Solving the nonlinear pendulum equation with friction and drag forces using the Finite Element Method


  • Youssef Hafez Nile Research Institute, Cairo, Egypt


Nonlinear pendulum, damped oscillations, Galerkin’s Finite Element Method, Maclaurin’s Power Series, incremental boundary conditions technique, friction and form drag resistance forces


Pendulum phenomenon plays an important, role in applied mathematics, earthquake engineering, vibrations mechanics and physics due to its dynamical nonlinear nature. The nonlinear pendulum governing differential-equation is numerically solved herein using the Finite Element Method for the first time. To overcome the nonlinearity resulting from the sine term, Maclaurin’s power-series expansion for the sine function with ten terms (odd powers up to the 19th power) is substituted into the differential equation. The nonlinearity is then shifted from the sine term to terms of powers of the angular displacement. The original initial value pendulum problem is converted into a two-point boundary value problem suitable for the Finite Element Method. This is possible owing to a power series solution that determines an explicit expression of the period of motion in terms of the initial displacement angle (amplitude). In this way the time of motion which is given by the period of motion is considered as a space domain. The periodic nature of the pendulum problem allows specifying the amplitude value at the two ends of the time domain (now becomes the space domain) as the two end boundary conditions. An incremental boundary condition technique and the Newton-Raphson method are used to solve the nonlinear system of equations. The solution is applicable up to maximum amplitude of 179º. The differential form of the pendulum equation facilities investigating damped motion by including the frictional and form drag resistance forces to the differential equation of the pendulum motion in fluids such as air and water. The ability to solve the differential equation form of the nonlinear pendulum allows ease in dealing with tangential forces to the pendulum bob’s direction of motion such as fluid resistance and electromagnetic forces. Comparison with existing analytical solution shows that the present approach succeeded in predicting the pendulum motion profile under damped conditions. The approach developed here could be used to investigate pendulum problems where electromagnetic forces exist.


HAFEZ, Y., Complete analysis of the nonlinear pendulum for amplitudes in all regimes using numerical integration, International Journal of Numerical Methods and Applications, 14, 1, pp. 53-76, 2015.

EDWARDS, C.H., PENNY, D.E., Elementary differential equations with boundary value problems, Sixth Edition, Pearson International Edition, pp. 137, 533-537, 2009.

FULCHER, L.P., DAVIS, B.F., Theoretical and experimental study of the motion of the simple pendulum, Am. J. Phys., 44, 1, pp. 51-55, 1976.

BENACKA, J., Power series solution to the pendulum equation, International Journal of Mathematical Education in Science and Technology, 40, 8, pp. 1109-1117, 2009.

BELENDEZ, A., PASCUAL, C., MENDEZ, D.I., BELENDEZ, T., NEIPP, C., Exact solution for the nonlinear pendulum, Revista Brasileira de Ensino de Fisica, 29, 4, pp. 645-648, 2007.

OCHS, K., A comprehensive analytical solution of the nonlinear pendulum, Eur. J. Phys., 32, 2, pp. 479-490, 2011.

BELENDEZ, A., ARRIBAS, E., ORTUNO, M., GALLEGO, S., MARQUEZ, A., PASCUAL, I., Approximate solutions for the nonlinear pendulum equation using a rational harmonic representation, Computers and Mathematics with Applications, 64, 6, pp. 1602-1611, 2012.

SINGH, I., ARUN, P., LIMA, F., Fourier analysis of nonlinear pendulum oscillation, Revista Brasileira de Ensino de Fisica, 40, 1, e1305, 2018.

THOMPSON, E.G., Introduction to the finite element method, theory, programming, and applications, fifth ed. John Wiley &Sons, Inc., 2005.

BURDEN, R.L., FAIRES, J.D., Numerical Analysis, PWS Publishers, Third Edition, pp.184-188, 1985.

BUTIKOV, I. E, The rigid pendulum- An antique but evergreen physical model, European Journal of Physics, 20, 6, 2003.

NRICh maths, Considering the simple pendulum with friction,

SWOKOWSKI, E.W., OLINICK, M., COLE, J.A., Calculus, Sixth Edition, PWS publishing company, Boston, USA, 1994.