Modal Analysis - Derivation of Mathematical Models
+Table of Contents
+ +1 Type of Model
++The model that we want to obtain is a multi-body model. +It is composed of several solid bodies connected with springs and dampers. +The solid bodies are represented with different colors on figure 1. +
+ ++In the simscape model, the solid bodies are: +
+-
+
- the granite (1 or 2 solids) +
- the translation stage +
- the tilt stage +
- the spindle and slip-ring +
- the hexapod +
+
Figure 1: CAD view of the ID31 Micro-Station
++However, each of the DOF of the system may not be relevant for the modes present in the frequency band of interest. +For instance, the translation stage may not vibrate in the Z direction for all the modes identified. Then, we can block this DOF and this simplifies the model. +
+ ++The modal identification done here will thus permit us to determine which DOF can be neglected. +
+2 TODO Extract Physical Matrices
++wang11_extrac_real_modes_physic_matric +
+ ++Let's recall that: +\[ \Lambda = \begin{bmatrix} + s_1 & & 0 \\ + & \ddots & \\ + 0 & & s_N +\end{bmatrix}_{N \times N}; \quad \Psi = \begin{bmatrix} + & & \\ + \{\psi_1\} & \dots & \{\psi_N\} \\ + & & +\end{bmatrix}_{M \times N} ; \quad A = \begin{bmatrix} + a_1 & & 0 \\ + & \ddots & \\ + 0 & & a_N +\end{bmatrix}_{N \times N}; \] +
+ + +\begin{align} + M &= \frac{1}{2} \left[ \text{Re}(\Psi A^{-1} \Lambda \Psi^T ) \right]^{-1} \\ + C &= -2 M \text{Re}(\Psi A^{-1} \Lambda^2 A^{-1} \Psi^T ) M \\ + K &= -\frac{1}{2} \left[ \text{Re}(\Psi \Lambda^{-1} A^{-1} \Psi^T) \right]^{-1} +\end{align} + +psi = eigen_vec_CoM; +a = modal_a_M; +lambda = eigen_val_M; + +M = 0.5*inv(real(psi*inv(a)*lambda*psi')); +C = -2*M*real(psi*inv(a)*lambda^2*inv(a)*psi')*M; +K = -0.5*inv(real(psi*inv(lambda)*inv(a)*psi')); ++
+From ewins00_modal +
+ +\begin{align} + [M] &= [\Phi]^{-T} [I] [\Phi]^{-1} \\ + [K] &= [\Phi]^{-T} [\lambda_r^2] [\Phi]^{-1} +\end{align} +3 Some notes about constraining the number of degrees of freedom
++We want to have the two eigen matrices. +
+ ++They should have the same size \(n \times n\) where \(n\) is the number of modes as well as the number of degrees of freedom. +Thus, if we consider 21 modes, we should restrict our system to have only 21 DOFs. +
+ ++Actually, we are measured 6 DOFs of 6 solids, thus we have 36 DOFs. +
+ ++From the mode shapes animations, it seems that in the frequency range of interest, the two marbles can be considered as one solid. +We thus have 5 solids and 30 DOFs. +
+ + ++In order to determine which DOF can be neglected, two solutions seems possible: +
+-
+
- compare the mode shapes +
- compare the FRFs +
+The question is: in which base (frame) should be express the modes shapes and FRFs? +Is it meaningful to compare mode shapes as they give no information about the amplitudes of vibration? +
+ + +| Stage | +Motion DOFs | +Parasitic DOF | +Total DOF | +Description of DOF | +
|---|---|---|---|---|
| Granite | +0 | +3 | +3 | ++ |
| Ty | +1 | +2 | +3 | +Ty, Rz | +
| Ry | +1 | +2 | +3 | +Ry, | +
| Rz | +1 | +2 | +3 | +Rz, Rx, Ry | +
| Hexapod | +6 | +0 | +6 | +Txyz, Rxyz | +
| + | 9 | +9 | +18 | ++ |
+ +
Bibliography
+- [wang11_extrac_real_modes_physic_matric] Tong Wang, Lingmi Zhang & Kong Fah Tee, Extraction of Real Modes and Physical Matrices From Modal Testing, Earthquake Engineering and Engineering Vibration, 10(2), 219-227 (2011). link. doi. +
- [ewins00_modal] Ewins, Modal testing: theory, practice and application, Wiley-Blackwell (2000). +