Bir yolcu vagonunun dinamik tasarımı ve titreşim konferunun analizi üzerine bir yaklaşım
Dynamic design verification and vibratory comfort analysis of a passenger coach by using the lumped and the continious systems models
Yazar: ERDAL ABA
Danışman: DOÇ. DR. VEDAT KARADAĞ
İstanbul Teknik Üniversitesi / Fen Bilimleri Enstitüsü
Derece: Doktora, 1994
Konu: Makine Mühendisliği = Mechanical Engineering
Dizin: Dinamik tasarım = Dynamic design ; Konstrüksiyon = Construction ; Titreşim analizi = Vibration analysis ; Yolcu vagonu = Passenger coach

Özet:
Raylı araç dinamiğinin matematik modellemesindeki gelişmeler, tasarım mühendisine gerçek çalışma şartlarına yakın şartlar altında aracın dinamik davranışını inceleme imkanı verir. Daha henüz ürünün ortada olmadığı imalattan önceki proje tasarım aşamasında, bir yolcu vagonunun araç seyir kalitesini ve yolcu konforu kalitesini optimize edebilmek için değişik etkileri de gözönüne alarak, aracın dinamik davranışını etkileyen sistem parametrelerinin önceden hesaplanmasının amaçlandığı bu çalışmada sistem davranışı hesabı için Modal Analiz-Sonlu Elemanlar kombine yöntemi kullanılmıştır. Süspansiyon tasarımı olarak da isimlendirilen bu çalışmada, aracın düşey titreşimlerini inceleyebilmek için gerçek sistemin özelliklerini en iyi şekilde yansıtacak bir dinamik eşdeğer sistem modeli geliştirilerek, toplu kütleli ve sürekli elastik çubuk sistem modelleri sönümlü ve sönümsüz haller için incelenmiştir. Literatürde belirtilen dinamik tasarım kriterleri ve standartlarda belirtilen konfor değerlerini elde edebilmek için geliştirilen etkin bilgisayar programları ile sistem davranışı hesaplanmış, sistemin fiziksel parametrelerinin sistem davranışı üzerindeki etkileri incelenmiştir. Akslardan harmonik tahrik halinde geçici rejim titreşimleri de dahil sistem davranışı sönümlü ve sönümsüz haller için hesaplanmıştır. Ayrıca istatistiki kuvvetlerle tahrik halinde aracın rastgele sistem davranışı da hesaplanmıştır. Sürekli elastik çubuk sistem modelle- mesi halinde, vagon gövdesi eşit boyda sonlu elemanlara bölünerek, düğüm noktalarının yer değişimleri ve kesit dönmeleri hesaplanmış tır. Bu hal için sistem matrisleri sonlu elemanlar yöntemi yaklaşımı kullanılarak elde edilmiştir. Problemin sönümlü olarak ele alınışın da dış viskoz sönüm elemanları yanısıra, vagon gövdesine kütle ve rijitlik matrisiyle orantılı iç sönüm uygulanmıştır. Boji ve akslar toplu kütleler olarak model lenmiştir. Vagon gövdesi dört adet elastik mesnet üzerinde boji şasilerine ve her boji şasisi dört adet yay ve dört adet amortisör ile iki aksa bağlanır. Hesaplarda ince çubuk etkileri yanısıra, dönme ataleti etkileri de gözönüne alınmıştır. Ayrıca aerodinamik kuvvetlerin sistem davranışı üzerindeki etkileri de incelenmiştir. Vagon gövdesinin birinci eğilme modu doğal frekansı fıalO Hz şartı sağlanmıştır. Hesaplanan ve ölçülen ivme değerleri konfor şartlarını sağlamaktadır. Sistemin doğal frekansları sönüm değerleri ile çok fazla değişmemektedir. Vagon gövdesinin yüksek eğilme ri- jitliğine sahip olması halinde, toplu kütleli sistem modelleri ile sürekli sistem modelleri yaklaşık aynı sonuçları vermektedir. Daha düşük eğilme rijitliği halinde vagon orta noktası ile uç noktalar arasında farklı değerler elde edilmektedir. Boji elastik mesnetleri nin daha yumuşak halleri için vagon gövdesi serbest çubuk gibi davranmaktadır. Aerodinamik etkiler süspansiyon tasarımı açısından ihmal edilebilecek düzeydedir. XIV

Abstract:
In this thesis, the transient response of a passenger coach due to harmonic excitations in finding the optimum solution of the vehicle riding quality and the passenger comfort is analyzed using both the lumped and the continuous vehicle models with damping employing combined Finite Element -Modal Analysis method. Determination of the white noise response due to railway excitation conditions, also, is incorporated into the analysis. Effectiveness of the use of the continuous system models in comparison with the lumped models, particularly, is discussed by using improved railway vehicle models as a new aspect of research on the area. Guidelines for the better vehicle riding quality and the passenger comfort and the dynamic design criteria are established. A real system example of a new generation passenger coach is used in the analysis. The displacements, the velocities and the accelerations of the system are calculated in the case of excitation by harmonic forces. A mean square response analysis of the displacements is carried out in the case of excitation by stochastic forces. Also, the effects of aerodynamic drag forces and crosswinds at large yaw angles on the transient response of the passenger coaches are included into the analysis. The dynamic behaviour of a