Authors:
Rabab Jarrar,Rabia Safdar,Noorhan F. AlShaikh Mohammad,Olivia Florea,Jihad Asad,DOI NO:
https://doi.org/10.26782/jmcms.2025.02.00001Keywords:
Mass-Spring System,Lagrangian,Euler-Lagrange equation,Laplace Transformation,Simulation,MATLAB Software,Abstract
In this research, we study the dynamical behaviors of a mass-spring system on a massless moving cart. The Lagrangian of the system was first constructed, which resulted in obtaining the Euler-Lagrange equation (ELE) of the system. As a next step, we used the Laplace transformation technique to attain an exact solution for ELE of the system. Furthermore, numerical and simulation techniques were applied with the help of MATLAB software, where we solved ELE numerically for some specified initial conditions. Simulation results indicate that they are in good agreement with the exact analytical solution. Finally, some simulation results were presented in this research.Refference:
I. Abergel, F., et al. “Differential Equations.” Sovremennye Problemy Matematiki, vol. 5, VINITI, Moscow, 1989.
II. Abdullah, M., A. R. Butt, and N. Raza. “Heat Transfer Analysis of Walters’-B Fluid with Newtonian Heating through an Oscillating Vertical Plate by Using Fractional Caputo–Fabrizio Derivatives.” Mechanics of Time-Dependent Materials, vol. 23, no. 2, 2019, pp. 133–151.
III. Alkhader, Taqwa, Dilip K. Maiti, Tapas Roy, Olivia Florea, and Jihad Asad. “Time Dependent Harmonic Oscillator via OM-HPM.” Journal of Computational Applied Mechanics, vol. 56, no. 1, 2025, pp. 264–275.
IV. Ali, A., et al. “Magnetohydrodynamic Oscillating and Rotating Flows of Maxwell Electrically Conducting Fluids in a Porous Plane.” Punjab University Journal of Mathematics, vol. 50, no. 4, 2020, pp. 61–71.
V. Bajkowski, J. M., and R. Zalewski. “Transient Response Analysis of a Steel Beam with Vacuum Packed Particles.” Mechanics Research Communications, vol. 60, 2014, pp. 1–6.
VI. Brenner, J. L. Problems in Differential Equations. Dover Publications, 2013.
VII. Brown, E., et al. “Universal Robotic Gripper Based on the Jamming of Granular Material.” Proceedings of the National Academy of Sciences of the United States of America, vol. 107, 2010, pp. 18809–18814.
VIII. Fowles, G. R., and G. L. Cassiday. Analytical Mechanics. Thomson Brooks/Cole, 2005.
IX. Goldstein, H., C. P. Poole, and J. Safko. Classical Mechanics. Vol. 2, Addison-Wesley, 1950.
X. Hand, L. N., and J. D. Finch. Analytical Mechanics. Cambridge University Press, 1998.
XI. Imran, M., et al. “The Solutions of Non-Integer Order Burgers’ Fluid Flowing through a Round Channel with Semi-Analytical Technique.” Symmetry, vol. 11, no. 8, 2019, p. 962.
XII. Landau, L. D., and E. M. Lifshitz. Mechanics: Volume 1. Butterworth-Heinemann, 1976.
XIII. Makowski, M., and L. Knap. “Reduction of Wheel Force Variations with Magnetorheological Devices.” Journal of Vibration and Control, vol. 20, no. 10, 2014, pp. 1552–1564.
XIV. Makowski, M., and R. Zalewski. “Vibration Analysis for Vehicle with Vacuum Packed Particles Suspension.” Journal of Theoretical and Applied Mechanics, vol. 53, no. 1, 2015, pp. 109–117.
XV. Marion, J. B. Classical Dynamics of Particles and Systems. Academic Press, 2013.
XVI. Sadiq, N., et al. “Analytic and Semi-Analytic Solution for Motion of Fractional Second Grade Fluid in a Circular Cylinder.” Journal of Mathematical Analysis, vol. 9, 2018, pp. 28–47.
XVII. Safdar, R., M. Imran, and C. M. Khalique. “Time-Dependent Flow Model of a Generalized Burgers’ Fluid with Fractional Derivatives through a Cylindrical Domain: An Exact and Numerical Approach.” Results in Physics, vol. 9, 2018, pp. 237–245.
XVIII. Spiegel, M. R. Laplace Transforms. McGraw-Hill, 1965.
XIX. Trench, W. Elementary Differential Equations. 1st ed., Brooks Cole, 1999.
XX. Zalewski, R. “Constitutive Model for Special Granular Structures.” International Journal of Non-Linear Mechanics, vol. 45, no. 3, 2010, pp. 279–285.
XXI. Zalewski, R., and M. Pyrz. “Experimental Study and Modeling of Polymer Granular Structures Submitted to Internal Underpressure.” Mechanics of Materials, vol. 57, 2013, pp. 75–85.
XXII. Zalewski, R., et al. “Dynamic Model for a Magnetorheological Damper.” Applied Mathematical Modelling, vol. 38, no. 9-10, 2014, pp. 2366–2376.
XXIII. Zalewski, R., and T. Szmidt. “Application of Special Granular Structures for Semi-Active Damping of Lateral Beam Vibrations.” Engineering Structures, vol. 65, 2014, pp. 13–20.
XXIV. Zhang, J. P., and J. L. Li. “Application Study on the Laplace Transformation and Its Properties.” Journal of Taiyuan University of Science and Technology, vol. 3, 2011, pp. 1–22.
XXV. Zill, D. G. First Course in Differential Equations: The Classic Fifth Edition.