Improved Model for Coupled Structural-Acoustic Modes of Tires · II. Literature Review 4 A coupled tire structure/acoustic cavity model Effects of rotation on the dynamics of a circular

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Publications of the Ray W. Herrick Laboratories School of Mechanical Engineering

6-2015

Improved Model for Coupled Structural-AcousticModes of TiresRui CaoRay W. Herrick Laboratories, [email protected]

J. Stuart BoltonRay W. Herrick Laboratories, [email protected]

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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

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IMPROVED MODEL FOR COUPLED STRUCTURAL-ACOUSTIC MODES OF TIRESRui Cao, J. Stuart Bolton, Ray W. Herrick Laboratories,

School of Mechanical Engineering, Purdue University

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I. Introduction

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Traffic noise

Vehicle noise In cabin

noise

• Power Unit noise

• Aerodynamic noise

• Tire/pavement noise

Transfer

paths

PassengersRoadside

residences

Dominant at high speed

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I. Introduction

3

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

Tire structureInternal air cavity

Fixed axle

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II. Literature Review

4

A coupled tire structure/acoustic cavity model

Effects of rotation on the dynamics of a circular

cylindrical shell with application to tire vibration

Structure-borne sound on a smooth tyre

Effects of Coriolis acceleration on the free and forced in-

plane vibrations of rotating rings on elastic foundation

The Influence of Tyre Air Cavities on Vehicle Acoustics

A wave model of a circular tyre. Part 1: belt modelling

Kropp

Huang & Soedel

Pinnington

Kim and Bolton

Molisani, Burdisso & Tsihlas

Fernandez

The wave number decomposition approach

to the analysis of tire vibration Bolton, Song, Kim & Kang

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Review of previous models

III. Analytical Model

5

string

air

* Cao & Bolton, NoiseCon 2013 * Cao & Bolton, NoiseCon 2014

Air CavityFlow

1 2

yx

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Fully coupled circular cavity model


6

Tire tread

Wheel rimAir cavity

Rotation

wu

kw

ku

R

θ

Ω

Y

X

sheared

air flow

r

θAir cavity

1. The tire rotates about a fixed axle

2. The wheel rim is rigid

3. Tire sidewall is represented by springs in radial and tangential directions

4. Ring structure includes flexural and longitudinal waves

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Rotating ring structure


7

11 12

21 22

0

0

M M

M M

jk j tw e e jk j tu e e

Assume harmonic solutions for displacements:

Substitution into rotating ring EOMs and write solutions in matrix form:

Where M11, M12, M21 and M22 are expressions of structure-related constants and

variables kθ and ω. For example:

33

11 4 2 22 2

12

Eh Eh hM j k k k h

R R R

32 2 2 20

12 4 2 22

12u

Eh Eh h pM k k k k h k h

R R R R

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Circular air cavity


8

sheared

air flow

r

θ

Air cavity

0vv r

R

2 2 2 2 2 2

0 0

2 2 2 2 2 2 2

0

1 1 12

v v

r r r r c t R t R

( , , ) ( ) jm j t

m mr t g r e e

( ) ( ) ( )m m m m m m m mg r A J r B Y r

Velocity of the flowing air is expressed as

By using velocity potential ψ, the wave

equation in the circular air cavity is

Harmonic solution of pressure is assumed in circumferential direction while

Bessel function is assumed in radial direction:

0 flowp v gradt

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Coupling relations


9

0

0r

r r

vr

r w

r R

wv v

r t

2 3

r=r0

pr=R

wf p

1

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Solving the coupled system


10

11 12

21 22

0

0

M M

M FL M

11 22 12 21( ) ( ) 0f M M M M FL

By supplying the mode number m, which is equivalent to wavenumber, we have

The values of ω that satisfy this equations are the natural frequencies of the

coupled model.

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:

0 0 A/ /A( ( ) ( )) j t

m m B m mjm

pFL j jmv C J r C Y r e

e

Where the fluid loading term FL can be expressed as

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Tire mobility measurement set up

IV. Testing

11

LDV Tire Tread

Shaker

AmplifierFilter

Data Acquisition Box

Signal Generator

Computer

Force Transducer

radial velocity

force

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Tire mobility measurement set up

IV. Testing

12

LDV Tire Tread

Shaker

AmplifierFilter

Data Acquisition Box

Signal Generator

Computer

Force Transducer

radial velocity

force

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Table of model parameters

Tire measured

V. Results

13

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

Goodyear 225/55 R17

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Dispersion relation (static case)

V. Results

14

Fre

quency [

Hz]

Mode number

1st structural wave

(slow flexural wave)

2nd structural wave

(fast extensional wave)

3rd acoustical wave

2nd acoustical wave

1st acoustical wave

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V. Results

15

Fre

quency [

Hz]

Mode number

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V. Results

16

Fre

quency [

Hz]

Mode number

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V. Results

17

Fre

quency [

Hz]

Mode number

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V. Results

18

Fre

quency [

Hz]

Mode number

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Fre

qu

en

cy [

Hz]

Mode number

Dispersion relation (no fluid loading)

V. Results

19

Fre

quency [

Hz]

Mode number

Acoustical waves disappear

Fluid loading has minor impact

on structural features

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Dispersion relation (rotating case)

V. Results

20

Mode number

Fre

quency [

Hz]

Natural frequencies split into

two at each mode of all waves

Airborne

WavesStructural

Waves

+ -

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Dispersion relation (experimental)

V. Results

21

fast extensional wave

slow flexural wave

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V. Results

22

circumferential acoustical modes

radial acoustical mode340 m/s line

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V. Results

23



+ -

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V. Results

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+

-

-

+

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V. Results

25



+

- +

+

-

-

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Phase speed (static)

V. Results

26

Phase S

peed [m

/s]

Frequency[Hz]

Phase S

peed [m

/s]

Frequency[Hz]

1st structural wave

2nd structural wave 2nd acoustical wave

1st acoustical wave

3rd acoustical wave

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Phase speed (rotating)

V. Results

27

Phase S

peed [m

/s]

Frequency[Hz]

Phase S

peed [m

/s]

Frequency[Hz]

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Radial pressure distribution in cavity

V. Results

28

1st structural wave 1st acoustical wave

Mode number is 2, at the natural frequencies of each wave

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Radial pressure distribution in cavity

V. Results

29

1st structural wave 1st acoustical wave


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Pressure distribution in cavity (static)

V. Results

30



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Pressure distribution in cavity (static)

V. Results

31



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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

VI. Conclusion

32Paper 2199

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Question?

Thank you

33Paper 2199

Improved Model for Coupled Structural-Acoustic Modes of Tires · II. Literature Review 4 A coupled tire structure/acoustic cavity model Effects of rotation on the dynamics of a circular

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