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# MECHANICAL VIBRATION MEEN 617

Texas A&M

GPA 3.58

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This 16 page Class Notes was uploaded by Orrin Weissnat on Wednesday October 21, 2015. The Class Notes belongs to MEEN 617 at Texas A&M University taught by Luis Sanandres in Fall. Since its upload, it has received 62 views. For similar materials see /class/225990/meen-617-texas-a-m-university in Mechanical Engineering at Texas A&M University.

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Date Created: 10/21/15

Handout 2c pp 5671 Interpretation of Forces for the Periodic Forced Response of a 2nd Order Mechanical System Transmissibili Analysis of forces transmitted to base or foundation Frequency Response Function for Support Or Ground Motion MEEN 617 Notes Handout 20 Luis San Andr s 2008 256 Interpretation of Forces for the Periodic Forced Response of a 2 Order Mechanical System Recall that the equation of motion for a 2nd order system forced into motion by a harmonic force is 61 or as a balance of forces FtFD FK FI 0 62 with solution XI XSS A sinQl 0 63 F and X 2 2 magmtude of external force K stlffness X t where Damping Force FD DX K D Elastic Force FK KX 64 Inertia Force E MX From Eq 63 DISPLACEMENT gt X1XSS A SinQt C0 VELOCITYgt X1XSSQA 003Q 0 ACCELERATION Then FKr F0A sinQI FD DXZ FO2 fA cosQt go 66 MEEN 617 Notes Handout 20 Luis San Andres 2008 257 Et MXFOf2AsinQt go 92 D Let s plot the external force and X X X as rotating phasors n F s inquot rot Note that the velocity X leads by 90 the displacement Xt and the acceleration Xleads by 180 the displacementXt The elastic damping and inertia forces are graphed as MEEN 617 Notes Handout 20 Luis San Andr s 2008 258 dZX M alt2 dX DWKXFasmQI 61 Balance of Forces for 2nd Order System FD FK P Fn I il Fswquot FK 39 I Velocity XSAOJ DisRI cement Acceleration X5sz quot MEEN 617 Notes Handout 20 Luis San Andr s 2008 259 Let s study the forced response at various excitation frequencies of lbw fregq uern Qltltwn gtfltlt1 Then gtA gt1go gt0 FKI FOA sinQt go z FOI sinQt Oz Ft FD zO FIzO At low frequencies the elastic force FK balances the external force Fm At high frequencies l Qgtgtwn fgtgt1then an gtoogtA gt20 go Mr 180 Af2 gt1Af2 gt1f gt0 FKIN 0 FOA sinQt go zFO 0 sinQt 0 z 0 cl an FDzO FIG FD fZAsinQr gpzFO1sinQt7 FO sinQt F1 MEEN 617 Notes Handout 20 Luis San Andr s 2008 260 At high frequencies the inertia force F balances the external force Fa 90quot gtf1 gtA gt go gtZ 90 FKt FO A sinltQt FO A cosQt 11711 FOAfZ sinQt FK FD FO 2610i cosQt Fo 005Qi FO sinQt Ft ie the viscous damping force balances the external force If no damping is present then the equilibrium of forces is not possible and the system develops amplitudes of motion increasing and leading to a catastrophic failure MEEN 617 Notes Handout 20 Luis San Andr s 2008 261 TRANSMISSIBILITY Analysis of forces transmitted to base or foundation The analysis of transmitted forces is important to determine the maximum stresses on the structural supports as well as to verify the isolation characteristics of the system from the base The equation of motion of a MKD system for a periodic force of constant magnitude F0 and frequency 0 is FIT 61 with solution Xt 2X A sinQt go 62 K D where BASE A 2 mg K 1 f22gf2 lf andf 96quot as the ratio of the excitation frequency to the system natural frequency The dynamic force transmitted to the base or K foundation is Substitution of Eq 62 into Eq 64 gives MEEN 617 Notes Handout 20 Luis San Andr s 2008 262 FBzKXmA sinQt DXSSAQ cosQt go 65 EB FOA sinQt goD KQ cosQt 0 K D 2 with w and f 90quot then 1 2gf a Jl2 f2 1a J12 f2 And write Eq 66 as Define cos FB FUA 12 f 2 cosa sinQt gosina cosQt FB FO Aquot12 f 2 sinQI 0 68 where 12 g 2W 1f22 f2w T 3 69 tan39112gfj 05 tan3912 f MEEN 617 Notes Handout 20 Luis San Andr s 2008 263 Define the transmissibility T as the ratio of force transmitted to base or foundation FB to the input excitation force Fo sinQt Le 71 Regimes of operation at low frequencies Qltltwn gtf gtO gtAT 1 at high frequencies Qgtgtwn gtf gtoo gtAT 2gf at resonance 22019le gtAT 14 2 NOTES At low frequencies fltJ the transmitted force to base is largerthan external force ie Tgt1 At f J5 the system shows the same transmissibility regardless ofthe damping value Operation abovef gt 5 determines the lowest transmitted forces ie mechanical system is ISOLATED from base foundation A desirable operating condition MEEN 617 Notes Handout 20 Luis San Andr s 2008 264 When operation at large frequencies f gt 2 viscous damping causes transmitted forces to be larger than wo damping Damping is NOT desirable for operation at high frequencies FRF 2nd order system damping ratio005 Periodic force Fo sinQt damping ratio01 100 damping ratio02 b damping ratio05 9 E 5 E m 9 1 E 2 m E l l I I I I I I I 0 025 05 075 1 125 15 175 2 frequency