Structural Dynamics of Roller Coaster Structures: Transient Analysis
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1 Faculty of Engineering Technology, Mechanics of Solids, Surfaces & Systems Structural dynamics of roller coaster structures: transient analysis R.L.J. Mekers M.Sc. Thesis January 2020 Supervisor: dr. ir. J.P. Schilder Document number ET.19/TM-5092 University of Twente P.O. Box 217 7500 AE Enschede The Netherlands Structural dynamics of roller coaster structures: transient analysis Faculty: Engineering Technology Programme: Mechanical Engineering Department: Mechanics of Solids, Surfaces & Systems Research Chair: Structural Dynamics, Acoustics & Control Author: Robert Leonardus Johannes Mekers Date: January 9, 2019 Thesis Committee: Prof. dr. ing. B. Rosic (Chairwoman) Dr. ir. J.P. Schilder (University supervisor) Dr. ir. P.C. Roos (External member) Ir. F. de Ruiter CEng (External member) R.L.J. Mekers, J.P. Schilder Preface During my study in mechanical engineering, I progressively developed more interest in dynamics and finite element applications. After I did an internship at Van Velzen Extern Engineering B.V. (now Intamin Holland B.V.), a company specialized in design and analysis of amusement rides, I was quite sure to do my master thesis about an amusement ride. Actually I am not that big of a fan of amusement rides, but the application of knowledge about subjects I really enjoy can wonderfully be used to design and especially analyse amusement rides. Unfortunately due to personal circumstances I have taken a little too much time for my personal liking to perform this research, but I am very pleased with the final result. Hopefully this research does contribute in making it easier and more predictable to design a roller coaster in the future. I would like to thank Jurnan Schilder for his supervision, without his help and knowledge this research was not possible. We always had very efficient meetings, helping me to get along. Furthermore I would like to thank my family, who stood by my side during tough times. At last I would like to thank Helmer van den Hoorn, who helped me by creating the multibody dynamic model used in the analysis. Rob Mekers Enschede, 3 January 2020 R.L.J. Mekers et al.: Preprint submitted to Elsevier Page 1 of 16 R.L.J. Mekers, J.P. Schilder Contents 1 Introduction 3 2 Moving load application 4 2.1 Equivalent nodal loads/moments........................................5 3 The Finite Element Model 6 3.1 Stiffness matrix.................................................6 3.2 Mass matrix...................................................8 4 Solving the equation of motion 8 5 Test case 10 5.1 Moving load application............................................ 10 5.2 The Finite Element Model........................................... 11 5.3 Solving the equation of motion......................................... 12 6 Conclusions 12 A Nodal loads 12 B Rotation matrix 13 Progress discussion 15 R.L.J. Mekers et al.: Preprint submitted to Elsevier Page 2 of 16 Structural dynamics of roller coaster structures: transient analysis R.L.J. Mekers< (Researcher), J.P. Schilder (Supervisor) University of Twente, Drienerlolaan 5, 7522 NB Enschede, The Netherlands ARTICLEINFO ABSTRACT Keywords: In this paper a method for the analysis of vibrations in roller coaster structures is described. An initial Roller coaster design is required, from which the dimensions, material properties and the time histories of contact Moving loads forces between vehicles and rails have to be retrieved as an input for the Finite Element model. The Structural analysis transient loads applied on the structure are obtained from a multibody dynamic model. The guiding Finite element analysis principle of the Finite Element model is the equation of motion, which is solved iteratively to obtain Transient analysis the displacements of the structure for each load case. The proposed method is applied to an existing roller coaster to clarify the essential steps in the model and to test its capabilities. In the end, a second roller coaster is treated where undesired vibrations showed up, which could potentially have been identified before building it using the method proposed in this paper. The method is developed such that existing Finite Element models and multibody dynamic models of roller coasters can be reused to perform a transient analysis of the structure. 