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@@ -221,7 +221,7 @@ This process itself falls into two stages:
Most of the effort goes into this second stage, which is widely referred to as "modal parameter extraction", or simply as "modal analysis".
We have seen that we can predict the form of the FRF plots for a multi degree-of-freedom system, and that these are directly related to the modal properties of that system.
The great majority of the modal analysis effort involves **curve-fitting** an expression such as equation <eq:frf_modal> to the measured FRF and thereby finding the appropriate modal parameters.
The great majority of the modal analysis effort involves **curve-fitting** an expression such as equation \eqref{eq:frf\_modal} to the measured FRF and thereby finding the appropriate modal parameters.
A completely general curve-fitting approach is possible but generally inefficient.
Mathematically, we can take an equation of the form
@@ -477,11 +477,11 @@ where \\(\eta\\) is the **structural damping loss factor** and replaces the crit
#### Alternative Forms of FRF {#alternative-forms-of-frf}
So far we have defined our receptance frequency response function \\(\alpha(\omega)\\) as the ratio between a harmonic displacement response and the harmonic force <eq:receptance>.
So far we have defined our receptance frequency response function \\(\alpha(\omega)\\) as the ratio between a harmonic displacement response and the harmonic force \eqref{eq:receptance}.
This ratio is complex: we can look at its **amplitude** ratio \\(|\alpha(\omega)|\\) and its **phase** angle \\(\theta\_\alpha(\omega)\\).
We could have selected the response velocity \\(v(t)\\) as the output quantity and defined an alternative frequency response function <eq:mobility>.
Similarly we could use the acceleration parameter so we could define a third FRF parameter <eq:inertance>.
We could have selected the response velocity \\(v(t)\\) as the output quantity and defined an alternative frequency response function \eqref{eq:mobility}.
Similarly we could use the acceleration parameter so we could define a third FRF parameter \eqref{eq:inertance}.
<div class="definition">
@@ -588,7 +588,7 @@ This type of display is not widely used as we cannot use logarithmic axes (as we
##### Real part and Imaginary part of reciprocal FRF {#real-part-and-imaginary-part-of-reciprocal-frf}
It can be seen from the expression of the inverse receptance <eq:dynamic_stiffness> that the Real part depends entirely on the mass and stiffness properties while the Imaginary part is a only function of the damping.
It can be seen from the expression of the inverse receptance \eqref{eq:dynamic\_stiffness} that the Real part depends entirely on the mass and stiffness properties while the Imaginary part is a only function of the damping.
[ 5](#org-target--fig-inverse-frf-mixed) shows an example of a plot of a system with a combination of both viscous and structural damping. The imaginary part is a straight line whose slope is given by the viscous damping rate \\(c\\) and whose intercept at \\(\omega = 0\\) is provided by the structural damping coefficient \\(d\\).
@@ -629,7 +629,7 @@ This makes the Nyquist plot very effective for modal testing applications.
#### Free Vibration Solution - The modal Properties {#free-vibration-solution-the-modal-properties}
For an undamped MDOF system, with \\(N\\) degrees of freedom, the governing equations of motion can be written in matrix form <eq:undamped_mdof>.
For an undamped MDOF system, with \\(N\\) degrees of freedom, the governing equations of motion can be written in matrix form \eqref{eq:undamped\_mdof}.
<div class="important">
@@ -645,7 +645,7 @@ where \\([M]\\) and \\([K]\\) are \\(N\times N\\) mass and stiffness matrices, a
We shall consider first the free vibration solution by taking \\(f(t) = 0\\).
In this case, we assume that a solution exists of the form \\(\\{x(t)\\} = \\{X\\} e^{i \omega t}\\) where \\(\\{X\\}\\) is an \\(N \times 1\\) vector of time-independent amplitudes.