ratio f MEEN 617 Notes Handout 20 Luis San Andr s 2008 Frequency Response Function for Support or Ground Motion Consider the motion of a M K D system with its base or s port moving with known or specified periodic displacement The dynamic response ofthis system is of particular importance for the correct design and performance of vehicle suspension systems Response to earthquake excitations as well The EOM is MYKY ZDY Z 0 71 Since Zb cos Qt is prescribed then 239 b Q sin Qt b Substitution on and dZdt into eqn 72 gives MYDYKY Kb cosQt DQb sinQt K b cosQt BQ sinQt K 73a 2 Kb cosQt 2 f sinQt MEEN 617 Notes Handout 20 Luis San Andr s 2008 266 D2 where E a and f Q0 The equation of motion is rewritten as 1 39 sinor ng t cos or e Wag2 12fgt2 and write Eq73b as 74 After all transients die out due to damping the system periodic steadystate response or FRF is 75 or MEEN 617 Notes Handout 20 Luis San Andr s 2008 267 i FRF of Base Motion 76 12 gff T2 212 3 a 1f22f with 77 0 tan391 a tan3912 f NOTE thatAB is identical to the amplitude of FRF for transmitted force ie the transmissibility ratio FRF 2nd order system damping ratio005 Support motion zb coswt damping ratioo1 100 39 damping ratio02 h a dampingratio05 8 s o c a o 10 quot quot3 o 39o E 3 t o 1 9 a E o N lt 3 I l I l l I I I 0 025 05 075 1 125 15 175 2 frequency ratio f MEEN 617 Notes Handout 20 Luis San Andr s 2008 268 EXAMPLE A 3000 lb empty automobile with a 10 wheelbase has wheels which weigh 70 lb each with tires Each tire has an effective stiffness contact patch to ground of 1000 lbin A static test is done in which 5 passengers oftotal weight 800 lb climb inside and the car is found to sag depress toward the ground by 2 a From the standpoint ofthe passenger comfort what is the worst wavelength in feet sine wave road which the car with all 5 passengers could encounter at 65 mph b For the worst case in a above what percent of critical damping is required to keep the absolute amplitude ofthe vertical heaving oscillations less than 12 of the amplitude ofthe undulated road c What is the viscous damping coef cient required for the shock absorber on each wheel assume they are all the same to produce the damping calculated in b above Give the physical units ofyour answer d State which modes ofvibration you have neglected in this analysis and give justi cations for doing so k V The wavelength is equal to 1 v T with T as the period of motion And the frequency 0 ofthe forced motion is 27239 27239 v a T A 2 b 603031 The system mass is Meq K 3000800 4x70 g g 2 K28001b2400 M2 3520 lbf 2 912 lbf sec 2 1n 1n q 3864 1nsec in MEEN 617 Notes Handout 20 Luis San Andr s 2008 269 12 d a 623 105112 Meq sec a For passenger comfort the worst wavelength in feet which the car could encounter at 65 mph is when the excitation frequency coincides with the system natural frequency ie 0 0n Thus from 2723927239v w aquot Then T 1 2 27239 65 mph 3530 C718 J 12 Sec 0 1 9047 ft 00171 miles 1 662 g sec b For the worst case what percent of critical damping is required to keep the absolute amplitude of the vertical heaving oscillations less than 12 of the amplitude ofthe Y undulated road ie What value of damping ratio Q makes 7 at 0202 Recall that at 0 0 the amplitude of the support FRF is from eqn 77 A 7 13 3 2w 2 4a 4 The solution indicates that the damping ratio Q is imaginary This is clearly impossible Note that the ampli cation ratio AB gt 1 at f 1 Le the amplitude of motion Y forthe system will always be larger than the amplitude of the base excitation b regardless of the amount of damping c What is the viscous damping coefficient required for the shock absorber on each wheel assume they are all the same to produce the damping calculated in b above No value of viscous damping ratio Q is available to reduce the amplitude of motion However if there should be one value then D 1b g ithen gtD1 M a f ZMEQ aquot ZMM aquot 2 q 1nsec MEEN 617 Notes Handout 20 Luis San Andr s 2008 270 d State which modes of vibration you have neglected in this analysis and give justifications for doing so Heaving up amp down motion is the most important mode and the one we have studied In this example pitching motion is not important because the road wavelength j is large We have also neglected yawing which is not important if the car cg is low One important mode to consider is the one related to tire bouncing ie the tires have a mass and spring coef cient of their own and therefore its natural frequency is given by a 7425 quot2 70 3864 sec However the car bouncing natural frequency is 662 radsec is much lower than the tire natural frequency ie mm 662 rads lt lt mm 7425 radsec Therefore it is reasonable to neglect the tire bouncing mode since its frequency is so high that it can not be excited by the road wavelength speci ed MEEN 617 Notes Handout 20 Luis San Andr s 2008 271

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