Frequency Characteristics Analysis of Vibration System for Planetary Wheeled Vehicle

Transcription

1 Sensos & Tansduces, Vol. 5, Special Issue, Decembe 03, pp. 9-6 Sensos & Tansduces 03 by IFSA Fequency Chaacteistics Analysis o Vibation System o Planetay Wheeled Vehicle Yuchuan Lu, Zhiqi Du, Bo Su, Lei Jiang, Guoxuan Lu, Zhijie Zhang China Noth Vehicle Reseach Institute Tel.: luych7050@6.com Received: 6 Septembe 03 /Accepted: 5 Octobe 03 /Published: 3 Decembe 03 Abstact: In ode to conside the sink eect that caused by the mental wheels when the planetay wheeled vehicle is diving on the sot soil, 7-DOF spatial vibation model o the whole vehicle is ebuilt. Dieential equations o this system ae obtained by Lagangian equations. Then the mathematical expessions o the equency chaacteistics ae deived, which espectively belong to the motion o the body cente-o-mass, the dynamic delections o the ou suspensions and the elative dynamic loadings o the ou equivalent wheels. Finally, the coesponding amplitude-equency chaacteistic cuves and phase-equency chaacteistic ones ae pesented by MATLAB, and the equency chaacteistics o the whole vehicle vibation system ae also analyzed. Simulation esults show that the designed paametes o the vehicle ae easonable. The analytic expessions solved above have a cetain guiding signiicance in theoy o vehicle vibation study, vibation contol and vehicle-elated paametes optimization. Copyight 03 IFSA. Keywods: Planetay, Spatial vibation, Dynamic delection, Relative dynamic loadings, Amplitude equency chaacteistic, Phase equency chaacteistic.. Intoduction Planetay wheeled vehicle has cetain advantages while opeating in the o-oad envionment []. In this pape, in ode to solve the sink eect o the mental wheels while the planetay wheeled vehicles ae opeating on the sot soil, the vibation model in the liteatue [] is impoved. In this model, the soil sot chaacteistics and the wheels sinking ae tanse to damping and each planetay gea tain wheels is equivalent to a single wheel with damping. Because o the sot soil and the igid metal wheels, the oad can be egaded as had-suace without any total eect changed. With the advantages o the equivalent and convesion, the oad oughness incentive is egaded as point incentive o each wheel which acilitates the analysis o the poblem, but also simpliies the mathematical pocessing.. Dieential Equations o Vibation System A new 7-DOF spatial vibation model o the planetay wheeled vehicle is shown in Fig.. The genealized coodinates o this vibation system is T deined as { Z } = { zm θ ϕ zwl zw zwl zw }. The physical meaning o each coodinate is stated as ollows: vetical displacement o the body cente-omass, the angula displacement accoding to positive x-axis otation o the body, the angula displacement accoding to positive y-axis otation o the body, the vetical displacements o the let ont equivalent wheel, ight ont equivalent wheel, let ea equivalent wheel, ight ea equivalent wheel. The genealized coodinate oigin is the static equilibium position o each DOF while the vehicle is on a Aticle numbe P_SI_434 9

2 Sensos & Tansduces, Vol. 5, Special Issue, Decembe 03, pp. 9-6 hoizontal static plane. Accoding to Lagange equations o the vibation system, d L L ψ ( ) + = 0, dt q q q i i i i =,,, 7 dieential equations in matix om o the vibation system is obtained: { } + { } + { } = { } [ M ] Z [ C] Z [ K] Z F, () whee the mass matix is: M Iθ I ϕ [ M] = mw mw mw m w the damping matix is: ( c + c) ( n m)( c + c) ( ac + bc) c c c c ( n + m )( c + c) ( n m)( ac + bc) nc mc nc mc ( a c + b c) ac ac bc bc [ C] = c + c w sym c + c 0 0 w c + c 0 w c + c the stiness matix is: w [ K] = sin α sin α sin α ( k + ( nm)( k ( ak sin sin sin sin k k k k sin β sin β sin β ) + ) ) + k k bk ( n + m)( k ( nm)( ak sin β ) sin α sin β + k + bk ) sin α ( ak sin + bk ) sym sin α sin sin sin sin ak ak bk bk β α α β β sin α sin α sin β sin β nk mk nk mk α α β β sin α k + k w sin α k + k 0 0 w sin β k + k 0 w sin β k + k w 0

