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The Ritz Method
Starting from Lagrange's equationsin N generalized coordinates,
(1) Qk = (d)/(dt)⎛⎝(∂L)/(∂qk)⎞⎠ − (∂L)/(∂qk), k = 1, 2, ..., N
the Lagrangian is expressed as the difference between kinetic energyand potential energy. The goal is to solve these discrete equations at the generalized coordinates qk. The generalized forceQk can account for the effects of any loads, but need to only include those forces that are not included in PE. The relationship between virtual work and the generalized forces is expressed by
(2) δW = n⎲⎳k = 1Qkδqk, 
where δqk is an increment of the generalized coordinate.
These equations need to be in terms of a series of basis functions φk(x) in order to apply the Ritz method. For a beam in bending, the deflection can be expressed as
(3) w(x, t) = n⎲⎳k = 1qk(t)φk(x).
Characteristics of Basis Functions
The series approximation resulting from basis functions allow us to reduce a continuous problem with infinite degrees of freedom to one with n degrees of freedom. These basis functions of the Ritz method must satify the following criteria in order for the approximate solution to converge to the continuous solution:
  • Be p-differentiable where p is the maximum order of the Lagrangian. For example, the beam in bending problem has an order p = 2 based on inspection of the strain energy equation.
  • Satisfy geometric or essential boundary conditions.
  • The set of functions must be complete for k = 1, ..., N. Examples include
    1. Polynomial: φk(x) = xk
    2. Harmonic: φk(x) = sin(kx ⁄ L)
  • The set of functions need to be linearly independent.
The equations of motion are expressed in the following generalized form:
(4) [M]q̈ + [C]q̇ + [K]q = F, 
where M is the mass matrix, C is the damping matrix, and K is the stiffness matrix of the system.