vehicle vibration

ME 360/390 – Prof. R.G. LongoriaVehicle System Dynamics and Control

Department of Mechanical EngineeringThe University of Texas at Austin

Vehicle Vibration and Ride – 2

R.G. Longoria

Spring 2012



Overview

• Pitch and bounce model

• A ½ car model and its simulation

• Comparison between ½ car model, CarSim, and

ADAMS implementation



Pitch and bounce ride models

Wong (2001)

Why and how can we study pitch

and bounce dynamics separately?

A full model will have 4 DOF.

Recall that for 1/4-car models the

natural frequencies of the sprung

and unsprung mass are ‘widely

separated’.

This assumption allows us to

conceptualize the model shown

here to understand the relationship

between pitch and bounce of the

vehicle body.

sk sk

,t tk b

,t tk btmtm

1z2z

cm,vm J

1L2L

V



Pitch and bounce modelWong (1993)

Write an equation for each mode using

Newton’s second law.

1 2

1 1 2 2

Bounce: ( ) ( ) 0

Pitch: ( ) ( ) 0

s f r

y f r

m z k z l k z l

I k l z l k l z l

θ θ

θ θ θ

+ − + + =

− − + + =

ɺɺ

ɺɺ

1 2

232

0

0y

z D z D

Dz D

r

θ

θ θ

+ + =

+ + =

ɺɺ

ɺɺ

1

2 2 1

2 2

3 1 22

1( )

1( )

1( )

f r

s

f r

s

f r

s y

D k km

D k l k lm

D k l k lm r

= +

= − +

= +2

y s yI m r=

Coupled Pitch and Bounce

NOTE: the damping is ignored here



Pitch and bounce model - uncoupled vs. coupled

1

3

0

0

z D z

Dθ θ

+ =

+ =

ɺɺ

ɺɺ

2 2 1

1( ) 0f r

s

D k l k lm

= − + =

2

1

2 2 2

3 1 22

1Bounce: ( )

1Pitch: ( )

f r nz

s

f r n

s y

D k km

D k l k lm r

θ

ω

ω

= + =

= + =

Uncoupled Pitch and Bounce

Uncoupled case

gives poor ride

quality. Why?

Coupled Pitch and Bounce

24 2 2

1 3 1 3 2

22 2 21,2 1 3 1 3 2

( ) ( ) 0 (C.E.)

1 1( ) ( )

2 4

n n

y

n

y

DD D D D

r

DD D D D

r

ω ω

ω

− + + − =

= + ± − +

1 2

uncoupled case

n nz n nθω ω ω ω> > >��

2

1 2

2232

( ) ( ) ( ) 0

( ) ( ) ( ) 0y

D Z s D s

DZ s D s

r

ω θ

ω θ

− + =

+ − =

Let , and, .d

s j sdt

ω= ≡

Assume sin , and thennx X tω=

Use this to get

eigenvalues

and

eigenvectors.

Eigenvalues



Pitch and bounce modes (eigenvectors)

22 2 21,2 1 3 1 3 2

1 1( ) ( )

2 4n

y

DD D D D

rω = + ± − +

2

1 2

2232

( ) ( ) ( ) 0

( ) ( ) ( ) 0y

D Z s D s

DZ s D s

r

ω θ

ω θ

− + =

+ − =

Assume sin , so we seek the value of .nx X t Xω=

1,2

2

2

1,2 1n n

Z D

Dωθ ω=

−

For each eigenvalue (or natural frequency) you

get a ratio of the amplitudes - these are the

eigenvectors or modes.

These modes can be shown to have opposite sign.

Gillespie (1992)



Pitch and bounce oscillation centers

An oscillation center is associated with each natural frequency.

21,2 2

1,2 1

o

n

Dl

Dω=

−

Comes from

amplitude

ratios.

0 O.C. to right of C.G.Z

θ< ⇒

Wong (2001)

0 O.C. to left of C.G.Z

θ> ⇒

An input at either wheel will induce

oscillation about both centers, since

the total response is a function of

both modes.

If O.C. is outside wheelbase it is called the

bounce center and is associated with a

bounce frequency (commonly ranges from

1 to 1.5 Hz).

If O.C. is inside wheelbase it is called the

pitch center and is associated with a pitch

frequency (usually higher than bounce).



Example 7.1 (Wong)



Pitch and bounce: locus of oscillation centers

(Gillespie, 1992)

Case 1

Case 2

Case 2 shows center locations

when the front has a lower

frequency, putting the bounce

behind the rear axle and pitch

center in front of the front axle.

This is recognized by Olley as

“achieving good ride”.

Olley’s guidelines can be found

in Gillespie (p. 176).

