Question 6.4: In the modeling of a certain structure, the natural frequenc......
In the modeling of a certain structure, the natural frequencies of the first three normal modes, as calculated, and then verified by tests, were
f_{1} = 10.1 Hz, f_{2} = 17.8 Hz \mathrm{and} f_{3} = 25.3 Hz
The measured non-dimensional viscous damping coefficients of these modes, expressed as a fraction of critical damping were
\gamma _{1} = 0.018, \gamma _{2} = 0.027, \mathrm{and} \gamma _{3} = 0.035
respectively. If the modal vectors were scaled to make them orthonormal (so that the mass matrix is a unit matrix), write down the equations of motion.
It is assumed that the system, in normal mode coordinates, can be treated as three uncoupled, single-DOF, equations as follows:
\underline{m}_{ii} \ddot{q}_{i} + \underline{c}_{ii} \dot{q}_{i} + \underline{k}_{ii} q_{i} = Q_{i} (i = 1, 2, 3) (A)
where
\underline{m}_{ii} = 1. (B)
Using Eq. (2.16) and Eqs (2.18–2.19):
\omega_{n} = \sqrt{\frac{k}{m}} (2.16) \\ c_{c} = 2m\omega_{n} (2.18) \\ \gamma = \frac{c}{c_{c}} (2.19) \\ \underline{k}_{ii} = \underline{m}_{ii} \omega_{i}^{2} = \underline{m}_{ii} (2\pi f_{i})^{2} = (2\pi f_{i})^{2} (C)\\ c_{ii} = 2\gamma_{ii}\underline{m}_{ii} \omega_{i} = 4\pi \gamma_{ii} f_{i} (i=1,2,3) (D)where ω_{i} is the (undamped) natural frequency of mode i in rad/s, and f_{i} the corresponding natural frequency in Hz.
Table 6.1 gives the numerical values.
The equations of motion are therefore:
\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{bmatrix} \begin{Bmatrix} \ddot{q}_{1} \\ \ddot{q}_{2} \\ \ddot{q}_{3} \end{Bmatrix} + \begin{bmatrix} 2.28 & 0 & 0 \\ 0 & 6.04 & 0 \\ 0 & 0 & 11.13 \end{bmatrix} \begin{Bmatrix} \dot{q}_{1} \\ \dot{q}_{2} \\ \dot{q}_{3} \end{Bmatrix} + \begin{bmatrix} 4027 & 0 & 0 \\ 0 & 12508 & 0 \\ 0 & 0 & 25269 \end{bmatrix} \begin{Bmatrix} q_{1} \\ q_{2} \\ q_{3} \end{Bmatrix} = \begin{Bmatrix} Q_{1} \\ Q_{2} \\ Q_{3} \end{Bmatrix} (E)
| Table 6.1 Calculation of Stiffness and Damping Terms from Measured Data | ||||||
| Mode number i | Mode frequency f_{i}(Hz) | Generalized mass \underline{m}_{ii} | Damping (% Critical) | Damping coefficient (Fraction of critical) \gamma_{ii} | Stiffness term \underline{k}_{ii} | Damping term \underline{c}_{ii} |
| 1 | 10.1 | 1 | 1.8 | 0.018 | 4027 | 2.28 |
| 2 | 17.8 | 1 | 2.7 | 0.027 | 12 508 | 6.04 |
| 3 | 25.3 | 1 | 3.5 | 0.035 | 25 269 | 11.13 |