Frequency analysis

The function freq() solves the eigenvalue problem \(([K] + \lambda^2 [M])\{u\} = \{0\}\), where \(\lambda^2 = -\omega_n^2\) and \(\omega_n\) is the natural frequency in rad/s. The code below is extracted from one of the structsolve unit tests:

def test_freq_simple():
    """Test frequency analysis with known natural frequencies

    For a diagonal system: [K]{u} + omega^2 [M]{u} = 0
    K = diag(k1, k2, ...), M = diag(m1, m2, ...)
    omega_i = sqrt(ki/mi)
    """
    n = 10
    k_vals = np.arange(1, n + 1, dtype=float) * 100
    m_vals = np.ones(n)
    K = csc_matrix(np.diag(k_vals))
    M = csc_matrix(np.diag(m_vals))

    eigvals, eigvecs = freq(K, M, silent=True, num_eigvalues=5,
                            sparse_solver=True)

    # freq returns lambda2 where [K] + lambda2 [M] = 0
    # so lambda2 = -omega^2 = -k/m
    # The sorted natural frequencies should match sqrt(k_vals)
    assert eigvecs.shape[0] == n