Purdue University

Purdue e-Pubs Publications of the Ray W. Herrick Laboratories

School of Mechanical Engineering

6-2015

Improved Model for Coupled Structural-Acoustic Modes of Tires Rui Cao Ray W. Herrick Laboratories, [email protected]

J. Stuart Bolton Ray W. Herrick Laboratories, [email protected]

Follow this and additional works at: http://docs.lib.purdue.edu/herrick Cao, Rui and Bolton, J. Stuart, "Improved Model for Coupled Structural-Acoustic Modes of Tires" (2015). Publications of the Ray W. Herrick Laboratories. Paper 114. http://docs.lib.purdue.edu/herrick/114

This document has been made available through Purdue e-Pubs, a service of the Purdue University Libraries. Please contact [email protected] for additional information.

IMPROVED MODEL FOR COUPLED STRUCTURAL-ACOUSTIC MODES OF TIRES Rui Cao, J. Stuart Bolton, Ray W. Herrick Laboratories, School of Mechanical Engineering, Purdue University

Company Logo Here

I. Introduction

Roadside residences

Traffic noise

Vehicle noise

Passengers

•

Power Unit noise

•

Aerodynamic noise

•

Tire/pavement noise

Transfer paths

In cabin noise

Dominant at high speed

SAE INTERNATIONAL

Paper 2199

2

I. Introduction

Fixed axle

Internal air cavity

Tire structure

Objective: 1. Build a model coupling the tire structure and air cavity 2. Identify tire structural vibration 3. Study sound characteristics in interior air cavity 4. Investigate spinning influence SAE INTERNATIONAL

Paper 2199

3

II. Literature Review

Structure-borne sound on a smooth tyre

Kropp

Effects of Coriolis acceleration on the free and forced inplane vibrations of rotating rings on elastic foundation A wave model of a circular tyre. Part 1: belt modelling

Effects of rotation on the dynamics of a circular cylindrical shell with application to tire vibration

Molisani, Burdisso & Tsihlas

The Influence of Tyre Air Cavities on Vehicle Acoustics

SAE INTERNATIONAL

Paper 2199

Pinnington Kim and Bolton

A coupled tire structure/acoustic cavity model

The wave number decomposition approach to the analysis of tire vibration

Huang & Soedel

Fernandez

Bolton, Song, Kim & Kang

4

III. Analytical Model

Review of previous models

string 1

2 Flow

air y

x

* Cao & Bolton, NoiseCon 2014

* Cao & Bolton, NoiseCon 2013 SAE INTERNATIONAL

Air Cavity

Paper 2199

5

III. Analytical Model

Fully coupled circular cavity model w

u

Y

ku

r

R

Tire tread Air cavity

θ Ω

X

kw

Rotation Wheel rim

1. 2. 3. 4.

θ

sheared air flow

Air cavity

The tire rotates about a fixed axle The wheel rim is rigid Tire sidewall is represented by springs in radial and tangential directions Ring structure includes flexural and longitudinal waves

SAE INTERNATIONAL

Paper 2199

6

III. Analytical Model

Rotating ring structure Assume harmonic solutions for displacements:

w   e jk  e jt

u   e jk  e jt

Substitution into rotating ring EOMs and write solutions in matrix form:  M11 M12    0 M      0 M   22     21

Where M11, M12, M21 and M22 are expressions of structure-related constants and variables kθ and ω. For example:  Eh3 3 Eh   h M11  j  k  k  2 k  2  h     4  2  2 12 R R R  

Eh3 2 Eh 2   h 2 p0 2 M12  k  k  k   k  2  h  k    h     u  12R 4 R2 R2 R SAE INTERNATIONAL

Paper 2199

7

III. Analytical Model r

Circular air cavity

Air cavity

Velocity of the flowing air is expressed as

v 

θ

v0 r R sheared

By using velocity potential ψ, the wave air flow equation in the circular air cavity is  2 1  1  2 1   2 v0  2 v02  2      2   r 2 r r r 2  2 c02  t 2 R t R 2  2  Harmonic solution of pressure is assumed in circumferential direction while Bessel function is assumed in radial direction:

 (r, , t )  gm ( mr )e jm e jt

gm ( mr )  Am J m ( mr )  BmYm ( mr )

   p   0   v flow  grad   t  SAE INTERNATIONAL

Paper 2199

8

III. Analytical Model

Coupling relations

fw  p

p r=R

 vr  r

rR

1

w  vw  t

2 SAE INTERNATIONAL

r=r0

 vr  r

0 r  r0

3 Paper 2199

9

III. Analytical Model

Solving the coupled system Substituting sound pressure as distributed load in radial direction into the characteristic equations of the ring structure and express p as function of α and β by using the boundary conditions: M12    0  M11  M  FL M      0   22     21

Where the fluid loading term FL can be expressed as p FL   jm    j0  jmv0  CA/ ( J m ( mr )  CB / AYm ( mr ))e jt e By supplying the mode number m, which is equivalent to wavenumber, we have

f ()  M11M 22  M12 (M 21  FL)  0 The values of ω that satisfy this equations are the natural frequencies of the coupled model.

