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Identification of flexible characteristics of a mechanical system based on “passive” experimentation
(1987) l(4), 377-387
Mechanical Systems and Signal Processing
IDENTIFICATION
OF FLEXIBLE
A MECHANICAL “PASSIVE” JERZY
CHARACTERISTICS
SYSTEM
BASED
OF
ON
EXPERIMENTATION
KISILOWSKI
AND
STEFAN
URBA~SKI
Institute of Transport, Warsaw University of Technology, 00-662 Warsaw, ul. Koszykowa
75, Poland
(Received December 1986, accepted April 1987) The paper presents a method of identifying the flexible non-linear characteristics of a mechanical system. For this purpose we used the Fokker-Planck equation and on this basis the criterion function has been built. The proposed method does not impose any restrictions on the character of non-linearities or the dimension of the system. The present method determines the non-linear characteristics of elasticity and damping of the objects while in use. The results obtained from identification of the non-linear dissipative force that is caused by the dry friction of a mechanical system such as a freight car are presented here as a computational example. The results, which are based on the passive experiment results, confirm the efficiency of the method. 1. INTRODUCTION
Mechanical systems often contain a number of non-linear elements which can be exemplified by the characteristics of the stiffness-damping connections. While analysing this kind of system, there is a tendency towards linearisation which significantly simplifies the problem. The introduction of equivalent linear coefficients leads to the building and analysing of linear mathematical models. At the same time the problem of identification while using these models is reduced to the identification of definite parameters. In applied mechanics the bulk of work on identification problems concentrates on parameteric identification. Linearity is a rare characteristic of mechanical systems. Forces of elasticity and damping usually depend non-linearly on displacements and velocities. Using linearisation we obtain linear mathematical models. Linearisation, however, frequently leads to excessive simplification which limit the possibility of carrying out a complete analysis of the system’s properties, and in some cases they lead to incorrect conclusions about the processes that occur within the system. A complete representation of the phenomena that occur in real dynamic systems requires the building of non-linear mathematical models. An analysis of non-linear systems creates mathematical problems. However, it lets us investigate a number of the systems properties such as for example the occurrence of various kinds of resonance frequencies, the occurrence of limit cycles, the influence of the initial conditions on the course of vibrations the problem of exceeding the trajectories of solutions, etc, which cannot be investigated by analysing linear models. Non-linear mathematical model building involves identifying the character of the modelled nonlinearities. 377 0888-3270/87/040371+11
%03.00/O
@ 1987 Academic Press Limited
378
J. KISILOWSKI
AND
S. URBAI;ISKI
There are only a few papers presenting effective methods of identifying the non-linear systems, eg. refs [l-4]. The first reference presents a method of reproducing the characteristics of non-linear elements on the basis of substitute linear coefficients that have been obtained by equivalent linearisation for different levels of the exciting signals. In refs [2] and [3] the Fokker-Planck equation was used to define non-linear characteristics, and Stepinski [2] reduces the characteristics of non-linearities to continuous functions. Masri and Caughey [4] present a method of identifying non-linear models by applying the regressive technique in conjunction with Chebyshev’s orthogonal polynomials. This method can be applied in identifying a wide class of non-linear systems. References [l-4] are based on active experimentation. This article presents both a method of reproducing the systems non-linearity characteristics by applying the Fokker-Planck equation [3] and the results obtained as applied to a mechanical system with a non-linear dissipative characteristic. Tests carried out on the “passive” experiment are the basis of the present method [5].
2. THE
MODEL
All considerations concerning the method of identifying non-linear characteristics of the stiffness-damping elements were carried out on mechanical systems such as railway vehicles. A railway vehicle is a mechanical system normally consisting of three or seven elements that are treated as rigid. To describe their position, we can use 18 or 42 co-ordinates. The number of these co-ordinates is reduced by geometrical and constructional constraints. Complete consideration of the systems of co-ordinates and nominal models can be found in Chudzikiewicz et al. [6]. In the case of railway vehicles it is usually assumed that there is a longitudinal plane of symmetry and that the system performs small vibrations. This assumption allows us to consider vibrations in the vertical longitudinal plane of symmetry and in the lateral plane separately. We shall consider a two-truck railway vehicle, the nominal model of which can be presented as in Fig. 1. This figure presents the model of vibrations in the vertical longitudinal plane of symmetry. It has been assumed that we shall search for the characteristics of the stiffness-damping connections between the wheel-sets and the frame of the truck. The vertical vibrations are excited due to the vertical irregularities of the track-the vertical displacement signals of wheel-set centres z,(r), z2( t), zj(t), zq(f) can be their
r
20
20 Figure
1. Model of the vehicle.
FLEXIBLE
CHARACTERISTICS
OF A MECHANICAL
SYSTEM
379
representations. So defined, the inputs are valid in the suspension modelling process over the frequency range of O-20 Hz this being the range of the dominating input components and the system’s natural frequencies. The resonant frequency of the system, non-sprung mass-track occurs somewhere between 60-80 Hz and therefore it is justifiable to ignore the track mass as well as the subgrade stiffness and damping [6,7]. Assuming linearity of both stiffness and damping coefficients, the motion of the masses of the truck frames can be described with a linear system of four equations which describe co-ordinates zbl, zb2, xbI , xb2 : m,+- rnNh2 1 z, +- 1.i 4aZ,
+‘tk,,Zb,+
44aZ, ( i
4
MIV) ib,+(4&z+&)ib+
Wlj.~----mNh2
4aZ,
1 &
)
..
4 4aZ,
_A..
