations and torsiona vibrations ike a beam or shaft. The hose wa can conduct osciating ongitudina forces that may interact with waves in the fuid inside as we as outside. Fexura forces and moments get couped to ongitudina forces and acoustic pressures because of bends in the piping system. If the fuid inside were a iquid, its inertia woud accentuate the roe of the bend. Wiggert et a. 1 considered Poisson couping in a pipe with two bends. Based on inearized assumptions and periodic motion, a simutaneous soution of the couped fuidstructure system equations in waveform were presented recenty by Lesmez et a. 2 They have derived a composite 1414 transfer matrix consisting of submatrices for Poissoncouped axia stress waves in the wa and pressure waves in the fuid coumn, fexura waves in the two norma panes, and torsiona waves about the axis of the pipe or hose see Appendix A for the transfer matrix reations and Fig. 1 for the forces and dispacements. In their anaysis, compiance or yieding of the hose wa is buit into a modified buk moduus for the fuid, and structura damping is accounted for by means of a modified buk moduus. 3,4 At higherfrequencies, however, inertia of the wa pays a significant roe and must be accounted for by means of wa impedance. The concept of wa impedance, incidentay, enabes one to predict transverse transmission oss and thence the breakout noise too. 5 The concept of transmission oss TL, which has been used extensivey as a measure of the performance of an acoustica fiter 6 or muffer, may aso be appied to the structure-borne sound that traves in the form of ongitudina waves and fexura waves. 7 TL of a one-dimensiona acoustica fiter has been expressed in terms of the eements of the overa transfer matrix of the fiter in the iterature see Ref. 6 for exampe, but no such expressions are to be found for structure-borne sound. The present paper seeks to fi this gap. Structure-borne ongitudina waves, fexura waves, and fuid-borne sound waves are dynamicay couped. Hence, a wave of any of these three types when incident on a panar hose-pipe system wi in genera resut in refection and transmission of a three types of waves. Thus, compete knowedge of the transmission characteristics of a hose system wi in genera be given by a 33 matrix of transmission oss TL vaues. In this paper, these TL vaues have been derived in terms of eements of the overa 88 matrix of the hose-pipe system, adapted from Ref. 2 see Appendix A wherein the effects of shear deformation, rotary inertia, and the fuid oad have been incorporated. These expressions have then been used for parametric studies reated to vibroacoustics of hoses. Some genera design guideines based on the parametric studies have been arrived at for visco-eastic hoses, with or without bends, with gases or iquids as the media of propagation. I. BASIC GOVERNING EQUATIONS Longitudina waves in the hose wa and onedimensiona sound waves in the medium inside are governed by the foowing couped equations:

2 FIG. 1. A straight hose-pipe reach with state variabes. f z A p r e pea u z p z 0, 1 FIG. 3. A hose-pipe with a smooth bend. f z z pa p 2 u z t 2 0, 2 shear ange: x u y z x, shear force: f y GA p k s x, 6 5 pk* v z 2K* u z z 0, 3 bending moment: m x EI p x z, 7 p 2 v z f t 2 0. Fexura or bending waves are governed by the foowing equations for a beam incorporating shear deformation, rotary inertia, and fuid inertia: 4 force equiibrium: moment equiibrium: f y z pa p f A f 2 u y t 2 0, m x z f y p I p f I f 2 t 2 0, where f z, u z, p, v, m x, x, f y, and u y are state variabes as shown in Fig. 1. Other notations are as foows: A p is the cross-sectiona area of the pipe, is the Poisson s ratio of the pipe materia, r is radius of the pipe cross section, e is the pipe wa thickness, E is Young s moduus generay compex, z is the axia coordinate as shown in Fig. 1, p is density of the pipe materia, f is density of the fuid inside the pipe, G is the shear moduus of the pipe materia, k s is the shear shape factor, I p is the moment of inertia of the pipe, I f is the moment of inertia of the fuid inside the pipe, K* is the modified buk moduus; for thin-waed pipes, K K* 1K1 2 r/ee, where K is the buk moduus of the fuid II. EVALUATION OF THE TL MATRIX FIG. 2. A sharp bend. For a one-dimensiona hose-ine system configured in the y-z pane the pane of the paper, fied matrices A6 for straight pipes Fig. 1 and point matrices A8 of Appendix A for sharp bands Fig. 2 may be mutipied successivey as per configuration so as to obtain the overa or product transfer matrix reation connecting the state vector 2