Substitution of this condition into <eq:undamped_mdof> leads to
Substitution of this condition into \eqref{eq:undamped\_mdof} leads to
\begin{equation} \label{eq:free\_eom\_mdof}
\left( [K] - \omega^2 [M] \right) \\{X\\} e^{i\omega t} = \\{0\\}
@@ -655,7 +655,7 @@ for which the non trivial solutions are those which satisfy
\\[ \det \left| [K] - \omega^2 [M] \right| = 0 \\]
from which we can find \\(N\\) values of \\(\omega^2\\) corresponding to the undamped system's **natural frequencies**.
Substituting any of these back into <eq:free_eom_mdof> yields a corresponding set of relative values for \\(\\{X\\}\\): \\(\\{\psi\\}\_r\\) the so-called **mode shape** corresponding to that natural frequency.
Substituting any of these back into \eqref{eq:free\_eom\_mdof} yields a corresponding set of relative values for \\(\\{X\\}\\): \\(\\{\psi\\}\_r\\) the so-called **mode shape** corresponding to that natural frequency.
<div class="important">
@@ -792,7 +792,7 @@ An alternative means of deriving the FRF parameters is used which makes use of t
\\[ [K] - \omega^2 [M] = [\alpha(\omega)]^{-1} \\]
Pre-multiply both sides by \\([\Phi]^T\\) and post-multiply both sides by \\([\Phi]\\) to obtain
\\[ [\Phi]^T ([K] - \omega^2 [M]) [\Phi] = [\Phi]^T [\alpha(\omega)]^{-1} [\Phi] \\]
which leads to <eq:receptance_modal>.
which leads to \eqref{eq:receptance\_modal}.
<div class="important">
@@ -802,7 +802,7 @@ Receptance FRF matrix - Modal Properties;
[\alpha(\omega)] = [\Phi] \left[ \bar{\omega}\_r^2 - \omega^2 \right]^{-1} [\Phi]^T \label{eq:receptance\_modal}
\end{equation}
Equation <eq:receptance_modal> permits us to compute any individual FRF parameters \\(\alpha\_{jk}(\omega)\\) using the following formula
Equation \eqref{eq:receptance\_modal} permits us to compute any individual FRF parameters \\(\alpha\_{jk}(\omega)\\) using the following formula
\begin{align}
\alpha\_{jk}(\omega) &= \sum\_{r=1}^N \frac{\phi\_{jr} \phi\_{kr}}{\bar{\omega}\_r^2 - \omega^2}\\\\
@@ -816,7 +816,7 @@ where \\({}\_rA\_{jk}\\) is called the **modal constant**.
<div class="important">
It is clear from equation <eq:receptance_modal> that the receptance matrix \\([\alpha(\omega)]\\) is **symmetric** and this will be recognized as the **principle of reciprocity**.
It is clear from equation \eqref{eq:receptance\_modal} that the receptance matrix \\([\alpha(\omega)]\\) is **symmetric** and this will be recognized as the **principle of reciprocity**.
This principle of reciprocity applies to many structural characteristics.
@@ -954,7 +954,7 @@ From this full matrix equation, we have:
Having derived an expression for the general term in the frequency response function matrix \\(\alpha\_{jk}(\omega)\\), it is appropriate to consider next the analysis of a situation where the system is **excited simultaneously at several points**.
The general behavior for this case is governed by equation <eq:force_response_eom> with solution <eq:force_response_eom_solution>.
The general behavior for this case is governed by equation \eqref{eq:force\_response\_eom} with solution \eqref{eq:force\_response\_eom\_solution}.
However, a more explicit form of the solution is
\begin{equation} \label{eq:ods}
@@ -977,7 +977,7 @@ The properties of the normal modes of the undamped system are of interest becaus
</div>
We are seeking an excitation vector \\(\\{F\\}\\) such that the **response** \\(\\{X\\}\\) **consists of a single modal component** so that all terms in <eq:ods> but one is zero.
We are seeking an excitation vector \\(\\{F\\}\\) such that the **response** \\(\\{X\\}\\) **consists of a single modal component** so that all terms in \eqref{eq:ods} but one is zero.