-$tibl+kb2)
zb2 2aZ, zb2
N
-2c,,(i,+i2)-2kzz(zI+zz)=O; mNh2 1 IN m,+ -+~~+am,),2+(4~~~+~)~b2+4k~~zb2+(~m~-~~-~) 4aZ,
.. XZb,-~i,,+~(~b,+~b2)-2c=l(i3+i.)-2k,,(Z1+Z4)=0; N
N
z~~b,f(C,+4a2E,,)41+4a2k,~b,+~(ib2-ibl)f2ac,,’(i2-i,)+2ak==(z2-z,)=0; N
z~~b2+(E,+4a2e,,)ib2+4a2k~~b2+~
(ib2
-ib,)+2aCz,(i,-i~)+2ak*~(z~-z~)=0;
N
(1)
where m, is the mass of the truck frame, m N is the mass of the car body, Zwyis the moment of inertia of the truck frame, ZN,,is moment of inertia of the car body, k,, is the coefficient of rigidity, c,, and c, are coefficients of damping and a, aN, h are geometrical parameters. Adding the sides of the two first equations (with zbI and zbz) and subtracting the equation with xbz from the equation with xbl, we obtain two independent equations. Assuming that the vibrations of both trucks can be considered separately, the first of the equations takes the following form: (mw+tmN)a,+2k,,[2zb,-(z,+z,)]+f[2ib,+(i,+i2)]=0.
(2)
This equation describes the vertical vibrations of the system with one degree of freedom with the kinematic input (z, + z2). 3. DESCRIPTION
OF THE
METHOD
Equation (2) in which the third element describing the damping in the system will be determined in the identification process, is the initial equation to present the method. Introducing the new variable y = 2zbl - (z, + z2) and denoting M = m, +$mN equation (2) is reduced to the following form: (3)
j;+pY+4i)=df) where
n(t) = - (i’, + i2) = the input function.
p
~5,
a(i)
c’s
380
J.
KISILOWSKI
AND
S. URBAI;ISKI
Relation (3) has been reduced to the system of equations of the first order by substituting 91 =Y 92 =.i
(4)
41= 42 42=-&1-~(42)+~(4,,
q2,
t)‘t(t).
The above equation is a stochastic differential equation in which the input 7(t) has been modelled as an interaction of a modulating function +!r(q,, q2, r) = + on the white noise l(t) with the average value equal to zero and with the unitary spectral density, and thus: 77(r) = @(41, q2,
t)
. 5(t).
For equations (4), we can write the Fokker-Planck
equation as follows [8]:
p( q, , q2, r) = p, density of the conditional probability a1 =
42
a2=-&,-4q2) b22=Mql,
q2,
01’
Assuming the solution of the above equation to be stationary, i.e. ap/ar = 0 we define a function of the following form
(6) where & = Jl(ql, q2, r); p’= ( ql, q2, r) is determined from the experimental data, on the basis of which we introduce the criterion function in the damping force identification process. The criterion function is defined by the relation m cc (7) J(P, a) = {F[q, >q2, P, 4d112 da de. I --ooI -aI Which satisfies the Fokker-Planck equation by parameters to be found which describe the characteristics being identified. The damping characteristic curve to be found can be approximated by a broken line described by the equation l+(k/2)
(y(q2) =
C i=l
[ciOi(q2)+
biRi(q2)l
where k/2
,s+,441-r)
Qi(q2)=~(1-i)q2+~(1-i)(i-1)a-a k/2
C 9(r),
Ri(S2)=~(i)q2-~(i)(i-1)a+a
r=i+l
Q k/2+1=
.R X(qpnin,O),
l+k/2
=
X(0,q2max)
i=l,...,-
k
2
FLEXIBLE
a=(hllax-
qzmin)/k=
CHARACTERISTICS
OF
A MECHANICAL
381
SYSTEM
the length of the section and k is the number of broken line sections f$(i)=H[q*-(i-l)a]-H(q,-i-a) 0
H(q,-
for for 1 for
d) = Heaviside’s function
tgri
Ci =
.... $
xcd
x=d x>d
= tangents of the sections slopes
bi = tgvi
Denoting c = [cl, The
. . . , Cl+k,Zl,
b =
criterion function is expressed in the form m cu J(B, b, c) = [F(q,, I --ooI --a3
[h , .
. . , h+k,21.
qz, P, b, cJ12dq, da
(9)
The condition necessary to minimise the criterion function for the parameters to be found is that the following equations WP, b, c) = o aa
aJ(B, b, 4 = o dCi (10)
aJUt
b,
cl
= o
i=l,...,l+k/2
abi
have to be satisfied. Solution of equations (lo), in the form presented here, would require determining a two-dimensional distribution of the conditional probability density p( q, , q2) the modulating function t,h(q, , q2, t), double integrals from the combinations of these functions as well as functions Ri, Qi, 4i. Considering the computational difficulties connected with this, the following simplifying assumptions have been introduced: 1. Displacements and velocities are statistically independent of each other, hence I%%, q2) =A(41)
’~*2(92)-
2. The modulating function is only the function of displacements &I,,
42,
t)
=
&sA.