3 U z P V U y x M x F y T 11 on the right-hand or downstream side with that on the eft hand or upstream side see Fig. 1. Here U, P, V, F,, and M denote compex ampitudes of dispacement, acoustic pressure, acoustic partice dispacement, force, fexura anguar rotation or sope, and bending moment, respectivey. Fied matrix of the constituent hose-pipe eements woud be made up of the 44 submatrices T fp and T yz ony. Before mutipying out, a transfer matrix eements have to be mutipied with the dimensiona factors associated with the state variabes A7, so that state variabes have their physica dimensions as in the state vector 11. It may be noted that for a smooth bend Fig. 3 the bent portion wi have to be partitioned and the anguar changes wi be umped at sections 2, 3, 4, and 5 15, 30, 30, 15 and the intermediate portions wi be treated as curved pipes with modified fexura rigidity EI. 2 In the absence of any bends, the resutant transfer matrix woud consist of two uncouped 44 submatrices, one representing the product of the individua T fp matrices and the other of the T yz matrices. Further, if we put Poisson s ratio hypotheticay equa to zero in the T fp matrix that is, if we negect Poisson couping, then it woud separate into two uncouped 22 submatrices corresponding to ongitudina waves in the hose wa and acoustic waves in the fuid medium inside the hose. Thus, and U z d P Vd T 11 T 14 T 41 T 44 U z u T 22 T 23 T 32 T 33 P Vu Uy T55 T56 T57 T58 x T 65 T 66 T 67 T 68 x M x T 75 T 76 T 77 T 78 M x F yd T 85 T 86 T 87 T 88Uy F yu 12 13, 14 where T is the overa transfer matrix, and subscripts d and u denote downstream and upstream variabes, respectivey. In genera, and particuary in the presence of bends, one type of input incident wave wi resut in an output transmitted wave of not ony the same kind, but aso of other kinds. So, one shoud try to predict a set of nine TLs corresponding to three types of incident waves and three types of transmitted waves, making use of the entire 88 transfer matrix of a panar hose-pipe system see Eqs As TL is a symmetric function for stationary medium, state vectors subscripted u and d may be interchanged without having to invert the overa transfer matrix T. Thus, Uz P V U y x M x F yu 88 overa transfer matrix of the hose-pipe system T Uz P V U y x M x F yd. 15 For an incoming or incident progressive wave, there wi be one refected wave and one transmitted wave in the case of sound waves and ongitudina waves in the hose-pipe wa, and two refected waves and two transmitted waves, one propagating and one evanescent exponentiay decaying in the case of fexura waves. Thus, in genera, there wi be four refected waves and four transmitted waves anechoic termination is presumed on the downstream side as impied in the definition of transmission oss. Substituting these into the state variabes in Eq. 15 woud yied a set of eight inhomogeneous equations. These may be soved to obtain the three propagating type progressive waves evanescent fexura wave carries no wave energy in terms of the incident wave, and thence the corresponding three transmission oss vaues corresponding to this particuar type of incident wave. The process can be repeated for the other two types of incident waves to obtain a nine eements of the desired 33 TL matrix. In the foowing anaysis, the known incident wave ampitude is denoted by A and the unknown refected and transmitted wave ampitudes compex in genera by B. Substituting u z, f z,p,vu z,,p,ve z/ e jt 16 into the homogeneous set of Eqs. 1 4, and appying the compatibiity criterion, yieds the characteristic equation 4 2 0, 17 where,, and are defined by the identities A1a to A1c of Appendix A. Equation 17 has roots j 1,j 2, 18 where 1 and 2 are given by Eqs. A1 and A1m, and represent wave numbers for the ongitudina wave and sound wave, respectivey. Therefore, for prediction of transmission oss of ongitudina waves and sound waves, the standing wave soutions may be written as u z A 1 e j 1 z/ B 1 e j 1 z/ e jt, 19 Q j 1 A 1 e j 1 z/ Q j 1 B 1 e j 1 z/1 e jt, 20 va 2 e j 2 z/ B 2 e j 2 z/ e jt, ph j 2 A 2 e j 2 z/ H j 2 B 2 e j 2 z/ e jt, where functions Q() and H() are given by Eqs. 2 and 4 as Q pa p 2, d dz j 1/jk 1, 23