passenger coach, especially, vertical vibrations, transverse vibrations, wheel-rail contact problems..etc., is a subject of extensive research for an half of century in the literature. The results obtained from the analysis give some explanatory information associated with the vibration character of the vehicle body and the coefficients of the vehicle suspension elements. Advances in the mathematical modelling of the railway vehicle dynamics enables design engineers to simulate vehicle behaviour for a variety of real world conditions. In this thesis, it is aimed to predict variations in design parameters to evaluate the sensitivity of these parameters in the design phase before manufacturing. For this purpose, several computer programs have been developed for vehicle behaviour evaluation both to assist in the design and the development phase as well as evaluate undesired behaviour in testing or in service. In the field of guided transports various criteria must be checked during the design phase of a vehicle in order to implement the requirements of the specifications. Among these criteria, the two of the most important criteria from the vibratory point of view XVare the dynamic behaviour of the body structure and the passenger comfort. For this reason, it is often necessary to develop the specific models and the computer programs to evaluate and solve these problems. Numerous solution models have been developed to estimate the vibratory behaviour of the vehicle body which was represented by a continuous, isotropic uniform beam with a constant cross section. Most of these models were an Euler-Bernoulli beam. Nowadays, the calculations codes by the finite element methods have been already proved highly efficient. In this thesis, the coach body is modeled as a deep beam which is represented by a continuous isotropic uniform beam with a constant cross section which is equivalent of actual system with respect to bending stiffness and mass per length. Both the properties of Euler-Bernoulli beam and the rotary inertia of the beam are incorporated into the analysis. This approach has not been used by researchers in the past due to difficulties to obtain the mathematical solutions. The effects of aerodynamic forces on the vehicle response has been also discussed in the analysis. The system dynamic model is shown below. The seats and the passengers can be modeled as a single degree of freedom system which vibrates independently each other on the vehicle body. The vehicle body is placed on two elastic supports situated at the same level with the bogies. Each bogie frames and axles are considered as a lumped mass. Each bogie frame is linked to the two axles by four springs and four dampers. The springs and the dampers are assumed to be linear. The rail can be modeled as a beam which is supported by the sleepers. The axle-track junction has a stiffness and a damping which represents the presence of load-carrying tires. A,EI.m1,L1.J1 COACH BODY 2c1,2k1 ^tjj 2c1,2kl BOGIE v__ __/ v. ' ?m2 »L 2 »^ 2 AXLE 2c2,2k2 AXLE 2=3. 2k3 BOGIE Dynamic Modelling Of A Passanger Coach. XVIIn the present analysis, the rail effects are taken into account as harmonic forces in the case of deterministic excitation and white noise excitation in the case of stationary stochastic excitation. In the case of usual passenger train, it can be demonstrated very easily that the connection between vehicles, constituted by the screw coupling and the side buffers, have little influence on the vibratory comfort of passengers. In these conditions, it is possible to isolate a coach and to study its vibratory behaviour with regard to the external stresses to which it is subjected. In the case of an articulated set, the high rigidities of the connections at the end of the coach bodies produce tight coupling between their vertical movements, which it is impossible to ignore. In this case, to construct a good model which is sufficiently representative of reality, it is essential to take into account these elasticities which have a great influence on the vertical comfort of passenger. Because of that the whole system in the case of seated coach is to be completely symmetrical about longitudinal axis and according to parameters of inertia, stiffness and damping, it can be possible to uncouple transversal dynamic motions from the vertical dynamic motions. The most important element in the model is the mathematical modeling of the body. For this purpose, a multi degrees of freedom deep beam model is developed. In the present analysis, the vertical bending displacements of the body and the bending slopes of its cross sections, the vertical displacements of the lumped bogies mass and its rotational angles, the vertical displacements