3. p = p0 const. After completing differentiation and grouping the dependent equations (10) can be expressed in the following form AK(q,)
f?C,
I
dq,
ABT(q2)Qj(qd
AC:(a) dq,
I
B@(q,) dq,
-cu
m
X
1 [I I m
ACXqdd41--i) dq2 +Po
-a
-cc
dq2
-52
m
cc
+
separately
00
@Q,,*b*c)=
2
functions
BDy(q2)Qj(q2)dq2+
I
02
CQAq,) dq,
--oo
aD
C4(q,M(l-j)
dq,
-02
I
BB,(qA
Wc(q,)dq,
I -co
* Qj(q2)Qi(q2)dq2-t j -cc
BCx(q,)dq, J_,BCv(q2)Qi(qJ+(l -j) dq2+
J CCAq,) --05
dq,
382
J. KISILOWSKI
AND
S. URBA\rcrSKI m
.x
00
a,BCy(q,M(l-_d + BCx(ql) dq, da J I=O, CC,(q2)4(1
-I -m
-_Ml-
4 dq2+
G+IG
[J -m
B&(a)
dq,
I -‘x
BBy(q2)Oj(q2)
dq2
cc
--co
1 ~J(~o,hc)= ’
5
1 [I
acl+k/2 0
.I
j=l,...,k/2
-m
n=,
O” AB:(q,) dq, --oo
J
0
Awe)da]+;
J -co
B’Xq,) dq,
-m
-02
[I m
+PO
J
1
0
m
BBy(q2)Qi(q2) dq2+
BCJq2b#41- i) dq2
--a
BR(qJ de
-02
-I 0
*m
BD,(q,) dqz+ Cl+k,Z
-co
J B&(q,) -03
a’
AB:(q,) dq,
-I 0
+Bo
BBy(q2)Ri(q2)dq2+
J -cc
B&(q,)
i=l
dq,
-CL!
J
BCy(q2M4i)de
J
BBy(qJdq, = 0
oz
BCx(q,)dq,
0
I
BR(q,) dq, m
BDy(q,) dqz+ b,+k,z
J
00
B&(q,) dq, --a0
0
1
J co 02 cc A’Xq,MO’ da BR(q,) ) dq, BDy(q*)&(qz) dq2 *J-m I‘PO [I-co J--05
1 aJ(Bo,b,c)= 2 i
J0
ao
bi[J
--03
00
-I 0
BBy(qi4dq, = 0
2
m
[I
J --m
k/2
cc
OD
1
ABFCq2)i da
abj
1
mAElq,)
r=, [I --oD
cc
dq,
J -co
ABT(q*)Rj(q,)
m
dq2+
-m
AC(q,)
da
o[) 03 +J--co CR(e) dq, CDytq,I&O’ dq2] +:[bi[JI BBx(q,) ) dq,J-1BBy(q2) J--cc m m *&(qdRi(qd dq,+ BCAq,) dq, BCy(qz)R(qz)+O’ dqz ) -m J J -m 00 Lx +J-cc CC,(a) dq, CCy(qzMo’ de M(~) --oc J 1 m co +bi+k/2 BWq,) dq, BB,(qz) -m [I-m J m m BCy(qh#4d dq, *Rj(q2) dq2+ BCAq,) dq, (11) J-02 J-cc I=a
After determining the limits of variation of the co-ordinates Ymin, ymaxr )imin, it,,,., and calculating the integrals, we finally obtain a linear system of algebraic equations
(12)
FLEXIBLE
CHARACTERISTICS
OF
A MECHANICAL
SYSTEM
383
systems with 1+ k/2 equations
which can be divided into two independent
A,.b=B, A,.c=
(13) B,.
(14)
Solving the above system, we obtain the directional coefficients bi, ci of the broken line that approximates to the damping characteristic which we are looking for. The following notation has been used in equations (11): A%%,)
= -p”:(qJ - $(a)
Mq,) *A(%) ABAd = - dq, AB;( qJ = --
1 d*h(qd 2 dqz
2
dP,(q,) dq2
Wqz) AB;(qz) = i&2(qz) dq
2
AC;(q,)
=F:(q,hhd
didq,) Pd41) AG(q,) = d
AG(q2)
q
1
=; h(q2)
d* &2(q2) d q;
AC;(q,) = -qzp:(qd BBAsJ = j:(qd
K(q,)
= -p”:(q,)
BDAq,) = q,&(%)
CCAS,) = 8%) CCy(q2) =iht) CD,(q,)
= -qd(qJ
Cry
=
d izz(qz)
-iz(q*)
dq’
2
(1%
384
J. KlSlLOWSKl
AND TABLE
Speed
I
interval
JWI)
1 2 3 4 5 6 7 8 9 10 11
0.0029 0.0058 0.0087 0.011 0.014 0.017 0.020 0.023 0.026 0.0
1 2 3 4 5 6 7 8 9 10 11
0~00000 -0.003477 -0.006955 -0.010432 -0.013909 -0.017387 -0.20864 -0.024341 -0.0277818 -0.031296 0.0
0~00000
1
XK(I)
Directional coefficient of the broken curve
0.0029 0.0058 0.0087 0.011 0.014 0.017 0.020 0.023 0.026 0.029 0.0
17.63 20.17 22.07 22.98 21.85 18.96 13.16 6.19 0.81 1.87 2447.53
-0.003477 -0.006955 -0.010432 -0.013909 -0.017387 -0.020864 -0.024341 -0.027818 -0.031296 -0.034773 0.0
4. COMPUTATIONAL 4.1.
S. URBAtiSKl
Force (NxlO’) 4.47889 4.47878 4.47866 4.47854 4.47842 4.47832 4.47825 4.47822 4.47822 4.47823 4.47898 -6.78611 -6.78610 -6.78609 -6.78609 -6.78609 -6.78609 -6.78610 -6.78610 -6.78610 -6.78611 -6.78610
-4.69 -3.96 -2.65 -2.46 -2.15 -3.10 -2.51 -0.76 9.09 14.67 8721.9
EXAMPLE
EXPERIMENT
Co-ordinates z,,, , zl , z2 that appear in equation (2) and characterise the displacements of both the frame of one of the car trucks and of the wheel-sets, have been measured under normal operating conditions. Measurements of acceleration of these co-ordinates have been carried out in the following conditions 1. Driving along a straight track; 2. At two travel speeds, u, 60 and 80 km/h. 3. At two car loading states (empty and full). Results obtained from the acceleration recordings are treated as a realisation of a stationary stochastic process with ergodic properties. Values of both masses and rigidity coefficients for both empty and loaded cars, in the equation have been determined on the basis of the standard tests [ 51. The recorded signals have been subjected to A/D conversion with a 12 bit convertor at a sampling frequency of 300 Hz. All the parameters that are used in equation (2), have been taken from ref. [6]. The passive experiment has been carried out using a freight car with Y 25/ Cs trucks. Thus we have obtained definite parameters as well as other random characteristics that are indispensable in carrying out numerical calculations. 4.2.