4 FIG. 4. Vibro-acoustic isoation of a composite rubber hose with iquid medium. H f 2, d dz j 2 /jk Putting z0 arbitrariy at the upstream as we as downstream end of the hose-pipe system, and assuming the downstream termination to be anechoic as required by definition of TL, one gets U z,u A 1 B 1, U z,d B 2, 25 V u A 2 B 3, V d B 4, 26 P u H jk u,2 A 2 H jk u,2 B 3,,u Q jk u,1 A 1 Q jk u,1 B 1,,d B 2 Q jk d,1. p d B 4 H jk d,2, The corresponding reations for fexura waves may be obtained as foows. Substituting u y, x,m x, f y U y, x,m x,f y e z/ e jt 29 into Eqs. 5 9, eiminating shear ange x, and appying the compatibiity criterion, yieds the characteristic equation 4 2 0, 30 where,, and are defined by identities A3a to A3c of Appendix A. Equation 28 has roots 3, j 4, 31 where 3 and 4 are given by Eqs. A3i and A3j of Appendix A, and represent evanescent waves and propagating waves, respectivey. Therefore, for prediction of fexura transmission oss, the genera soutions may be written in terms of 3 and j 4 as a sum of four progressive waves two evanescent and two propagating simiar to Eqs. 19 and 20. Putting z0 arbitrariy at the upstream as we as downstream end of the hose-pipe system, and assuming the downstream termination to be anechoic as required by definition of TL, yieds U y,u A 3 B 5 B 6, U y,d B 7 B 8, 32 x,u E jk u,4 A 3 E jk u,4 B 5 Ek u,3 B 6, x,d E jk d,4 B 7 Ek d,3 B 8, M x,u F jk u,4 A 3 F jk u,4 B 5 Fk u,3 B 6, M x,d F jk d,4 B 7 Fk d,3 B 8, F y,u G jk u,4 A 3 G jk u,4 B 5 Gk u,3 B 6, F y,d G jk d,4 B 7 Ek d,3 B 8, where E(), F(), and G() are cacuated from Eqs. 5 9 as 2 C/ E EI 2 D 2, F EI2 /C EI 2 D 2, GC 2 /, d dz 3/, j 4 /k 3,jk 4,

5 FIG. 5. Effect of the wa storage moduus of a hose on its vibration isoation. C p A p f A f, 40 S 1 A 1, S 2 H jk u,2 A 2, S 3 A 2, D p I p f I f. 41 Substituting Eqs and into the transfer matrix equations 15, and rearranging, yieds a set of eight inhomogeneous equations for the eight unknowns B 1 to B 8 : ABS, 42 where BB 1 B 2 B 3 B 4 B 5 B 6 B 7 B 8 T, A 11 1, A 23 Hjk u,2, A 33 1, A 41 Qjk u,1, A 55 1, A 56 1, A 65 Ejk u,4, A 66 Ek u,3, A 75 Fjk u,4, A 76 Fk u,3, A 85 Gk u,4, A 86 Gk u,3, A i2 T i1 Q jk d,1 T i4, i1,2,...,8, A i4 T i2 Qjk d,2 T i3, i1,2,...,8, A i7 T i5 Ejk d,4 T i6 F jk d,4 T i7 G jk d,4 T i8, i1,2,...,8. The rest of the eements of matrix A are equa to zero: S 4 Q jk u,1 A 1, S 5 a 3, S 6 H jk u,4 A 3, S 7 F jk u,4 A 3, S 8 G jk u,4 A 3. The matrix equations 42 are soved by means of a standard subroutine for vector B for given say unity vaues of the incident waves A 1, A 2, and A 3 representing ongitudina, acoustic, and fexura waves, respectivey. Now, the expressions for power fux associated with different types of waves can be given by ongitudina waves: ongitudina waves: W 1 2 Re U z *, 43 sound waves: W S 2 A f ReP V* 44 fexura waves: W f 4 ReF y U y *M x x *, 45 where the superscript asterisk indicates compex conjugate. Appying these expressions to incident waves and transmitted propagating waves of the three types, yieds W 1,i W,i 2 ReQ jk u,1a 1 2, W 1,t W,t 2 ReQ jk d,1b 2 2, 46 47

