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I'm using modal decomposition to predict the steady state response of a beam structure to harmonic loading. The structure itself is very lightly damped, but we know from experiments that the aerodynamic drag force as it moves through the air is very significant.

I have the mass and stiffness matrices for the beam ($\boldsymbol{[M]}$ and $\boldsymbol{[K]}$) and so I am able to obtain the eigenvectors and eigenvalues ($\boldsymbol{[V]}$ and $\boldsymbol{[D]}$). The eigenvalues yields a vector containing the natural frequencies of the system (the square root of the diagonal entries of $\boldsymbol{[D]}$, I'll call it $\boldsymbol{\omega}$.

The system is excited by a force vector in the global reference frame $\boldsymbol{F_0}$ at an excitation frequency $\omega_F$.

There is a rotation matrix which rotates the beams from the local to the global reference frame $\boldsymbol{[R]}$.

I have successfully used the procedure to calculate the steady state response without aerodynamic damping:

  • Get eigenvalues and eigenvectors $\boldsymbol{[V]}$ and $\boldsymbol{[D]}$
  • Get generalised force vector $\boldsymbol{Q}=\boldsymbol{[V]}^T \boldsymbol{F_0}$
  • Calculate modal participation factor vector $\boldsymbol{q}$. The individual elements are calculated as follows:

$$q_i=\frac{Q_i}{{\omega_i}^2}\frac{1}{\sqrt{[(1-(\omega_F/\omega_i)^2)^2+(2\zeta_i\omega_F/\omega_i)^2]}}$$

  • Calculate response from $\boldsymbol{x}=\boldsymbol{[V]q}$

Of course, this drastically overpredicts the amplitude of the oscillations because there is no aerodynamic damping. I've found a few examples for single d.o.f. systems in which the energy lost from the system is calculated (by integrating the drag force x the velocity). The integration results in the following equation for energy loss: $$\Delta E=\frac{8}{3}\frac{1}{2}\rho A C_D \omega^2 X^3$$ This can then be converted to an equivalent damping coefficient as follows: $$C_{eq}=\frac{\Delta E}{\pi \omega X^2}$$ Which can be converted to a damping coefficient by doing: $$\zeta=\frac{C_{eq}}{2 \omega_n}$$ I've modelled a simple spring-mass system with air resistance and it works fine (I have to iterate to find the final position because the amplitude calculation requires knowledge of the damping which requires knowledge of the amplitude so there is a circular reference, but that's no biggy!)

Could someone please explain the equivalent for a multi-d.o.f. system? My initial forays have been unsuccessful, with much higher drag coefficients than would really occur.

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