53/58:153 Lecture 19 Fundamental of Vibration ______________________________________________________________________________ - 1 - Lecture 19: Continuous Systems Reading materials: Sections 7.2, 7.3, and 7.1 1. Introduction Discrete systems vs Continuous systems We have so far dealt with discrete systems where mass, damping, and elasticity were assumed to be present only at certain discrete points in the system. In continuous systems it is not possible to identify discrete masses, damping, or springs. We must consider the continuous distribution of the mass, damping, and elasticity and assume that each of the infinite number of points of system can vibrate. If a system is modeled as a discrete one, the governing equations are ODEs. If a system is modeled as a continuous one, the governing equations are PDEs. 2. Axial Vibration of Bars Governing Equation53/58:153 Lecture 19 Fundamental of Vibration ______________________________________________________________________________ - 2 - Initial conditions and boundary conditions53/58:153 Lecture 19 Fundamental of Vibration ______________________________________________________________________________ - 3 - Free vibration solution for uniform bars53/58:153 Lecture 19 Fundamental of Vibration ______________________________________________________________________________ - 4 - Free vibration of a fixed-free bar53/58:153 Lecture 19 Fundamental of Vibration ______________________________________________________________________________ - 5 - Natural frequencies of a bar carrying a mass53/58:153 Lecture 19 Fundamental of Vibration ______________________________________________________________________________ - 6 - Vibrations of a bar subjected to initial force53/58:153 Lecture 19 Fundamental of Vibration ______________________________________________________________________________ - 7 - 3. Torsional vibration of a shaft or rod Governing Equation Initial conditions and boundary conditions Solution53/58:153 Lecture 19 Fundamental of Vibration ______________________________________________________________________________ - 8 - 4. Transverse vibration of beams Governing Equation53/58:153 Lecture 19 Fundamental of Vibration ______________________________________________________________________________ - 9 - Initial conditions and boundary conditions53/58:153 Lecture 19 Fundamental of Vibration ______________________________________________________________________________ - 10 - Free vibration solution for uniform beams Equation of motion Separation of variable method The original PDE becomes two ODEs53/58:153 Lecture 19 Fundamental of Vibration ______________________________________________________________________________ - 11 - Solution for the first ODE Solution for the second ODE or General solution of the original PDE Free vibration solution for a simply supported beams53/58:153 Lecture 19 Fundamental of Vibration ______________________________________________________________________________ - 12 - Boundary conditions so Apply BCs To obtain non-trivial solution, we have Therefore, The coefficients are53/58:153 Lecture 19 Fundamental of Vibration ______________________________________________________________________________ - 13 - and the mode shapes are The first four mode shapes Complete free vibration solution If the following initial conditions are given We have53/58:153 Lecture 19 Fundamental of Vibration ______________________________________________________________________________ - 14 - 5. Modal superposition approach (optional) Orthogonality property of free vibration mode shapes The free vibration mode shapes of continuous systems satisfy orthogonality condition. Modal superposition Modal equations53/58:153 Lecture 19 Fundamental of Vibration ______________________________________________________________________________ - 15 -