, 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:
- there are at least as many vectors as there are forces: \\(p^\prime > p\\)
- the individual force vectors are linearly independent of each other
A second matrix is also constructed containing the response vectors \\([X]\_{n\times p^\prime} = [\\{X\\}\_1, \dots, \\{X\\}\_{p^\prime}]\\).
Now, these two collections of measured data can be used to determine the required FRF matrix:
\begin{equation}
[H(\omega)]\_{n\times p} = [X]\_{n\times p^\prime} [F]\_{p^\prime \times p}^+
\end{equation}
where \\(+\\) denotes the generalized inverse of the forcing matrix.
#### Multi-point random (MPR) testing {#multi-point-random--mpr--testing}
##### Concept {#concept}
In this method, advantage is taken of the incoherence of several uncorrelated random excitations which are applied simultaneously at several points.
Then, the need to repeat the test several times, as was necessary for the MPSS method, is avoided.
The purpose of this methods is to obtain the FRF data in an optimal way and to reduce the probability of introducing systematic errors to the FRF measurements.
Let's consider the simplest form of a multi excitation as that of a system excited by two simultaneous forces \\(f\_1(t)\\) and \\(f\_2(t)\\) where the response \\(x\_i(t)\\) is of particular interest.
We can derive expressions for the required FRF parameters functions of the auto and cross spectral densities between of three parameters of interest:
\begin{equation}
\begin{aligned}
H\_{i1}(\omega) &= \frac{S\_{1i}S\_{22} - S\_{2i}S\_{12}}{S\_{11}S\_{22} - S\_{12}S\_{21}}\\\\
H\_{i1}(\omega) &= \frac{S\_{2i}S\_{11} - S\_{1i}S\_{21}}{S\_{11}S\_{22} - S\_{12}S\_{21}}
\end{aligned}
\end{equation}
These expressions can be used provided that \\(S\_{11}S\_{22}\neq |S\_{12}|^2\\) which is equivalent of that the two excitation forces must not be fully correlated.
##### General formulation {#general-formulation}
In practice, the method is applied using different numbers of exciters, and several response points simultaneously.
We have that
\begin{equation}
[H\_{xf}(\omega)]\_{n\times p} = [S\_{xf}(\omega)]\_{n\times p} [S\_{ff}(\omega)]\_{p\times p}^{-1}
\end{equation}
where it can be seen that the matrix of spectral densities for the forces \\([S\_{ff}(\omega)]\_{p\times p}\\) must be non singular.
Thus, care must be taken in practice to ensure this condition, noting that it is the applied forces and not the signal sources which must meet the requirement.
In practice, this is difficult to obtain as even if the input signals to the exciters' amplifiers are uncorrelated, the forces applied to the structure will certainly not be. This is particularly true near the resonances as the dynamic response is dominated by the one mode which is independent of the actual force pattern.
##### Coherence in MPR measurements {#coherence-in-mpr-measurements}
In a similar way in which we defined coherence for the SISO system, we can make use of the same concepts for a MIMO system.
During a MIMO test, we basically measure three matrices:
\\[ [S\_{ff}(\omega)]; \ [S\_{xx}(\omega)]; \ [S\_{fx}(\omega)] \\]
Then, we can derive an estimate for the FRF matrix:
\\[ H\_1(\omega)^T = [S\_{ff}(\omega)]^{-1} [S\_{fx}(\omega)] \\]
and then compute an estimate for the autospectrum of the response from:
\begin{align\*}
[\tilde{S}\_{xx}(\omega)] &= [H\_1^\*(\omega)] [S\_{fx}(\omega)] \\\\
&= [S\_{xf}(\omega)] [S\_{ff}(\omega)]^{-1} [S\_{fx}(\omega)]
\end{align\*}
Now, by comparing the estimated response spectrum \\([\tilde{S}\_{xx}(\omega)]\\) with the actual measurement \\([S\_{xx}(\omega)]\\), we obtain a formula for the multiple coherence between the two parameters \\(\\{f(t)\\}\\) and \\(\\{x(t)\\}\\):
\begin{equation\*}
\tcmbox{[\gamma^2(\omega)] = [S\_{xx}(\omega)]^{-1} [S\_{xf}(\omega)] [S\_{ff}(\omega)]^{-1} [S\_{fx}(\omega)]}
\end{equation\*}
#### Multiple-reference impact tests {#multiple-reference-impact-tests}
This class of hammer excitation is referred to as **Multi-reference Impact Tests** (**MRIT**).
