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Natural Science Forum / Physics / Acoustics / May 2007multi DOF eigenvalues
Say I have a 3 DOF system and I find the eigenvalues of the 3x3 stiffness/mass matrix (no damping). The eigenvalue solution gives me 3 complex numbers. As I understand it the square root of the imaginary part of the solution gives me the natural frequency (in radians/sec). Do these map back to the physical system in the obvious way, i.e. the first eigenvalue is the DOF described by the top and leftmost mass and stiffness in the matrix? 2nd eigenvalue is the natural frequency of the middle row/column of my 3x3 matrix? On 5/4/07 7:15 AM, in article EdH_h.6$s7.5@trndny05, "zed" <zed@nospam.org> wrote: > Say I have a 3 DOF system and I find the eigenvalues of the 3x3 > stiffness/mass matrix (no damping). The eigenvalue solution gives me 3 [quoted text clipped - 4 lines] > in the matrix? 2nd eigenvalue is the natural frequency of the middle > row/column of my 3x3 matrix? There are a number of problems with your post. You do not explain what you are getting at very well. It took me a while to figure out that DOF stood for degrees of freedom. You said that your eigenvalues were complex. An engineer or physicist would probably not use that terminology if the eigenvalues were pure real or imaginary even thought those are special cases of complex numbers. If there is no damping, there would be no pure real or imaginary eigenvalues. My background is in electrical engineering rather than acoustics. Maybe there is some jargon in acoustics that is different. I presume that your matrix is a symmetrical impedance matrix with purely imaginary terms. When you go through the process in detail, you will probably end up with a characteristic equation for which the roots are negative squares of frequency. That will give you eigenvectors of the form exp(jwt). Each eigenvalue will give a resonant frequency while the eigenvector will tell you the relative amplitude of the vibration at that frequency. Bill -- Fermez le Bush--about two years to go. > Say I have a 3 DOF system and I find the eigenvalues of the 3x3 > stiffness/mass matrix (no damping). The eigenvalue solution gives me [quoted text clipped - 4 lines] > leftmost mass and stiffness in the matrix? 2nd eigenvalue is the > natural frequency of the middle row/column of my 3x3 matrix? The eigenvalue solution gives me > 3 complex numbers. it shouldn't the square root of the > imaginary part of the solution gives me the natural frequency (in > radians/sec) it shouldn't Either you have formulated your problem very strangely, or you have done something wrong. The eigenvalues should be real for this problem. >> Say I have a 3 DOF system and I find the eigenvalues of the 3x3 >> stiffness/mass matrix (no damping). The eigenvalue solution gives me [quoted text clipped - 20 lines] > > The eigenvalues should be real for this problem. The characteristic equation should be [lambda K - M]X=0, usually reaaranged to det[lambda I-D]=0 Rao section 6.8 |
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