This article addresses the exact solution of free vibration of a beam with a concentrated mass within its intervals when the beam is subjected to axial loadings. This problem is frequently encountered in the design and modeling of resonant micro-sensors, where an exact model is required to calibrate the frequency output with a physical measurand. The significance of this approach is its ability to determine the exact mode shapes of vibration, which are necessary in the study of the time-domain response of sensors and determination of stability regions. The effects of the concentrated mass ratio, its location on the beam, and the applied axial force on the natural frequencies and the mode shapes of the resonant beam were studied in detail. The results of this study reveal how the mass ratio and location can change the sensitivity of the sensor to the input axial force. As an application of the presented approach, the modeling of a fabricated micro-electro-mechanical system (MEMS) has been discussed.
The orthogonality of the modes of vibration of a distributed parameter system plays an important role in the study of the dynamic behavior of that system. The definition of orthogonality for beams with classical boundary conditions is well known. However, it has been shown that the exact mode shapes of beams that carry one or more attached lumped masses are not orthogonal to each other under the classic condition. In this paper, the effect of axial force on the orthogonality condition of the exact mode shapes of beams with several attached lumped masses, as well as translational and rotational springs, is investigated. It has been shown that, in contrast to the mode shapes themselves, the orthogonality condition remains unchanged when an axial force is applied. Furthermore, the generalized orthogonality condition is employed to study the dynamic behavior of a beam—mass system under different boundary conditions. It has been shown that for the precise investigation of the dynamic response, application of exact mode shapes is not adequate enough, and the orthogonality condition associated with the problem must also be used. Finally, a discussion of the effect of the orthogonality condition on the damping matrices is presented. It has been shown that using the exact mode shapes and generalized orthogonality condition may result in non-modal and non-diagonal damping matrices, which, in turn, may increase the computation time.
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