Advanced Dynamics and Vibrations: Modal analysis of mdof systems with generalized coordinates, semi-definite systems, repeated roots
Introduction to Normal Mode Analysis
Our analysis of MDOF systems has shown that regardless of the coordinates used that there are specific properties of the system(natural frequencies, mode shapes) that do not depend on our choices. These properties can be used to uncouple the equations of motion into equations that can be easily solved and then transformed back to the original coordinates which are the most useful form to use the results. The process uses the modal shapes, in the form of vectors, to build a modal matrix. Consider the modal vectors . The modal matrix is formed by making these vectors the columns in the matrix:
but instead of using we normalize it to using the mass matrix so that:
To start, consider the undamped forced 2DOF system:
Normalize and using where . Therefore:
Modal vectors are orthogonal with respect to or . Thefore the modal matrix is:
Now transform the equations of motion using by setting then .
Now multiply the equations of motion by to get:
Therefore, the transformed equations become
We can now solve this set of uncoupled SDOF equations for the free, transient on steady-state conditions. In the transformed state the stiffness matrix is now the matrix of natural frequencies:
After solving we transform back to the original condition.
Normal analysis of MDOF systems
The equations of motion for viscously damped MDOF systems are in general
For up to 2 DOF we can easily use a complex analysis (transform methods). However for larger system we often use modal analysis.
In this approach we start with the homogeneous equations and we will also start with the undamped case. (We will return to it later!) Therefore,
We look for so-called synchronous motions SSHM (Simultaneous Simple Harmonic motion) SSHM by assuming
This is true for only special say and special associated with
We can find the & the within a multiplicative constant. Each and its associated is called a mode.
If the are distinct then we can easily show that the are orthogonal to one another in some sense. To see this consider the two modes say & , then
Multiply the first by
and the second by
Since we can transpose the first equation so that the left hand side are the same. It suggests that
Therefore, the modal vector are orthogonal with respect to the mass and stiffness matrices.
These eigenvectors form a linearly independent set i.e. any motion of the system can be thought of as a linear combination of these modes. Physically this implies that any motion can be thought of as a superposition of these normal modes.
Linear independence means that we cannot write any vector as a combination of the others. If they were linearly dependent there should be some way to combine them to produce one from the others. This means there should be:
But if we premultiply this equation by we get:
but all the terms are zero (orthogonality) except for and therefore . The same is value for each term therefore
as a result any motion can be written as a linear combination of the modal vectors. They are a set of basis vectors in the same way as are for positions. Therefore,
Where is the modal participation factor and
This is called the expansion theorem in vibration where the are functions of time.
It turns out that if we have repeated eigenvalue we can still find a set of eigenvectors that span the system.
We can use these properties of the system to solve these equations in general. The process uses the modal vectors put together in a so-called modal matrix. Before starting we will normalize the modal vectors into as they are only known to within a multiplicative constant.
The normalization is:
Now form the modal matrix
Now to solve the original system we transform the equations to a set of generalized coordinates. That is we set:
putting this into the original set of equation gives:
now premultiply by
This uncouples the equations so that each looks like:
The general solution of this equation is:
We can then transform the solution back to the original coordinates using:
This expression contains the initial conditions in the transformed coordinates and are related to the original values by:
We can also solve the steady-state forced problem by simply considering the particular integral instead of the total solution.
Summary (Free Vibration)
We have been considering the solution of
Where we transform the equation using
If we premultiply by and substitute into we get:
The solution to these equation is
Where & are constants, Therefore
If and are the initial displacements and velocities then
since , therefore
This solves every free vibration problem for any initial conditions.
Characteristic equation is:
now find the force vector in the normal coordinates
Since there are initial conditions with no displacement and no velocity we can simply integrate the convolution integrals to give
We can now find the response in terms of the original coordinates using:
Note that the first mode is most pronounced in the overall response.
Check static deflection:
Consider a second example with certain initial conditions:
A static force is suddenly released for the system of 3 masses mounted on a tightly stretched wire with tension . Determine the resulting free vibration using normal analysis.
While the mass matrix is easy we must find the stiffness matrix using say
Apply unit def’l to m,
now find the ‘s & , assume to get:
The characteristic equation is
now find , choose
now normalize the to get where , therefore as
Also need the initial conditions
We can simplify put this into the general solution:
Again the first mode is the dominant one and the second mode unsymmetric mode is not present if we had chosen different initial condition say, then
All three modes would now participate in the motion. To find the response simply insert the new and calculate the .
e.g Steady State
The equations of motion are
Divide by and set
Therefore, from and
The transformed forces are:
Therefore transformed equations are:
Steady State Solutions of these equations
For the case in which we have repeated eigenvalues we can still have orthogonal eigenvectors using a process called successive orthogonalization.
If the eigenvalues is repeated p times let be an arbitrary set of modal vectors that satisfy the degenerate set of equations. We construct an orthoganal set using
To prove take
The characteristic equation is
For the repeated root
Therefore there are an infinite number of solutions so choose one which satisfies the equations. e.g.:
Now we make them orthogonal
Now and are orthonormal for therefore:
and this is our basis.
Now consider the addition of damping to the system
Now transform this to normal coordinates
and premultiply by
but there is no a priori reason that is a diagonal matrix and therefore the equations are coupled. If it was, we could use the same approach as before. One condition that is often used is the so-called proportional damping where
so that the transformed equations are necessarily diagonal. We can then set
then each transformed equation is
The general solution of this equation is
Now consider that the are related
Therefore we can only choose the damping independently for 2 modes. If then so that the damping in each mode is proportional to the frequency of that mode. The higher the frequency the higher the damping so that higher modes damp out more rapidly. [Note: Often called relative damping.]
On the other hand, setting implies the damping is proportional to the mass matrix which damps out the lower modes more than the higher ones. [Note: Often called absolute damping.]
Return to a simple problem we have done previously
and the transformal equations for free vibration are
We transform these back to the original coordinates
Assume the initial conditions are:
We can find the transformed initial conditions from
Now add proportional damping
Select relative damping (proportional to stiffness)
The transformed equations will be
but we called
The solution to the initial value problem for the case discussed is
We can now consider the decay of each of these two components using the idea of the so-called logarithmic decrement. Recall from the SDOF case that
where is the number of cycles to decay to the ratio selected for
Once is chosen decreases linearly with .
If we set
and increases linearly with .
Consider a steady-state damped problem
The equations of motion are:
Consider the undamped problem to find the and
In the same manner,
Therefore undamped differential equations are
The undamped steady state solution is
As a result the undamped solution is
As was shown previously at
Now lets add damping
Therefore the equations of motion are:
Consider the standard form where the coefficient of is