Typically, three response references are measured (often, the \\(x\\), \\(y\\) and \\(z\\) components at the response measurement location) every time a hammer blow is applied to the structure.
FRF data collected by performing a test in this way will be the equivalent of exciting the structure at three points simultaneously while measuring the response at each of the \\(n\\) points of interest.
Thus, in the same sense that a multiple-input test is a multi-reference measurement (measuring several columns of the FRF matrix), so too is the MRIT since it provides a multi-reference measurement including several rows of the same FRF matrix.
## Modal Parameter Extraction Methods {#modal-parameter-extraction-methods}
### Introduction {#introduction}
#### Introduction to the concept of modal analysis {#introduction-to-the-concept-of-modal-analysis}
This section describes some of the many procedures that are used for **Modal Analysis** and attempts to explain their various advantages and limitations.
These methods generally consists of **curve-fitting a theoretical expression for an individual FRF to the actual measured data**.
Degree of complexity of curve-fitting:
1. **part of single FRF curve**
2. **complete curve** encompassing several resonances
3. **a set of many FRF plots** all on the same structure
In every case, the task is basically to **find the coefficients in a theoretical expression** for the FRF which then most closely **matches the measured data**.
This phase of the modal test procedure is often referred to as **modal parameter extraction** or **modal analysis**.
#### Types of modal analysis {#types-of-modal-analysis}
A majority of current curve-fitting methods operate on the response characteristics in the frequency domain, but there are other procedures which perform a curve-fit in the time domain. These latter methods are based on the fact that the Impulse Response Function is another characteristic function of the system.
Modal analysis methods can be classified into a series of different groups.
**Classification - Analysis Domain**:
It depends on the **domain in which the analysis is performed**:
- Frequency domain of FRFs
- Time domain of IRFs
**Classification - Frequency range**:
Next, it is appropriate to consider the **frequency range** over which each individual analysis will be performed.
Either a single mode is to be extracted at a time, or several:
- **SDOF methods**
- **MDOF methods**
**Classification - Number of FRFs**:
A further classification relates to the **number of FRFs** which are to be included in a single analysis:
- **SISO**: the FRF are measured individually
- **SIMO**: a set of FRF are measured simultaneously at several response points but under the same single-point excitation.
This describes the FRFs in a column or row of the FRF matrix
- **MIMO**: the responses at several points are measured simultaneously while the structure is excited at several points, also simultaneously
#### Difficulties due to damping {#difficulties-due-to-damping}
Many of the problems encounter in practice are related to the difficulties associated with the **reliable modeling of damping effects**.
In practice, we are obliged to make certain assumptions about what model is to be used for the damping effects.
Sometimes, significant errors can be obtained in the modal parameter estimates (and not only in the damping parameters), as a result of a conflict between the assumed damping behavior and that which actually occurs in reality.
Another difficulty is that of **real modes and complex modes**.
In practice, all modes of practical structures are expected to be complex, although in the majority of cases, such complexity will be very small, and often quite negligible.
#### Difficulties of model order {#difficulties-of-model-order}
One problem is determining **how many modes are there in the measured FRF**.
This question is one of the most difficult to resolve in many practical situations where a combination of finite resolution and noise in the measured data combined to make the issue very unclear.
Many modern modal analysis curve-fitters are capable of fitting any FRF of almost any order, however, it might fit **fictitious modes** introduced in the analysis process.
Correct **differentiation between genuine and fictitious modes** remains a critical task in many modal tests.
### Preliminary checks of FRF data {#preliminary-checks-of-frf-data}
#### Visual Checks {#visual-checks}
Before starting the modal analysis of any measured FRF data, it is always important to do a few **simple checks** in order to ensure that no obvious error is present in the data.
Most of the checks are made using a **log log plot of the modulus of the measured FRF**.
