Mathematical Model for Vibrations Analysis of The Tram Wheelset

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1 TH ARCHIVS OF TRANSPORT VOL. XXV-XXVI NO - 03 Mathematical Moel for Vibrations Analysis of The Tram Wheelset Anrzej Grzyb Karol Bryk ** Marek Dzik *** Receive March 03 Abstract This paper presents a mathematical moel for the analysis of the 05Na tram trolley small vibrations. In the paper there is also presente a physical moel of the analyse tram trolley, accoring to which the mathematical moel was create. The leas were mae sing the methos of analytical mechanics. For the moel, the Lagrange II type eqations were etermine in matrices form. Using the moel it is possible to create simlations of the 05Na tram trolley motion.. Introction In railway vehicle's ynamic moeling the Lagrange II type eqations are wiely se. These eqations allow obtaining a set of ifferential eqations, which escribes the physical moel withot internal reactions occrring between the objects of the moel. To escribe the physical moels sing the Lagrange eqations, it is necessary to introce the concept of generalize coorinates. The generalize coorinates are the least nmeros set of variables, which clearly escribes the position of an element of the physical moel. These variables' nmber is efine as a nmber of egrees of freeom of the analyze moel. Therefore, geometrical coorinates of each object forming part of the analyze moel can be represente by generalize coorinates an written as: Taesz Kosciszko Cracow University of Technology, Instytt Pojazów Szynowych, Jana Pawła II 37 Street, 3-864, Cracow, Polan **IPS TABOR, Warszawska 8 Street, 6-055, Poznan, Polan ***Taesz Kosciszko Cracow University of Technology, Katera Trakcji i Sterowania Rchem, Warszawska 4 Street, 3-55, Cracow, Polan

2 56 Anrzej Grzyb, Karol Bryk, Marek Dzik x x ( q,, q, t) m x x ( q,, q, t) n n m (.) x,,xn geometrical coorinates of each object forming part of the analyze physical moel n nmber of geometrical coorinates of the physical moel q,, q m generalize coorinates of the physical moel t time For the physical moel generalize coorinates were efine the Lagrange II type eqations can be written as: U D Q t q q q q (.) U D Q t qm qm qm qm m kinetic energy of the analyze moel escribe in generalize coorinates U potential energy of the analyze moel escribe in generalize coorinates D issipation fnction of the analyze moel's energy escribe in generalize coorinates m nmber of generalize coorinates Q,,Qm external generalize forces escribe as [5]: x x Q P P n n q q x Q P P x n m n qm qm (.3) P,...,Pn external forces relate to movements or external torqes relate to rotation angle for geometrical coorinates: x,,xn.

3 Mathematical Moel for Vibrations Analysis of The Tram Wheelset 57 For the prpose of this sty the set of eqations (.) of the 05Na tram trolley was written assming linearity an stationarity of the analyze physical objects. These eqations were transforme into a set of eqations escribe by generalize coorinates an time (.4) an (.5), store sing qaratic forms (.6) [5]. U U x q,, qm, t,, xn q,, qs, t,,,,,,,,,,,,,, x q qm q qm t xn q qm q qm t T x x T D x D x T U x U x (.4) (.5) (.6), D, U qaratic form matrices of: kinetic energy, issipation fnction an potential energy. x,x geometrical an velocity coorinates vectors escribe as (.7) x x x, x x x n n (.7) For the stationary, linear set of eqations of geometric coorinates an their velocities, their relationship to general coorinates is presente in (.8). x W x, x W x (.8) q q T,, qn x q general coorinates vector (.9) q W constraints matrix,

4 58 Anrzej Grzyb, Karol Bryk, Marek Dzik x xn x xn q q q q T W 0 x x x x n n qm qm qm qm (.0) When the matrices in eqation (.) are invariant in time, the Lagrange II type eqations can be written as: W T Wx W T U Wx W T D Wx WP q q q (.) P external forces vector T,, Pn P P (.), D, U graient matrices of kinetic energy, issipation fnction an potential energy x x, (.3) x x x x n n Wx x (.4) x x D = D U = U Wx Wx

