A Time Integration Scheme for Dynamic Problems

A Time Integration Scheme for Dynamic Problems A Thesis Submitted In Partial Fulfillment of the Requirements for the Degree of Master of Technology by Sandeep Kumar Roll No. 134103123 to the DEPARTMENT OF MECHANICAL ENGINEERING INDIAN INSTITUTE OF TECHNOLOGY GUWAHATI May, 2015 CERTIFICATE It is certified that the work contained in the thesis entitled “A Time Integration Scheme for Dynamic Problems”, by “ Mr. Sandeep Kumar” (Roll No. 134103123), has been carrie d out under my supervision and that this work has not been submitted elsewhere for a degree. Dr. S. S. Gauta m May, 2015. Department of Mecha nical Engineering, I.I.T. Guwahati. Dedicated to My Parents and to Dr. Pankaj Biswas and Dr. Rashmi Ranjan Das Teachers and Friends Acknowledgement The most important lesson I have learned during the course of my work is that failures are part of life and they are the best teachers who can guide one to success. Any work of this stature has to have contributions of many people. During the course of this work, I have been supported by many people. First of all, I would like to express my gratitude to thesis supervisor, Dr. S. S. Gautam, for his guidance in completing the first phase of my project. The technical and personal lessons tha t I learned by working under him are now foundation pillars for the rest of my life. I am specially grateful to Prof. Pankaj Biswas for his support, encouragement and inspiring advices which will guide me all my life. I also extend my gratitude to Prof. Debabrata Chakraborty, Prof. A. K. De, Prof. Karuna Ka lita, Prof. Poonam kumari, Prof. G. Madhusudhana, Prof. K. S. R. Krishna Murthy, Prof. Deepak Sharma a nd all other faculty members of the Department of Mecha nical Engineering for imparting me knowledge of various subjects and helping me at the time of difficulty in solving any problem. I am grateful to Prof. Trupti Ranjan Mahapatra, Prof. Rashmi Ranjan Das, Prof. A. K. Sahoo of KIIT University, Bhubaneswar and Prof. Subrata Panda of NIT Rourkela for their motivation and support. I am thankful to my parents, Shri A. Mohan Rao and Smt. A. Sarita, for providing me support and encouragement at every step of my life. I am also thankful to my seniors, Dipendra Kumar Roy, Vinay Mishra, Sibananda Mohanty, Manish Kumar Dubey, Sunil Kumar Singh, Debabrata Gayen, Susanta Behera and Parag Kamal Talukdar. Further, I am also thankful to all my friends at IIT Guwahati - Sandeep Kumar, Ashish Gajbhiye, Ashish Rajak, Nishiket Pandey, Soumya Ranjan Nanda, Anurag Mishra, Nikhil Sharma, Abhishek Yadav, Sateesh Kumar, Vivek Badhe and Susobhan Patra. Finally, I express my thanks to all those who have helped me directly or indirectly for successful completion of this work. Sandeep Kumar IIT Guwahati May, 2015 i Conference Publications • S. Kumar and S. S. Gautam, Extension of A Composite Tim e Integration Scheme for Dynamic Problems, Indian National Conference on Applied Mechanics (INCAM 2015), July 13-15, 2015, New Delhi, India, (acce pted). • S. Kumar and S. S. Gautam, Analysis of A Composite Time Integration Scheme, Indian National Conference on Applied Mec hanics (INCAM 2015), J uly 13-15, 2015, New Delhi, India, (accepted). Contents List of Figures xiii List of Tables xv Nomenclature xvi 1 Introduction 1 1.1 Need for Direct Time Integration Sc hemes . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Mode Superposition Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.2 Direct Time Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Objectives of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Structure of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Review of Direct Time Integra tion Schemes 4 2.1 Classification of Direct Time Integration Schemes . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Classification of Collocation-Based Time Integration Schemes . . . . . . . . . . . . . . . 5 2.2.1 Explicit Time Integration Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2.2 Implicit Time Integration Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2.2.1 Literature Review on Implicit Time Integration Schemes . . . . . . . . . 8 2.2.2.2 Details of Some Implicit Time Integration Schemes . . . . . . . . . . . . 10 2.2.3 Selection of E xplicit or Implicit S cheme . . . . . . . . . . . . . . . . . . . . . . . . 