##### Low-frequency asymptotes {#low-frequency-asymptotes}
If the structure is **grounded**, then we should clearly see a **stiffness-like** characteristic, appearing as asymptotic to a stiffness line at the lowest frequencies (below the first resonance) and the magnitude of this should correspond to that of the static stiffness of the structure at the point in question.
If the structure has been tested in a free condition, then we should expect to see a **mass-line** asymptote where its magnitude may be deduced from purely rigid-body considerations.
Deviations from this expected behavior may be caused by the frequency range of measurement not extending low enough to see the asymptotic trend, or they may indicate that the required support conditions have not in fact been achieved.
##### High-frequency asymptotes {#high-frequency-asymptotes}
In the upper end of the frequency range, is it sometimes found (especially on point mobility measurements), that the curve becomes asymptotic to a **mass line** or, more usually to a **stiffness line**.
Such situation can result in considerable difficulties for the modal analysis process and reflects a situation where the excitation is being applied at a point of very high mass of flexibility.
Then, modal parameters are difficult to extracts as they are overwhelmed by the dominant local effects.
##### Incidence of anti-resonances {#incidence-of-anti-resonances}
For a **point FRF**, there must be **antiresonance after each resonances**, while for transfer FRFs between two points well-separated on the structure, we should expect **more minima than antiresonances**.
A second check to be made is that the resonance peaks and the antiresonances exhibit the **same sharpness on a log-log plot**:
- Frequency resolution limitation will cause blunt resonances
- Inadequate vibration levels results in poor definition of the antiresonance regions
##### Overall shape of FRF skeleton {#overall-shape-of-frf-skeleton}
The relative position of the resonance, antiresonances and ambient levels of the FRF curve can give information on the validity of the data.
This will be further explained.
##### Nyquist plot inspection {#nyquist-plot-inspection}
When plotting the FRF data in a Nyquist format, we expect that each resonance traces out at least **part of a circular arc**, the extent of which depends largely on the interaction between adjacent modes.
For a system with well-separated modes, it is to be expected that each resonance will generate the major part of a circle, but when modal interference increases, only small segments will be identifiable.
However, within these bounds, the Nyquist plot should ideally exhibit a smooth curve, and failure to do so may be an indication of a poor measurement technique.
#### Assessment of multiple-FRF data set using SVD {#assessment-of-multiple-frf-data-set-using-svd}
When several FRFs are acquired (either from SIMO or MIMO data), the **Singular Value Decomposition** has proved to be a very useful tool to check the **quality**, **reliability** and **order** of the data.
The set of FRF which are to be assessed is stored in a series of vectors \\(\\{H\_{jk}(\omega)\\}\\) each of which contains the values for one FRF at all measured frequencies \\(\omega = \omega\_1, \dots, \omega\_L\\).
These vectors are assembled into a matrix
\\[ [A]\_{L\times np} = [\\{H\_{11}(\omega)\\}\_{L\times 1} \\{H\_{21}(\omega)\\}\_{L\times 1} \dots \\{H\_{np}(\omega)\\}\_{L\times 1} ] \\]
where \\(n\\) and \\(p\\) represent the number of measured DOFs and the number of excitation points.
\\(L\\) represents the number of frequencies at which the FRF data are defined.
**Singular Value Decomposition**:
\begin{equation}
[A]\_{L\times np} = [U]\_{L\times L} [\Sigma]\_{L\times np} [V]\_{np\times np}^T
\end{equation}
Interpretation of SVD:
- The **singular values** \\(\sigma\_1, \dots, \sigma\_w\\) describes the **amplitude** information
- Number of **non-zero singular values** represents the **order of the system** (i.e. the number of independent modes of vibration which effectively contribute to the measured FRFs)
- The columns of \\([U]\\) represent the **frequency distribution** of these amplitudes
- The columns of \\([V]\\) represent their **spatial distribution**
From the SVD, we can compute a new matrix \\([P]\_{L\times np}\\) which is referred to as the **Principal Response Function** (**PRF**) matrix.
Each column of the PRF contains a response function corresponding to one of the original FRFs:
\begin{equation}
[U]\_{L\times L} [\Sigma]\_{L\times np} = [P]\_{L\times np}
\end{equation}
Then, each PRF is, simply, a particular combination of the original FRFs, and thus each FRF contains all the essential information included in those FRFs (eigenvalues for instance).
