On the Existence of Normal Modes of Damped Discrete-Continuous Systems

Abstract

In this paper we investigate a class of combined discrete-continuous mechanical systems consisting of a continuous elastic structure and a finite number of concentrated masses, elastic supports, and linear oscillators of arbitrary dimension. After the motion equations for such combined systems are derived, they are formulated as an abstract evolution equation on an appropriately defined Hilbert space. Our main objective is to ascertain conditions under which the combined systems have classical normal modes. Using the sesquilinear form approach, we show that unless some matching conditions are satisfied, the combined systems cannot have normal modes even if Kelvin-Voigt damping is considered.

Nature of coupling in non-conservative distributed parameter systems attached to external damping sources

“…Nevertheless, as was justified recently [10], assuming damping proportional to mass leads to significant errors. On the other hand, authors have proved in the past that continuous structures with appendages rarely have normal modes even when the base structure does [5]. Their result dispels the commonly quoted statement that light damping results in normal modes.…”

“…The actual form of damping is not always clearly known in distributed parameter systems. In general, the form of L 1 is an elusive topic of much perpetual research [2][3][4][5][6][7][8]. The problem becomes even more complex when operator L 1 contains additional damping terms accounting for lumped viscous damping sources attached externally (see Figure 1.).…”

Multi-Point Hybrid Vibration Absorption in Flexible Structures

“…This is a deliberate e!ort to avoid the extreme di$culty investigated by Banks et al [18] when the inner products involve all displacement variables simultaneously. By avoiding the discrete variables in deriving equation (2a), one can still apply the classical modal theory without contradicting Banks et al [18]. As a matter of fact, every mode co-ordinate G (t) is strongly cross-coupled with all discrete variables +w…”

“…In equation (1a) [18], it is almost impossible to "nd a complete set of orthonormal mode functions to decompose w in modal space if the actuators are all shut-o! ( f ?H "0 for 1)j)n).…”

LQR Control of Thin Shell Dynamics: Formulation and Numerical Implementation

“…To resolve the uncontrolled and controlled shell dynamics, the Fourier index m was taken from -4 to 4 for a total of 9 Fourier coe cients. The axial components of the longitudinal and circumferential displacements were approximated using 13 modi ed cubic splines while 11 splines were employed in the transverse displacement (two degrees of freedom are lost to accommodate the additional zero-slope boundary conditions indicated in (6)). This yielded N u = N v = 117; N w = 99 in (19) and a total of N = 666 coe cients in the ODE system (20).…”