of the lumped axle masses are considered as degrees of freedom of the system. In the case of the continuous system model, an example of a complete system (a seat + body + two bogies + four axles).in the 7 finite element solution of the coach body comprises a total of 25 degrees of freedom in undamped case and the dimensions of matrices become 50 in the damped case. In the case of lumped mass model, the complete system comprises of a total of 10 DOF's in the undamped case and the matrices dimensions become 20 in the damped case. The system dynamic equations of motion are derived by using the potential and the kinetic energy equations and the elements of the system matrices are calculated by the Lagrange's equations which have found widespread applications in various field of science and engineering. In the analysis, the coach body is divided by the finite elements of equal length. For each finite element, vertical bending displacements and bending slopes of the coach body and the rotary inertia effects are considered. Thus, the strain energy stored in each of the beam finite element is the energy due to bending, which is given by, XVIIu = ı _1_ 2 I "2 2 EI (_!*) dx y axz (1) Under the effects of aerodynamic drag forces, the coach body is stiffned due to additional stresses created by drag forces. These forces create normal stresses in the cross section of coach body. The strain energy stored in the element due to aerodynamic drag forces is as follows, U = 2 2 f\ ( dV\ A (2) The kinetic energy of the beam finite element consists of the kinetic energy due to the bending deformation and the bending slopes which are expressed as, T = i 2 I i pk (-^-) dx at 2 r a2 2 ply{~d^t] dx (3) In the case of crosswinds at large yaw angles, pressure distribution around the coach body due to crosswinds far from the nose of the train set tend to acquire an approximately constant value. For this reason, the crosswinds effects on the coach body can be modelled as a constant distributed load. Hence, it is taken into account as added-mass effect of the crosswinds in the analysis as follows, T = 3 2 r 2 v dx py (4) o The stiffness and the mass matrices of the beam are determined as in the following form by making use of above equations in FEM processes as, [V EI "3 12 -12 -Bl 6£ 2ZZ Sym. 12 -Bİ + [KKz}={F(t)} (15) The response {z> of eq. (15) consists of two parts: the first part, {z } is the transient response, i.e., the complementary h solution of the equations of motion, and the second part {z } is forced response, i.e., the particular solution of the equations of motion. In the case of undamped systems, The general equations of motion for an undamped mult i degrees of freedom system in matrix form are, [MHz-> + [K]{z}={F(t)} (16) By using a linear transformation together with the initial conditions to replace {z} by {z} = [0Kp> or {p}=[0]_1{z} (17) The uncoupled equations of motion of the system in the modal coordinates are obtained by using the orthogonality condition of the modal vectors in the following form, pj+ (f pt= Nt, (i = 1,2...n) (18) The solutions of eq. (18) are obtained by using the initial conditions and the transient response is found by transforming the solutions from the modal coordinates to the generalized coordinates. In the present study, the eigenvalue problem which appears in the XXcase of modal analysis has been solved by QR method which results in good accuracy in mult i degrees of freedom systems. In the case of damped system, the general equations of motion for a damped mult i degree of freedom system in matrix form are like in eq. (15). We define a vector as follows, {q>={z z}T (19) and use an identity ( [M]{z}-[MHz}=0) to obtain [EKq> + [BHq}={Q(t)} (20) We use a linear transformation together with the initial conditions to replace {q} by {q} = [«/r]{p} or {p} = [i/»]"1{q} (21) The uncoupled equations of motion of the system in the modal coordinates are obtained by using the orthogonality condition of the modal vectors in the following form, İ>i+7l P^FNj. ( i = 1.2 2n) (22) The solution of eq. (22) is obtained by using the initial conditions and the transient response is found by transforming the solutions from the modal coordinates to the generalized coordinates. In the present study, the eigenvalue problem which appears in the case of damped modal analysis has been solved by QR methods. The QR method results in good accuracy in the cases of reasonably high mult i degrees of freedom damped systems. In the case of random vibrations, the excitation is not deterministic but a random process, therefore the question arises as to how to calculate the response of the system. In addition, there remains the question as to how to interpret the results. Hence, a more