On example of this form of pre-processing is shown on figure [19](#table--fig:PRF-numerical) for a numerically-simulation test data, and another in figure [20](#table--fig:PRF-measured) for the case of real measured test data.
The second plot [19](#org-target--fig:PRF_numerical_svd) helps to determine the true order of the system because the number of non-zero singular values is equal to this parameter.
The third plot [19](#org-target--fig:PRF_numerical_PRF) shows the genuine modes distinct from the computational modes.
It can be seen that the PRFs tend to fall into **two groups**:
- The most prominent are a set of response function, each of which has a small number of dominant peaks.
It represents the **physical modes** of the system.
- The lower group shows less distinct and clear-cut behavior.
It represents the **noise or computational modes** present in the data.
The two groups are usually separated by a clear gap (depending of the noise present in the data):
- If such gap is present, then is will be possible to extract the properties of the \\(m\\) modes which are active in the measured responses over the frequency range covered.
- If not, then it may be impossible to perform a successful modal parameter extraction.
Table 19:
FRF and PRF characteristics for numerical model
|  |  |  |
|-----------------------------------------------------------------------------|-----------------------------------------------------------------------------------------|-----------------------------------------------------------------------------|
| FRF | Singular Values | PRF |
| width=\linewidth | width=\linewidth | width=\linewidth |
Table 20:
FRF and PRF characteristics for measured model
|  |  |  |
|----------------------------------------------------------------------------|----------------------------------------------------------------------------------------|----------------------------------------------------------------------------|
| FRF | Singular Values | PRF |
| width=\linewidth | width=\linewidth | width=\linewidth |
#### Mode Indicator Functions (MIFs) {#mode-indicator-functions--mifs}
##### General {#general}
The Mode Indicator Functions are usually used on \\(n\times p\\) FRF matrix where \\(n\\) is a relatively large number of measurement DOFs and \\(p\\) is the number of excitation DOFs, typically 3 or 4.
In these methods, the frequency dependent FRF matrix is subjected to an eigenvalue or singular value decomposition analysis which thus yields a small number (3 or 4) of eigen or singular values, these also being frequency dependent.
These methods are used to **determine the number of modes** present in a given frequency range, to **identify repeated natural frequencies** and to pre process the FRF data prior to modal analysis.
##### Complex mode indicator function (CMIF) {#complex-mode-indicator-function--cmif}
The Complex Mode Indicator Function is defined simply by the SVD of the FRF (sub) matrix.
The **Complex mode indicator function** (CMIF) is defined as
\begin{align\*}
[H(\omega)]\_{n\times p} &= [U(\omega)]\_{n\times n} [\Sigma(\omega)]\_{n\times p} [V(\omega)]\_{p\times p}^H\\\\
[CMIF(\omega)]\_{p\times p} &= [\Sigma(\omega)]\_{p\times n}^T [\Sigma(\omega)]\_{n\times p}
\end{align\*}
The actual mode indicator values are provided by the squares of the singular values and are usually plotted as a function of frequency in logarithmic form as shown in figure [27](#figure--fig:mifs):
- **Natural frequencies are indicated by large values of the first CMIF** (the highest of the singular values)
- **double or multiple modes by simultaneously large values of two or more CMIF**.
Associated with the CMIF values at each natural frequency \\(\omega\_r\\) are two vectors:
- the left singular vector \\(\\{U(\omega\_r)\\}\_1\\) which approximates the **mode shape** of that mode
- the right singular vector \\(\\{V(\omega\_r)\\}\_1\\) which represents the approximate **force pattern necessary to generate a response on that mode only**
{{< figure src="/ox-hugo/ewins00_mifs.png" caption="Figure 27: Complex Mode Indicator Function (CMIF)" >}}
In addition to identifying all the significant natural frequencies, the CMIF can also be used to **generate a set of enhanced FRFs** from the formula:
\begin{equation} \label{eq:efrf}
[EFRF(\omega)]\_{n\times p} = [H(\omega)]\_{n\times p} [V(\omega)]\_{p\times p}
\end{equation}
There is one non-trivial EFRF for each mode, the result of which is an almost **SDOF characteristic** response function which is then readily amenable to modal analysis by the simplest of methods.