This can be attained if \\(\\{F\\}\\) is chosen such that
\\[ \\{\phi\_r\\}^T \\{F\\}\_s = 0, \ r \neq s \\]
@@ -1059,7 +1059,7 @@ where \\(\omega\_r\\) is the **natural frequency** and \\(\xi\_r\\) is the **cri
When the modes \\(r\\) and \\(q\\) are a complex conjugate pair:
\\[ s\_r = \omega\_r \left( -\xi\_r - i\sqrt{1 - \xi\_r^2} \right); \quad \\{\psi\\}\_q = \\{\psi\\}\_r^\* \\]
From equations <eq:viscous_damping_orthogonality>, we can obtain
From equations \eqref{eq:viscous\_damping\_orthogonality}, we can obtain
\begin{align}
2 \omega\_r \xi\_r &= \frac{\\{\psi\\}\_r^H [C] \\{\psi\\}\_r}{\\{\psi\\}\_r^H [M] \\{\psi\\}\_r} = \frac{c\_r}{m\_r} \\\\
@@ -1352,7 +1352,7 @@ One these two series are available, the FRF can be defined at the same set of fr
##### Analysis via Fourier transform {#analysis-via-fourier-transform}
For most transient cases, the input function \\(f(t)\\) will satisfy the **Dirichlet condition** and so its Fourier Transform \\(F(\omega)\\) can be computed from <eq:fourier_transform>.
For most transient cases, the input function \\(f(t)\\) will satisfy the **Dirichlet condition** and so its Fourier Transform \\(F(\omega)\\) can be computed from \eqref{eq:fourier\_transform}.
\begin{equation} \label{eq:fourier\_transform}
F(\omega) = \frac{1}{2 \pi} \int\_{-\infty}^\infty f(t) e^{i\omega t} dt
@@ -1500,10 +1500,10 @@ However, the same equation can be transform to the frequency domain
\tcmbox{ S\_{xx}(\omega) = \left| H(\omega) \right|^2 S\_{ff}(\omega) }
\end{equation}
Although very convenient, equation <eq:psd_input_output> does not provide a complete description of the random vibration conditions.
Although very convenient, equation \eqref{eq:psd\_input\_output} does not provide a complete description of the random vibration conditions.
Further, it is clear that **is could not be used to determine the FRF** from measurement of excitation and response because it **contains only the modulus** of \\(H(\omega)\\), the phase information begin omitted from this formula.
A second equation is required and this may be obtain by a similar analysis, two alternative formulas can be obtained <eq:cross_relation_alternatives>.
A second equation is required and this may be obtain by a similar analysis, two alternative formulas can be obtained \eqref{eq:cross\_relation\_alternatives}.
<div class="important">
@@ -1517,8 +1517,8 @@ A second equation is required and this may be obtain by a similar analysis, two
##### To derive FRF from random vibration signals {#to-derive-frf-from-random-vibration-signals}
The pair of equations <eq:cross_relation_alternatives> provides the basic of determining a system's FRF properties from the measurements and analysis of a random vibration test.
Using either of them, we have a simple formula for determining the FRF from estimates of the relevant spectral densities <eq:H1> <eq:H2>.
The pair of equations \eqref{eq:cross\_relation\_alternatives} provides the basic of determining a system's FRF properties from the measurements and analysis of a random vibration test.
Using either of them, we have a simple formula for determining the FRF from estimates of the relevant spectral densities \eqref{eq:H1} \eqref{eq:H2}.
<div class="important">
@@ -1555,7 +1555,7 @@ Then in [ 13](#org-target--fig-frf-feedback-model) is given a more detailed and
| width=\linewidth | width=\linewidth |
In this configuration, it can be seen that there are two feedback mechanisms which apply.
We then introduce an alternative formula which is available for the determination of the system FRF from measurements of the input and output quantities <eq:H3>.
We then introduce an alternative formula which is available for the determination of the system FRF from measurements of the input and output quantities \eqref{eq:H3}.
