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<ul><li><p>Acta Mechanica 61, 89--107 (1986) ACTA MECHANICA 9 by Springer-Verlag 1986 </p><p>Chaos in a Railway Bogie </p><p>By </p><p>Ch. Kaas-Petersen, Lyngby, Denmark </p><p>With 10 Figures </p><p>(Received September 13, 1985; revised October ld, 1985) </p><p>Summary </p><p>Railway bogies may perform sustained lateral oscillations -- the hunting motion -- when the speed reaches a certain value. We examine the hunting motion in the complex Cooperrider bogie and find that the wheel flanges have a strong influence on the behaviour. When the bogie has flangeless wheels we have found a symmetry breaking bifurcation, by which we mean, that there is a transition from a symmetric periodic oscillation to an asymmetric periodic oscillation. The periodic motions are examined using the residual map. When the bogie has flanged wheels (the flange is represented as a dead band spring), we do instead find chaotic behaviour confirmed by a positive Liapunov number. The dynamical equations for the bogie model are strongly nonlinear, and we use computer methods to examine the dynamical behaviour. </p><p>1. Introduction </p><p>We examine the dynamical motion of a railway bogie running with constant velocity V along a straight, horizontal and perfect track. The resulting dynamical motion is a coupling between seven basic motions: The lateral motion (i.e. dis- placement transversal to the track) and yaw motion (i.e. rotation around a vertical axis) of the front wheel axle, the lateral and yaw motions of the rear wheel axle, and the lateral, yaw, and roll motions (i.e. rotation around a horizontal axis parallel to the track) of the bogie frame. The equations of motion are nonlinear because of nonl inear wheel-rail contact forces and nonlinear wheel flange-rail forces. I f the motion of the bogie is a periodic oscillation we have the socalled hunting behaviour. We are interested in the asymptot ic motion, i.e. the motion when the transient has died away. </p><p>We state the equations decribing the dynamical motion of the bogie. The equations are due to Cooperrider [1]. The equations are formulated as a system of 14 first order autonomous nonlinear ordinary differential equations. We deter- mine the steady state periodic motion by solving these equations numerically </p></li><li><p>90 Ch. Kaas-Petersen: </p><p>and use the residual map to eliminate the transient. We describe how we define and use the residual map. We next examine the steady state periodic motion in dependence of the velocity V of the bogie. </p><p>When the wheels are flangeless we have found only periodic motions of which some are symmetric and some are asymmetric. These two types of motions have been found to be connected to each other in a symmetry breaking bifurcation. We show this in a solution diagram and we show both types of periodic motions. </p><p>With flanged wheels we have found periodic motions and chaotic motions. A motion is said to be chaotic when the maximal Liapunov number is positive. We describe what the Liapunov number means geometrically and how to determine it. </p><p>Some preliminary results have already been presented. In [2] we have dealt with the Hopf bifurcation where the periodic motion is bifurcating from the stationary solution. In [3], [4] some further results on the periodic and chaotic motions have been given. But because of some errors in the equations we used, we have redone and extended the analysis, the results of which are presented in this paper. </p><p>2. The Bogie Model </p><p>The bogie we shall describe is the complex bogie due to Cooperrider [1]. The bogie consists of three stiff elements: Two wheel axles and a bogie frame. The wheels are rolling on the rails. The wheel axles are connected to the frame with springs and dampers. The frame is connected to the car body with springs and dampers. The car body is moving with constant velocity V along the track. </p><p>We