Advancing Modal Analysis of Mechanical Linkages via Differential Algebraic Equations

Year 2024, EARLY VIEW, 1 - 1

Can Ulaş Doğruer , Can Barış Toprak , Bora Yıldırım

Abstract

In this study, we delve into the modal analysis of mechanical linkages across arbitrary configurations. Leveraging linear elasto-dynamic modeling, we successfully decouple rigid body motion from the flexible motion associated with pseudo-dynamic equilibrium configurations. These equilibrium configurations are determined through the solution of nonlinear differential equations. Our analytical investigation reveals that the modal analysis of a mechanical linkage at arbitrary configurations is intricately tied to the position parameters of the linkage. Mode shapes and the spectrum of natural frequencies dynamically evolve as the mechanical linkage traverses space. To validate our findings, we present numerical results for a slider-crank mechanism, which are further corroborated through analytical simulations.

Keywords

Coordinate partitioning, Embedding technique, Kinematic chain, Modal analysis

Diferansiyel Denklemler Yoluyla Mekanik Bağlantıların Modal Analizinin Geliştirilmesi

Abstract

Bu çalışmada, farklı konfigürasyonlara sahip mekanik bağlantıların modal analizi derinlemesine incelenmiştir. Doğrusal elasto-dinamik modellemeden yararlanarak, katı cisim hareketi, dinamik denge konfigürasyonlarıyla ilişkili esnek hareketten ayrıştırılmıştır. Bu denge konfigürasyonları doğrusal olmayan diferansiyel denklemlerin çözümü yoluyla belirlenmiştir. Analitik yöntemlerle farklı konfigürasyonlardaki mekanik bağlantının modal analizinin, bağlantının konum parametrelerine karmaşık bir şekilde bağlı olduğu ortaya konmuştur. Buna göre mod şekilleri ve doğal frekansların spektrumu, mekanik bağlantıların farklı konfigürasyonlarında dinamik olarak değişmektedir. Sonuçların doğrulanması için, simülasyonlarla desteklenen krank - biyel mekanizmasının sayısal sonuçları sunulmuştur.

