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Chapter 8 Application of Second-order Differential ...

Chapter 8 Application of Second-order Differential equations in Mechanical Engineering Analysis( Chapter 8 second order DEs) Tai-Ran Hsu*Based on the book of Applied Engineering Analysis , by Tai-Ran Hsu, published byJohn Wiley & Sons, 2018 (ISBN 9781119071204) Applied Engineering Analysis- slides for class teaching*1 Chapter Learning Objectives Refresh the solution methods for typical Second-order homogeneous and non- homogeneous Differential equations learned in previous math courses, Learn to derive homogeneous Second-order Differential equations for free vibration analysis of simple mass-spring system 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 under in resonant vibration.

8.2 Typical form of second-order homogeneous differential equations (p.243) ( ) 0 2 2 bu x dx du x a d u x (8.1) where a and b are constants The solution of Equation (8.1) u(x) may be obtained by ASSUMING: u(x) = emx (8.2) in which m is a constant to be determined by the following procedure: If the assumed solution u(x) in Equation (8.2) is a valid solution, it must …

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