On approach for obtaining approximate solution to highly nonlinear oscillatory system with singularity

Abstract

This paper deals with further investigations of recently introduced so-called low-frequency pendulum mechanism, which represents an extended form of classical pendulum. Exact equation of motion, which is in Eksergian’s form, is a singular and highly nonlinear second order differential equation. It is transformed by suitable choice of a new “coordinates” into classical form of nonlinear conservative oscillator containing only inertial and restoring force terms. Also, due to the singularity of coefficient of governing equation that shows hyperbolic growth, Laurent series expansion was used. Using these, we derived a nonsingular nonlinear differential equation, for which there exists an exact solution in the form of a Jacobi elliptic function. By using this exact solution, and after returning to the original coordinate, both explicit expression for approximate natural period and solution of motion of mechanism were obtained. Comparison between approximate solution and solution is obtained by numerical integration of exact equation shows noticeable agreement. Analysis of impact of mechanism parameters on period is given.

Appendix

Coefficients of Laurent series:

$$ a_{k} = \frac{1}{2{\uppi}}\oint\limits_{{K_{1} }} {\frac{{f\left( \zeta \right){\text{d}}\zeta }}{{\left( {\zeta - a} \right)^{k + 1} }}} ,\quad i = { 1},\; 2, \ldots $$

(A1)

$$ b_{k} = \frac{1}{2{\uppi}}\oint\limits_{{K_{2} }} {\left( {\zeta - a} \right)^{k - 1} } f\left( \zeta \right){\text{d}}\zeta ,\quad i = 1,\; 2, \ldots $$

(A2)

$$ a_{1} = \frac{{m\,L^{2} \,H^{3} }}{{4\,l^{3} }} - \frac{5}{6}\,\frac{{m\,H\,L^{2} }}{l}, $$

(A3)

$$ a_{2} = \frac{{m\,L^{2} \,H^{4} }}{{8\,l^{4} }} - \frac{1}{2}\,\frac{{m\,H^{2} \,L^{2} }}{{l^{2} }}. $$

(A4)