Keywords

Yerleştirme tekniği, Koordinat bölümlendirme, modal analiz, Kinematik zincir

References

• [1] Shabana, A., and R. A. Wehage. "Variable degree-of-freedom component mode analysis of inertia variant flexible mechanical systems." J. Mech., Trans., and Automation (1983).
• [2] Erdemir F. And Özkan M. T., "Modal Analysis of a Snap-fits Joint Model for Plastic Parts," Journal of Polytechnic-Politeknik Dergisi , 22(4): (2019).
• [3] Tekeci, U., & Yildirim, B. (n.d.). “Predicting Fatigue Life of A Mount of A Device with Shock Absorber.” Journal of Polytechnic -Politeknik Dergisi (2023).
• [4] Ward H., Lammens S., and Sas P. “Modal analysis theory and testing”. Katholieke Universiteit Leuven, Belgium 200(7): (1997).
• [5] Javier L.M., et al. "Design and analysis of a flexible linkage for robot safe operation in collaborative scenarios." Mechanism and Machine Theory, (2015).
• [6] Dupac M., and Beale D. "Dynamic analysis of a flexible linkage mechanism with cracks and clearance." Mechanism and Machine Theory 45(12): (2010).
• [7] Zhang, Xianmin, and Erdman A. "Dynamic responses of flexible linkage mechanisms with viscoelastic constrained layer damping treatment." Computers & Structures 79(13): (2001).
• [8] Liou, Frank W., and Erdman A. "Analysis of a high-speed flexible four-bar linkage: Part I—formulation and solution." J. Vib., Acoust., Stress, and Reliab. (1989).
• [9] Sayahkarajy, Mostafa, Mohamed Z., and Faudzi A. "Review of modelling and control of flexible-link manipulators." Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering 230(8): (2016).
• [10] Wang, Xiaoyun, and Mills J. "Experimental modal analysis of flexible linkages in a smart parallel platform." Proceeding of the 7th Cansmart Meeting-International Workshop on Smart Materials and Structures. (2004).
• [11] Li, Haihong, Yang Z., and Huang T. "Dynamics and elasto-dynamics optimization of a 2-DOF planar parallel pick-and-place robot with flexible links." Structural and Multidisciplinary Optimization 38: (2009).
• [12] Saadé, C., et al. "Space-time isogeometric analysis for linear and non-linear elastodynamics." Computers & Structures 254: (2021).
• [13] Idesman, Alexander. "Accurate time integration of linear elastodynamics problems." CMES-Computer Modeling in Engineering and Sciences 71(2): (2011) .
• [14] Andersen, L. "Linear elastodynamic analysis." (2006).
• [15] Burel, Aliénor, Imperiale S., and Joly P. "Solving the homogeneous isotropic linear elastodynamics equations using potentials and finite elements. the case of the rigid boundary condition." Numerical Analysis and Applications 5: (2012).
• [16] Gulseven H. C. And Özdemir V., "Design and analysis of four cylinder diesel engine balancer," Journal of Polytechnic-Politeknik Dergisi , (2022).
• [17] Rizky A. et al. "Experimental investigation of vibration response of a flexible coupler in a four bar mechanism due to varying crank length and crank speed." MATEC Web of Conferences. 248: (2018).
• [18] Erkaya, Selçuk, and Uzmay İ. "Experimental investigation of joint clearance effects on the dynamics of a slider-crank mechanism." Multibody system dynamics 24:(2010).
• [19] Chu, S-C., and K. C. Pan. "Dynamic response of a high-speed slider-crank mechanism with an elastic connecting rod." (1975).
• [20] Khemili, Imed, and Romdhane L. "Dynamic analysis of a flexible slider–crank mechanism with clearance." European Journal of Mechanics-A.Solids 27(5): (2008).
• [21] Wang X., Mills J. K. and Guo S., "Experimental Identification and Active Control of Configuration Dependent Linkage Vibration in a Planar Parallel Robot," in IEEE Transactions on Control Systems Technology, 17(4): (2009).
• [22] Mehta, Manish, George P. M., and Patolia H. P. "Effect of Rocker Length on the Dynamic Behavior of a Coupler Link in Four Bar Planar Mechanism." Procedia Technology 14: (2014).
• [23] Palmieri, G., et al. "Configuration-dependent modal analysis of a Cartesian parallel kinematics manipulator: numerical modeling and experimental validation." Meccanica 49: (2014).
• [24] Wang, Xiaoyun, and Mills J.K. "Experimental modal identification of configuration-dependent vibration using smart material transducers with application to a planar parallel robot." 2005 IEEE International Conference on Robotics and Biomimetics-ROBIO. (2005).
• [25] bdeslam A. "Kineto-elastodynamic analysis of high-speed four-bar mechanism." PhD thesis (1997).
• [26] Midha A. , Karam, D, & Thompson, B.S. "The Elastic Slider-Crank Mechanism: A Study of the Intrinsic Configuration-Dependent Modal Properties." Proceedings of the ASME 1992 Design Technical Conferences. (1992).
• [27] Eberhard P. and Schiehlen W. Computational Dynamics of Multibody Systems: History, Formalisms, and Applications, Journal of Computational and Nonlinear Dynamics 1(1): (2005) .
• [28] Golebiewski, Eugene P., and Sadler J. "Analytical and experimental investigation of elastic slider-crank mechanisms." J. Eng. Ind. (1976).
• [29] Matekar, S. B., and Fulambarkar A. M. "Displacement analysis of slider in slider-crank mechanism with joint clearance." Australian Journal of Mechanical Engineering 20(4): (2022).
• [30] Khemili, Imed, and Romdhane L. "Dynamic analysis of a flexible slider–crank mechanism with clearance." European Journal of Mechanics-A/Solids 27.5 (2008).
• [31] Chai, Kai, Lou J., and Yang Y.. "Mechanical Performance Analysis and Experimental Study of Four-Star-Type Crank-Linkage Mechanism." Applied Sciences 13(14): (2023).
• [32] Bonisoli, Elvio, Lisitano D. , and Dimauro L. "Detection of critical mode-shapes in flexible multibody system dynamics: The case study of a racing motorcycle." Mechanical Systems and Signal Processing 180: (2022).
• [33] Meirovitch L. “Computational Methods in Structural Dynamics.” Springer Science & Business Media, (1980).
• [34] Shabana A.A. “Computational dynamics.” John Wiley & Sons, (2009).
• [35] Geradin M. and Rixen D. J., “Mechanical vibrations: theory and application to structural dynamics.” John Wiley & Sons, (2014).
• [36] A. Palazzolo, Vibration Theory and Applications with Finite Elements and Active Vibration Control. John Wiley & Sons, (2016).
• [37] L. Meirovitch, Computational Methods in Structural Dynamics. Springer Science & Business Media, (1980).
• [38] Catbas, F. N., et al. "A correlation function for spatial locations of scaled mode shapes-(COMEF)." Society for Experimental Mechanics, Inc, 16 th International Modal Analysis Conference.(1998).