17 3 Proposed Time Integration Scheme 18 4 Analysis of Proposed Time Integration Scheme 22 4.1 Characteristics of Time Integration Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4.1.1 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4.1.2 Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 4.1.2.1 Amplitude Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.1.2.2 Period Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.1.3 Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.2 Stability and Accuracy Analysis of the Proposed Scheme . . . . . . . . . . . . . . . . . . 26 4.2.1 Amplification Matrix for the Proposed Scheme . . . . . . . . . . . . . . . . . . . . 26 5 Results and Discussi on 35 5.1 Numerical example: Flexible Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 5.2 Numerical example: Stiff Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 xi 6 Conclusions and Scope for the Future Work 5 2 6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 6.2 Scope of the Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 References 56 xii List of Figures 3.1 Proposed Composite Scheme. The time step is denoted by t n + 1 − t n = h. . . . . . . . . 18 4.1 Variation of spectral radii, amplitude error and period error for γ t = 0.2. . . . . . . . . . 30 4.2 Variation of spectral radii, amplitude error and period error for γ t = 0.4. . . . . . . . . . 31 4.3 Variation of spectral radii, amplitude error and period error for γ t = 0.5. . . . . . . . . . 32 4.4 Variation of spectral radii, amplitude error and period error for γ t = 0.6. . . . . . . . . . 33 4.5 Variation of spectral radii, amplitude error and period error for γ t = 0.8. . . . . . . . . . 34 5.1 Flexible pendulum. Data and initial conditions. . . . . . . . . . . . . . . . . . . . . . . . 36 5.2 Variation of energy-momentum with time for h = 0.01 s and γ t = 0.2. . . . . . . . . . . 37 5.3 Variation of energy-momentum with time for h = 0.01 s and γ t = 0.5. . . . . . . . . . . 37 5.4 Variation of energy-momentum with time for h = 0.01 s and γ t = 0.9. . . . . . . . . . . 38 5.5 Variation of energy-momentum with time for h = 0.05 s and γ t = 0.2. . . . . . . . . . . 38 5.6 Variation of energy-momentum with time for h = 0.05 s and γ t = 0.5. . . . . . . . . . . 39 5.7 Variation of energy-momentum with time for h = 0.05 s and γ t = 0.9. . . . . . . . . . . 39 5.8 Variation of energy-momentum with time for h = 0.0001 s and γ t = 0.2. . . . . . . . . . 40 5.9 Variation of energy-momentum with time for h = 0.0001 s and γ t = 0.5. . . . . . . . . . 40 5.10 Variation of energy-momentum with time for h = 0.0001 s and γ t = 0.9. . . . . . . . . . 40 5.11 Variation of trajectory of the pendulum for h = 0.01 s. . . . . . . . . . . . . . . . . . . . . 41 5.12 Variation of trajectory of the pendulum for h = 0.05 s. . . . . . . . . . . . . . . . . . . . . 42 5.13 Variation of trajectory of the pendulum for h = 0.0001 s. . . . . . . . . . . . . . . . . . . 43 5.14 Variation of strain with time for h = 0.0001 s. . . . . . . . . . . . . . . . . . . . . . . . . . 44 5.15 Variation of strain with time for h = 0.01 s. . . . . . . . . . . . . . . . . . . . . . . . . . . 45 5.16 Variation of energy-momentum with time for h = 0.1 s and γ t = 0.2. . . . . . . . . . . . 45 5.17 Variation of energy-momentum with time for h = 0.1 s and γ t = 0.5. . . . . . . . . . . . 46 5.18 Variation of energy-momentum with time for h = 0.1 s and γ t = 0.9. . . . . . . . . . . . 46 xiii 5.19 Variation of energy-momentum with time for h = 0.0001 s and γ t = 0.2. . . . . . . . . . 46 5.20 Variation of energy-momentum with time for h = 0.0001 s and γ t = 0.5. . . . . . . . . . 47 5.21 Variation of energy-momentum with time for h = 0.0001 s and γ t = 0.9. . . . . . . . . . 47 5.22 Variation of axial strain with time for h = 0.0001 s. . . . . . . . . . . . . . . . . . . . . . . 48 5.23 Variation of axial strain with time for h = 0.1 s. . . . . . . . . . . . . . . . . . . . . . . . . 49 5.24 Variation of trajectory of the pendulum for h = 0.0001 s. . . . . . . . . . . . . . . . . . . 50 5.25 Variation of trajectory of the pendulum for h = 0.1 s. . . . . . . . . . . . . . . . . . . . . 51 xiv List of Tables 4.1 Newmark parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 xv [...]