5 Mathematical Moel for Vibrations Analysis of The Tram Wheelset 59. Description of the Analyze Tram Trolley, its Kinematic Analysis an Physical Moel The analyze tram trolley is a part of 05Na tram's power transmission system. It was being proce in by Konstal instries in Chorzów. Nowaays it is the most nmerosly exploite tram trolley in Polan. The characteristic featre of the escribe tram trolley is a frame consisting of two parts joint connecte. Figre. presents fnctional moel of the analyze tram trolley. In this moel zero egree sspension was omitte. To escribe the motion of each of the objects in the moel there were se two sets of coorinates (figre.). The first set of coorinates (with the sperscript " ' "), which is rigily connecte to an object an its coorinate origin is sitate in the object's centre of inertia. The secon set of coorinates (withot the sperscript " ' "), moves with the nistrbe movement of the vehicle. When the object moves only with the nistrbe movement, both sets of coorinates are the same. The position of any point of the object in the secon set for small range of rotation angles changes is escribe by approximate relations (.): x x x ' y ' z ' C z y y y x ' y ' z ' C z x z z x ' y ' z ' C y x (.) In the analyze tram trolley the frame is constrcte from two parts, which are connecte by joints A an B (figre.). These joints limit mtal movement of the two parts of the frame. The frame's system has 7 egrees of freeom. As the generalize coorinates we can choose six coorinates escribing the position of the right-sie frame in 3D coorinate system an rotation angle between the frame's halves (the halves can rotate relatively to each other only aron the straight line connecting the joints A an B). Since both parts of the frame, consiere as inepenent objects, have egrees of freeom, an after combining them by bons the nmber of egrees of freeom ecrease to 7. Therefore five bon eqations were written. In orer to etermine these eqations there were set A an B points locations in the secon set of coorinates CiP, xip, yip, zip.

6 60 Anrzej Grzyb, Karol Bryk, Marek Dzik x x k h A ip ip zip yip y y k h A ip ip zip xip z z k h A ip ip yip xip x x k h B ip ip zip yip y y k h B ip ip zip xip z z k h B ip ip yip xip (.) The same points in the set of coorinates relate to the left sie frame are escribe: x x k h A il il zil yi L y y k h A il il zil xil z z k h A il il yil xil x x k h B il il zil yi L y y k h B il il zil xil z z k h B il il yil xil (.3) The location of a point in the set of coorinates CiP, xip, yip, zip, can be escribe by se of the location in the set of coorinates CiL, xil, yil, zil, sing their relation (these sets of coorinates are mtally shifte figre 3.): x y z ip ip ip x z il y il ip k k (.4) Since both parts of the frame have together seven egrees of freeom, in the obtaine relations there are seven inepenent variables: xip, yip, zip, φxip, φyip, φzip, φyil.

7 Mathematical Moel for Vibrations Analysis of The Tram Wheelset 6 Fig... 05Na tram trolley's moel

8 6 Anrzej Grzyb, Karol Bryk, Marek Dzik Fig... Sets of coorinates se in the moel In the consiere moel there are 30 geometric coorinates (the nmber of rigi blocks mltiplie by 6) an 3 general coorinates (the nmber of geometrical coorinates mins nmber of inepenent bons). x = W x q (.5) :

9 Mathematical Moel for Vibrations Analysis of The Tram Wheelset 63 W bons matrix x geometrical coorinates vector x x ip y ip z ip xip yip zip x il y il z il xil yil zil x ib y ib z ib,,,,,,,,,,,,,,, xib, yib, zib, xi, yi, zi, xi, zi, zi, xi, yi, zi, xi, yi, zi T (.6) x q generalize (inepenent) coorinates vector x q x ip y ip z ip xip yip zip yil y ib z ib xib yib,,,,,,,,,,, xi, yi, zi, xi, zi, zi, xi, yi, zi, xi, yi, zi T (.7) 3. The Movement qation For the linear an stationary moel the Lagrange II type eqations of tram trolley can be written as: W T W x W T D W x W T U W x W T F (3.) q q q x generalize (inepenent) coorinates vector, q W bons matrix energy graient matrix, etermine by the relation: = x (3.) X D energy issipation graient matrix: D = D x (3.3) X U potential energy graient matrix: U = U x X (3.4)

