In linear system, in-plane motions are decoupled from out-of-plane motions for planar frame structures. A theoretical method is proposed that permits the efficient calculations of modal characteristics of planar multi-story frame structures. There are 3 × m beam components for a planar m-story frame structure. By analyzing the transverse and longitudinal motions of each component simultaneously and considering the compatibility requirements across each frame joint, the undetermined variables of the entire m-story frame structure system can be reduced to six, regardless of the number of stories, and that can be determined by the application of the boundary conditions. The main feature of this method is to decrease the dimensions of the matrix involved in the finite element methods and certain other analytical methods.

In this paper, we present the static and dynamic structural characterisation of alow-mobility parallel kinematic manipulator, involving analysis of its stiffness andvibrational dynamic behaviour. The study starts by building numerical models of thebehaviour of the manipulator to be compared to experimental measurements from a prototype.For the case study, we consider a four-degree-of-freedom(x, y, z, θz) manipulatorwith prismatic actuators designed by the COMPMECH research group at the University of theBasque Country. The characterisation allows the behaviour of the static and dynamicstiffness, as well as the natural frequencies of the manipulator, to be mapped in themanipulator workspace. These maps together with kinematic, static and dynamic constraintslead to the definition of operational, static, dynamic and structural workspaces,respectively. Further, we analyse the modes of the manipulator to determine dynamicdisplacements, these being key in the performance in the machining tasks for which therobot was designed.

This paper deals with the transverse free vibrations of a system in which two beams are coupled with a spring-mass device. The dynamics of this system are coupled through the motion of the mass. The entire system is modeled as two two-span beams and each span of the continuous beams is assumed to obey the Euler-Bernoulli beam theory. Considering the compatibility requirements across each spring con-nection position, the eigensolutions (natural frequencies and mode shapes) of this system can be obtained for different boundary conditions. Some numerical results and experimental validations are presented to demonstrate the method proposed in this article.