Asymptotic Approach to Motion of Physical Pendulum with an Extended Model of Damping

In the paper, the plane motion of a physical pendulum involving the interactions with the surrounding air is considered. These interactions are described employing the model consisting of three components. The linear and quadratic terms are proportional to the magnitude of the velocity and its square, respectively. The last component is proportional to the tangential component of the acceleration. According to the semi-empirical Morison equation, the quadratic term and acceleration dependent component depict the total force exerted on the body i.e. the drag force and inertia force including the concept of mass added. The multiple scales method (MSM) is used to obtain the approximate asymptotic solution to the problem. A slight change in the natural frequency is caused by the inertial component of the total damping force. In turn, the occurrence of the absolute value of velocity in the damping model complicates the solving procedure. The derived asymptotic solution for the experimentally verified model of the air resistance force is a good basis for further qualitative analysis of the dynamic behaviour of the system. The model of interactions and the presented methodology of the asymptotic approach can be used to study the dynamics of other mechanical systems, including multibody systems. Two methods of assessing asymptotic solutions have been proposed. Both show that the applied MSM solves the governing equation to a high degree of accuracy.