53/58:153 Lecture 21 Fundamental of Vibration ______________________________________________________________________________ - 1 - Lecture 21: Energy Method (Con’t) Reading materials: Sections 8.4 and 8.5 1. Rayleigh’s energy method for MDOF systems Discrete MDOF systems can be reduced to equivalent SDOF systems by using the Rayleigh’s energy method. The assumption needed is regarding the deformation shape of the system. Equations of motion Assume the system vibrates as the follows, then () ()0=+ tutu kψmψ&& multiplying by where53/58:153 Lecture 21 Fundamental of Vibration ______________________________________________________________________________ - 2 - Natural frequency, Example, Assume the deformation shape is obtained from static deflection due to the weights. Static deflection53/58:153 Lecture 21 Fundamental of Vibration ______________________________________________________________________________ - 3 - 2. Rayleigh’s energy method for continuous systems Take Beams as an example The kinetic energy of the beam ()∫∫==lldxxAydmyT02022121ρ&& The maximum kinetic energy can be found by assuming a harmonic variation () ()txYtxyωcos, = () ()∫=ldxxYxAT022max2ρω The potential energy of the beam is the same as the work done in deforming the beam. By neglecting the work done by the shear forces, we have ∫∫∂∂∂∂==lldxxyxyEIMdU022220)(2121θ and ()()∫∂=ldxxxYdxEIU0222max)(21 By equating maxT to maxU, we have ()()() ()∫∫∂=lldxxYxAdxxxYdxEIw0202222)(ρ53/58:153 Lecture 21 Fundamental of Vibration ______________________________________________________________________________ - 4 - Example: Fundamental frequency of a tapered beam 3. Rayleigh-Ritz method For continuous systems. The basis of the FEM. Rayleigh’s method approximates a continuous system by an equivalent SDOF system via assuming a single deformation shape. In Rayleigh-Ritz method, a continuous system is reduced to a discrete MDOF system. The number of DOF is equal to the number of Ritz modes chosen. Considering the case of beam bending. The beam deflection is approximated by53/58:153 Lecture 21 Fundamental of Vibration ______________________________________________________________________________ - 5 - where Other expressions Potential due to strain energy energy Stiffness Work done by the external force where53/58:153 Lecture 21 Fundamental of Vibration ______________________________________________________________________________ - 6 - Kinetic energy where Equations of motion53/58:153 Lecture 21 Fundamental of Vibration ______________________________________________________________________________ - 7 -