Find damping coefficient at a given vibration mode

[highroller]

I am working on a motorcycle dynamics problem. I have written the equations of motion in matrix form in MATLAB (and mathcad). It is a 4 degree of freedom system, so I have a mass, damping and stiffness matrix all 4X4. I am able to find from these matrices the natural frequencies, and damping ratios. However I need to find the damping coefficients for each of the natural frequencies. Can I somehow convert my ratios to coefficients? Is there more matrix math that I need to do?

Any help would be greatly appreciated.

Thanks

 

[AlephZero]

I'm not sure what you mean by "damping coefficients" and "damping ratios". They sound like two different names for the same concept. If you look at the equations that use each of them, it should be clear how to get one from the other.

However, note that if you started with general mass stiffness and damping matrices (for example if the damping is from a device like a shock absorber), in general the damped mode shapes will be different from the undamped mode shapes, and different points in the damped modes will move with different relative phases.

Most of the "simple" theory of damped oscillations assumes the damping matrix has a mathematically convenient form where this doesn't happen. Effectively, that is equivalent to speciifying the damping coefficient (or ratio) for each undamped mode shape, rather than setting up a damping matrix. That means you only have 4 "independent" parameters to model the damping, not the 10 independent entries in a symmetrical 4x4 damping matrix.

 

[highroller]

Thanks for the reply.

I am trying to recreate the results of a paper and the author has specified damping coefficients for each mode rather than ratios, which is why I am interested in finding the coefficients.

Is there a way to simplify the equations in order for me to do what you described in your last paragraph.

 

[AlephZero]

The easiest way is use the mode shapes as generalized coordinates. Then the mass stiffness and damping matrices are all diagonal.

If the matrix of mass-normalized eigenvectors is $\Phi$, you have $$C_{\text{modal}} = \Phi^T C\, \Phi$$
so you can transform the modal damping matrix back to physical coordinates with $$C = \Phi^{-T} C_{\text{modal}} \Phi^{-1}$$
$C_{\text{modal}}$ is a diagonal matrix with entries like $2\beta_i \omega_i$ (where $\beta = 1$ for critical damping) and -T means the transpose of the inverse matrix.

 

[highroller]

Ok, I think that's what I'm looking for. Could you help me a little with finding the matrix of mass normalized eigenvectors. When I find the eigenvalues I get a vector of 8 values, some zero, some positive, some negative and some complex. How would I find the 4 vectors from them. What I have so far is shown in the attached pdf.

 

[AlephZero]

I don't understand your PDF. Why is the stiffness matrix not symmetric?

A non-symmetric stiffness means that if the system moves around a closed path back to the start point, energy is not conserved. That could mean your "stiffness" also includes some damping effects ...

The non-symmetric damping matrix is also non-physical - but that may or may not matter, depending what you plan to do with it.

Without seeing the original paper, I think it's hard to make any useful comments.

 

[highroller]

The equations were written by applying Lagrange's equation. There are 4 generalized coordinates and therefore 4 equations. I took these and collected terms of the generalized coordinates and their derivatives. I then put these coefficients into the corresponding matrices. Energy will not be conserved as I am dealing with frictional forces from the tire, which depend on positions of some coordinates, velocities of others, and accelerations too if relaxation length is considered. All I want to do is find the natural frequencies and the corresponding damping coefficients for each of the modes of the bike.

Here is a link to the paper... https://www3.imperial.ac.uk/pls/portallive/docs/1/42149703.PDF

 

[AlephZero]

Some people sure know how to make a "simple" modeling problem complicated!

I suspect this would be much easier if they had set up a FE model rather than working out everything with a symbolic algebra package, but (from what I've heard talking to a prof at Imperial, as it happens!) using a standard software package doesn't count for much as "original research".

Assuming they got the algebra right, this might help understand the quadratic eigenproblem
http://eprints.ma.man.ac.uk/466/01/38198.pdf

 

[gopi9]

I have a question related to this. I know that d^2<4mk for underdamped, d^2>4mk for overdamped and d^2=4mk for critically damped. This is true if there is only 1 mass and spring and damper. How to use this equation if I have 2 mass, 3 spring and 3 dampers. That is d,m,k are in 2x2 matrices. Please some one help me with this.