Dynamics of 3D Slender Blocks: A Mathematical Model

Abstract

The dynamics of three-dimensional (3D) slender systems such as statues, obelisks, monuments, museums, and household objects are highly nonlinear. In this study, a numerical model for rigid blocks in 3D, referred to as the physical model (PM), is developed to explore and view the motion of the slender blocks to base excitations. The model can represent the dynamics of both symmetric and mass-eccentric 3D blocks for unidirectional and bidirectional excitations. Keeping the kinematics of the system in view, the PM is developed as an assemblage of different subsystems in the Simscape Multibody Library. The rigid block supported by four tiny spheres at the corners is resting on a rigid table. Physical interaction between the table and the rigid body is simulated by introducing a virtual plane characterized by a spring and a damper that follows the visco-elastic Kelvin model. The values of the depth of the virtual plane, parameters of the Kelvin model, and the dimension of the tiny spheres are suitably chosen for such interaction. Adequately large values for kinetic and static friction are used in the virtual plane to prevent the body from sliding during its rocking motion. The motion of the table is described by choosing the inertial reference frame with origin at a fixed point and another parallel frame fixed with the tabletop. The response of the rigid body is described by the asymmetric Euler sequence. A third set of the frame, i.e. the body-fixed reference with the origin at the centre of mass of the corresponding symmetric body is introduced by a Bushing joint. A calculated amount of mass-eccentricity is generated by adding very small square blocks to the side-faces of the 3D symmetric block. The response of mass eccentric systems to trigonometric base excitations applied in-plane and out-of-plane of the eccentricity is computed using the PM, and a comparison of the same to the results derived analytically confirms the adequacy of the model. In the sequel, representative case studies are presented and interpreted to uncover the complex dynamics of the mass eccentric rocking systems. The PM can also serve as the basis of analyzing and viewing the motion of blocks with arbitrary geometry.