振动信号处理模态分析频响函数生成参数后处理

# Experimental Modal Analysis (Impact Testing) * <table><tr><td>（编辑公式时，如果公式中存在下划线前有空格，那么必须统一都包含空格，否则会出现无法显示的情况。）</td></tr></table> * <table><tr><td>（想显示 $ \lbrace \rbrace $，得用 \lbrace \rbrace 命令）</td></tr></table> Measured from the tool tip in axial $(z)$ direction, the approximate impact locations of the $5$ points are $0$, $55$, $90$, $130$ and $165$ mm. The vibration is measured at the tool tip (Point $1$) by an accelerometer. <div align = "center"> <img src = "Tool.png" width = "250" height = "200" alt = "Points on the Tool" title = "Points on the Tool"> </div> <p align = "center"><b>Points on the Tool</b></p> In the five .txt files, the first column of each file is the frequency $\text{[Hz]}$, second column is the real part of the computed $FRF$ $\text{[m/N]}$, and the third column is the imaginary part of the computed $FRF$ $\text{[m/N]}$. For multi-degree freedom systems, the equation of motion in local coordinate system is given as: $$m\ddot x + c\dot x + kx = F $$ or in matrix from: $${[{M_x}]{\lbrace \ddot x \rbrace } + {[{C_x}] \lbrace \dot x \rbrace} + [ {{K_x}} ]\lbrace x \rbrace = \lbrace F \rbrace}$$ These equations are coupled and are cumbersome to solve for more than two degree of freedom systems, thus they are converted to modal coordinates as: $$[ {{M_q}} ]\lbrace {\ddot q} \rbrace + [ {{C_q}} ]\lbrace {\dot q} \rbrace + [ {{K_q}} ]\lbrace q \rbrace = \lbrace R \rbrace$$ Where the modal mass, modal damping and model stiffness are separately: $${[ M _q] _k} = \lbrace P \rbrace _k^T[ {{M_x}} ]{\lbrace P \rbrace _k}$$ $${[ C _q] _k} = \lbrace P \rbrace _k^T[ {{C_x}} ]{\lbrace P \rbrace _k}$$ $${[ K _q] _k} = \lbrace P \rbrace _k^T[ {{K_x}} ]{\lbrace P \rbrace _k}$$ And $\lbrace P \rbrace _k $ is the Eigenvector (mode shapes) of $k^{th}$ order with a structure like: $$ \lbrace P \rbrace _k = \begin{pmatrix} {p _{1}}\\\\ {p _{2}}\\\\ {p _{3}}\\\\ {\vdots }\\\\ {P _{n}} \end{pmatrix} _k $$ The Frequency Response Function ($FRF$) of the system can be write as: $$ [ {H(s)} ] = \frac{{\lbrace {X(s)} \rbrace}}{{\lbrace {F(s)} \rbrace}} $$ The element in row $i$ and column $l$ of $\left[ {H\left( s \right)} \right]$ matrix may be given as residues form: $$ {h_{il}} = \sum\limits_{k = 1}^n {\left( {\frac{{{\alpha _{il,k}} + {\beta _{il,k}}s}}{{{s^2} + 2{\xi _k}{\omega _{n,k}}s + \omega _{n,k}^2}}} \right)} $$ Where $n$ – number of modes; $\omega _{n,k}$ and $\zeta _k$ are the undamped natural frequencies and the modal damping ratio for mode $k$ of the system. The mode shapes are found from the estimated residues. Displacement vector can be expressed by its mode shapes $\lbrace P \rbrace_k $ and modal transfer functions $ \Phi _{q,k} $. $$\lbrace X \rbrace = \left( {\sum\limits_{k = 1}^n {{{\lbrace P \rbrace}_k}\lbrace P \rbrace_k^T {\Phi _{qk}}} } \right)\lbrace F \rbrace$$ Thus, $$ {[H(s)]} = {\sum\limits _{k = 1}^n {{\frac{{\lbrace P \rbrace _k} {\lbrace P \rbrace _k^T}}{m _{q,k}}}{\frac{1}{{{s^2}+ {2{\zeta _k}{\omega _{n,k}}s} + {\omega _{n,k} ^2}}}}}} = {\sum\limits _{k = 1}^n {\frac{[R] _k}{s^2 + 2{\zeta _k}{\omega _{n,k}}s + \omega _{n,k}^2 }}} $$ Note that the modal mass for mode $k$ using the unscaled modal matrix is: $$ {m _{q,k}} = \lbrace P \rbrace _k^T[ {{M _x}} ]{\lbrace P \rbrace _k} $$ Thus $ ({\lbrace P \rbrace _k^T{\lbrace P \rbrace _k}})/{m _{q,k}} $ represents the normalization of each eigenvector with the square root of the modal mass. $$ {\frac{\lbrace P \rbrace _k^T}{\sqrt {m _{q,k}}}}{[M _x]}{\frac{\lbrace P _k \rbrace}{\sqrt {m _{q,k}}}} = 1 $$ $$ \lbrace u \rbrace _k^T[ {{M _x}} ]{\lbrace u \rbrace _k} = 1 $$ This is a convenient way to identify the modal parameters, i.e. mode shapes, modal stiffness and modal damping of the structure. Where ${\lbrace u \rbrace _k}$ corresponds to the normalized mode shape giving a unity modal mass, whit a structure like: $$ \lbrace u \rbrace _k = \begin{pmatrix} {u _{1}}\\\\ {u _{2}}\\\\ {u _{3}}\\\\ {\vdots }\\\\ {u _{n}} \end{pmatrix} _k $$ $$ \frac{\lbrace P \rbrace _k \lbrace P \rbrace _k^T }{m _{ q,k}} = \lbrace u \rbrace _k \lbrace u \rbrace _k^T = [R] _k $$ Where, the residue matrix for a particular mode $k$ can be expressed in the following general form: $$[ R ]_k = \begin{bmatrix}{u_1}{u_1} & {u_1}{u_2} & \cdots & {u_1}{u_l} & \cdots & {u_1}{u_n}\\\\ {{u_2}{u_1}} & {{u_2}{u_2}} & \cdots & {u_2}{u_l} & \cdots & {{u_2}{u_n}}\\\\ \vdots & \vdots & \ddots & \vdots & \vdots & \vdots\\\\ {{u_l}{u_1}} & {{u_l}{u_2}} & \cdots & {u_l}{u_l} & \cdots & {{u_l}{u_n}}\\\\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots\\\\ {{u_n}{u_1}} & {{u_n}{u_2}} & \cdots & {u_n}{u_l} & \cdots & {{u_n}{u_n}}\end{bmatrix}_k$$ If we take row $l$ of the residue matrix for mode $k$, we have: $$ \begin{pmatrix} {R_{l1}}&{R_{l2}}& \cdots &{R_{ll}}& \cdots &{R_{ln}} \end{pmatrix}_k = \begin{pmatrix} {u_l}{u_1}& {u_l}{u_2}& & \cdots & {u_l}{u_l}& & \cdots & {u_l}{u_n} \end{pmatrix}_k $$ Where, $k=1,2,…, n$ for n number of modes. In our case, when we choose move the hammer to impact all the point on the tool-holder combination and measure the vibration at point $1$ where the accelerometer is mounted. The residue matrix for specific modes will be of the form: $$ \begin{pmatrix} {R_{11}}& {R_{12}}& {R_{13}}& {R_{14}}& {R_{15}} \end{pmatrix}_k = \begin{pmatrix} {u_1}{u_1}& {u_1}{u_2}& {u_1}{u_3}& {u_1}{u_4}& {u_1}{u_5} \end{pmatrix}_k $$ The transfer function ${\Phi _{11}}$is measured by hitting the structure at point $1$ and measuring at point $1$, i.e. where the accelerometer is mounted. This is known as the direct transfer function. The transfer function ${\Phi _{12}}$ is measured by hitting the structure at point $2$ and measuring at point $1$. This is a cross transfer function. $$ \Phi = G + jH $$ <b> Where, the $u_{11}u_{11}$ means the direct displacement response residues of point $1$ contributed by the first mode. E.g. for $u_{11}$, First subscript denote the measured (for left $u$) or impact point (for right $u$), and Second subscript denote the modal number.</b> If let $ s \to j\omega $ $$ {\Phi _{11}} = {\left( {\frac{{{u _{11}}{u _{11}}}}{{{\omega ^2} + j2{\zeta _1}{\omega _{n1}}\omega - \omega _{n1}^2}}} \right) _{mode\ 1}} + {\left( {\frac{{{u _{12}}{u _{12}}}}{{{\omega ^2} + j2{\zeta _2}{\omega _{n2}}\omega - \omega _{n2}^2}}} \right) _{mode\ 2}} + {\left( {\frac{{{u _{13}}{u _{13}}}}{{{\omega ^2} + j2{\zeta _3}{\omega _{n3}}\omega - \omega _{n3}^2}}} \right) _{mode\ 3}} $$ When $ω=ω_{n1}$ , leads to a negligible contribution from $ω_{n2}$ and $ω_{n3}$, and the first part approximately equal to: $$ {\Phi _{11}}\left( {\omega = {\omega _{n1}}} \right) \approx {\left( {\frac{{{u _{11}}{u _{11}}}}{{\omega _{n1}^2 + j2{\zeta _1}{\omega _{n1}}{\omega _{n1}} - \omega _{n1}^2}}} \right) _{mode\ 1}} $$ So we have: $$ {\Phi _{11}}\left( {\omega = {\omega _{n1}}} \right) \approx j{H _{11,1}} = {\left( {\frac{{{u _{11}}{u _{11}}}}{{j2{\zeta _1}\omega _{n1}^2}}} \right) _{mode\ 1}} $$ $$ {u _{11}} = \sqrt { - 2{\zeta _1}\omega _{n1}^2{H _{11,1}}} $$ Similarly, when $ω=ω_{n2}$ and $ω=ω_{n3}$, $$ {\Phi _{11}}\left( {\omega = {\omega _{n2}}} \right) \approx j{H _{11,2}} = {\left( {\frac{{{u _{12}}{u _{12}}}}{{j2{\zeta _2}\omega _{n2}^2}}} \right) _{mode\ 2}} $$ $$ {u _{12}} = \sqrt { - 2{\zeta _2}\omega _{n2}^2{H _{11,2}}} $$ $$ {\Phi _{11}}\left( {\omega = {\omega _{n3}}} \right) \approx j{H _{11,3}} = {\left( {\frac{{{u _{13}}{u _{13}}}}{{j2{\zeta _3}\omega _{n3}^2}}} \right) _{mode\ 3}} $$ $$ {u _{13}} = \sqrt { - 2{\zeta _3}\omega _{n3}^2{H _{11,3}}} $$ Similarly, $$ {\Phi _{12}} = {\left( {\frac{{{u _{11}}{u _{21}}}}{{{s^2} + 2{\zeta _1}{\omega _{n1}}s + \omega _{n1}^2}}} \right) _{mode\ 1}} + {\left( {\frac{{{u _{12}}{u _{22}}}}{{{s^2} + 2{\zeta _2}{\omega _{n2}}s + \omega _{n2}^2}}} \right) _{mode\ 2}} + {\