Point Excitation of a Coupled Structural-Acoustical Tire Model with Experimental Verification

Transcription

1 Purdue University Purdue e-pubs Publications of the Ray W. Herrick Laboratories School of Mechanical Engineering Point Excitation of a Coupled Structural-Acoustical Tire Model with Experimental Verification J Stuart Bolton Purdue University, bolton@purdue.edu Rui Cao Purdue University, cao101@purdue.edu Follow this and additional works at: Bolton, J Stuart and Cao, Rui, "Point Excitation of a Coupled Structural-Acoustical Tire Model with Experimental Verification" (2015). Publications of the Ray W. Herrick Laboratories. Paper This document has been made available through Purdue e-pubs, a service of the Purdue University Libraries. Please contact epubs@purdue.edu for additional information.

2 Point excitation of a coupled structural-acoustical tire model with experimental verification Rui Cao, J. Stuart Bolton Ray W. Herrick Laboratories, School of Mechanical Engineering, Purdue University

3 I. Introduction Traffic noise Roadside residences Passengers Vehicle noise Power Unit noise Aerodynamic noise Tire/pavement noise Transfer paths In cabin noise 2

4 I. Introduction Structural Waves Airborne Waves Objective: 1. Build a model coupling the tire structure and air cavity 2. Identify tire structural vibration and acoustical modes 3. Create forced response model incorporating coupled modes 4. Investigate the dynamics property of the tire structure 3

5 II. Literature Review Structure-borne sound on a smooth tyre Kropp A wave model of a circular tyre. Part 1: belt modelling Pinnington Vibrations of Shells and Plates Soedel A coupled tire structure/acoustic cavity model Molisani, Burdisso & Tsihlas The influence of tyre air cavities on vehicle acoustics Fernandez The wave number decomposition approach to the analysis of tire vibration Bolton, Song, Kim & Kang 4

6 III. Model description Build an analytical model Free vibration analysis Forced vibration analysis w u The wheel rim is rigid and fixed Tire sidewall is represented by springs in radial and tangential directions Ring structure allows for flexural and longitudinal waves Harmonic point input excitation at arbitrary angle is applied 5

7 III. Model description Build an analytical model Free vibration analysis Forced vibration analysis Solving for natural frequencies Harmonic displacements are assumed as: jk jt w e e jk jt u e e Substitution into the static coupled ring EOMs and write solutions in matrix form: M11 M12 p M21 M 22 0 p is the acoustic pressure disturbance in the air cavity, which is assumed to be the distributed load in radial direction p 0 vflow t By applying boundary conditions at two air-structure contact surfaces, we can obtain the pressure p as a function of radial coefficient α 6

8 III. Model description Build an analytical model Free vibration analysis Forced vibration analysis Solving for natural frequencies So the coupled equations can be expressed in homogeneous form M11 FL M12 0 M M Where M 11, M 12, M 21, M 22 and FL are expressions in terms of structure-related constants and the variables k θ and ω. Setting the determinant to zero gives us a characteristic equation: f ( ) ( M FL) M M M The natural frequencies can be solved for each mode number k θ. 7

9 III. Model description Build an analytical model Free vibration analysis Forced vibration analysis Obtaining forced response With the derived natural frequencies, the modal summation method is adopted here to predict the response due to harmonic point excitation. θ* Generalized loads: Assumed displacement: Fr ( *) j q t r e R F q ( *) e j t R 5 u U ( ) e jt r ni rni i1 n0 5 u U ( ) e jt i1 n0 ni ni Number of modes considered Number of wave types considered η ni is the modal participation factor, U ni (θ) is the mode shape 8

10 III. Model description Build an analytical model Free vibration analysis Forced vibration analysis. Assuming displacements To obtain the complete solution, two sets of orthogonal mode shapes need to be considered: ( ) Bn U sin n(1) n A U ( ) cos rn(1) An n ( ) Bn U cos n(2) n A U ( ) sin rn(2) An n The ratio of amplitudes can be found from the characteristic matrix from the previous modal analysis. The derivation process is the same for the two sets and the complete results would be the sum of the two sets of solutions: Bn M FL M A M M n n n (1) (2) 9

