Modal analysis of an elastic structure

Introduction

This program performs a dynamic modal analysis of an elastic cantilever beam represented by a 3D solid continuum. The eigenmodes are computed using the SLEPcEigensolver and compared against an analytical solution of beam theory. The corresponding file can be obtained from cantilever_modal.py.

The first four eigenmodes of this demo will look as follows:

The first two fundamental modes are on top with bending along the weak axis (left) and along the strong axis (right), the next two modes are at the bottom.

Implementation

After importing the relevant modules, the geometry of a beam of length \(L=20\) and rectangular section of size \(B\times H\) with \(B=0.5, H=1\) is first defined:

from fenics import *
import numpy as np

L, B, H = 20., 0.5, 1.

Nx = 200
Ny = int(B/L*Nx)+1
Nz = int(H/L*Nx)+1

mesh = BoxMesh(Point(0.,0.,0.),Point(L,B,H), Nx, Ny, Nz)

Material parameters and elastic constitutive relations are classical (here we take \(\nu=0\)) and we also introduce the material density \(\rho\) for later definition of the mass matrix:

E, nu = 1e5, 0.
rho = 1e-3

# Lame coefficient for constitutive relation
mu = E/2./(1+nu)
lmbda = E*nu/(1+nu)/(1-2*nu)

def eps(v):
    return sym(grad(v))
def sigma(v):
    dim = v.geometric_dimension()
    return 2.0*mu*eps(v) + lmbda*tr(eps(v))*Identity(dim)

Standard FunctionSpace is defined and boundary conditions correspond to a fully clamped support at \(x=0\):

V = VectorFunctionSpace(mesh, 'Lagrange', degree=1)
u_ = TrialFunction(V)
du = TestFunction(V)


def left(x, on_boundary):
    return near(x[0],0.)

bc = DirichletBC(V, Constant((0.,0.,0.)), left)

The system stiffness matrix \([K]\) and mass matrix \([M]\) are respectively obtained from assembling the corresponding variational forms:

k_form = inner(sigma(du),eps(u_))*dx
l_form = Constant(1.)*u_[0]*dx
K = PETScMatrix()
b = PETScVector()
assemble_system(k_form, l_form, bc, A_tensor=K, b_tensor=b)

m_form = rho*dot(du,u_)*dx
M = PETScMatrix()
assemble(m_form, tensor=M)

Matrices \([K]\) and \([M]\) are first defined as PETSc Matrix and forms are assembled into it to ensure that they have the right type. Note that boundary conditions have been applied to the stiffness matrix using assemble_system so as to preserve symmetry (a dummy l_form and right-hand side vector have been introduced to call this function).

Modal dynamic analysis consists in solving the following generalized eigenvalue problem \([K]\{U\}=\lambda[M]\{U\}\) where the eigenvalue is related to the eigenfrequency \(\lambda=\omega^2\). This problem can be solved using the SLEPcEigenSolver.

eigensolver = SLEPcEigenSolver(K, M)
eigensolver.parameters['problem_type'] = 'gen_hermitian'
eigensolver.parameters["spectrum"] = "smallest real"
eigensolver.parameters['spectral_transform'] = 'shift-and-invert'
eigensolver.parameters['spectral_shift'] = 0.

The problem type is specified to be a generalized eigenvalue problem with Hermitian matrices. By default, SLEPc computes the largest eigenvalues, here we instead look for the smallest eigenvalues (they should all be real). To improve convergence of the eigensolver for finding the smallest eigenvalues (by default it computes the largest ones), a spectral transform is performed using the keyword shift-invert i.e. the original problem is transformed into an equivalent problem with eigenvalues given by \(\dfrac{1}{\lambda - \sigma}\) instead of \(\lambda\) where \(\sigma\) is the value of the spectral shift. It is therefore much easier to compute eigenvalues close to \(\sigma\) i.e. close to \(\sigma = 0\) in the present case. Eigenvalues are then transformed back by SLEPc to their original value \(\lambda\).

We now ask SLEPc to extract the first 6 eigenvalues by calling its solve function and extract the corresponding eigenpair (first two arguments of get_eigenpair correspond to the real and complex part of the eigenvalue, the last two to the real and complex part of the eigenvector):

N_eig = 6   # number of eigenvalues
print "Computing %i first eigenvalues..." % N_eig
eigensolver.solve(N_eig)

# Exact solution computation
from scipy.optimize import root
from math import cos, cosh
falpha = lambda x: cos(x)*cosh(x)+1
alpha = lambda n: root(falpha, (2*n+1)*pi/2.)['x'][0]

# Set up file for exporting results
file_results = XDMFFile("modal_analysis.xdmf")
file_results.parameters["flush_output"] = True
file_results.parameters["functions_share_mesh"] = True

# Extraction
for i in range(N_eig):
    # Extract eigenpair
    r, c, rx, cx = eigensolver.get_eigenpair(i)

    # 3D eigenfrequency
    freq_3D = sqrt(r)/2/pi

    # Beam eigenfrequency
    if i % 2 == 0: # exact solution should correspond to weak axis bending
        I_bend = H*B**3/12.
    else:          #exact solution should correspond to strong axis bending
        I_bend = B*H**3/12.
    freq_beam = alpha(i/2)**2*sqrt(E*I_bend/(rho*B*H*L**4))/2/pi

    print("Solid FE: {0:8.5f} [Hz]   Beam theory: {1:8.5f} [Hz]".format(freq_3D, freq_beam))

    # Initialize function and assign eigenvector (renormalize by stiffness matrix)
    eigenmode = Function(V,name="Eigenvector "+str(i))
    eigenmode.vector()[:] = rx

The beam analytical solution is obtained using the eigenfrequencies of a clamped beam in bending given by \(\omega_n = \alpha_n^2\sqrt{\dfrac{EI}{\rho S L^4}}\) where \(S=BH\) is the beam section, \(I\) the bending inertia and \(\alpha_n\) is the solution of the following nonlinear equation:

\[\cos(\alpha)\cosh(\alpha)+1 = 0\]

the solution of which can be well approximated by \((2n+1)\pi/2\) for \(n\geq 3\). Since the beam possesses two bending axis, each solution to the previous equation is associated with two frequencies, one with bending along the weak axis (\(I=I_{\text{weak}} = HB^3/12\)) and the other along the strong axis (\(I=I_{\text{strong}} = BH^3/12\)). Since \(I_{\text{strong}} = 4I_{\text{weak}}\) for the considered numerical values, the strong axis bending frequency will be twice that corresponsing to bending along the weak axis. The solution \(\alpha_n\) are computed using the scipy.optimize.root function with initial guess given by \((2n+1)\pi/2\).

With Nx=400, we obtain the following comparison between the FE eigenfrequencies and the beam theory eigenfrequencies :

Mode	Eigenfrequencies
#	Solid FE [Hz]	Beam theory [Hz]
1	2.04991	2.01925
2	4.04854	4.03850
3	12.81504	12.65443
4	25.12717	25.30886
5	35.74168	35.43277
6	66.94816	70.86554