IMNTIFICAllON
AND
STRUCTURAL
OF A NONLINEAR
Xiong
Shibo
Professor
Institute
of
Shanxi
Measurement
and Dynamic
mining
College,Taiyuan
ik
c*
fi
Ck=l,*),stiffness
Zhang Dihuan
Professor
frequencies
coefficients
(k=l,2),damping
coefficients
xu,&,&
(~=1,2),displacement,velocity,
acceleration
at the kth
~k=1,2),masses
mC
lNTRoD"CTlON
System Analysis
030024,P.R.China
Technique
the
differential
equation
Most of these
techniques
are
usually
based
on single
degree
of freedomCSD3F)
models.Extension
to
multy
degree
of freedom
(MDOF) system
is
difficult.
Identification
of the
suspension
system
that
is nonlinear
complicated
MDOF systems
is very difficult
through
the "se of above
method
. Authors
have
measured
statical
stiffness
curve
of
the
front
wheels
,
relation
to suspension
mae
and
frequency
response
function
that
is
obtained
by
exciting
suspension
system
through
the
"se varied
amplitude
sinusoidal
signals.
Experimental
results
show
clearly
that
suspension
system
display
apparently
nonlinear
property
only
in ultralow-frequency
band lower
than 5HZ . In
this
paper
"sing
random signal
excition
and averaging
a great number
of
measured
data
groups
are
proposed
to obtain
optimal
linea+ization
estimation
of
the frequecy
response
function
for weak nonlinear
systems.
we can
consider
that
independent
suspension
systems
are weak nonlinear
systems,therefore
nonlinear
problems
may change
into
linear
problems
process
of measurement
through
using excition
of random
signals
.Of
course
amplitude
probabilistic
density
function
of
the
exciting
signals
m"3t
presents
Gauss
distribution.
Then modal parameters
ai-e obtained
by means of Chebyshe"
Polyno-'
mial
curve-fit
algorithm
Lastly
dynamics
modification
scheme
of
this
independent
suspension
system
is proposed
on the basis
of
modal analysis.
NONENCLANRE
exciting
force
time
(k-1.2)natural
MOOIFICAl!ON
Cheng Hang
Engineer
Independent
suspension
system of automobile
front
wheels
are a nonlinear
system
that
consists
of swing am mechanism,torsion
IO*
spring
and
damper etc
This
is a system
incorporating
stiffness
nonlinearities
and
having
damping
nonlinearities
e.g.Co"lomb
friction.
This
paper
presents
"IeaSurement
Of
the
frequency
response
functionlFRF)and
parametric
identification
which
uses
average
linearization
method
under
random
signal
excition
condition
Authors
have analysed
ultralow
frequency
characteristics
of
this
vibration
system and given
relation
between
FRF curves
of various
exciting
singnal
levls
and average
times,and
modal parameter
idetification
results
. In addition
, structural
dynamics
modification
scheme is provide.
qltl
t
fl
DYNAMICS
SYSTEM
coordinate.
Independent
suspension
system of automobile
front
wheels
are a nonlinear
system that
consists
of
swing
arm
mechanism,
torsion
rod
spring
and
damping
element.
Dynamic
characteristics
of suspension
systems
under
various
loading
conditions
greatly
affect
the
vehicle's
handling,stability
and tire
wear
Thus
identification
for suspension
system parameters
and its
structural
dynamics modification
are important
problem.
some techniques
for
identifying
nonlinear
vibrating
systems
thro"gh
the
use
Of
experimental
data have
been
developed
in
recent
years
Among
them
exist
a number
of
techniques
for
the
identification
of
special
classes
of
nonliner
system
, i.e.
Classes
of differential
equation
are known.
Thus identification
procedure
is reduced
to
the
determination
of
the
coeficients
of
Studied
suspension
system
is
system
that
incorporating
stiffness
nonlinearities
and
having
damping
nonlinearities
e.g.co"lomb
friction.Statical
stiffness
c"rve(s"spension
spring
Stiffness
1 of
the
front
wheels
in relation
to
vertical
displacement
of
274
suspension
mean~ of
suspension
40
u.
