On the eigenvalues of a uniform cantilever beam carrying any number of spring-damper-mass systems

Abstract

SUMMARYThe eigenvalues of a uniform cantilever beam carrying any number of spring}damper}mass systems with arbitrary magnitudes and locations were determined by means of the analytical-and-numerical-combined method (ANCM). First of all, each spring}damper}mass system was replaced by a massless e!ective spring with spring constant k , which is the main point that the ANCM is available for the present problem. Next, the equation of motion for the &constrained' beam (with spring}damper}mass systems attached) was derived by using the natural frequencies and normal mode shapes of the &unconstrained' beam (without carrying any attachments) incorporated with the expansion theorem. Finally, the equation of motion for the &constrained' beam in &complex form' is separated into the real and the imaginary parts. From either part, a set of simultaneous equations were obtained. Since the simultaneous equations are in &real form', the eigenvalues of the &constrained' beam were determined with the conventional numerical methods. To con"rm the reliability of the presented theory, all the numerical results obtained from the ANCM were compared with the corresponding ones obtained from the conventional "nite element method (FEM) and good agreement was achieved. Because the order of the property matrices for the equation of motion derived by using the ANCM is much lower than that by using the conventional FEM, the storing memory and the CPU time required by the ANCM are much less than those required by the FEM.Besides, the solution of the equation of motion derived from the ANCM can always be obtained with the general personal computers, but that from the FEM can sometimes be obtained only with the computers of workstations or main frames when the total degrees of freedom exceeding a certain limit.

Dynamic Response Evaluation of Damped-Outrigger Systems with Various Heights

“…Other approaches to evaluate mode damping based on the principle of virtual work have also been applied (Chen et al 2009, Jeremiah 2006. Further relevant studies using the continuous cantilever model can be found in beam vibration control theory from the applied mechanics field (Wu et al 1998(Wu et al , 1999Laura et al 1977). These methods require validation against a range of building heights and further simplification to be of practical use.…”

A beam with arbitrarily placed lateral dampers: Evolution of complex modes with damping

“…Accordingly, Gürgöze (1998) studied the sensitivities of the eigenvalues of a cantilever beam with a viscous damper in span and a tip mass at the free end. Wu et al. (1999) investigated the eigenvalues of a cantilever beam with arbitrarily attached dampers by using the analytical-and-numerical-combined method (ANCM).…”

Damping augmentation of a rotating beam-tendon system via internally placed spring-damper elements

“…It is found that the damper attached to a cable or beam can effectively increase the damping ratio and thus suppress the vibration. There are also some publications investigating the free vibration of a beam carrying spring-damper-mass systems [19][20][21], and some spring-damper-mass systems are designed to behave as vibration absorbers to mitigate the vibrations of a beam [22,23]. However, to the best of the authors' knowledge, the free vibration of a beam-tendon system interconnected by intermediate spring-damper elements and in a rotating configuration has not been studied so far.…”