1. Introduction multibody dynamic (MBD) model; a complete roller coaster ride is simulated during one lap. The loads and positions of Roller coasters are designed to be fun and safe at the the wheels are retrieved for each time step, where each time same time. International norms dictate, therefore there are step represents a slightly different position of the train on the restrictions on the amount of G-forces on passengers and track, so each time step can also be seen as a new load case. safety of the train, track and supporting structure. These re- The obtained loads retrieved from the MBD-model are used strictions directly influence the maximum allowable stresses, as external loads in FEM, after which the stiffness and mass often by defining safety factors to multiply the calculated matrix can be determined. stresses with. These safety factors cover the uncertainties The FEM-model is based on beam elements. The ap- present in the stresses. Dynamic effects are considerably propriate local properties are retrieved from CAD-models, responsible for this, as these effects are not modelled pre- from which the stiffness and mass matrix of each element can cisely in industry practice. Hence if these dynamic effects be obtained. These local matrices are rotated to the global can be modelled (more) accurately, safety factors can be re- coordinate system using rotation matrices, after which the duced and roller coasters could be more fun without sacri- stiffness and mass matrices are collected in two uncoupled ficing structural integrity. global matrices. The constraint stiffness and mass matrix are The current state of the art and international norms are obtained after coupling the degrees of freedom by means of tailored to the quasi-static analysis of the structure of roller a boolean matrix. The resulting equation of motion is solved coasters. Earlier papers like [1,2] are assuming the struc- iteratively using the Newmark-Beta integration method. ture to be "sufficiently stiff". The first paper [1] considers a Throughout this paper the reference roller coaster is con- procedure to properly and quickly size the rails, whereas the sidered based on Zierer’s Donderstenen, however it must be second paper [2] presents a tool for the design of support- noted the method proposed is not limited to this track. A ing structure characteristics. With the help of these two pa- schematic overview of the track is shown in Fig. 1. The pers roller coaster structures can be modelled properly on a displacements obtained from the transient analysis are com- global scale, however problems occur when dynamic effects pared with the static solution for each load case to show the are underestimated (Lost Gravity [3]) or when a structure is benefits of this method. designed more lightweight (Karacho). In both roller coasters In the end, a second analysis is performed on a roller tuned mass dampers are installed after opening to minimize coaster where undesired transverse vibrations showed up. the (unexpected) structural vibrations. Therefore, in this pa- This model is inspired by MACK’s Lost Gravity roller coaster per a method is proposed to predict these vibrations before and analysed as a test case for the proposed FEM-model to building the actual roller coasters, so the general design of test its capabilities. roller coasters can be improved. The proposed method in this paper is developed such that In this paper a Finite Element Method (FEM) (now also existing FEM- and MBD-models can be reused as an input readily available) is proposed as a fundamental concept to to the developed transient analysis method. Therefore this solve the equation of motion. Performing a transient analy- method can be thought of as an additional analysis to already sis gives the displacements of the structure while a train is existing methods (e.g. static calculations), with the intention running. to make the method as efficient as possible to limit compu- The representative loads on the track are obtained from a tational time. <Corresponding author [email protected] (R.L.J. Mekers) R.L.J. Mekers et al.: Preprint submitted to Elsevier Page 3 of 16 R.L.J. Mekers, J.P. Schilder Fig. 1: Reference roller coaster track layout, the driving direction is indicated by the arrow. 2. Moving load application A total of ten carts make up for the train, where each cart has four guide wheels and four running wheels, except The equation of motion to be solved is defined as: the front cart, which has eight guide wheels and eight run- ning wheels. Hence in total 88 wheels make contact with the M q̈ + C q̇ + K q = f (1) track (which is rigid) at each time step resulting in 88 loads, c c c consisting of components in orthogonal (y and z) direction x With Mc the constraint mass matrix, Cc the constraint as well as some tangent ( ) components (friction force) ac- damping matrix, Kc the constraint stiffness matrix, q the dis- cording to Fig. 2. The wheels are connected to the track placement vector and f the force vector. using point on curve constraints, from which the position Throughout this paper it is assumed that the damping (parametrized relative to the track) and loads can be retrieved matrix Cc is equal to zero, since it is not required to show for each time step. the advantages of the proposed method and due to the fact The initial position of the train in the MBD-model is that knowledge about the damping parameters of the con- shown in Fig. 3. After positioning the train in the station, sidered roller coaster is lacking.