0
IO
obtained
mass is
statical
measurement,
mass,
20
1
30
Fig
nonsuspension
curve
of the
1’
40
SO
I
mass
tyre(
and
Fig.2
C Fig.
1 ) by
in addition,
’ -_L
60
70
--I
80 87
&mm)
stiffness
statical
)are measured
too.
Fig.3
A front
wheel
is excited
by
exciter
under
wheel
. Exciting
force
that
is measured
by
piezoeletric
force
transducer
is
random
signals
. Response
signals
are measured
at
appropriate
points
of the suspension
system
and automobile
~tnxtuzze
using
piezoelectric
accelerometers
and eddy
current
relative
displacement
transducers.
The key to success
in
system linearization
is using
appropriate
exciting
signals
and
sufficient
average
times
to obtain
optimal
linear
estimation
of weak nonlinear
systems.
Fig.4
and fig.5
show
three
frequency
response function
c*rves
using
three
different
exciting
level.
Fig.4
shows condition
using
insufficient
average
times
Fig.5
ShO"S
condition
using
sufficient
average
times.
Fig.2
Suspension
spring
stiffness
appears
StrO"g
( multiple
cubic
nonlinearity
stiffness
nonlinearities)under
light
loading
condition
however
it appear
weak
nonlinearity
under
"OlTlal
loading
condition
This point
has
been verified
through
step sine
excitation
test
under
various
excitation
amplitude
condition.
275
Fig. 4
Fiy
6
Fi). 7
difference
at
low frequency
section
in Fig.*.It
illustrates that
system presents
nonlinear
property
only at ultralow
frequency
section
. Three
curve.3 present
small
difference
at ultzalow
frequency
section
in Fig.5.
It illustrates
the
that
optimal
linear
estimation
of
frequency
response
function
is
obtained
under this
condition.
Fig.6
shows
independent
front
"heels.
optimal
suspension
estimation
system
of
of
FRP
Fig.7
shows Mechanical
model of this
system.
Its input(excition)is
q(t),response
is x.,;C
frequency
response
Of
the
or XI . CUrYe
function
is fitted
using
chebyshev
polynomial fit
algorithm.&
obtain
modal parameters
shown in following
table.
for
automobile
276
We
assume
that
system
shown
linear
system
. then following
parameters
are
determined
statical
meaS"reme"t
results:
m,=25kg
m.=24zfg
k. =2.8*lo6N,m
t =4.02+1$
N/m
Motion
differential
approximate
linear
equations
system are
in
Fig.7
approximate
by
"lean.9
is
of
c,=1697.7N.s/m
c2 =626ON s/m
of
this
m,~~+c,(jr,-x,)+L,(X,--X,)=O
m,i(,-c,li-x,)-k,(X,-X,)=q(t)-c,X,-k,X,
(1)
(2)
We perform
laplace
transform
of equations(l)
obtain
frequency
response
function
of
the
suspension
mass
acceleration
in
relation
to excitina
force
HLiwl.
Natural
frequency
of nonsuspAsion
mass is
f, =16.86
HZ
Natural frequency of suspension
mass is
f, =l. 626 xz
It
is
obvious
that
these
tow
natu+al
frequency
are included
in modal parameters
(fit
results).
Dynamics
modification
should
reduce
f, to
ensure
riding
comfort.
and(2)to
Linearization
of
veak
nonlinear
systems
should
use appropriate
exciting
signals
and
sufficient
average
times
to obtain
optimal
linear
estimation.
[I]
[2]
[3]
(41
Y.benhafsi
" A Method
of
System
ldentificatlon
for Nonlinear
Vibrating
St~"Ct"TeS",
8th IMRC,l990.
J.L.Adcock,"Curve
Fitter
for Pole-Zero
AnalySis",HeHlett-Packard
Jounal",Jan,
1987.
R.C.St=oud,"Excitation,Measurement,and
Analysis
Methods
for Modal Testing"
Sound
and
Vibration,August,l987.
A.Tondl,"the
Application
of Skeleton
Curves
and
Limit
Envelopes
to Analysis
of Nonlinear
Vibration",Shock
and
Vibration
Digest
7(7),1975.
277