Advanced Dynamics and Vibrations: Transfer matrices and periodic structures
Transfer Matrices – State Vectors for Vibrating Systems
For many MDOF systems, it is often more efficient to use a “shooting” technique rather than solve a boundary value problem. Essentially, we employ a series of initial value problems to find the solution to the boundary value problem. In vibrations, several numerical procedures have led to the “transfer matrix/state vector” approach. This includes the methods of Holzer and Myklestad’s for beams. This technique is an iterative technique where an estimate of the natural is used at one boundary then checked at the other and if not correct an iterative procedure is used to “honeinon” the answer (eg. Newton and Raphson). After the correct answer is found the mode is also known as the “states” have been calculated.
Transfer Matrices – State Vectors
In the transfer matrix approach, we follow the idea of the Holzer method and guess the eigenvalue (natural frequency). This approach can be extended to continuous systems.
We will introduce this for torsional and spring systems. We define the state of a position in the system as the force and displacement at that point.
Transfer Matrix – relates the state vectors at two points in the system
Therefore:
This is called a field transfer matrix.
If we have a discontinuity of mass at a point:
Assume (SHM). Therefore:
Therefore:
Where the point transfer matrix is . We can combine these to generate a transfer matrix from one position to another which are at like locations.
Therefore:
Example 1
Therefore:
A long way to get this simple result.
Consider a torsional system:
Then the positive directions are similar to the analogous linear case. The field equations are then:
The point matrix is:
It is often useful to divide the torques by a to make them nondimesional as well.
Example 2
Note that and are not required. It is sometimes better to nondimensionalize the equations by dividing the torques by . Therefore:
Now we have only to choose the value of . Assume:
Then:
This is not a natural frequency as and it should be zero. Also, look at the mode shape:
The correct answer is .
Non Dimensionalized Matrices
Assume . Then:
Therefore, close at . Modal vector is:
Where to start, use Dunkerley’s technique.
Therefore:
Therefore:
Therefore:
Release Point
Consider that there are masses at the ends of the lever .
Again, we wish to go from to .
Therefore:
Consider a branched system:
The transfer matrix will include the effect of the branch.
Therefore:
Therefore:
This now has the “effect” of the branch in it.
Consider the linear system analog of the torsional one, but with the mass on the other side.
Therefore:
Therefore:
Let :
Therefore:
Therefore:
Therefore:
Therefore:
By analogy, the torsional system is:
The transfer matrix approach can be extended to handle geared and branched systems. We can derive the results from scratch or use the analog of the translational systems.
We wish to find in terms of :
Therefore:
set :
We can also include the effect of the inertia of the gears:
Therefore:
Again we assume . Therefore:
Therefore:
Therefore:
To determine the relationship between and , consider the branch (B).
Therefore:
If we take counterclockwise as positive with increasing numbers as shown above then viewing the gears from the left gives.
Therefore:
Consider a free body diagram of the gears viewed from the left:
Therefore:
Therefore:
Therefore:
Therefore the effect of the branch is embodied in the term.
Example 3
Again write down the state vectors and transfers matricies also nondimensionalize by dividing the torques by .
We need this only to calculate the mode shape.
For this case :
Now begin by selecting :
Therefore:
Now try :
Now try :
Now try :
The coefficient as . This is because of the transfer function where the branch occurs. This hold for any number of branches. If the main shaft is excited at the natural frequency of any one of the branches, the amplitude at the branch point is zero and the branch acts as a vibration absorber.
We can extend the transfer matrices to beams and to include a more realistic effect for shafts etc.
Shafts
at therefore
at
Therefore:
Therefore:
evaluate these at
Therefore:
If the moment of inertia of the shaft is small and the field matrix becomes
which is what we had previously.
If the stiffness is very high
and the transfer matrix becomes
then the rod is simply a mass.
Beams
The idea of transfer matrices can also be used to handle these complicated continuous system problems. For flexural vibrations the beam is modelled as massless section with a certain stiffness and with discrete masses. We can derive the equations for these various components from elementary strength of materials.
Consider first a section of beam without mass.
The state vector at position is
Therefore
Now must relate the slope and deflection at to
Deflection (L)  Angle (L)  

 


These are the socalled Myosotis Palustris formulae.
Therefore:
but
Therefore:
Also the slope
Therefore:
If we consider a point mass
If we have a spring
Therefore:
Therefore:
Therefore: is the equation of interest
Note that is the same
Therefore:
Therefore:
Where
let
Therefore:
Compare with a beam
Consider a Rayleigh approximation
assume
For forced steady state, assume all masses at
picture
Therefore acceleration is
Therefore:
Therefore:
picture
picture
Therefore:
Therefore:
Therefore:
Therefore:
When :
This approach to formulation and solution to vibration problems is generally called the impedance approach and is commonly used in electrical current systems. It can be further expanded to include damping. The approach is identical except that the matrices are complex as this allows the phase relations to be handled. (In the undamped case displacement/forces are either in phase or 180˚ out of phase).