Chapter 8Application of Second-order Ordinary Differential Equations in Mechanical Vibration Analysis
Chapter Learning Objectives
- Refresh your knowledge of the solution methods for typical second-order homogeneous and nonhomogeneous differential equations learned in previous mathematics courses.
- Learn to derive homogeneous second-order differential equations for free vibration analysis of simple mass–spring systems with and without damping effects.
- Learn to derive nonhomogeneous second-order differential equations for forced vibration analysis of simple mass–spring systems.
- Learn to use the solution of second-order nonhomogeneous differential equations to illustrate the resonant vibration of simple mass–spring systems and estimate the time for the rupture of the system in resonant vibration.
- Learn to use second-order nonhomogeneous differential equations to predict the amplitudes of the vibrating mass in near-resonant vibration and the physical consequences to the mass–spring systems.
- Learn the concept of modal analysis of machines and structures and the consequence of structural failure under the resonant and near-resonant vibration modes.
8.1 Introduction
Like the first-order differential equations presented in Chapter 7, second-order differential equations are also derived from the laws of physics. These equations are used to solve a variety of engineering problems relating to heat conduction in solids, stress analysis of structures such as bending of beams and buckling ...
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