10 64 Anrzej Grzyb, Karol Bryk, Marek Dzik graient operator In the analyze case it refers to ifferentiation by geometrical coorinates or geometrical velocities: x x x i T (3.5) F external forces vector After the calclations the matrices The matrix an D has similar form to the matrix instea of elasticity coefficients. U take the (3.7) an (3.8) forms. U, however it has attenation factors In each of the matrices, to the elements of the i row or the j colmn there were assigne: i, j xip, yip, zip, xip, yip, zip, xil, yil, zil, xil, yil, zil, xib, yib, zib, xib, yib, zib, xi, yi, zi, xi, yi, zi, xi, yi, zi, xi, yi, zip h h h *k / k k h *k / k3 k4 k / k / h a / / h a / W a h / a h / (3.6) The non zero elements of the matrix i, j (3.7) are:

11 Mathematical Moel for Vibrations Analysis of The Tram Wheelset 65 m,, 3,3 ip J 4,4 xip J 4,5 5,4 xyip J 4,6 6,4 zxip J 5,5 yip J 5,6 6,5 yzip J 6,6 zip m 7,7 8,8 9,9 il J 0,0 xil J 0,,0 xyil J 0,,0 zxil J, yil J, yzil J, zil m 3,3 4,4 5,5 ib =J 6,6 xib J 7,7 yib J 8,8 zib m 9,9 ib m 0,0, i J, xi J 3,3 yi J 4,4 zib = =m 5,5 6,6 7,7 i J 8,8 xi J 9,9 yi J 30,30 zi The non zero elements of the matrix i,j U (3.8) U are:

12 66 Anrzej Grzyb, Karol Bryk, Marek Dzik k, x k H,5 5, x k,6 6, x k,9 9, x,4 4, k x k,5 5, x,30 30, k x k k, y y k H k a,4 4, y y k (p p ) k f,6 6, y y k,4 4, y,6 6, k y (a a ) k,0 0, y k,30 30, y k k 3,3 z z k k 3,4 4,3 z z k (p p ) k f 3,5 5,3 z z k 3,5 5,3 z k 3,6 6,3 z k 3,,3 z 3,,3 k z k 3,7 7,3 z 3,8 8,3 k z 4,4 k z yh k z k a k y x k (p p ) 4,5 5,4 z k f z k H(p p ) k 4,6 6,4 y a f y k 4,0 0,4 x k a 4,4 4,4 y k 4,5 5,4 z 4,6 6,4 kz k a (a a ) y k H 4,0 0,4 k 4,,4 z 4,,4 kz 4,6 6,4 4,7 7,4 z 4,8 8,4 kz 5,5 z x z y 5,6 6,5 x 5,,5 y k H k k (p p ) k H k f k k H 5,5 5,5 z 5,6 6,5 k zf 5,9 9,5 x 5,,5 z 5,,5 kzp 5,4 4,5 k xh 5,5 5,5 x 5,7 7,5 z 5,8 8,5 kzp 5,30 30,5 k xh 6,6 x k k f k H k p k H k p k k yf k z k y 6,,6 z 6,4 4,6 y k (p p ) k f k f (a a ) 6,6 6,6 y k 6,9 9,6 x k p 6,0 0,6 y 6,4 4,6 k x k 6,5 5,6 x k p 6,6 6,6 y 6,30 30,6 k x k 7,7 x k H 7,,7 x k 7,,7 x k 7,9 9,7 x 7,4 4,7 x k 7,5 5,7 x 7,30 30,7 k x k k 8,8 y y 8,0 0,8 y y 8,,8 y y 8,4 4,8 y 8,6 6,8 y 8,0 0,8 y 8,30 30,8 y 9,9 z z k k H k a k ( p p ) k f k k (a a ) k k k k 9,0 0,9 k z k z k ( p p ) k f 9,,9 z z k 9,5 5,9 z 9,6 6,9 kz k 9,,9 z 9,,9 kz

