Lagrangian Multiplier Method
Sinniah Ilanko, Luis E. Monterrubio, Yusuke Mochida · 2014
The difficulty in choosing admissible functions has been the most significant drawback of the Rayleigh-Ritz method. One way to relax the admissibility requirement may be found by utilizing a constrained optimization technique dating back to the 18th Century. That is by using the Lagrangian multiplier method, in conjunction with the Rayleigh–Ritz method. To illustrate how the Lagrangian multiplier method works, this chapter formulates the procedure for calculating the frequencies of a propped cantilever. It should be noted that compared to the standard Rayleigh–Ritz method where individual functions satisfy all geometric constraint conditions, in the Lagrangian multiplier method, one need to choose extra terms in the series, equal in number to the constraints to be introduced. There are two problems with the Lagrangian multiplier method. One is that it requires a different formulation of the stiffness and mass matrices with the addition of each constraint. Another problem involves continuous constraints.