NUMERICAL
CALCULATIONS
Based on the equations presented in point 3, the following to calculate the dissipative characteristic of the system.
algorithm
has been suggested
FLEXIBLE CHARACTERISTICS
OF
MECHANICAL
A
385
SYSTEM
TABLE 2
Speed interval I
XP(I)
J=(Z)
1 2 3 4 5 6 7 8 9 10
0~00000 0.003 1 0.0062 0.0093 0.012 0.015 0.018 0.021 0.024 0.028
11
0.0
1 2 3 4 5 6 7 8 9 10 11
Directional coefficient of the broken curve
Force (Nx 103)
1334.11 868.77 425.45 27.29 -193.90 -306.67 -154.05 92.55 561.4 638.03
1.54811 1.54811 1.54812 1.54812 1.54811 1.54811 l-54810 1.54810 1.54810 1.54811
om3 1 0.0062 0.0093 0.012 0.015 0.018 0.021 0.024 0.028 0.03 1
0.0
o*ooo -0.0029 -0.0059 -0.0089 -0.0119 -0.0149 -0.0179 -0.0209 -0.0239 -0.0268 0.0
483086.7
-0.029 -0.0059 -0.0089 -0.0119 -0.0149 -0.0179 -0.0209 -0.0239 -0.0268 -0.0298 0.0
1.54810
955.6 575.7 144.3 -202.9 -399.6 -507.4 -404.1 -231.9 119.1 405.7 -3099511.7
-1.87197 -1.87197 -1.87197 -1.87197 -1.87199 -1.87199 -1.87198 -1.87198 -1.98199 -1.87200 -1.87196
Determination of velocities and displacements of the relative deflections yi(t) = 2&,, - ( il + &), yi( t) = 2zh1 - (z, + z2) by double integration of the digitised values of the recorded signals. Determination of the empirical densities of the conditional probability distributions of displacements b(q,) and velocities i2(q2). Determination of the modulating functionTin the form of a polynomial that approximates a set of points with co-ordinates ( t,ki,yi) in the least square sense
1.
2. 3.
where
N,
i=l,...,
pseudorandom
N
noise
is the sample with
a normal
size and
pi is the generated
distribution,
zero
mean
vector
value
and
of the unitary
variance. 4. Calculation defined 5. Solution
(15)
of the system of equations
and their integration
of the system (13) and (14),
we obtain 4.3.
of the coefficients
by relations directional
coefficients
(13) and (14) i.e. expressions
according
to (11).
where the Gauss method
of a broken
is applied.
curve (~(4~) - bi and
As a result
Ci.
RESULTS
Figures 2 and 3 present tions)
that we obtained.
damping
examples
They
force for u > 0 is different
In Tables
1 and 2 we have given
curve directional
coefficients
of the damping
confirm
the occurrence
force characteristics of dry friction
(vertical
vibra-
in the object.
The
from the force for u < 0. detailed
values
of the force
as well as the broken
of each segment of the velocity variations.
In our calculations
386
AND
J. KISILOWSKI
t
S.
UREAE;ISKI
F(N)
4.48.103’
h
1--3.5.10-Z
2.9.10-z 1 F2
J* I
’
-6.70
. IO3
v=EO(km/h
Figure 2. Damping
j hk.ec)
force F us. relative
velocity
) j for a fully-loaded
vehicle.
we have assumed that the number of the broken curve segments that approximates the characteristic was k = 20 after k = 10 for the range of v > 0 and u < 0. The force mean values obtained for 18 analysed examples are F, = 7562 N, F2 = 9150 N for a laiden car and F, = 3 154 N, F2 = 2720 N for an empty car.
F(N)
t
I
-3.104
3.1 * 10-Z
’
3(m/sec)
F2
’ v=60
Figure
3. Damping
force F VS.relative
(km/h)
velocity
j for an empty vehicle.
FLEXIBLE CHARACTERISTICS
OF A MECHANICAL
SYSTEM
387
REFERENCES
1. A. BARWICKI and L. PLONECKI 1977 Doctoral Thesis, Technical University of Kielce. ‘On certain methods of identification of a non-linear mechanical system’. (In Polish). 2. B. STEPHQSKI 1979 Doctoral Thesis, Warsaw University. Identification of certain non-linear discrete dynamic systems. (In Polish). 3. Z. CHEOPEK 1982 Doctoral Thesis, Warsaw University. Identification of a non-linear dynamic system by applying the Fokker-Planck equation and modulating functions. (In Polish). 4. S. F. MASRI and T. K. CAUGHEY 1979 Journal of Applied Mechanics, 46, 433-447. A nonparametric identification technique for non-linear dynamic problems. 5. S. URBAI&KI 1984 Doctoral Thesis, Warsaw University. On certain methods of mechanical system identification taking a freight car as an example. (In Polish). 6. A. CHUDZIKIEWICZ, J. DRO~DZIEL, J. KISILOWSKI and A. ZOCHOWSKI 1982 Mode/Zing and Analysing a Track-railway Vehicle Mechanical System. Warsaw: PWN (In Polish). 7. S. I. SOKOLOV 1976 Zssljedowanije Dinamiki i Procnosti Passaiirskich Wagonow: Moscow: Masinostroenie (In Russian). 8. K. SOBCZYK 1973 Methods of Statistical Dynamics. Warsaw: PWN (In Polish).
