Short version:

Suppose I excite a vibrational response in some structure and measure its mode shapes and natural frequencies. Is it possible to create a structural dynamic model of the structure using only that measured data?

Long version:

A structural finite element model of a structure can be represented with the following equation (neglecting damping).

$$ M\ddot{x} + Kx = F $$

where $M$ is the mass matrix, $K$ is the stiffness matrix, and $F$ is the applied force.

Solving the eigenvalue problem to find the natural frequencies and mode shapes, making the substitution $x=\phi q$, and pre-multiplying by $\phi^T$ gives the following.

$$ \phi^T M \phi \ddot{q} + \phi^T K \phi q = \phi^T F $$

where $\phi$ is the matrix of mode shapes and $q$ is the modal or generalized displacement.

It's common to mass-normalize the mode shapes such that $\phi^T M \phi$ equals the identity matrix $(I)$ and $\phi^T K \phi$ contains the square of the natural frequencies on the diagonal, with zero terms everywhere else. This results in an uncoupled system of equations, where each mode responds as a single degree of freedom mass/spring system. This equation is easy to numerically integrate to calculate the transient response to an arbitrary applied force.

Suppose I don't have a finite element model of the structure, but instead experimentally measure the mode shapes and natural frequencies. I can easily form $\phi^T K \phi$ by plugging in the measured frequencies, and the $\phi^T$ term on the right hand side is measured directly.

I'm still missing the $\phi^T M \phi$ mass term. When starting with a finite element model, it's only equal to $I$ because of the way the mode shapes were scaled. Is there a way it can be calculated experimentally when no model of the structure is available?

I have an additional experimental data point - the force applied when exciting the structure to measure the modes. Can that be used to calculate the modal mass terms?

  • 1
    $\begingroup$ Sort of relevant and interesting: can you hear the shape of a drum? $\endgroup$
    – DanielSank
    Aug 4 '15 at 17:28
  • $\begingroup$ Isn't $\phi^\intercal K \phi$ a matrix with the natural frequencies in the diagonals. So you can construct this experimentally by finding said natural frequencies. $\endgroup$ Aug 4 '15 at 18:38
  • $\begingroup$ @ja72 That's correct, but I'm asking about the mass matrix ($\phi^T M \phi$). $\endgroup$
    – joshayers
    Aug 4 '15 at 20:07
  • $\begingroup$ I think I did this in college with an impact hammer connected to a computer which reconstructed the modal shapes. See youtube.com/watch?v=tBRjPN8m6zE $\endgroup$ Aug 4 '15 at 20:10

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