13 Mathematical Moel for Vibrations Analysis of The Tram Wheelset 67 k 9,7 7,9 z 9,8 8,9 kz 0,0 k z yh k z k a k y x k (p p ) k f 0,,0 z z k H(p p ) k a f 0,,0 y y k a 0,4 4,0 y k 0,5 5,0 z k y a (a a ) 0,6 6,0 k z k H 0,0 0,0 y k 0,,0 z 0,,0 k z k H 0,6 6,0 y k 0,7 7,0 z 0,8 8,0 k z k (p p ) k H, z x k f z k y k H,, x k f,5 5, z,6 6, k zf,9 9, x,, z,, kzp,4 4, k xh,5 5, x,7 7, z,8 8,5 kzp,30 30, k xh, x y k H k p k H k p k k (p p ) k f y k z k f (a a ),6 6, y k,9 9, x k p,0 0, y,4 4, kx k,5 5, x k p,6 6, y,30 30, k x 4,4 y 4,6 6,4 y 5,5 z 9,9 x 0,0 y, z, z 4,4 x 5,5 k k (a a ) k 6,6 kz k y(a a ) k k k k k k k x 6,6 y k 7,7 z k 8,8 z k 30,30 x k = k + k = + k + k -k 3 = k - + k + k -k 4 = -k + + (3.9) (3.0) xib yib zib xib yib zib xip xil yip yil F 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, F, F, F, M,M, M,(F F ), ( F F ), ( F F ), ( F s F s F r F r), (F r F r),( F s F s),(f F ),( F F ), ( F zip zil zip zil yip yil xip xil xip xil xip xil yip yil T zip F zil),( FziP s FziL s FyiP r FyiL r),(fxip r FxiL r),( FxiPs FxiLs) (3.)

14 68 Anrzej Grzyb, Karol Bryk, Marek Dzik The information on kinematic excitations, which may be the basis to complete the elements of the F matrix, can be se on the basis of the materials containe in the paper []. 4. Smmary In the paper there was presente the mathematical moel of 05Na tram trolley sing the Lagrange II type eqations' matrix form. For this prpose there were etermine bons occrring in the analyze moel. There were also set kinematic energy, potential energy an energy issipation fnction eqations, which serve to create wante mathematical moel's eqations. These eqations after connecting with box's or torsion pivot's eqations an with eqations moeling the wheel-rail contact forces can be written after transforming to the set of ifferential eqations form an then can be se to the analysis of small vibrations an stabilization of the vehicle's movement in mathematical programs sch as Matlab or Mathematica. References. Gąsowski W., Marciniak Z.: Constrction an moels of trolleys an sspension systems for wagons an locomotives for high-spee rail. Poznan University of Technology Pblishing Hose, Poznan Grzyb A., Czaerna T.: Research an analysis of trams' kinematic vibration excitations. TTS Rail Transport Technology, no 9, 64 68, Leitner R.: Overview of higher mathematics for stents. Scientific an Technical Pblishing Hose, Warsaw Leyko J.: General Mechanics. PWN Scientific Pblishing Hose, Warsaw Sobczyk T.J.: Methoical aspects of inction machines' mathematical moeling. Scientific an Technical Pblishing Hose, Warsaw 004.

2.13 Variation and Linearisation of Kinematic Tensors

2.13 Variation and Linearisation of Kinematic Tensors Section.3.3 Variation an Linearisation of Kinematic ensors.3. he Variation of Kinematic ensors he Variation In this section is reviewe the concept of the variation, introce in Part I, 8.5. he variation