6 FIG. 6. Effect of the structura damping oss factor of a hose on its vibration isoation. W 2,i W s,i 2 A f,u ReH jk u,2 A 4 2, W 2,t W s,t 2 A f,d ReH jk d,2 B 4 2, W 3,i W f,i 4 ReG jk u,4f jk u,4 E* jk u,4 ]A 3 2, W 3,t W f,t 4 G jk d,4f jk d, TLM1,1TL LTL TLM2,2TL s STL TL of ongitudina waves in the hose wa, TL of ongitudina waves in the fuid inside, TLM3,3TL f FTL TL of fexura waves In the presence of bends, however, couping terms of the TL matrix may aso become reevant because of cross-mode energy transfer. E*jk d,4 ]B Finay, eements of the 33 matrix of transmission oss may be cacuated by means of the equation: TLM j1,j210 og W j1,i W j2,t db, j1, j21, 2, and For straight hose-pipe systems, there is no couping between fexura waves and the other two waves. Therefore, for systems without any bends, TLM1,3TLM2,3TLM3,1 TLM3,20. Simiary, if one negects Poisson couping between ongitudina waves in the hose wa and the sound waves in the fuid inside, then TLM1,2TLM2,10. For most practica appications, ony the diagona eements of the TL matrix are of reevance. These are III. TYPICAL RESULTS AND CONCLUSIONS A comprehensive computer program has been prepared in FORTRAN for prediction of the transmission oss vaues for ongitudina waves, sound waves, and fexura waves LTL, STL, and FTL, making use of the expressions derived above. Parametric studies were made for different materias and geometries as isted in Appendix B, for fexura waves (TL f or FTL. Typica resuts are shown in Fig. 4 for the defaut configuration of a 500-mm ong, 19-mm nomina diameter, composite-rubber hose with stee-pipe terminations, but with a iquid medium power-steering system oi: see Appendix B for specifications. Dips in the TL curves correspond to resonance frequencies that roughy correspond to kn, n1,2,3,..., where k K*/ f 1/2 for sound waves, 56

7 FIG. 7. Effect of the materia of a hose on its vibration isoation. FIG. 8. Effect of the ength of a hose on its vibration isoation.

8 FIG. 9. Effect of the number of hose pieces within the same overa ength on vibration isoation. and k E r / p 1/2 for ongitudina waves. 57 The observed vaues are within 20% of those given by the approximate reationships 56 and 57. However, this simpe reation (kn) does not hod for fexura waves for which k pa p 2 E r I p 1/4. 58 If the medium were a gas, LTL and FTL curves woud re- FIG. 10. Effect of wa thickness of a hose on its vibration isoation.

9 FIG. 11. Effect of the inner radius of a hose on its vibration isoation. FIG. 12. Effect of bend ange on vibration isoation of a hose.

