Modal analysis of continuous systems by replacing displacement excitation with equivalent excitation force and fixed boundary

Abstract

This paper describes the theoretical modal analysis of continuous systems that are subjected to displacement excitation. Because vibration modes of continuous systems whose boundaries actively move are unknown, we proposed an equivalent replacement method of displacement excitation with an excitation force and fixed boundary. We can easily derive the vibration modes of the continuous system because the excitation boundaries are fixed by this replacement method. Using the proposed replacement method and modal analysis, we can derive a certain vibration mode independently. In other words, the number of degrees of freedom can be decreased using the proposed method even when continuous systems are subjected to displacement excitation. The equivalent replacement methods for ‘one-dimensional and multi-dimensional continuous systems whose equation of motion is a wave equation’, and ‘one-dimensional and two-dimensional continuous systems whose equation of motion is a fourth-order partial differential equation’ were proposed in this research. As a representative, an acoustic tube and a cuboidal room were used as the analytical models for the one-dimensional and multi-dimensional continuous systems whose equation of motion is a wave equation. In contrast, an Euler-Bernoulli beam and a Kirchhoff-Love plate were used as the analytical models for the one-dimensional and two-dimensional continuous systems whose equation of motion is a fourth-order partial differential equation. The correctness of the proposed methods was mathematically proved. In addition, the effectiveness of the proposed method was verified through numerical simulations.

Keywords

• continuous system
• displacement excitation
• modal analysis
• wave equation
• bending vibration
• acoustic field
• beam
• plate