The front and rear natural frequencies are defined by

1 1 and

2 2

f rf r

f r

k g k gf f

W Wπ π≡ ≡



Pitch and bounce models – wheelbase filtering

Gillespie (1992)

Understanding pitch and bounce

dynamics provides insight into how the

vehicle responds to road profile.

Bounce motion can be excited when the

road has wavelength equal to wheelbase

(WB) and for much longer multiples and

shorter with integer multiples.

Pitch motion can be excited by

wavelengths that are twice the WB, and

by shorter wavelengths that are odd

integer multiples of this value.

So pitch and bounce are each filtered

from certain excitations.

2πγ

λ=

Vω γ=



Example: Wong, Problem 7.2



Example: Wong, Problem 7.2 (cont.)



Wong, Problem 7.3



½ car (2D ride) simulation models

• Direct model (by hand)

• CarSimEd – 2D Ride

• ADAMS – 2D Ride

sk sk

,t tk b

,t tk btmtm

1z2z

cm,vm J

1L2L

Passive or Active

Force Elements

V



½ car simulation model

This ½ car model combines the ¼ car model with the

pitch and bounce model (see Wong, Sec. 7.2.3).

Three bodies, 4 DOF: each tire has vertical motion,

vehicle body has vertical (heave) and rotational

motion (pitch). The ‘passive’ (since we’ll add active

suspension later) dynamic model will require 8

ordinary differential equations.

For the diagram given, assume we are provided input in the form of a terrain profile, zg(x). The vehicle

has a forward velocity, V.

Develop the differential equations that model this system. Model the ‘force generating’ element with a

force that is a function of the relative velocity of its ends. This force generating element at the front and

rear axle will be used to study both passive and active suspension performance for this vehicle model.

At this stage, we are considering V constant so there is no need to consider the longitudinal dynamics.

We may revisit this later to see what it would take to add these dynamics as well as any traction effects.

sk sk

,t tk b

,t tk btmtm

1z2z

cm,vm J

1L2L

Passive or Active

Force Elements

V

Refer to Examples in CarSim and ADAMS



½ car with passive suspension

In a preliminary evaluation (passive suspension) use

a simulation model to solve for the following

quantities:

a. Body motion: vertical acceleration and pitch of the

center of mass (C.M.)

b. Forces in the suspension springs

c. Forces at the tire-surface contact

d. Deflection of the suspension

sk sk

,t tk b

,t tk btmtm

1z2z

cm,vm J

1L2L

Passive or Active

Force Elements

V

Complete the following :

1. Complete the equations of motion (help on next two pages)

2. Show for proper the initial conditions

3. Complete a simulation for the vehicle going over the bump (see Parameter Data plot), and plot

the quantities listed above (in a, b, c, and d). Compare with results given on subsequent slides

from CarSim and from Matlab at V = 40 km/hr (partial Matlab files will be provided).

4. Design an ADAMS model of this transient vehicle vibration, and compare simulation results with

those from step #3.



Example: Tractor ride model

A slightly different pitch and bounce model is required in tractor

dynamics.

Here the model focuses on the stiffness and damping of the tires, the

only suspension typically found on most tractors.

Liljedahl, et al (1996)



½ car model bond graph

2 1

cm r f v

cm r f

sr tr r

sf tf f

tr tr r tr

tf tf f tf

tr r tr

tf f tf

p F F m g

h L F L F

x V V

x V V

p F F m g

p F F m g

x z V

x z V

= + −

= − +

= −

= −

= − −

= − −

= −

= −

ɺ

ɺ

ɺ

ɺ

ɺ

ɺ

ɺ ɺ

ɺ ɺ

If you allow for large angle,

then you need to include,

cmcm

cm

hJ

θ ω= =ɺ

And the velocity and torque

relations are affected, since it is

assumed here that pitch angle is

less than about 10 degrees.



½ car velocities and forces

2

1

( )

( )

( )

( )

spe

cm cm v

cm cm cm

r cm cm

f cm cm

tr tr tr

tf tf tf

r sr sbr sr sr sr tr r

f sf sbf sf sf sf tf f

tr tsr tbr tr tr tr r tr

tf tsf tbf tf tf tf f tf

f

V p m

h J

V V L

V V L

V p m

V p m

F F F k x b V V

F F F k x b V V

F F F k x b z V

F F F k x b z V

z

ω

ω

ω

=

=

= − ⋅

= + ⋅

=

=

= + = + −

= + = + −

= + = + −

= + = + −

=

ɺ

ɺ

ɺ cified ground input at front =

specified ground input at rear = ( ) (lags behind front wheel)r f

dzV

dx

z z x L

⋅

= −ɺ ɺ

θ1L

2LfF

rF

fV

rV

cmωcmV

Small angle approximation implied.