10 main more or ess unchanged, but STL woud reduce to zero ess than 0.5 db. This is because ongitudina and fexura waves trave in the wa materia whereas sound waves trave in the fuid medium inside. For air medium, impedance mismatch between air and hose materia is so strong that the wa behaves more or ess ike a rigid boundary. For iquid medium, however, the impedance mismatch is rather weak. Poisson couping is not of much significance because its negect putting 0 inthet fp transfer matrix aters LTL and STL by ess than 0.5 db even at 500 Hz; the difference is much ess at ower frequencies. Therefore, use of uncouped transfer matrices for evauation of LTL and STL woud suffice. FTL vaues computed with the cassica beam assumptions negecting shear deformation, rotary inertia, and fuid inertia turn out to be substantiay ower by 4 5 db at higher frequencies 500 Hz. Therefore, the exact theory that is, without the cassica beam assumptions has been used here for the parametric studies. These studies for gaseous medium refrigerant vapor indicate that vibration isoation represented by fexura transmission oss FTL woud improve with: i Softer hoses ower storage moduus, E r as indicated by Fig. 5, ii ossier hoses higher oss factor, as is obvious from Fig. 6, iii rubber hoses instead of auminum as is cear from Fig. 7, iv onger hoses as is borne out by Fig. 8, v more smaer hoses in series with metaic stee joints as indicated by Fig. 9, vi thinner was as is obvious from Fig. 10, and vii ower interna radius as is borne out by Fig. 11. In particuar, Fig. 9 shows that the effect of a mutipe hose is simiar to that of a mutipe expansion chamber muffer: better at higher frequencies, but worse at ower frequencies. Bends have a mixed but generay beneficia effect on vibration isoation see Fig. 12, and therefore may be used as necessitated by ogistics. A these observations, viz., i vii, are in fact independent of the fuid medium inside the hose pipe, inasmuch as they have been found to hod for iquid medium we. Finay, it may be noted that this paper has been concerned primariy with vibration isoation of hoses. For appications deaing primariy with sound waves aong hoses or hose muffers ike power-steering systems, 3,4,8,9 the theory deveoped in Ref. 5 with that of Ref. 6 as the base woud be more usefu inasmuch as it woud hep in evauating the breakout noise or transverse transmission oss TLtp that imposes a imit on effectiveness of axia sound TL STL reaization. ACKNOWLEDGMENTS The work reported here was done at Ford Motor Company when the first author was on sabbatica eave from Indian Institute of Science, Bangaore. The authors thankfuy acknowedge the support and permission to pubish by the Cimate Contro Operations of the Automotive Components Division of Ford Motor Company. APPENDIX A: TRANSFER MATRICES The fied transfer matrix of a beam eement is composed of submatrices representing different types of waves. For Poisson couped axia stress waves in the hose wa and pressure waves in the fuid inside, the nondimensiona representation of the fied transfer matrix and state vector are 2 C 2 C 0 T fp C 1 C 3 b h C 1C 3 b h C 2 C 1 C 3 2C 3 C 2 C 0 C 3 C 1 2C 2 1 2C 2 C 1 2, A1 C 3 C 2 C 0 2C 3 C 1 b h C 2 b h C 3 C 2 C 0 where subscript fp indicates sound waves in the fuid and ongitudina waves aong the pipe wa, 2 2 a f 2, A1a 2 2 a p 2, A1b b e, d p f, h E K*, A1d A1e A1f 2 2 b d, A1c C cos cos 2, A1g

11 C sin sin 1 2, 2 C 2 cos 1 cos 2, A1h A1i t 2 4 1/2, A1m and the nondimensiona state vector at ocation i the righthand end in Fig. 1 is C sin sin 2, , /2, A1j A1k A1 Z i U z p V K* T i A p E. A2 For fexura vibration of the hose in the yz pane, the nondimensiona representation of the fied transfer matrix and state vector are 2 T yz C 0 C 2 C 1 C 3 C 2 1 C 1 2 C 3 C 3 C 0 C 2 C 1 C 3 C 2, A3 C 2 2 C 3 C 1 C 0 C 2 C 1 C 3 C 1 C 3 C 2 C 3 C 0 C 2 pa p f A f GA p k s 2 2, A3a pi p f I f EI p 2 2, A3b pa p f A f EI p 2 4, A3c C cosh cos 4, A3d 1. Genera fied transfer matrix The fied transfer matrix for a singe straight pipe reach shown in Fig. 1 is composed of two submatrices: ongitudina vibration of the iquid and pipe wa and transverse vibration in the y-z pane negecting torsion. Their expressions were given in Eqs. A1 and A3, respectivey. The state vectors have eight dependent variabes: two for each of the forces, moments, dispacements, and rotations of the pipe wa, and one each for acoustic pressure and partice dispacement of the fuid inside. The equation beow shows these arrangements: C sinh sin 3 4, 4 C 2 cosh 3 cos 4, A3e A3f Z i T L Z i1, A5 where T L is the fied transfer matrix for a pipe reach of ength in the oca coordinate system. This 88 matrix may be partitioned as 2 C sinh sin 4, , /2 1 2, / A3g A3h A3i A3j The state vector in the y-z pane at ocation i right-hand end in Fig. 1 is Z i U y M x F y 2 T x. A4 EI x EI p i T L T fp 0 A6 0 T yz. The state vector at ocation i in Fig. 1 is Z i U z P V K* 2. Transfer matrix for a bend U y M x F y 2 T A p E x EI p EI p. A7 The point transfer matrix reation for a sharp bend see Fig. 2 is 2

