vehicle vibration
DESCRIPTION
vehicle vibrationTRANSCRIPT
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
ME 360/390 – Prof. R.G. LongoriaVehicle System Dynamics and Control
Department of Mechanical EngineeringThe University of Texas at Austin
Overview
• Pitch and bounce model
• A ½ car model and its simulation
• Comparison between ½ car model, CarSim, and
ADAMS implementation
ME 360/390 – Prof. R.G. LongoriaVehicle System Dynamics and Control
Department of Mechanical EngineeringThe University of Texas at Austin
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
ME 360/390 – Prof. R.G. LongoriaVehicle System Dynamics and Control
Department of Mechanical EngineeringThe University of Texas at Austin
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
ME 360/390 – Prof. R.G. LongoriaVehicle System Dynamics and Control
Department of Mechanical EngineeringThe University of Texas at Austin
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
ME 360/390 – Prof. R.G. LongoriaVehicle System Dynamics and Control
Department of Mechanical EngineeringThe University of Texas at Austin
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)
ME 360/390 – Prof. R.G. LongoriaVehicle System Dynamics and Control
Department of Mechanical EngineeringThe University of Texas at Austin
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).
ME 360/390 – Prof. R.G. LongoriaVehicle System Dynamics and Control
Department of Mechanical EngineeringThe University of Texas at Austin
Example 7.1 (Wong)
ME 360/390 – Prof. R.G. LongoriaVehicle System Dynamics and Control
Department of Mechanical EngineeringThe University of Texas at Austin
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≡ ≡
ME 360/390 – Prof. R.G. LongoriaVehicle System Dynamics and Control
Department of Mechanical EngineeringThe University of Texas at Austin
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=
ME 360/390 – Prof. R.G. LongoriaVehicle System Dynamics and Control
Department of Mechanical EngineeringThe University of Texas at Austin
Example: Wong, Problem 7.2
ME 360/390 – Prof. R.G. LongoriaVehicle System Dynamics and Control
Department of Mechanical EngineeringThe University of Texas at Austin
Example: Wong, Problem 7.2 (cont.)
ME 360/390 – Prof. R.G. LongoriaVehicle System Dynamics and Control
Department of Mechanical EngineeringThe University of Texas at Austin
Wong, Problem 7.3
ME 360/390 – Prof. R.G. LongoriaVehicle System Dynamics and Control
Department of Mechanical EngineeringThe University of Texas at Austin
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
ME 360/390 – Prof. R.G. LongoriaVehicle System Dynamics and Control
Department of Mechanical EngineeringThe University of Texas at Austin
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
ME 360/390 – Prof. R.G. LongoriaVehicle System Dynamics and Control
Department of Mechanical EngineeringThe University of Texas at Austin
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.
ME 360/390 – Prof. R.G. LongoriaVehicle System Dynamics and Control
Department of Mechanical EngineeringThe University of Texas at Austin
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)
ME 360/390 – Prof. R.G. LongoriaVehicle System Dynamics and Control
Department of Mechanical EngineeringThe University of Texas at Austin
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.
ME 360/390 – Prof. R.G. LongoriaVehicle System Dynamics and Control
Department of Mechanical EngineeringThe University of Texas at Austin
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.
ME 360/390 – Prof. R.G. LongoriaVehicle System Dynamics and Control
Department of Mechanical EngineeringThe University of Texas at Austin
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
ME 360/390 – Prof. R.G. LongoriaVehicle System Dynamics and Control
Department of Mechanical EngineeringThe University of Texas at Austin
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
− = −
−
ME 360/390 – Prof. R.G. LongoriaVehicle System Dynamics and Control
Department of Mechanical EngineeringThe University of Texas at Austin
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
ME 360/390 – Prof. R.G. LongoriaVehicle System Dynamics and Control
Department of Mechanical EngineeringThe University of Texas at Austin
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.
ME 360/390 – Prof. R.G. LongoriaVehicle System Dynamics and Control
Department of Mechanical EngineeringThe University of Texas at Austin
Results from CarSim
ME 360/390 – Prof. R.G. LongoriaVehicle System Dynamics and Control
Department of Mechanical EngineeringThe University of Texas at Austin
Results from ADAMS Model
ME 360/390 – Prof. R.G. LongoriaVehicle System Dynamics and Control
Department of Mechanical EngineeringThe University of Texas at Austin
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
ME 360/390 – Prof. R.G. LongoriaVehicle System Dynamics and Control
Department of Mechanical EngineeringThe University of Texas at Austin
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.