allow 7 degrees of freedom in this model, namely lateral and yaw motion of the front axle, the rear axle, and the frame, and roll motion of the frame. By lateral motion we mean displacement in. a horizontal plane orthogona] to the track direction; by yaw motion we mean rotation in a horizontal plane around a vertical axis; and by roll motion we mean rotation in a vertical plane around a horizontal axis parallel to the track. In a coordinate system moving with the bogie these 7 state variables are assumed to be small and will be denoted ql . . . . . q7; see Fig. 1. We want to examine ql, ..., q7 as functions of time t. In order to do this we for- mulate the dynamical equations using Newton's second law. The mass of the wheel axle is mw ~ 1022 kg and of the frame it is ms -~ 2918 kg. The moment of inertia for the yaw motion of the wheel axle is I~y ~ 678 kg-m 2 and of the frame it is Is~ ~ 6780kg-m ~. The roll motion inertia of the frame is I Ir ~ 6780 kg-m 2. </p><p>The geometry of the bogie is described by the parameters a =- 0.716 m being half the track gauge, b ---- 1.074 m being half the axle distance and the distances dl ~ 0.620 m, d~ ~ 0.680 m, h~ ~- 0.0762 m and h2 -~ 0.6584 m from the springs or dampers to u centre of gravity; see Fig. 1. </p><p>There are three types of forces acting on this bogie: The wheel-rail contact forces, the wheel flange-rail forces and the spring and damper forces, </p></li><li><p>Chaos in a Railway Bogie 91 </p><p>f </p><p>2D~ k 6 2k5 ~I/11/ / / / / / / t l / / / / / I / / / / / / / / / </p><p>V > </p><p>~V </p><p>| </p><p>2d 2 </p><p>~f~111111III//IIf1111I///I/1111///////////////////////////////////////~ </p><p>DI~ k5 :~k6 k5*"~Dt ~ ~k4 </p><p>kz, k , D2 h2 </p><p>ql </p><p>Fig. 1. The bogie model with the notation used. The wheels are shown flangeless </p></li><li><p>92 Ch. Kaas-Petersen: </p><p>The wheels and the rails are in contact and create contact forces - - here we describe how these forces are determined. The wheels and the rails are made of steel with shear modulus G = 8.08. l0 s N/m 2. The wheels are conical with eonicity ~ = 0.05 and the radius r0 = 0.4572 m (when the wheel is centred on the track). The railhead is circular with radius 0.21 m. Hertz theory then tells, that the contact area is elliptical, and we have found the major and minor semi- axes of that ellipse to be 6.578 mm and 3.934 mm respectively, when the vertical load (equal to one eights of the total weight of one railway vehicle i.e. two bogies and a car body) in one contact area is 66 670 N. Then we can use Vermeulen and Johnson's theory [5] to the non-ideal rolling motion. They show, that in the contact area the wheel slides relative to the rail. This sliding is denoted creep and results in creep forces acting on the wheel. I f $, is the creep lateral to the track and ~v is the creep longitudinal to the track, then Vermeulen and Johnson derive, that </p><p>the creep force lateral to the track: Fx ----- - - - - </p><p>the creep force longitudinal to the track : Fy = - - - - </p><p>Here ~:~ is the resulting creep </p><p>~ FR </p><p>~ = V(~/~,) ~ +($~/~)~, </p><p>and we have found T1 : 0.54219 and ~ : 0.60252, cf. [1], [5]. The resulting force F~ is given by </p><p>1 1 u ~-4- u 3 for u 3, /~N </p><p>and here ~ = 0.15 is the friction coefficient, 2t is the vertical toad in the contact: area, and a, b are the semiaxes of the ellipse. (Thus G:~ab-= 6.563 MN and ~N = 10 kN.) </p><p>The flanges of the wheels will contact the rail when the lateral excursion exceeds the clearance 6 = 0.0091 m, and usually we have a two point contact problem [6]. However, Cooperrider [1] reduces this to a one point contact problem by using a stiff spring with a dead band. The flange force FT is then </p><p>[ k0 . (u - -8 ) for 8</p></li><li><p>Chaos in a l~ailway Bogie 93 </p><p>The spring (damper) forces depend linearly on the relative displacements </p><p>(velocities). We have used </p><p>kl = 1.823 MN/m, </p><p>k4 = 0.1823 MN/m, </p><p>D1 = 20.0 kN-s/m, </p><p>k~ = 3.646 M_N/m, </p><p>k~ = 0.3333 MN/m, </p><p>D~ = 29.2 kN-s/m. </p><p>k3 = 3.646 MN/m, </p><p>ks = 2.710 MN.m, </p><p>The governing equations are listed below. (Here a = 0.716 m and b = 1.074 m.) </p><p>m,,(~ + Ax + 2F~f + Fr(ql) = 0 </p><p>I,~,,~ + A~ + 2aNy! = 0 </p><p>m~3 + A~ + 2F~,+ Fr(q3) = 0 </p><p>Iwv~, + A4 + 2aF~ = 0 </p><p>mill5 - - AI - - A2 + As = 0 </p><p>Itv~s - - bA~ + bA~ - - Aa - - A , -+-As =0 </p><p>where </p><p>A1 = 2kt(ql - - q5 -- bqs - - htqT) </p><p>A~ = 2kl(q~ - - qs + bq6 - - hlq,) </p><p>A3 = 2k2dle(q2 - - q6) </p><p>A4 = 2k2d12(q~ -- qs) </p><p>A~ = 2D~(~5 - - h.i~,) + 2~,(q5 - - h~qT) </p><p>A, = ksqs </p><p>A~ = 2Dld2~ + 2ksd2~q~ + 4kadl2q~ </p><p>Fxt, Fvl are creep forces for front axle, and the creep terms are ~e = ~I/V - - q2, </p><p>~v = a~/V + ~q~/ro </p><p>F~,, Fu, are creep forces for rear axle, and the creep terms are ~:x = ~3/V - - q4, </p><p>~, = ~. lV + ~q~/~o </p><p>z~T is the flange force. </p><p>We note in passing, that Cooperrider's simple bogie model [1] - - a two degrees of freedom bogie with lateral and yaw motion of the full bogie as one stiff system - - can be derived from the equations above by the substitution </p><p>(q, q2, q~, q,, qs, qs, qT) --> (q5 + bqs, qs, q5 - - bqs, q~, qs, qs, 0). </p></li><li><p>94: Ch. Kaas-Petersen: </p><p>With the substitution x l = ql, x2 ~ ~[1, xa ~ q~, x4 = ~'2 . . . . we obtain a system of 14 ordinary differential equations (ODEs) of first order. We shall use the velocity V as parameter and take all other geometrical and physical quantities as constants. The dynamical system above can therefore be rewritten on the form </p><p>dx - -=- -2=/ (x ;V) , O</p></li><li><p>Chaos in a Railway Bogie 95 </p><p>d v </p><p>t=o </p><p>v d - </p><p>d 6 </p><p> Fig. 2. A piece o~ a chaotic solution starting in x~ at time t = 0. The chaotic solution remains bounded in the Xl,X2--x3-state space. Nearby solutions (initially the distance away but after a time interval ~ a distance d away from the chaotic solution) are shown. </p><p>The unit vectors v are shown; the vectors w are not shown </p><p>time, where we compute the Vector </p><p>w = [ r x0 + ~v) - r Xo)]l~. </p><p>Since 6 is Small, then w is an approx imat ion to the image of v for the flow l inearized </p><p>around the chaotic solution, w is now renormal ized to a unit vector </p><p>d-= llwll, </p><p>v = w/d , </p><p>where d is the factor with which v has been magnif ied. A f irst approx imat ion to </p><p>the growth rate is therefore </p><p>~u-] = _1 In (d). </p><p>We now take the point ~b(~; x0) as init ial point, and solve the ODEs for a new time </p><p>nterva l of length v, then take a point close to it, ~b(v; x0) -k 8v where v is the </p><p>vector determined after the f irst renormal izat ion, f ind the new vector w, the growth factor d, the vector v and the second approx imat ion to the exponential </p><p>divergence: ~[2] = 1 In (d). And so on and so forth. T </p><p>The largest L iapunov number can then be computed as the averaged value of </p><p>al l the growth rates </p><p>1 ~r </p><p>.Y--*co ~' j= l </p><p>The number of renormal izat ion steps needed depends on the convergence rate of the series, of how close x0 is to the chaotic solution, and the vector v we choose to point in the direct ion of exponent ia l divergence. The renormal izat ion t ime </p></li><li><p>96 Ch. Kaas-Petersen: </p><p>should be short enough to catch the exponent ia l growth, and long enough to avoid cancel lat ion of digits in the computat ions of w. </p><p>A periodic solution will have a ---- 0 within computat ional accuracy, and the </p><p>vector v will converge to the velocity vector of the solution. A quasi periodic </p><p>solution will also have a ~ 0 and the vector v will converge to a vector, tangent </p><p>to the corresponding toms surface in the state space. </p><p>4. Per iod ic Bchav iour </p><p>For a periodic solution we can do much better than comput ing the L iapunov </p><p>number. The characterist ic feature of a periodic solution is that the solution </p><p>repeats itself after a t ime interval T, the period. And since the dynamica l system </p><p>is autonomous, if we know one point of the solution, we can take that point as </p><p>init ial po int a t t ime t ----- 0 and solve the ODEs to generate all of the periodic </p><p>solution. </p><p>..