Motion analysis of a suspended mass attached to a crane

Abstract

This paper presents a qualitative study of the pendulum motion of the load suspended from the jib of a crane. The objective is to investigate the effect of various jib motions on the trajectory of the suspended load. The system equations are formulated such that the mass of the load has no effect on the trajectory. Two types of jib slewing motion are studied independently: rotation of the jib about r-axis, rotation of the cab about the y-axis, and translation of the crane along the x-axis. Solving for the y-axis rotation and using state space plots show that the load motion is a two-dimensional motion which can be classified into three types: (1) limit cycles; (2) rotating plots; and (3) rotating non-symmetrical plots. In the absence of control and damping, the most favourable response is a limit cycle with small amplitudes in two angular directions. For a given cable length, this can be achieved by the use of a smooth velocity profile like a half-sine instead of a trapezoidal type. In the z-axis rotation, the load trajectory is generally the same for the two types of jib input profiles at high slewing speed. At a slower input speed, however, the half-sine gives smaller swing angle of the load. For the x-axis crane translation over a short distance, the pendulum motion of the load can be reduced substantially if the cable length is adjusted to suit the input. For a triangular input with acceleration time t a = 0.5 τ, where τ is the time of the travel, the cable length ( L) must be such that the resulting period of swing ( T) is equal to t a . For a trapezoidal input L must be such that T = ( τ - t a ).