MASS MATRIX ESTIMATION FOR THE DISCRETIZED DYNAMIC MODEL OF RAILWAY CAR BODIES
TRA.N Quoc KHANH
Department of Transport Engineering .Mechanics, Technical University of Budapest, H-1521
Received: February 25, 1988.
Presented by Prof. Dr. Pal Michelberger
Abstract
A method has been developed for estimating the mass matrix for the dynamical model with discretized masses of railway car bodies. This method relies on the comparison between vibrations of real beams and of those with discretized. section-wise uniform mass distribution.
The objective function relying on the least squares m~thod has been minimized by the SU.~IT method. Implementation of the computation method is illustrated on hand of analyzing a railway luggage truck body.
1. Introduction
This scope is strictly related to algorithmic modelling replacing heuristic modelling in the strength analysis of railway car bodies exposed to vertical, symmetric, dynamic loads. The continuum model "will be replaced hy a discre- tized model relying on data of the continuum model, and the algorithm is expected to suit mass and stiffness matrices [4, 5].
The method presented here lends itself to estimate the mass matrix of the discretized model of railway car bodies.
2. :ffIethod for estimating the discretized mass distrihution
This method reli,,;s on the comparison bet·ween ·vihrations of real beam::- and of those with discretized masses, namely:
In the analysis of bending vibrations, the car body is considered as a heam conform to [3]. In estimating the mass distribution m, the car body is considered as a beam of mass sections [1], differing from the former heam:::
with continuous mass distribution by its sectional mass distribution. Masses c:ssigned to beam sections are expected to exhibit as small deviations hetween yihration patterns of real and discretized-mass beams during test period T as possihle. Comparison refers to the case where both beams are hinged at both ends.
In estimating the discrete mass distribution, in compliance with the car body symmetry, the model heam is assumed to suffer linear bending alone.
Bending axes of every cross section are normal to the dravving plane. The beam
IS only exposed to forces in the drawing plane, nor mal to the axis of the no- 3
i02 TiLL\ QL'OC KJU.Yll
load beam and applied in the principal plane normal to the bending axis. In addition, angular rotation of the displacing bar section is assumed to he small enough, so that components normal to the axis of the beam at Test of the shear forces tilted togethcr with the beam section are equal to the shear fOTces them- selves. Cross sections of bar sections are supposed to he constant, and 'within bar sections, no extern~l forces are assumed to act.
In applying the method, Tegularities of the free vibTations of the con- tinuum heam aTe assumed to be known. For that, knowledge of continuum beam data El' Ii , Ai' Hi' iJi' Xi and Fi is sufficient.
Here: Ei elasticity constant of har section i of the continuum heam (i
=
1, 2, ... , 1\1);Ii second-order moment of inertia of har section i of the continuum beam (i = 1, 2, ... , 1\1);
Ai cross section of har section i of the continuum heam (i = 1, 2, ... , 1\1);
Hi length of har section i of the continuum heam (i
=
1, 2, ... , 1\1);Qi specific density of har section i of the continuum heam (i =
=
1, 2, ... , 1\1);Xi ahscissa of point i of the continuum heam (i = 1, 2, ... , p);
Fi deflection of the continuum heam at Xi at time t = 0 (i
=
1, 2, ... , p).With these notations and under these conditions, the prohlem of estimat- ing discrete mass distribution is as follows:
Let us find m
=
(ml' ln2, • • • , TnN*) such as to minimize scalar vcctor functionT
g:(m) =
.:E J (z(S7,t)-.:(m)(57,t))2dt ( 1)
i E J ,1* o under conditions
where Z(5t, t)
o
Ko >
iE
J*(i = 1, 2, ... , N5) i },I - ~ Tni i
< ()
iE! JI"
(2)
(3) ( 4)
flexural displacement of the cross-sectional cintroid of the con- tinuum heam at a fixed spot
51
at time t.57
is the ahscissa of the centroid of har section i of the beam of sectional IllS':;S(i
E
I / 1*);JIASS JIATRI.\: ESTIJIATIOS OF IUIlVAY CAN BODIES 103
zm(S1,
t) - flexural displacement of the cross-sectional centroid - again atS1 -
of the beam of discretized mass distribution ill. at time t (iEJjI*);~\m)
"'/
NS N*
i-th circular eigenfrequency of the continuum beam (i = 1, 2, ... , NS);
i-th circular eigenfrequency of the beam of mass distribution m (i
=
1,2, ... , NS);number of circular eigenfrequencies reckoned ·with;
numeral of the discretized beam section;
fixed positive constants;
c,Kand 0 -
l'vI total mass of the continuuIll beam;
I = {1, 2, ... , N*}
1* = {i
El!
no-mass, elastic har section i of the discretized beam.}Problems (1) to (4) haye heen solved hy optimization method SUMT [2].
3. Practical computation outcomes
This practical computation method has hp-en applied for the railway luggage truck hody discrihl'd in (3). TlH' SUlVIIT method leads to the optimum mass distribution:
1
.,
7 9 11 13 16 19
Si'" (cm)
10 352.66 708.86 106·1.66 ] ·120.66 l-J·16.66 ::132.66 ::481,4
References
6.0ll 3.209 5.539 6.385 6.303 6.909 ::.Oll ::.057
l. Boszi.\"AY. _.\..: }iechanical "Vibrations." }IUszaki K. BudapesL 1965.
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3. TRAi.\" Quoc KH . .\i.\"H-}iIcHELBERGER. P.: Stiffness Estimation }Iethod for the Discretized Dynamic xlodel of Railway Car Bodies. Periodica Polytechnica, Transp. Eng. \'01. 16 (1988) :No. l.
4. ZOBORY. 1.-GyORIK. A.-SZABO. A.: Dvnamic Loads in the Driye SYstem of Railwav Tra~tion Yehicle~ Due to Tr~ck Uneyennesses. Periodica Polytech~lica Transp. Eng.
Yo1. 15 (1987) No. 1.
5. ZOBORY, 1.: Dynamic Processes in the Drive Systems of Railway Traction Vehicles in the Presence o'f Excitation caused by Uncycnuesses in the Track. Vehicle System Dynamics.
198.~. :\'0. 1-3,1985. pp. 33-39.
TRAN QUOC KHANII H-1521, Budapest
* In Hungarian.
3*
