Equilibria and stability of a rigid body suspended by a flexible string: analyzing two suspension systems

Subject of investigation are equilibrium positions and their stability of a rigid body suspended by a massless, flexible, inextensible string of given length, the endpoints of which are attached to two points of the body. Two suspensions are investigated. In Suspension I, the string is passed over two frictionless hooks fixed on a horizontal line a given distance apart. In Suspension II, the string is passed over a frictionless pulley of given radius, the center of which is a fixed point. The center of mass is an arbitrarily given point of the body. Suspension I: Equilibrium positions for a given center of mass are determined by the positive roots of two 8th-order polynomial equations. A bifurcation curve divides a body-fixed plane into domains differing in the number of equilibrium positions depending on the location of the center of mass. The total number of equilibrium positions is between four and eight, depending on the parameters of the system. Stability and instability criteria are formulated. By the results obtained, the special case of the single-hook suspension is covered. Suspension II: Every mathematical relationship describing suspension I is valid, in modified and more complex form, for suspension II.