11 III. Model description Build an analytical model Free vibration analysis Forced vibration analysis. Finding modal participation factors The next step is to solve for the modal participation factors η n. From Love s equations, for each type of wave, we have n0 n0 From eigenvalue analysis, we have L { U, U } h U q n r n rn n rn r L { U, U } h U q n n rn n n n0 h U q 2 2 n n rn r 2 2 n n r n0 h U q 10

12 III. Model description Build an analytical model Free vibration analysis Forced vibration analysis. Finding modal participation factors By multiplying orthogonal modes U rm and U θm respectively and integrating around the ring circumference, we have * hn n UrnU rmrd Fr ( ) UrmRd 0 0 n0 R * hn n UnU mrd F ( ) UmRd 0 0 n0 R Perform the integration and add the above two equations, to give 2 2 * * * * h ni n Nn Fr ( ) Urn( ) F( ) Un( ) 2 2 Bn / An 1 R n 0 Nn U 0 rnu rn UnUn Rd 2 R n 0 So the modal participation factors η n are found for each frequency. 11

13 III. Model description Build an analytical model Free vibration analysis Forced vibration analysis. Obtaining displacements The last step is to substitute those factors back into the assumed displacement solutions. u r 5 F n F n hn f * * rcos ( ) tsin ( ) i1 n0 ni u 5 B ni A ni i1 n0 ni F n F n * * rsin ( ) tcos ( ) hn f As for the numerical calculation, we ll choose a limited numbers of terms for each wave considered. 12

14 III. Model description Build an analytical model Free vibration analysis Forced vibration analysis. Calculating mobilities The frequency-velocity function can be easily calculated as r, v j u v j u r If we apply unit excitation forces F r and F θ, the mobility function will have the same numerical value as the velocity function. Radial mobility = v F r r Circumferential mobility = v F 13

15 IV. Experimental Set up Computer LDV Data Acquisition Box radial velocity force Tire Tread Force Transducer Signal Generator Shaker Filter Amplifier 14

16 V. Results Dispersion relations 3 rd acoustical wave 2 nd acoustical wave 1 st acoustical wave 2 nd structural wave (fast extensional wave) 1 st structural wave (slow flexural wave) 15

17 V. Results Tire material properties Parameter Value Parameter Value Young s Modulus [Pa] Pressure [bar] 3 Density [kg/m 3 ] 1200 Inner Radius [m] Thickness [m] Outer Radius [m] Radial stiffness [N/m] Tangential stiffness [N/m] Natural frequencies [Hz] n\name slow acoustical 1 fast extensional acoustical 2 nd acoustical flexural (circumferential) (radial) (radial) N/A

18 V. Results Mobility functions P 1 Point 1 is input point, Point 2 is π/2 away from input point Radial Tangential P 2 The high amplitude peaks are due to the neglect of damping, which would smooth the peaks otherwise 17

19 V. Results Dispersion relations Analytical Experimental 18

20 V. Results Dispersion relations 19

21 V. Results Dispersion relations 215/60 R16 Dispersion Coherence (velocity & force) Cavity depth: 5 in 20

22 V. Results Dispersion relations 225/45 R18 Dispersion Coherence (velocity & force) Cavity depth: 4 in 21

23 VI. Conclusion The depth-direction acoustical modes in a tire s air cavity were analytically predicted The frequency-mobility functions were derived under point harmonic excitation Experimental data confirmed the analytical results, although disparities exist due to the difference between analytical and actual tire material parameters For precise radial mobility measurement, the laser should be pointed toward the tire in the radial direction. 22

24 Previous Papers on this topic R. Cao and J. S. Bolton. "Effect of rotation on the natural frequencies of coupled tire structuralacoustic modes." NoiseCon 2013, Denver CO. Vol No. 1. R. Cao. "Investigation of a fully coupled spinning tire-wheel model." M.S.M.E thesis, Purdue University (2014). R. Cao and J. S. Bolton. "Improved Model for Coupled Structural-Acoustic Modes of Tires." SAE International Journal of Passenger Cars-Mechanical Systems 8, (2015). 23

25 Thank you Questions? Thanks to for providing various tires for testing 24