Mechanical Systems and Signal Processing
IDENTIFICATION
OF FLEXIBLE
A MECHANICAL “PASSIVE” JERZY
CHARACTERISTICS
SYSTEM
BASED
OF
ON
EXPERIMENTATION
KISILOWSKI
AND
STEFAN
URBA~SKI
Institute of Transport, Warsaw University of Technology, 00-662 Warsaw, ul. Koszykowa
75, Poland
(Received December 1986, accepted April 1987) The paper presents a method of identifying the flexible non-linear characteristics of a mechanical system. For this purpose we used the Fokker-Planck equation and on this basis the criterion function has been built. The proposed method does not impose any restrictions on the character of non-linearities or the dimension of the system. The present method determines the non-linear characteristics of elasticity and damping of the objects while in use. The results obtained from identification of the non-linear dissipative force that is caused by the dry friction of a mechanical system such as a freight car are presented here as a computational example. The results, which are based on the passive experiment results, confirm the efficiency of the method. 1. INTRODUCTION
Mechanical systems often contain a number of non-linear elements which can be exemplified by the characteristics of the stiffness-damping connections. While analysing this kind of system, there is a tendency towards linearisation which significantly simplifies the problem. The introduction of equivalent linear coefficients leads to the building and analysing of linear mathematical models. At the same time the problem of identification while using these models is reduced to the identification of definite parameters. In applied mechanics the bulk of work on identification problems concentrates on parameteric identification. Linearity is a rare characteristic of mechanical systems. Forces of elasticity and damping usually depend non-linearly on displacements and velocities. Using linearisation we obtain linear mathematical models. Linearisation, however, frequently leads to excessive simplification which limit the possibility of carrying out a complete analysis of the system’s properties, and in some cases they lead to incorrect conclusions about the processes that occur within the system. A complete representation of the phenomena that occur in real dynamic systems requires the building of non-linear mathematical models. An analysis of non-linear systems creates mathematical problems. However, it lets us investigate a number of the systems properties such as for example the occurrence of various kinds of resonance frequencies, the occurrence of limit cycles, the influence of the initial conditions on the course of vibrations the problem of exceeding the trajectories of solutions, etc, which cannot be investigated by analysing linear models. Non-linear mathematical model building involves identifying the character of the modelled nonlinearities. 377 0888-3270/87/040371+11
%03.00/O
@ 1987 Academic Press Limited
378
J. KISILOWSKI
AND
S. URBAI;ISKI
There are only a few papers presenting effective methods of identifying the non-linear systems, eg. refs [l-4]. The first reference presents a method of reproducing the characteristics of non-linear elements on the basis of substitute linear coefficients that have been obtained by equivalent linearisation for different levels of the exciting signals. In refs [2] and [3] the Fokker-Planck equation was used to define non-linear characteristics, and Stepinski [2] reduces the characteristics of non-linearities to continuous functions. Masri and Caughey [4] present a method of identifying non-linear models by applying the regressive technique in conjunction with Chebyshev’s orthogonal polynomials. This method can be applied in identifying a wide class of non-linear systems. References [l-4] are based on active experimentation. This article presents both a method of reproducing the systems non-linearity characteristics by applying the Fokker-Planck equation [3] and the results obtained as applied to a mechanical system with a non-linear dissipative characteristic. Tests carried out on the “passive” experiment are the basis of the present method [5].
2. THE
MODEL
All considerations concerning the method of identifying non-linear characteristics of the stiffness-damping elements were carried out on mechanical systems such as railway vehicles. A railway vehicle is a mechanical system normally consisting of three or seven elements that are treated as rigid. To describe their position, we can use 18 or 42 co-ordinates. The number of these co-ordinates is reduced by geometrical and constructional constraints. Complete consideration of the systems of co-ordinates and nominal models can be found in Chudzikiewicz et al. [6]. In the case of railway vehicles it is usually assumed that there is a longitudinal plane of symmetry and that the system performs small vibrations. This assumption allows us to consider vibrations in the vertical longitudinal plane of symmetry and in the lateral plane separately. We shall consider a two-truck railway vehicle, the nominal model of which can be presented as in Fig. 1. This figure presents the model of vibrations in the vertical longitudinal plane of symmetry. It has been assumed that we shall search for the characteristics of the stiffness-damping connections between the wheel-sets and the frame of the truck. The vertical vibrations are excited due to the vertical irregularities of the track-the vertical displacement signals of wheel-set centres z,(r), z2( t), zj(t), zq(f) can be their
r
20
20 Figure
1. Model of the vehicle.
FLEXIBLE
CHARACTERISTICS
OF A MECHANICAL
SYSTEM
379
representations. So defined, the inputs are valid in the suspension modelling process over the frequency range of O-20 Hz this being the range of the dominating input components and the system’s natural frequencies. The resonant frequency of the system, non-sprung mass-track occurs somewhere between 60-80 Hz and therefore it is justifiable to ignore the track mass as well as the subgrade stiffness and damping [6,7]. Assuming linearity of both stiffness and damping coefficients, the motion of the masses of the truck frames can be described with a linear system of four equations which describe co-ordinates zbl, zb2, xbI , xb2 : m,+- rnNh2 1 z, +- 1.i 4aZ,
+‘tk,,Zb,+
44aZ, ( i
4
MIV) ib,+(4&z+&)ib+
Wlj.~----mNh2
4aZ,
1 &
)
..
4 4aZ,
_A..