--N </p><p>"/qJffR:u) "~/ </p><p>~ig. 3. Illustration of construction of the residual map Q in R a. In the hypersurface H we take u as an initial point at time t = 0. The solution ~b(t; u) starting in u generates the curve shown. At time t ~ T that curve hits H again. The difference vector q is different from the zero vector so u is not a point on the periodic solution. If H (i.e. x~ and np) and u are chosen suitably, then Newton's method can be used to find a point x/, such that ~b(TR; xi,) ~ xp and xp is then a point on the periodic solution, xiD is not shown on the </p><p>sketch </p><p>We will now introduce (Fig. 3) the residual map Q, which makes use of the feature above. Assume for a moment, that we know xg to be a point close to the periodic solution, and let r~p be a unit vector paral lel to the veloci ty vector / (x~) . Then we </p><p>can define the hypersufface H through xH with np as normal. Let u be a point in H. (We could take u ~ xH.) The solut ion of the ODEs start ing in u will after a certain t ime interval Ti~ (this return t ime will be close to the per iod T) again be </p><p>in H ; the return point is r u). Let q be the difference vector </p><p>q = ~(T~; u) - - u . </p></li><li><p>Chaos in a Railway Bogie 97 </p><p>q is a vector entirely in H, so we can define the residual map Q mapping points in </p><p>H into vectors in H: </p><p>Q: H ~-~ H, Q(u) = q. </p><p>We see, that if xp is a point in H on the periodic solution, then Te '= T, and Q(xp) = ~b(T; xp) -- xp = 0 and thus a periodic solution is a zero point of Q. Note that Q is defined on H, and we have </p><p>H = x~r -~ span {el . . . . . e-in} </p><p>where e~ .. . . . e,_~ is an orthonormal basis orthogonal to np. Another way to represent a point u in H is by the equation </p><p>(~ - x . , ~p) = 0 , </p><p>where (-, .) is the usual euclidean scalar product. Usually we do not know XP, but if q~ is a sufficiently good approximation then </p><p>we can use Newton's method to find xp. We than need the derivative DQ of Q. Now DQ is an (n -- 1) (n -- 1) matrix, where the i 'th element of the ]'th column is given by the expression </p><p>([Q(u ~- ~ei) - Q(u)]/d, @, </p><p>l</p></li><li><p>98 Ch. Kaas-Pe~ersen: </p><p>Continuation methods and stability analysis can be combined to determine critical points or bifurcation points, where the periodic solution changes its stability. Changes in the stability can come about in several ways. If one real eigenvalue lies on the unit circle in the point + 1, we may either have a symmetry breaking bifurcation or a saddle node bifurcation or no periodic solution any more (we are then in a ttopf bifurcation point). If one real eigenvalue lies in --1 we have a period doubling bifurcation. If one pair of complex conjugated eigenvalues lies on the unit circle we have bifurcation to a torus, also denoted a bi-periodie solution. When we have bifurcation to a torus we can determine the stability of the bifurcating hi-periodic solution by the method described in [10]. All these types of bifurcations appear in the bogie model. Further details on the different types of bifurcations can be found in [11]. Bi-periodic solutions can also be reformulated as zero point problems of a certain residual map [10]. However, we have not yet any results for the torus solutions in the bogie model treated in this paper. </p><p>5. Numerical Details </p><p>We have solved the ODEs with the routine LSODA [12] whose most important feature is that it automatically switches between stiff and no...</p></li></ul>