<div class="important">
@@ -1694,7 +1694,7 @@ First, if we have a **modal incompleteness** (\\(m<N\\) modes included), then we
However, if we have **spatial incompleteness** (only \\(n<N\\) DOFs included), then we cannot express any orthogonality properties at all because the eigenvector matrix is not commutable with the system mass and stiffness matrices.
In both reduced-model cases, it is not possible to use equation <eq:spatial_model_from_modal> to re-construct the system mass and stiffness matrices.
In both reduced-model cases, it is not possible to use equation \eqref{eq:spatial\_model\_from\_modal} to re-construct the system mass and stiffness matrices.
First of all because the eigen matrices are generally singular and even if it is not, the obtained mass and stiffness matrices produced have no physical significance and should not be used.
@@ -2699,7 +2699,7 @@ Then, the full \\(6 \times 6\\) mobility matrix can be measured, however this pr
Other methods for measuring rotational effects include specially developed rotational accelerometers and shakers.
However, there is a major problem that is encountered when measuring rotational FRF: the translational components of the structure's movement tends to overshadow those due to the rotational motions.
For example, the magnitude of the difference in equation <eq:rotational_diff> is often of the order of \\(\SI{1}{\\%}\\) of the two individual values which is similar to the transverse sensitivity of the accelerometers: potential errors in rotations are thus enormous.
For example, the magnitude of the difference in equation \eqref{eq:rotational\_diff} is often of the order of \\(\SI{1}{\\%}\\) of the two individual values which is similar to the transverse sensitivity of the accelerometers: potential errors in rotations are thus enormous.
### Multi-point excitation methods {#multi-point-excitation-methods}
@@ -2738,7 +2738,7 @@ The two vectors are related by the system's FRF properties as:
\\{X\\}\_{n\times 1} = [H(\omega)]\_{n\times p} \\{F\\}\_{p\times 1}
\end{equation}
However, it is not possible to derive the FRF matrix from the single equation <eq:mpss_equation>, because there will be insufficient data in the two vectors (one of length \\(p\\), the other of length \\(n\\)) to define completely the \\(n\times p\\) FRF matrix.
However, it is not possible to derive the FRF matrix from the single equation \eqref{eq:mpss\_equation}, because there will be insufficient data in the two vectors (one of length \\(p\\), the other of length \\(n\\)) to define completely the \\(n\times p\\) FRF matrix.
What is required is to make a series of \\(p^\prime\\) measurements of the same basic type using different excitation vectors \\(\\{F\\}\_i\\) that should be chosen such that the forcing matrix \\([F]\_{p\times p^\prime} = [\\{F\\}\_1, \dots, \\{F\\}\_p]\\) is non-singular.
This can be assured if:
@@ -3226,13 +3226,13 @@ For any frequency \\(\omega\\), we have the following relationship:
\end{aligned}
\end{equation}
From <eq:modal_circle_tan>, we obtain:
From \eqref{eq:modal\_circle\_tan}, we obtain:
\begin{equation} \label{eq:modal\_circle\_omega}
\omega^2 = \omega\_r^2 \left(1 - \eta\_r \tan\left(\frac{\theta}{2}\right) \right)
\end{equation}
If we differentiate <eq:modal_circle_omega> with respect to \\(\theta\\), we obtain:
If we differentiate \eqref{eq:modal\_circle\_omega} with respect to \\(\theta\\), we obtain:
\begin{equation}
\frac{d\omega^2}{d\theta} = \frac{-\omega\_r^2 \eta\_r}{2} \frac{\left(1 - (\omega/\omega\_r)^2\right)^2}{\eta\_r^2}
@@ -3317,10 +3317,10 @@ The sequence is:
Then we obtain the **center** and **radius** of the circle and the **quality factor** is the mean square deviation of the chosen points from the circle.
3. **Locate natural frequency, obtain damping estimate**.
The rate of sweep through the region is estimated numerically and the frequency at which it reaches the maximum is deduced.
At the same time, an estimate of the damping is derived using <eq:estimate_damping_sweep_rate>.