... proposed scheme Zhou and Zhou [36] proposed an implicit time integration scheme which has two control parameters to vary the accuracy To capture the high oscillatory modes accurately Liang [37] proposed a time integration scheme where acceleration within a particular time step is assumed to vary in a sinusoidal manner Gholampour et al [38–40] have proposed an unconditionally stable time integration scheme. .. earliest work on the spectral stability and accuracy analysis of direct time integration schemes has been done by Bathe and Wilson [52] Also, Bathe [2] has discussed the stability and accuracy characteristics of several direct time integration schemes (both implicit and explicit time integration schemes) In linear dynamic analysis, the spectral stability is sufficient condition for unconditional stability...Nomenclature Latin Symbols h u ˙ u ¨ u Time step size Displacement Velocity Acceleration Greek Symbols α Parameter for Chung and Hulbert (Generalised-α) scheme αg Parameter for Gohlampour composite scheme β γ Parameter for Newmark scheme Parameter for Newmark scheme γt θ Time step ratio for the proposed scheme Parameter for Wilson-θ scheme λ ξ Eigenvalue Modal damping ratio φ ω Mode shape Vibration... (2.13) (ii) Newmark Scheme: Newmark Scheme [4] is the most widely used family of direct time integration schemes This scheme can be used as a single-step or a multi-step algorithm For a single-step three-stage algorithm, un , un , and un have to be calculated at each time step For a single-step ˙ ˙ two-stage algorithm, un , and un have to be calculated at each time step It can be also used as a ˙ predictor-corrector... chapter, first, some properties that a time integration scheme should possess i.e., stability, accuracy, and high frequency damping are discussed Then, the stability and accuracy characteristics of the proposed scheme, presented in Chapter 3, is studied for various values of parameters 4.1 Characteristics of Time Integration Schemes The trapezoidal scheme is unconditionally stable for the linear dynamic. .. contact-impact as in for e.g., automobile crash simulation, design of landing gears for airplanes and space crafts, tire wear simulation or adhesion simulation These problems are mostly solved by direct time integration of the equations of motion Direct time integration schemes are considered as the only general methods to calculate the response of dynamic systems under any arbitrary loading They are... acceleration during a time interval h 2 Satisfy the equation of motion at constant time interval h to maintain static equilibrium between the inertia, the damping and the restoring forces and the applied dynamic loading at multiples of the time step h 2.1 Classification of Direct Time Integration Schemes Traditionally, the direct time integration schemes for the nonlinear equation of motion have been... Newmark scheme, algorithmic damping is introduced [29] • For different values of β and γ different schemes are obtained Some are given below Constant(average) acceleration scheme: For β = 1 4 and γ = 1 2, Newmark scheme gives average acceleration scheme, which is implicit, second order-accurate and unconditionally stable 10 Linear acceleration scheme: For β = 1 6 and γ = 1 2, Newmark scheme yields linear... non-dissipative time integration scheme The method combines the trapezoidal rule and the three-point backward Euler scheme to yield a composite scheme for numerical integration of nonlinear dynamical system of equations The method, unlike for e.g., Newmark scheme, has no parameter to choose or adjust The method is shown to be second order accurate and remains stable for large deformation and long time response... numerical dissipations are also large Klarmann and Wagner [48] have further analyzed the Bathe composite scheme for variable step sizes and have shown that at a particular value of the step size the period elongation is minimum and the numerical dissipation is maximum 9 2.2.2.2 Details of Some Implicit Time Integration Schemes In the current section, details of some implicit time integration schemes . also thankful to my seniors, Dipendra Kumar Roy, Vinay Mishra, Sibananda Mohanty, Manish Kumar Dubey, Sunil Kumar Singh, Debabrata Gayen, Susanta Behera and Parag Kamal Talukdar. Further, I am. I am also thankful to all my friends at IIT Guwahati - Sandeep Kumar, Ashish Gajbhiye, Ashish Rajak, Nishiket Pandey, Soumya Ranjan Nanda, Anurag Mishra, Nikhil Sharma, Abhishek Yadav, Sateesh. shown that for too large time step, the scheme remains sta b le but numerical dissipations are also large. Klarmann and Wagner [48] have further analyzed the Bathe composite scheme for variable