SAE INTERNATIONAL

Paper 2199

10

IV. Testing

Tire mobility measurement set up

Computer radial velocity

Tire Tread

LDV force

Data Acquisition Box

Signal Generator

Shaker Filter

SAE INTERNATIONAL

Force Transducer

Paper 2199

Amplifier

11

IV. Testing

Tire mobility measurement set up

Computer radial velocity

Tire Tread

LDV force

Data Acquisition Box

Signal Generator

Shaker Filter

SAE INTERNATIONAL

Force Transducer

Paper 2199

Amplifier

12

V. Results

Table of model parameters Ring density

ρ = 1200 kg/m3

Air density

ρ0 = 1.24 kg/m3

Outer radius

r1 = 0.3 m

Inner radius

r2 = 0.2 m

Ring thickness

h = 0.008 m

Tire inflation pressure

p0 = 20600 Pa

Radial stiffness

kw = 1×105 N/m

Tangential stiffness

ku = 1×105 N/m

Young’s modulus

E = 4.8×108 Pa

Tire measured Goodyear 225/55 R17 SAE INTERNATIONAL

Paper 2199

13

V. Results

Dispersion relation (static case) 3rd acoustical wave

Frequency [Hz]

2nd structural wave (fast extensional wave) 2nd acoustical wave 1st acoustical wave 1st structural wave (slow flexural wave)

Mode number

SAE INTERNATIONAL

Paper 2199

14

V. Results

Frequency [Hz]

Dispersion relation (static case)

Mode number

SAE INTERNATIONAL

Paper 2199

15

V. Results

Frequency [Hz]

Dispersion relation (static case)

Mode number

SAE INTERNATIONAL

Paper 2199

16

V. Results

Frequency [Hz]

Dispersion relation (static case)

Mode number

SAE INTERNATIONAL

Paper 2199

17

V. Results

Frequency [Hz]

Dispersion relation (static case)

Mode number

SAE INTERNATIONAL

Paper 2199

18

V. Results

Dispersion relation (no fluid loading)

[Hz] Frequency Frequency [Hz]

Acoustical waves disappear

Fluid loading has minor impact on structural features

Mode number number Mode

SAE INTERNATIONAL

Paper 2199

19

V. Results

Dispersion relation (rotating case)

Frequency [Hz]

Natural frequencies split into two at each mode of all waves

+

Mode number

SAE INTERNATIONAL

Paper 2199

Structural Waves

Airborne Waves

20

V. Results

Dispersion relation (experimental) fast extensional wave

slow flexural wave

SAE INTERNATIONAL

Paper 2199

21

V. Results

Dispersion relation (experimental) radial acoustical mode 340 m/s line

circumferential acoustical modes SAE INTERNATIONAL

Paper 2199

22

V. Results

Dispersion relation (experimental) radial acoustical mode 340 m/s line

++

circumferential acoustical modes

SAE INTERNATIONAL

Paper 2199

23

V. Results

Dispersion relation (experimental) radial acoustical mode 340 m/s line

-

+

+

-

SAE INTERNATIONAL

+-

circumferential acoustical modes Paper 2199

24

V. Results

Dispersion relation (experimental) radial acoustical mode 340 m/s line

+

+-

-

-

+ +

-

SAE INTERNATIONAL

circumferential acoustical modes Paper 2199

25

V. Results

Phase speed (static)

3rd acoustical wave

1st acoustical wave

Frequency[Hz]

2nd acoustical wave

Phase Speed [m/s]

Phase Speed [m/s]

2nd structural wave

Frequency[Hz]

1st structural wave

SAE INTERNATIONAL

Paper 2199

26

V. Results

Phase Speed [m/s]

Phase Speed [m/s]

Phase speed (rotating)

Frequency[Hz]

SAE INTERNATIONAL

Frequency[Hz]

Paper 2199

27

V. Results

Radial pressure distribution in cavity Mode number is 2, at the natural frequencies of each wave

1st structural wave

SAE INTERNATIONAL

Paper 2199

1st acoustical wave

28

V. Results

Radial pressure distribution in cavity Mode number is 2, at the natural frequencies of each wave

1st structural wave

SAE INTERNATIONAL

Paper 2199

1st acoustical wave

29

V. Results

Pressure distribution in cavity (static) Mode number is 2, at the natural frequencies of each wave

2nd structural wave

SAE INTERNATIONAL

Paper 2199

2nd acoustical wave

30

V. Results

Pressure distribution in cavity (static) Mode number is 2, at the natural frequencies of each wave

2nd structural wave

SAE INTERNATIONAL

Paper 2199

2nd acoustical wave

31

VI. Conclusion

 The ring model allows for motions in radial and circumferential directions, which are associated with flexural waves and longitudinal waves, respectively  The air cavity acts as a fluid loading on the ring structure  Rotation of tire causes frequency split phenomenon  Acoustical wave in tire radial directions exist – “depth modes” detectable in tire surface vibration  In circular air cavity, phase speed of circumferential acoustical wave varies with radius due to planar nature of waves

SAE INTERNATIONAL

Paper 2199

32

Thank you

Question?

SAE INTERNATIONAL

Paper 2199

33