-$tibl+kb2)
zb2 2aZ, zb2
N
-2c,,(i,+i2)-2kzz(zI+zz)=O; mNh2 1 IN m,+ -+~~+am,),2+(4~~~+~)~b2+4k~~zb2+(~m~-~~-~) 4aZ,
.. XZb,-~i,,+~(~b,+~b2)-2c=l(i3+i.)-2k,,(Z1+Z4)=0; N
N
z~~b,f(C,+4a2E,,)41+4a2k,~b,+~(ib2-ibl)f2ac,,’(i2-i,)+2ak==(z2-z,)=0; N
z~~b2+(E,+4a2e,,)ib2+4a2k~~b2+~
(ib2
-ib,)+2aCz,(i,-i~)+2ak*~(z~-z~)=0;
N
(1)
where m, is the mass of the truck frame, m N is the mass of the car body, Zwyis the moment of inertia of the truck frame, ZN,,is moment of inertia of the car body, k,, is the coefficient of rigidity, c,, and c, are coefficients of damping and a, aN, h are geometrical parameters. Adding the sides of the two first equations (with zbI and zbz) and subtracting the equation with xbz from the equation with xbl, we obtain two independent equations. Assuming that the vibrations of both trucks can be considered separately, the first of the equations takes the following form: (mw+tmN)a,+2k,,[2zb,-(z,+z,)]+f[2ib,+(i,+i2)]=0.
(2)
This equation describes the vertical vibrations of the system with one degree of freedom with the kinematic input (z, + z2). 3. DESCRIPTION
OF THE
METHOD
Equation (2) in which the third element describing the damping in the system will be determined in the identification process, is the initial equation to present the method. Introducing the new variable y = 2zbl - (z, + z2) and denoting M = m, +$mN equation (2) is reduced to the following form: (3)
j;+pY+4i)=df) where
n(t) = - (i’, + i2) = the input function.
p
~5,
a(i)
c’s
380
J.
KISILOWSKI
AND
S. URBAI;ISKI
Relation (3) has been reduced to the system of equations of the first order by substituting 91 =Y 92 =.i
(4)
41= 42 42=-&1-~(42)+~(4,,
q2,
t)‘t(t).
The above equation is a stochastic differential equation in which the input 7(t) has been modelled as an interaction of a modulating function +!r(q,, q2, r) = + on the white noise l(t) with the average value equal to zero and with the unitary spectral density, and thus: 77(r) = @(41, q2,
t)
. 5(t).
For equations (4), we can write the Fokker-Planck
equation as follows [8]:
p( q, , q2, r) = p, density of the conditional probability a1 =
42
a2=-&,-4q2) b22=Mql,
q2,
01’
Assuming the solution of the above equation to be stationary, i.e. ap/ar = 0 we define a function of the following form
(6) where & = Jl(ql, q2, r); p’= ( ql, q2, r) is determined from the experimental data, on the basis of which we introduce the criterion function in the damping force identification process. The criterion function is defined by the relation m cc (7) J(P, a) = {F[q, >q2, P, 4d112 da de. I --ooI -aI Which satisfies the Fokker-Planck equation by parameters to be found which describe the characteristics being identified. The damping characteristic curve to be found can be approximated by a broken line described by the equation l+(k/2)
(y(q2) =
C i=l
[ciOi(q2)+
biRi(q2)l
where k/2
,s+,441-r)
Qi(q2)=~(1-i)q2+~(1-i)(i-1)a-a k/2
C 9(r),
Ri(S2)=~(i)q2-~(i)(i-1)a+a
r=i+l
Q k/2+1=
.R X(qpnin,O),
l+k/2
=
X(0,q2max)
i=l,...,-
k
2
FLEXIBLE
a=(hllax-
qzmin)/k=
CHARACTERISTICS
OF
A MECHANICAL
381
SYSTEM
the length of the section and k is the number of broken line sections f$(i)=H[q*-(i-l)a]-H(q,-i-a) 0
H(q,-
for for 1 for
d) = Heaviside’s function
tgri
Ci =
.... $
xcd
x=d x>d
= tangents of the sections slopes
bi = tgvi
Denoting c = [cl, The
. . . , Cl+k,Zl,
b =
criterion function is expressed in the form m cu J(B, b, c) = [F(q,, I --ooI --a3
[h , .
. . , h+k,21.
qz, P, b, cJ12dq, da
(9)
The condition necessary to minimise the criterion function for the parameters to be found is that the following equations WP, b, c) = o aa
aJ(B, b, 4 = o dCi (10)
aJUt
b,
cl
= o
i=l,...,l+k/2
abi
have to be satisfied. Solution of equations (lo), in the form presented here, would require determining a two-dimensional distribution of the conditional probability density p( q, , q2) the modulating function t,h(q, , q2, t), double integrals from the combinations of these functions as well as functions Ri, Qi, 4i. Considering the computational difficulties connected with this, the following simplifying assumptions have been introduced: 1. Displacements and velocities are statistically independent of each other, hence I%%, q2) =A(41)
’~*2(92)-
2. The modulating function is only the function of displacements &I,,
42,
t)
=
&sA.
