On approach for obtaining approximate solution to highly nonlinear oscillatory system with singularity
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RESEARCH PAPER
On approach for obtaining approximate solution to highly nonlinear oscillatory system with singularity Zvonko Rakaric1 · Boris Stojic2 Received: 15 December 2019 / Revised: 19 January 2020 / Accepted: 16 March 2020 © The Chinese Society of Theoretical and Applied Mechanics and Springer-Verlag GmbH Germany, part of Springer Nature 2020
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. Keywords Pendulum mechanism · Low frequency · Singularity · Laurent series · Jacobi elliptic function
1 Introduction Recently, Starossek [1] investigated a conservative mechanism which can perform an oscillatory motion about its equilibrium. In Ref. [2] the analysis is further extended to forced oscillations. This mechanism with one degree of freedom (DOF) represents, in essence, an extended version of a simple pendulum mechanism (Fig. 1). The mechanism consists of three main kinematic parts, rigid bars OA and BD, as well as the slider D that is connected with rod BD by a rotational joint. The slider can move along the vertical guide. Hinge O is motionless and another cylindrical hinge is located at point A. All three elements are supposed to be massless. Beside these three massless elements, two concentrated masses are attached to the rod BD, symmetrically with respect to the point A. It is assumed that the mechanism is
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Zvonko Rakaric [email protected]
1
Department of Mechanics, Faculty of Technical Sciences, University of Novi Sad, 21000 Novi Sad, Serbia
2
Department of Mechanization and Design Engineering, Faculty of Technical Sciences, University of Novi Sad, 21000 Novi Sad, Serbia
frictionless and that motion is in the vertical plane only. This mechanism is conservative and has one degree of freedom. For this one DOF mechanism, there are a few options for choice of generalized coordinates. In Refs. [1, 2] the relative coordinate ν for slider motion with respect to the vertical position of hinge A was used. Using such a generalized c
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