3 Sensos & Tansduces, Vol. 5, Special Issue, Decembe 03, pp. 9-6 φ θ Fig.. The new 7-DOF spatial vibation model o the planetay wheeled vehicle. The oad excitation vecto is { F} = { F} { F } + whee q B qa F [ CW] q cw = =, q C 0 c w 0 0 qd 0 0 cw c w { } { } qb qa F = [ KW] q = kw qc 0 k w 0 0 qd 0 0 kw k w { } { } Accoding to liteatue [], qi + qi qi =, i= B, AC,, D qi and qi ae the oad oughness unctions that espectively belong to the ont and ea gounded gea o the i th planetay wheel, which satisy the ollowing elationship:, qi( t) = qi ( t 3R ) v The Fouie tansom o the above expession is: q j π 3R / v + e ) = qi ( ), i = B, A, C, D i ( 3. The Fequency Chaacteistics o Vibation System 3.. The Fequency Chaacteistics o Body Cente-o-mass By the Fouie tansom o both sides o the omulation (), the equency esponse unction matix o the system esponse displacement vecto elating to the equivalent oad excitation vecto is obtained: Z~ q { π π } [ H( )] = ( ) [ M] + j [ C] + [ K] i {[ K ] + j π [ C ]} W W () Theeoe, the equency esponse unction matix o the system esponse velocity vecto elating to the equivalent oad excitation vecto is obtained: [ H ( )] = jπ [ H ( )] ~ Z q Z q (3) The equency esponse unction matix o the system esponse acceleation vecto elating to the equivalent oad excitation vecto is obtained:

4 Sensos & Tansduces, Vol. 5, Special Issue, Decembe 03, pp. 9-6 [ H ( )] Z ~ = ( jπ ) [ H ( )] q Z q (4) So the equency esponse unction vecto o the displacement o the body cente-o-mass elating to the equivalent oad excitation vecto is the ist ow o matix (). Similaly, the equency esponse unction vecto o the velocity o the body cente-omass elating to the equivalent oad excitation vecto is the ist ow o matix (3), the equency esponse unction vecto o the acceleation o the body centeo-mass elating to the equivalent oad excitation vecto is the ist ow o matix (4). Since the symmety o the vehicle stuctue, the equency chaacteistics o the motion o the body cente-omass elating to the equivalent oad excitation ae consistent. Without loss o geneality, with the equency chaacteistics o the motion o the body cente-o-mass elating to the oad excitation o the equivalent ight ont wheel in mind, as shown in Fig.. When the equency is 5.4 Hz in the igue o amplitude-equency chaacteistics, the system eaches esonance peaks. The steady-state gain o displacement o body cente-o-mass elating to oad excitation o equivalent wheel is always less than, similaly, the steady-state gain o velocity and acceleation o body cente-o-mass elating to oad excitation o equivalent wheels each esonance peaks which is the maximum. In the igue o phaseequency chaacteistic, the phase values o displacement, velocity and acceleation incease successively by the step o 0.5π, which indicates that phase o acceleation advances 0.5π than phase o velocity, and phase o velocity advances 0.5π than phase o displacement. Duing (0.00, 90) Hz equency inteval, the phase-equency cuves ae acoss.5π, cuves tuning points ae also at the 5.4 Hz. In conclusion o equency chaacteistics o the motion o body cente-o-mass, vibation paametes o this vehicle avoid the equencysensitive inteval, which indicates that the paametes ae easonable. Fig.. Bode diagam o Body centoid motion o the oad suace excitation o the equivalent ight ont wheel. 3.. Fequency Chaacteistics o Suspension Dynamic Delection Suspension dynamic delection is expessed as: = z z = z aϕ+ nθ z = z z = z aϕmθ z B B wl M wl A A w M w = z z = z + bϕ+ nθ z C C wl M wl = z z = z + bϕmθ z D D w M w The Fouie tansom o above expession divides by the Fouie tansom o the equivalent oad excitation to obtain the equency esponse unction matix o suspension dynamic delection elating to equivalent oad excitation, as shown as ollowing: [ H~ q( )] = [ A][ H( )] = n a m a = [ H( )] n b m b (5)