3. p = p0 const. After completing differentiation and grouping the dependent equations (10) can be expressed in the following form AK(q,)
f?C,
I
dq,
ABT(q2)Qj(qd
AC:(a) dq,
I
B@(q,) dq,
-cu
m
X
1 [I I m
ACXqdd41--i) dq2 +Po
-a
-cc
dq2
-52
m
cc
+
separately
00
@Q,,*b*c)=
2
functions
BDy(q2)Qj(q2)dq2+
I
02
CQAq,) dq,
--oo
aD
C4(q,M(l-j)
dq,
-02
I
BB,(qA
Wc(q,)dq,
I -co
* Qj(q2)Qi(q2)dq2-t j -cc
BCx(q,)dq, J_,BCv(q2)Qi(qJ+(l -j) dq2+
J CCAq,) --05
dq,
382
J. KISILOWSKI
AND
S. URBA\rcrSKI m
.x
00
a,BCy(q,M(l-_d + BCx(ql) dq, da J I=O, CC,(q2)4(1
-I -m
-_Ml-
4 dq2+
G+IG
[J -m
B&(a)
dq,
I -‘x
BBy(q2)Oj(q2)
dq2
cc
--co
1 ~J(~o,hc)= ’
5
1 [I
acl+k/2 0
.I
j=l,...,k/2
-m
n=,
O” AB:(q,) dq, --oo
J
0
Awe)da]+;
J -co
B’Xq,) dq,
-m
-02
[I m
+PO
J
1
0
m
BBy(q2)Qi(q2) dq2+
BCJq2b#41- i) dq2
--a
BR(qJ de
-02
-I 0
*m
BD,(q,) dqz+ Cl+k,Z
-co
J B&(q,) -03
a’
AB:(q,) dq,
-I 0
+Bo
BBy(q2)Ri(q2)dq2+
J -cc
B&(q,)
i=l
dq,
-CL!
J
BCy(q2M4i)de
J
BBy(qJdq, = 0
oz
BCx(q,)dq,
0
I
BR(q,) dq, m
BDy(q,) dqz+ b,+k,z
J
00
B&(q,) dq, --a0
0
1
J co 02 cc A’Xq,MO’ da BR(q,) ) dq, BDy(q*)&(qz) dq2 *J-m I‘PO [I-co J--05
1 aJ(Bo,b,c)= 2 i
J0
ao
bi[J
--03
00
-I 0
BBy(qi4dq, = 0
2
m
[I
J --m
k/2
cc
OD
1
ABFCq2)i da
abj
1
mAElq,)
r=, [I --oD
cc
dq,
J -co
ABT(q*)Rj(q,)
m
dq2+
-m
AC(q,)
da
o[) 03 +J--co CR(e) dq, CDytq,I&O’ dq2] +:[bi[JI BBx(q,) ) dq,J-1BBy(q2) J--cc m m *&(qdRi(qd dq,+ BCAq,) dq, BCy(qz)R(qz)+O’ dqz ) -m J J -m 00 Lx +J-cc CC,(a) dq, CCy(qzMo’ de M(~) --oc J 1 m co +bi+k/2 BWq,) dq, BB,(qz) -m [I-m J m m BCy(qh#4d dq, *Rj(q2) dq2+ BCAq,) dq, (11) J-02 J-cc I=a
After determining the limits of variation of the co-ordinates Ymin, ymaxr )imin, it,,,., and calculating the integrals, we finally obtain a linear system of algebraic equations
(12)
FLEXIBLE
CHARACTERISTICS
OF
A MECHANICAL
SYSTEM
383
systems with 1+ k/2 equations
which can be divided into two independent
A,.b=B, A,.c=
(13) B,.
(14)
Solving the above system, we obtain the directional coefficients bi, ci of the broken line that approximates to the damping characteristic which we are looking for. The following notation has been used in equations (11): A%%,)
= -p”:(qJ - $(a)
Mq,) *A(%) ABAd = - dq, AB;( qJ = --
1 d*h(qd 2 dqz
2
dP,(q,) dq2
Wqz) AB;(qz) = i&2(qz) dq
2
AC;(q,)
=F:(q,hhd
didq,) Pd41) AG(q,) = d
AG(q2)
q
1
=; h(q2)
d* &2(q2) d q;
AC;(q,) = -qzp:(qd BBAsJ = j:(qd
K(q,)
= -p”:(q,)
BDAq,) = q,&(%)
CCAS,) = 8%) CCy(q2) =iht) CD,(q,)
= -qd(qJ
Cry
=
d izz(qz)
-iz(q*)
dq’
2
(1%
384
J. KlSlLOWSKl
AND TABLE
Speed
I
interval
JWI)
1 2 3 4 5 6 7 8 9 10 11
0.0029 0.0058 0.0087 0.011 0.014 0.017 0.020 0.023 0.026 0.0
1 2 3 4 5 6 7 8 9 10 11
0~00000 -0.003477 -0.006955 -0.010432 -0.013909 -0.017387 -0.20864 -0.024341 -0.0277818 -0.031296 0.0
0~00000
1
XK(I)
Directional coefficient of the broken curve
0.0029 0.0058 0.0087 0.011 0.014 0.017 0.020 0.023 0.026 0.029 0.0
17.63 20.17 22.07 22.98 21.85 18.96 13.16 6.19 0.81 1.87 2447.53
-0.003477 -0.006955 -0.010432 -0.013909 -0.017387 -0.020864 -0.024341 -0.027818 -0.031296 -0.034773 0.0
4. COMPUTATIONAL 4.1.
S. URBAtiSKl
Force (NxlO’) 4.47889 4.47878 4.47866 4.47854 4.47842 4.47832 4.47825 4.47822 4.47822 4.47823 4.47898 -6.78611 -6.78610 -6.78609 -6.78609 -6.78609 -6.78609 -6.78610 -6.78610 -6.78610 -6.78611 -6.78610
-4.69 -3.96 -2.65 -2.46 -2.15 -3.10 -2.51 -0.76 9.09 14.67 8721.9
EXAMPLE
EXPERIMENT
Co-ordinates z,,, , zl , z2 that appear in equation (2) and characterise the displacements of both the frame of one of the car trucks and of the wheel-sets, have been measured under normal operating conditions. Measurements of acceleration of these co-ordinates have been carried out in the following conditions 1. Driving along a straight track; 2. At two travel speeds, u, 60 and 80 km/h. 3. At two car loading states (empty and full). Results obtained from the acceleration recordings are treated as a realisation of a stationary stochastic process with ergodic properties. Values of both masses and rigidity coefficients for both empty and loaded cars, in the equation have been determined on the basis of the standard tests [ 51. The recorded signals have been subjected to A/D conversion with a 12 bit convertor at a sampling frequency of 300 Hz. All the parameters that are used in equation (2), have been taken from ref. [6]. The passive experiment has been carried out using a freight car with Y 25/ Cs trucks. Thus we have obtained definite parameters as well as other random characteristics that are indispensable in carrying out numerical calculations. 4.2.