efficient and more meaningful way of describing the excitation and response random process appears highly desirable in the vehicle dynamic analysis. In this work, a consistent approach given by Meirovitch [1] and Dukkipati [2] are discussed to calculate the mean square response to the white noise excitation associated with two random processes by the modal analysis. The stochastic response of a damped mult i -degree-of -freedom system due to white noise excitation can be obtained by using the modal analysis. But, the difficulty lies in the fact that the modal analysis can not generally be used to uncouple the system of equations. However, in the special case in which the damping matrix is a linear combination of the inertia and the stiffness matrices, the modal matrix associated with the undamped linear system can be used as a linear transformation in uncoupling of the system equations. Similarly, when damping is light, a reasonable approximation can be obtained by simply ignoring the coupling terms in the transformed equation. For simplicity, we shall confine ourselves to the case in which the modal matrix associated with the XXIundamped system can be used as a transformation matrix in uncoupling the set of equations of motion, either exactly or approximately. By using the orthogonality condition of the modal vectors, the set of independent equations of motion in the modal coordinates are obtained in the following form, p'+ 2ÇiUi Pı+ oZ Pi = [0]T {FCt)}, i = 1,2....n (23) To this end, let us consider a vehicle moving at constant speed in order to compute its response due to white noise excitation. The response correlation matrix [R (t)] of the system z between the two response process z (t) and z (t) is given as, T/2 [R (T)]=lim -J- J {z(t)> {z(t+x)}Tdt (24) z 1 T-*» -T/2 The response vectors {z(t)} and {p(t)} are related by Eq. (17). Moreover, we can write, {z(t+x)}T = {p(t+x)}T [<£]T (25) so that, inserting Eqs.(17) and (25) into Eq. (24), and after some elementary manipulations, the mean square value matrix of the system is obtained in matrix form as follows, [RzCO)] = ^ [ ]T[H(w)] dw [ f (26) -CD where, the excitation spectral matrix associated with the actual coordinates [S (a>)] is taken into account as ideal white F noise power spectral density of the wheels-track interaction. Ideal white noise is a uniform power density spectrum, corresponding to a sample function in which all the frequencies are equally represented. In Eq. (26), the matrix [H(w)] is the frequency response function matrix associated with the natural modes. The rth member of [H(w)] is as follows, [H^w)!2 = {(w2 - w2)2+ (2Çiuıco)2>~1, i = l,2..,n (27) [H (w)] is the complex conjugate of the [H(w)]. This procedure can be employed to obtain the response of a multi degrees of freedom system to a random input. In this thesis, the numerical results were calculated associated with the transient response in the case of harmonic excitation in which the axles are subject to sine wave excitation of XXIIamplitude 0.02 m and the stochastic response in the case of white noise excitation. The results obtained from analysis for the lumped and the continuous system models compared with each other in damped and undamped vehicle models cases. A brief survey of the literature shows that the only two of the important dynamic design criteria are mostly used in the design phase of a passenger coach. The first criterion is to design the natural frequency of first bending mode of a passenger coach body as f ı a 10 Hz and the second is the necessity of not exceeding the safe value of acceleration level to satisfy the passenger comfort and the riding quality. Because of that the vertical suspension elements of the vehicle and the bending stiffness of the body had to be designed with care. It is seen that the damping effects have a little influenceon the natural frequencies of the system. Approximately same results were obtained in the cases of the different values of the dampings. For a high value of the coach body bending stiffness EI, the continuous system model gives the same results of the lumped model. It was also calculated that the coach body vibrates like a free-free beam in the case of soft vertical suspension. In addition, it can be said that the bending stiffness of coach body have a greater influence than vertical suspension elements of the vehicle on the natural frequencies of the coach body. It was calculated that aerodynamic drag force effects can be neglected in scope of vehicle suspension design. The lumped and the continuous system models give aproximately same results in the case of high bending stiffness of the coach body. Damping effects have a very little influence on system natural frequencies. XXIII