½ car example parameter data

L = 2.7 m

h = 0.55 m

mv = 1700 kg

Ixx = 400 kg-m2

Iyy = 2704 kg-m2

Izz = 3136 kg-m2

Preliminary Evaluation

Passive suspension response

Base vehicle velocity:

V = 40 km/h

Rear suspension:

ks = 20 N/mm

bs = 0.75 N-s/mm

Rear tires:

mt = 80 kg

kt = 200 N/mm

Front suspension:

ks = 30 N/mm

bs = 0.75 N-s/mm

Rear tires:

mt = 100 kg

kt = 200 N/mm

Bump:

xg = [0,5,6,10,11,15] m

zg = [0,0,0.1,0.1,0,0] m

Tire rolling radius:

rw = 285 mm

Tire spin inertia:

rw = 1.1 kg-m2



½ car model initial conditions – critical!

2 1 2 1

0

0 0

0

0

0

0

0

0

cm r f v r f v

cm r f r f

sr tr r

sf tf f

tr tr r tr tr r tr

tf tf f tf tf f tf

tr r tr

tf f tf

p F F m g F F m g

h L F L F L F L F

x V V

x V V

p F F m g F F m g

p F F m g F F m g

x z V

x z V

= + − = ⇒ + =

= − + = ⇒ − + =

= − =

= − =

= − − = ⇒ − =

= − − = ⇒ − =

= − =

= − =

ɺ

ɺ

ɺ

ɺ

ɺ

ɺ

ɺ ɺ

ɺ ɺ

2 1 0

sr sr sf sf v

sr sr sf sf

tr tr sr sr tr

tf tf sf sf tf

k k m g

L k L k

k k m g

k k m g

δ δ

δ δ

δ δ

δ δ

+ =

− + =

− =

− =

0

0

0

0

( )

( )

( )

( )

r sr sbr sr sr sr tr r

f sf sbf sf sf sf tf f

tr tsr tbr tr tr tr r tr

tf tsf tbf tf tf tf f tf

F F F k b V V

F F F k b V V

F F F k b z V

F F F k b z V

δ

δ

δ

δ

=

=

=

=

= + = + −

= + = + −

= + = + −

= + = + −

��

��

ɺ��

ɺ��

2 1

0 0

0 0 0

0 0

0 0

sr sf sr v

sr sf sf

sr tr tr tr

sf tf tf tf

k k m g

L k L k

k k m g

k k m g

δ

δ

δ

δ

− = −

−



½ car example results for IC calculation

2 1

0 0

0 0 0

0 0

0 0

sr sf sr v

sr sf sf

sr tr tr tr

sf tf tf tf

k k m g

L k L k

k k m g

k k m g

δ

δ

δ

δ

− = −

−

% initial conditions for springs

Kmatrix = [ksr ksf 0 0;-L2*ksr L1*ksf 0 0;-ksr 0 ktr 0;0 -ksf 0 ktf];

Bloads = [mv*g;0;mtr*g;mtf*g];

delta_values=inv(Kmatrix)*Bloads;

delta_sr = delta_values(1);

delta_sf = delta_values(2);

delta_tr = delta_values(3);

delta_tf = delta_values(4);

delta_values =

0.1717

0.1635

0.0211

0.0294



½ car results from ‘direct model’

0 1 2 3 40

5000

10000Tire Forces, N

0 1 2 3 4-5

0

5

10Pitch, deg

0 1 2 3 40

5000

10000Spring Forces, N

0 1 2 3 4-1

-0.5

0

0.5

1Acceleration of cm, g

Note that the

tire forces for

left and right

side are

assumed equal,

and the forces

shown are ½ of

total on each

axle.



Results from CarSim



Results from ADAMS Model



Summary

• Pitch and bounce models build on our

understanding of ride dynamics, and are

especially important for considering the

influence of surface characteristics.

• Building a 2D ride model can be useful,

especially for building up later to study

controlled suspension systems.

• 3 different models for the 2D ride are compared



References

1. W.T. Thomson, “Theory of Vibration with Applications”, Prentice-Hall, 1993.

2. Gillespie, T.D., Fundamentals of Vehicle Dynamics, SAE, Warrendale, PA, 1992.

3. Liljedahl, et al, “Tractors and their power units,” ASAE, St. Joseph, MI, 1996.

4. Wong, J.Y., Theory of Ground Vehicles, John Wiley and Sons, Inc., New York, 2001.

5. Karnopp, D. and G. Heess, “Electronically Controllable Vehicle Suspensions,” Vehicle System Dynamics, Vol. 20, No. 3-4, pp. 207-217, 1991.

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