Consider a clamped-free beam in bending, where the kinetic energy is defined as

where the w is the bending deflection and m is the mass per unit length of the beam. The total potential energy PE is the sum of the internal strain energy U and additional potential energy. The term Q_k should comprise all other loads like damping and aerodynamic loads. The strain energy for a beam in bending is defined as

The virtual work resulting from an applied distributed load per unit length f(x, t) is

The beam itself will have potential energy (strain energy) and kinetic energy associated with it. All potential energies will populate the [K] (or stiffness) matrix, and all kinetic energies will populate the [M] (or mass) matrix.

The strain energy of the beam is

Starting from the general solution of the beam in bending, apply the boundary conditions:

The equation for strain energy can be derived using knowledge about the mode shapes and can be further simplified by orthogonality to the following:

The kinetic energy for the beam can be written as

Again, knowing the mode shapes from above, and φ_i(ℓ) and φ_j(ℓ) are also known from the beam bending problem as φ_i(ℓ) = 2( − 1)^i + 1. Removing common constants, and plugging in these values, the following expression is obtained:

where δ_ij is 1 only for i = j and is otherwise 0. Now looking at Lagrange’s equations,

Since the only energy contributing to the [K] matrix is beam strain energy, it will be a diagonal matrix with elements

The beam will contribute only along the diagonal of the mass matrix:

Finally, assuming q = qe^iωt and plugging in the [K] and [M] matrices to solve the eigenvalue problem

This solution is usually computed using a program such as mathematica or matlab, and a demo of this has been done to show the solutions of the modal frequencies (see Ritz Method Demo).