Introduction to Modal Analysis
MCE371: Vibrations
Prof. Richter
Department of Mechanical Engineering
Handout 14
Fall 2011
Introduction to Modal Analysis
Coupled vs. Decoupled Coordinates
Follow Palm, Sect. 8.3
Consider the multi-DOF vibration equation:
M Ẍ + KX = F
If M and K are not diagonal matrices, coupling of variables exists
(x1 appears in the equation for ẍ2 and x2 appears in the equation for
ẍ1 , etc.
Coupling makes analysis difficult (finding solutions, determining how
force components affect the vibration of an individual coordinate,
etc.)
Introduction to Modal Analysis
Coupled vs. Decoupled Coordinates ...
In modal analysis, we transform the coordinate X so that the
equations are completely decoupled. The new coordinates are called
modal coordinates.
This is possible due to the symmetry properties of M and K
Introduction to Modal Analysis
Seeking Orthonormal Coordinates
It is a fact of linear algebra (no proof here) that symmetric matrices
possess a set of orthogonal eigenvectors and real eigenvalues. Check
numerically by generating a large random matrix in Matlab. Make it
symmetric by adding it to its transpose. Find the eigenvectors and check
they are orthogonal to each other.
An orthonormal set of eigenvectors has the additional property that each
vector has unit length.
We saw that the free multi-dof vibration equation leads to the following
eigenvalue problem:
M −1 KX = w 2 X
Even if M and K are symmetric, M −1 K doesn’t have to be symmetric.
We are interested in transforming the eigenvalue problem to the form
AX = w 2 X
with A symmetric so that an orthonormal eigenvector set is obtained.
This will lead to coordinate decoupling, as seen next.
Introduction to Modal Analysis
First Transformation: Using the Mass-Normalized Stiffness
Matrix
Since M is symmetric and positive-definite, a matrix
√ L always exists such
that L′ L = M. When M is diagonal, just take L = M. If M is not
diagonal, use the Cholesky decomposition. In Matlab, use L=chol(M)’.
Define a coordinate transformation as X = L−1 q. Substituting into the
equation of motion and multiplying from the left by L−1 gives
q̈ + Hq = L−1 F
where L−1 ML−1 = I has been used and H = L−1 KL−1 .
Introduction to Modal Analysis
First Transformation: Orthonormal Eigenvectors
At this point, the equation of motion has been transformed to
q̈ + Hq = L−1 F
H is symmetric, but not necessarily diagonal, so the equations are still
coupled. However, following the same steps as in Handout 13, the
eigenvalue problem in these coordinates takes the form:
Hu = w 2 u
Since H is symmetric, an orthonormal set of eigenvectors exists. In this
introductory course, you will solve the eigenvalue problem using Matlab:
[P,D]=eig(H). Matrix P contains the eigenvectors in its columns, and
the eigenvalues are the diagonal entries of D.
Note that the eigenvectors u are related to the mode shape vectors by
u = LX .
Introduction to Modal Analysis
Second Transformation: Modal Coordinates
Define a third set of coordinates by r = P ′ q, where P is the matrix of
orthonormal eigenvectors from the previous step. Note that P has the
property P ′ P = I , so that q = Pr . That is, P is an orthonormal matrix,
satisfying P −1 = P ′ .
Substitute q = Pr in the equation of motion from the previous step and
multiply from the left by P ′ to get:
r̈ + Λr = P ′ L−1 F
where Λ = P ′ HP is called spectral matrix. The spectral matrix is
always diagonal, so decoupling is finally achieved.
Introduction to Modal Analysis
Second Transformation: Modal Coordinates...
Each modal equation has the form
r̈i + wi2 ri = gi (t)
where gi (t) is the i-th element of P ′ L−1 F . Modal equations have a direct
solution (the same as the 1-DOF forced solution). To find the overall
solution, assemble the ri solutions into vector r and use X = L−1 Pr .
Each mode may be analyzed using all of the tools studied for 1-DOF
systems: transfer functions, frequency response, etc.
Introduction to Modal Analysis
Example
Study Example 8.3-1 carefully, replacing manual computations by the
corresponing Matlab commands. Continue with Examples 8.3-2 and
8.3-3.
The summary shown on page 521 of Palm is very useful for problem
solving.