At the same time, an estimate of the damping is derived using \eqref{eq:estimate\_damping\_sweep\_rate}.
A typical example is shown on [Figure 29](#figure--fig:circle-fit-natural-frequency).
4. **Calculate multiple damping estimates, and scatter**.
A set of damping estimates using all possible combination of the selected data points are computed using <eq:estimate_damping>.
A set of damping estimates using all possible combination of the selected data points are computed using \eqref{eq:estimate\_damping}.
Then, we can choose the damping estimate to be the mean value.
We also look at the distribution of the obtained damping estimates as is permits a useful diagnostic of the quality of the entire analysis:
- Good measured data should lead to a smooth plot of these damping estimates, any roughness of the surface can be explained in terms of noise in the original data.
@@ -3426,7 +3426,7 @@ we now have sufficient information to extract estimates for the four parameters
3. Plot graphs of \\(m\_R(\Omega)\\) vs \\(\Omega^2\\) and of \\(m\_I(\Omega)\\) vs \\(\Omega^2\\) using the results from step 1., each time using a different measurement points as the fixing frequency \\(\Omega\_j\\)
4. Determine the slopes of the best fit straight lines through these two plots, \\(n\_R\\) and \\(n\_I\\), and their intercepts with the vertical axis \\(d\_R\\) and \\(d\_I\\)
5. Use these four quantities, and equation <eq:modal_parameters_formula>, to determine the **four modal parameters** required for that mode
5. Use these four quantities, and equation \eqref{eq:modal\_parameters\_formula}, to determine the **four modal parameters** required for that mode
This procedure which places more weight to points slightly away from the resonance region is likely to be less sensitive to measurement difficulties of measuring the resonance region.
@@ -3491,7 +3491,7 @@ From the sketch, it may be seen that within the frequency range of interest:
- the first term tends to approximate to a **mass-like behavior**
- the third term approximates to a **stiffness effect**
Thus, we have a basis for the residual terms and shall rewrite equation <eq:sum_modes>:
Thus, we have a basis for the residual terms and shall rewrite equation \eqref{eq:sum\_modes:}
\begin{equation}
H\_{jk}(\omega) \simeq -\frac{1}{\omega^2 M\_{jk}^R} + \sum\_{r=m\_1}^{m\_2} \left( \frac{{}\_rA\_{jk}}{\omega\_r^2 - \omega^2 + i \eta\_r \omega\_r^2} \right) + \frac{1}{K\_{jk}^R}
@@ -3551,7 +3551,7 @@ We can write the receptance in the frequency range of interest as:
In the previous methods, the second term was assumed to be a constant in the curve-fit procedure for mode \\(r\\).
However, if we have good **estimates** for the coefficients which constitutes the second term, for example by having already completed an SDOF analysis, we may remove the restriction on the analysis.
Indeed, suppose we take a set of measured data points around the resonance at \\(\omega\_r\\), and that we can compute the magnitude of the second term in <eq:second_term_refinement>, we then subtract this from the measurement and we obtain adjusted data points that are conform to a true SDOF behavior and we can use the same technique as before to obtain **improved estimated** to the modal parameters of more \\(r\\).
Indeed, suppose we take a set of measured data points around the resonance at \\(\omega\_r\\), and that we can compute the magnitude of the second term in \eqref{eq:second\_term\_refinement}, we then subtract this from the measurement and we obtain adjusted data points that are conform to a true SDOF behavior and we can use the same technique as before to obtain **improved estimated** to the modal parameters of more \\(r\\).
This procedure can be repeated iteratively for all the modes in the range of interest and it can significantly enhance the quality of found modal parameters for system with **strong coupling**.
@@ -3610,7 +3610,7 @@ If we further increase the generality by attaching a **weighting factor** \\(w\_
is minimized.