5 Sensos & Tansduces, Vol. 5, Special Issue, Decembe 03, pp. 9-6 [ H ~ ( )] whee matix q is given in the peceding [ H ~ ( )] paagaphs. q is a 4 th -ode squae matix. The i th ow o this matix is the equency esponse unction vecto o i th suspension dynamic delection elating to the oad excitation vecto (i =,,3,4 coesponding espectively to the B, A, C, D suspension). Taking the ight ont suspension as an example, Bode diagams o the equency esponse unctions o these dynamic delections ae shown in Fig. 3. Fig. 3(b) shows the dynamic delection elating to equivalent oad excitation on itsel wheel. On the amplitude-equency cuve, the steady-state gain is at 0.00 Hz, with equency inceasing successively, the value emains unchanged. Thee is the peak at 5.7 Hz with the value o 3.94, then, as the equency incement, it apidly decays to -5.3 (90 Hz); on the phase-equency cuve, the phase stats with the value o π, inally educed to 0, while thee is the tuning point at the same equency o 5.7Hz. The equency esponse cuves o this dynamic delection elating to equivalent oad excitation o the othe wheels ae simila, shown in (a), (c) and (d). With (a) in mind, on the amplitudeequency cuve, the steady-state gain is at 0.00 Hz, while at 3.4 Hz, it eaches to as a little peak as the body s inetia o being uplited by the oad shock, thee is a negative peak that apidly tansits to -93. at 5.64 Hz, which is caused by the compessed suspension when the diagonal cone o body is uplited, and late, it etuns to at 7.76 Hz, with the equency inceasing, the gain educes, at 90 Hz, it eaches to -38.; on the phaseequency cuve, the phase is 0 at 0.00 Hz, and as the same, at 5.64 Hz thee is a saltus, which jumps to a highe phase value, then, with equency inceasing, the phase educes to -.5π. (a) (b) (c) (d) Fig. 3. Bode diagam o the ight ont suspension dynamic delection o equivalent oad excitation vecto. 3

6 Sensos & Tansduces, Vol. 5, Special Issue, Decembe 03, pp. 9-6 In conclusion o equency chaacteistics o the suspensions dynamic delection, vibation paametes o this vehicle also avoid the equency-sensitive inteval, which indicates that the paametes ae easonable Fequency Chaacteistics o Relative Dynamic Loadings o the Equivalent Wheels Static loadings o equivalent wheels satisy the ollowing equilibium equation: ' ' ( GC + GD)( a+ b) = Mga ' ' ( GC + GB)( m+ n) = Mgm ' ' ( GA + GB)( a+ b) = Mgb ' ' ( GA + GD)( m+ n) = Mgn Theeoe, static loadings o the equivalent wheel ae solved: b n = + = b m = + = a m = + = a n = + = + ' GA mwg GA mwg Mg a b m n ' GB mwg GB mwg Mg a b m n ' GC mw g GC mw g Mg a b m n ' GD mw g GD mwg Mg a + b m + n (6) The expession o dynamic loadings is descibed as ollowing: FdA = mw zw + c [ z w (z M a φ m θ )] + sin z α w zm aφmθ + k [ ] F = m z + c [ z (z a φ+ n θ )] + db w wl wl M sin α wl M φ+ + z z a nθ k [ ] FdC = mw zwl + c [ z wl ( z M + b φ+ n θ )] + sin β zwl zm + bφ+ nθ + k [ ] F = m z + c [ z (z + b φ m θ )] + z z b mθ k [ ] dd w w w M sin β w M + φ + By the omulation (6), (7), we can obtain the elative Fdi G dynamic loading i o each equivalent wheel, and then obtain the equency esponse unctions o the elative dynamic loadings o equivalent wheels (7 ) elating to the oad excitations by Fouie tansom. Witten in matix om is as ollowing: [ H ( )] = [ A ( )][ H( )] Fdi, (8) ~ q Fdi Gi [ H ( )] = [ H ( )] whee ecoded as: [ A ( )] = F di G i Gi Z q the coeicient matix is Ω / GB nω / GB aω / GB / GB Ω / GA mω / GA aω / GA 0 / GA 0 0 = Ω / GC n Ω / GC b Ω / GC 0 0 / GC 0 Ω/ GD mω / GD bω / GD / GD Among them, [ H Fdi ~ q Gi = + + sin ( i π ) mw ( i π ) c k = + + sin α Ω = ( i π ) c + k sin β Ω = ( i π ) c + k α sin ( i π ) mw ( i π ) c k ( )] is a 4th-ode squae matix, The ith ow o this matix is the equency esponse unction vecto o elative dynamic loading o i th equivalent wheel elating to the oad excitation vecto (i =,, 3, 4 coesponding espectively to the B, A, C, D wheel). Taking the ight ea equivalent wheel as an example, Bode diagams o the equency esponse unctions o the elative dynamic loadings o this wheel elating to the ou oad excitations ae shown in Fig. 3. The Fig. 3(d) shows the elative dynamic loading elating to equivalent oad excitation on itsel wheel. On the amplitude-equency cuve, the steadstate gain is at 0.00 Hz, with equency inceasing successively, the value emains unchanged. At 0. Hz, the steady-state gain stats to ise. Thee is the peak at 5.53 Hz with the value o 4., then, as the equency incement, it apidly decays to (90 Hz); on the phase-equency cuve, the phase stats with the value o 0, and then gows along with the equency, thee is a tuning point at 0. Hz, late eaches a steady state with the value o π at.86 Hz, inally educed to 0, while thee is also a tuning point at the same equency o 5.53 Hz. The equency esponse cuves o the elative dynamic loadings o this wheel elating to equivalent oad excitations o the othe wheels ae simila, shown in (a), (b) and (c). With (a) in mind, on the amplitude-equency cuve, the steady-state gain is at 0.00 Hz, with equency β 4

