Dynamic responses of a three-dimensional framework due to a moving carriage hoisting a swinging object

The literature regarding the three-dimensional responses of a structure due to moving loads(s) is rare and that due to the moving swinging object has not been found yet. The purpose of this paper is to provide some information in this area, where the axial ( ¯ x), vertical ( ¯ y) and horizontal (¯z) dynamic responses of the three-dimensional framework of a tyred overhead crane under the action of a moving trolley (mT) hoisting a swinging object (msw) were calculated using the finite element method and the direct integration method. Instead of the conventional moving force problem where only the vertical ( ¯ y) inertia effect of the moving trolley was considered, the three-dimensional inertial effects due to the masses of both the moving trolley and the swinging object have been considered in this paper. To this end, an equivalent moving mass matrix has been presented and which is dependent on both the instantaneous swinging angle of the hoisted object (msw) and the instantaneous position of the moving trolley so that the contribution of the moving mass on the overall mass matrix of the entire structure itself is easily tackled. Finally, the title problem was solved by calculating the forced vibration responses of the three-dimensional framework with time-dependent overall mass and damping matrices and subjected to an equivalent moving force. Some factors pertinent to the title problem, such as the magnitude, velocity and acceleration of the moving trolley and the swinging angles of the hoisted object, were studied. Numerical results reveal that, in addition to the conventional dynamic responses in the vertical ( ¯ y) direction, the other two components in the axial ( ¯ x) and horizontal (¯z) directions for a three-dimensional structure are also significant