This is achieved by differentiating <eq:error_weighted> with respect to each unknown in turn, thus generating a set of as many equations as there are unknown:
This is achieved by differentiating \eqref{eq:error\_weighted} with respect to each unknown in turn, thus generating a set of as many equations as there are unknown:
\begin{equation}
\frac{d E}{d q} = 0; \quad q = {}\_1A\_{jk}, {}\_2A\_{jk}, \dots
@@ -3661,7 +3661,7 @@ leading to the modified, but more convenient version actually used in the analys
\end{equation}
In these expressions, only \\(m\\) modes are included in the theoretical FRF formula: the true number of modes, \\(N\\), is actually one of the **unknowns** to be determined during the analysis.
Equation <eq:rpf_error> can be rewritten as follows:
Equation \eqref{eq:rpf\_error} can be rewritten as follows:
\begin{equation}
\begin{aligned}
@@ -3715,7 +3715,7 @@ where \\([X], [Y], [Z], \\{G\\}\\) and \\(\\{F\\}\\) are known measured quantiti
\end{equation}
Once the solution has been obtained for the coefficients \\(a\_k, \dots , b\_k, \dots\\) then the second stage of the modal analysis can be performed in which the required **modal parameters are derived**.
This is usually done by solving the two polynomial expressions which form the numerator and denominator of equations <eq:frf_clasic> and <eq:frf_rational>:
This is usually done by solving the two polynomial expressions which form the numerator and denominator of equations \eqref{eq:frf\_clasic} and \eqref{eq:frf\_rational:}
- the denominator is used to obtain the natural frequencies \\(\omega\_r\\) and damping factors \\(\xi\_r\\)
- the numerator is used to determine the complex modal constants \\(A\_r\\)
@@ -3943,7 +3943,7 @@ First, we note that from a single FRF curve, \\(H\_{jk}(\omega)\\), it is possib
Now, although this gives us the natural frequency and damping properties directly, it does not explicitly yield the mode shape: only a modal constant \\({}\_rA\_{jk}\\) which is formed from the mode shape data.
In order to extract the individual elements \\(\phi\_{jr}\\) of the mode shape matrix \\([\Phi]\\), it is necessary to make a series of measurements of specific FRFs including, especially, the point FRF at the excitation position.
If we measure \\(H\_{kk}\\), then by using <eq:modal_model_from_frf>, we also obtain the specific elements in the mode shape matrix corresponding to the excitation point:
If we measure \\(H\_{kk}\\), then by using \eqref{eq:modal\_model\_from\_frf}, we also obtain the specific elements in the mode shape matrix corresponding to the excitation point:
\begin{equation}
H\_{kk}(\omega) \longrightarrow \omega\_r, \eta\_r, {}\_rA\_{jk} \longrightarrow \phi\_{kr}; \quad r=1, m
@@ -4417,7 +4417,7 @@ In this respect, the demands of the response model are more stringent that those
#### Synthesis of FRF curves {#synthesis-of-frf-curves}
One of the implications of equation <eq:regenerate_full_frf_matrix> is that **it is possible to synthesize the FRF curves which were not measured**.
One of the implications of equation \eqref{eq:regenerate\_full\_frf\_matrix} is that **it is possible to synthesize the FRF curves which were not measured**.
This arises because if we measured three individual FRF such as \\(H\_{ik}(\omega)\\), \\(H\_{jk}(\omega)\\) and \\(K\_{kk}(\omega)\\), then modal analysis of these yields the modal parameters from which it is possible to generate the FRF \\(H\_{ij}(\omega)\\), \\(H\_{jj}(\omega)\\), etc.
However, it must be noted that there is an important **limitation to this procedure** which is highlighted in the example below.
@@ -4541,7 +4541,7 @@ from which is would appear that we can write
\end{aligned}
\end{equation}
However, equation <eq:m_k_from_modes> is **only applicable when we have available the complete \\(N \times N\\) modal model**.
However, equation \eqref{eq:m\_k\_from\_modes} is **only applicable when we have available the complete \\(N \times N\\) modal model**.
It is much more usual to have an incomplete model in which the eigenvector matrix is rectangle and, as such, is non-invertible.
One step which can be made using the incomplete data is the construction of "pseudo" flexibility and inverse-mass matrices.