12 Uz P K* V A p E U y x M x EI p F y 2 R cos sin cos sin bq1cos 0 cos sin g sin cos gqb sin 0 g sin cos i Uz 1 P K* V A p E U y x M x EI p F y 2 L A8 EI pi EI pi or Z i R P L B i Z i L, A9 where superscripts R, B, and L denote right, bend, and eft respectivey, and g A p 2 I p, A9a q A f A p, b K* E. A9b A9c APPENDIX B: HOSE DIMENSIONS AND MATERIAL PROPERTIES USED IN THE PARAMETRIC STUDIES Inside fuid: defaut: HFC-134a refrigerant vapor density22 kg/m 3, buk moduus Pa aternative: power-steering system oi density834 kg/m 3, buk moduus Pa Hose materia: defaut: 3/4 in. hose used in cimate contro system Eastic moduus EE r (1j) Storage moduus, E r (1f/1000), Pa Loss factor, 0.2(10.1f /1000) Poisson s ratio, 0.48 Interna radius, r9.5 mm Wa thickness, e4 mm Density, 1196 kg/m 3 aternative: auminum hose Storage moduus, E r Pa Loss factor, Poisson s ratio, 0.33 Density, 2710 kg/m 3 Interna radius, r8 mm Wa thickness, e2 mm Termina materia: mid stee Storage moduus, E r Pa Loss factor, Poisson s ratio, n0.29 Density, 7800 kg/m 3 Interna radius, rhose radius Wa thickness, ehose wa thickness Fexura transmission oss FTL has been seected for parametric studies on vibration isoation. 1 D. C. Wiggert, R. S. Otwe, and F. J. Hatfied, The effect of ebow restraint on pressure transients, ASME J. Fuids Eng. 107, M. W. Lesmez, D. C. Wiggert, and F. J. Hatfied, Moda anaysis of vibrations in iquid-fied piping systems, ASME J. Fuids Eng. 112, L. Suo and E. B. Wyie, Compex wave speed and hydrauic transients in viscoeastic pipes, ASME J. Fuids Eng. 112, M. C. Hastings and Chuan-Chiang Chen, Anaysis of tuning cabes for reduction of fuid borne noise in automotive power steering hydrauic ine, SAE paper No , Noise and Vibration Conference in Traverse City, M. L. Munja and P. T. Thawani, A simpe mode for wave propagation aong and across a hose, Proceedings of Inter-Noise , pp M. L. Munja, Acoustics of Ducts and Muffers Wiey Interscience, New York, L. Cremer, M. Heck, and E. E. Ungar, Structure-borne Sound Springer- Verag, Berin, 1988, 2nd ed. 8 D. K. Longmore and A. Schesinger, Transmission of Vibration and pressure fuctuations through hydrauic hoses, Proc. Inst. Mech. Eng. 205, Jean Botti, G. Venizeos, and N. Benkaza, Optimization of power steering systems vibration reduction in passenger cars, in SAE Proceedings of the 1995 Noise and Vibration Conference, pp , paper No

Lecture 6: Moderately Large Deflection Theory of Beams

Lecture 6: Moderately Large Deflection Theory of Beams Structura Mechanics 2.8 Lecture 6 Semester Yr Lecture 6: Moderatey Large Defection Theory of Beams 6.1 Genera Formuation Compare to the cassica theory of beams with infinitesima deformation, the moderatey