NUMERICAL
CALCULATIONS
Based on the equations presented in point 3, the following to calculate the dissipative characteristic of the system.
algorithm
has been suggested
FLEXIBLE CHARACTERISTICS
OF
MECHANICAL
A
385
SYSTEM
TABLE 2
Speed interval I
XP(I)
J=(Z)
1 2 3 4 5 6 7 8 9 10
0~00000 0.003 1 0.0062 0.0093 0.012 0.015 0.018 0.021 0.024 0.028
11
0.0
1 2 3 4 5 6 7 8 9 10 11
Directional coefficient of the broken curve
Force (Nx 103)
1334.11 868.77 425.45 27.29 -193.90 -306.67 -154.05 92.55 561.4 638.03
1.54811 1.54811 1.54812 1.54812 1.54811 1.54811 l-54810 1.54810 1.54810 1.54811
om3 1 0.0062 0.0093 0.012 0.015 0.018 0.021 0.024 0.028 0.03 1
0.0
o*ooo -0.0029 -0.0059 -0.0089 -0.0119 -0.0149 -0.0179 -0.0209 -0.0239 -0.0268 0.0
483086.7
-0.029 -0.0059 -0.0089 -0.0119 -0.0149 -0.0179 -0.0209 -0.0239 -0.0268 -0.0298 0.0
1.54810
955.6 575.7 144.3 -202.9 -399.6 -507.4 -404.1 -231.9 119.1 405.7 -3099511.7
-1.87197 -1.87197 -1.87197 -1.87197 -1.87199 -1.87199 -1.87198 -1.87198 -1.98199 -1.87200 -1.87196
Determination of velocities and displacements of the relative deflections yi(t) = 2&,, - ( il + &), yi( t) = 2zh1 - (z, + z2) by double integration of the digitised values of the recorded signals. Determination of the empirical densities of the conditional probability distributions of displacements b(q,) and velocities i2(q2). Determination of the modulating functionTin the form of a polynomial that approximates a set of points with co-ordinates ( t,ki,yi) in the least square sense
1.
2. 3.
where
N,
i=l,...,
pseudorandom
N
noise
is the sample with
a normal
size and
pi is the generated
distribution,
zero
mean
vector
value
and
of the unitary
variance. 4. Calculation defined 5. Solution
(15)
of the system of equations
and their integration
of the system (13) and (14),
we obtain 4.3.
of the coefficients
by relations directional
coefficients
(13) and (14) i.e. expressions
according
to (11).
where the Gauss method
of a broken
is applied.
curve (~(4~) - bi and
As a result
Ci.
RESULTS
Figures 2 and 3 present tions)
that we obtained.
damping
examples
They
force for u > 0 is different
In Tables
1 and 2 we have given
curve directional
coefficients
of the damping
confirm
the occurrence
force characteristics of dry friction
(vertical
vibra-
in the object.
The
from the force for u < 0. detailed
values
of the force
as well as the broken
of each segment of the velocity variations.
In our calculations
386
AND
J. KISILOWSKI
t
S.
UREAE;ISKI
F(N)
4.48.103’
h
1--3.5.10-Z
2.9.10-z 1 F2
J* I
’
-6.70
. IO3
v=EO(km/h
Figure 2. Damping
j hk.ec)
force F us. relative
velocity
) j for a fully-loaded
vehicle.
we have assumed that the number of the broken curve segments that approximates the characteristic was k = 20 after k = 10 for the range of v > 0 and u < 0. The force mean values obtained for 18 analysed examples are F, = 7562 N, F2 = 9150 N for a laiden car and F, = 3 154 N, F2 = 2720 N for an empty car.
F(N)
t
I
-3.104
3.1 * 10-Z
’
3(m/sec)
F2
’ v=60
Figure
3. Damping
force F VS.relative
(km/h)
velocity
j for an empty vehicle.
FLEXIBLE CHARACTERISTICS
OF A MECHANICAL
SYSTEM
387
REFERENCES
1. A. BARWICKI and L. PLONECKI 1977 Doctoral Thesis, Technical University of Kielce. ‘On certain methods of identification of a non-linear mechanical system’. (In Polish). 2. B. STEPHQSKI 1979 Doctoral Thesis, Warsaw University. Identification of certain non-linear discrete dynamic systems. (In Polish). 3. Z. CHEOPEK 1982 Doctoral Thesis, Warsaw University. Identification of a non-linear dynamic system by applying the Fokker-Planck equation and modulating functions. (In Polish). 4. S. F. MASRI and T. K. CAUGHEY 1979 Journal of Applied Mechanics, 46, 433-447. A nonparametric identification technique for non-linear dynamic problems. 5. S. URBAI&KI 1984 Doctoral Thesis, Warsaw University. On certain methods of mechanical system identification taking a freight car as an example. (In Polish). 6. A. CHUDZIKIEWICZ, J. DRO~DZIEL, J. KISILOWSKI and A. ZOCHOWSKI 1982 Mode/Zing and Analysing a Track-railway Vehicle Mechanical System. Warsaw: PWN (In Polish). 7. S. I. SOKOLOV 1976 Zssljedowanije Dinamiki i Procnosti Passaiirskich Wagonow: Moscow: Masinostroenie (In Russian). 8. K. SOBCZYK 1973 Methods of Statistical Dynamics. Warsaw: PWN (In Polish).