7 Sensos & Tansduces, Vol. 5, Special Issue, Decembe 03, pp. 9-6 inceasing successively, the value emains unchanged. while at 0. Hz, it stats to ise, thee is the ist tuning point at 0.45 Hz in the ising pocess, thee is the second tuning point at 4.6 Hz whee the gain eaches a steady state, then it apidly ises to 6.64 as a peak at 5.53 Hz, late it apidly educes to (90 Hz); on the phase-equency cuve, the phase is 0 at 0.00 Hz, with equency inceasing successively, the value ises. at 0. Hz, thee is the ist tuning point, at.3 Hz, it eaches the maximum o π, then educes with the equency inceasing, and as the same, at 5.53 Hz thee is a apid tuning point, inally the phase educes to -π. In conclusion o equency chaacteistics o the elative dynamic loadings, vibation paametes o this vehicle also avoid the equency-sensitive inteval, which indicates that the paametes ae easonable. (a) (b) (c) (d) Fig. 4. Bode diagam o the ight ea equivalent wheel elative dynamic load o equivalent oad excitation vecto. 4. Conclusion Based on the 7-DOF spatial vibation model o the whole vehicle, dieential equations ae obtained by Lagangian equations. Then, the mathematical expessions o the equency chaacteistics ae deived, the simulation esults show that the designed paametes o the vehicle avoid the equencysensitive inteval, which indicates that the paametes ae easonable, in conclusion o equency chaacteistics o the motion o body cente-o-mass and the suspensions dynamic delection and the elative dynamic loadings o the equivalent wheels. Reeences []. Weisbin C. R., Rodiguez G., Autonomous ove technology o mas sample etun, in Poceedings o the 5 th Intenational Symposium on Atiicial 5

8 Sensos & Tansduces, Vol. 5, Special Issue, Decembe 03, pp. 9-6 Intelligence, Robotics and Automation in Space, 999, pp. -9. []. Gao Hai-Bo, Deng Zong-Quan, Wang Shao-Chun. Dynamics analysis o the luna ove with planetay wheel, Habin Institute o Technology Univesity, 37,, 005, pp [3]. Yu Zhisheng, Vehicle Theoy, 4th edition, Mechanical Industy Pess, Beijing, 000. [4]. Li Jie, Qin Yuying, Zhao Qi, The Application o Multi-point Pseudo Excitation Method to Vehicle Random Vibation Analysis, Automotive Engineeing, 3, 3, 00, pp Copyight, Intenational Fequency Senso Association (IFSA). All ights eseved. ( 6