As in the previous case, these modified FRFs are simply linear combinations of the original measured data and, as such, contain no more and no less information than in their original form.
However, such an approach lends itself to a very reliable extraction of the global properties (eigenvalues) for the measured FRF data set which can then be re-visited in a second stage to determine the local properties (mode shapes) for all the measured DOFs.
##### Other MIFs {#other-mifs}
There are multiple variants on the mode indicator function concepts. Some use the eigenvalue decomposition instead of the singular value decomposition.
Two are worth mentioning: the Multivariable Mode Indicator Function (MMIF) and the Real Mode Indicator Function (RMIF).
### SDOF Modal Analysis Methods {#sdof-modal-analysis-methods}
#### Review of SDOF modal analysis methods {#review-of-sdof-modal-analysis-methods}
The "SDOF" approach does not imply that the system being modeled is reduced to a single degree of freedom, that that just **one resonance is considered at a time**.
There are limitations to such simple approach, the principal one being that **very close modes cannot easily be separated**.
There are several implementations of the basic concept of SDOF analysis, ranging from the simple **peak-picking method**, through the classic **circle-fit approach** to more automatic algorithms such as the **inverse FRF "he-fit" method** and the general **least-squares methods**.
As the name implies, the method exploits the fact that in the vicinity of a resonance, the behavior of the system is dominated by a single mode (the magnitude is dominated by one of the terms in the series).
The general expression of the receptance FRF
\begin{equation}
\alpha\_{jk}(\omega) = \sum\_{s=1}^N \frac{{}\_sA\_{jk}}{\omega\_s^2 - \omega^2 + i \eta\_s \omega\_s^2}
\end{equation}
can be rewritten as:
\begin{equation}
\alpha\_{jk}(\omega) = \frac{{}\_rA\_{jk}}{\omega\_r^2 - \omega^2 + i \eta\_r \omega\_r^2} + \sum\_{\substack{s=1\\\s \neq r}}^N \frac{{}\_sA\_{jk}}{\omega\_s^2 - \omega^2 + i \eta\_s \omega\_s^2}
\end{equation}
Now, the SDOF assumption is that for a small range of frequency in the vicinity of the natural frequency of mode \\(r\\), \\(\alpha\_{jk}(\omega)\\) can be approximated as
\begin{equation}
\alpha\_{jk}(\omega)\_{\omega\approx\omega\_r} = \frac{{}\_rA\_{jk}}{\omega\_r^2 - \omega^2 + i \eta\_r \omega\_r^2} + {}\_rB\_{jk}
\end{equation}
This does not mean that the other modes are unimportant or negligible (their influence can be considerable), but rather that their combined effect can be represented as a **constant term** around this resonance.
#### SDOF Modal Analysis I - Peak-Amplitude method {#sdof-modal-analysis-i-peak-amplitude-method}
In this method, it is assumed that close to one local mode, any effects due to the other modes can be ignored.
This is a method which works adequately for structures whose FRF exhibit **well separated modes**.
This method is useful in obtaining initial estimates to the parameters.
The peak-picking method is applied as follows (illustrated on figure [28](#figure--fig:peak-amplitude)):
1. First, **individual resonance peaks** are detected on the FRF plot and the maximum responses frequency \\(\omega\_r\\) is taken as the **natural frequency** of that mode
2. Second, the **local maximum value of the FRF** \\(|\hat{H}|\\) is noted and the **frequency bandwidth** of the function for a response level of \\(|\hat{H}|/\sqrt{2}\\) is determined.
The two points thus identified as \\(\omega\_b\\) and \\(\omega\_a\\) are the "half power points"
3. The **damping** of the mode in question can now be estimated from of the following formulae:
\begin{equation}
\begin{aligned}
\eta\_r &= \frac{\omega\_a^2 - \omega\_b^2}{2 \omega\_r^2} \approx \frac{\Delta\omega}{\omega\_r} \\\\
2\xi\_r &= \eta\_r
\end{aligned}
\end{equation}
4. We now obtain an estimate for the **modal constant** of the mode being analyzed by assuming that the total response in this resonant region is attributed to a single term in the general FRF series:
\begin{equation}
|\hat{H}| = \frac{A\_r}{\omega\_r^2 \eta\_r} \Leftrightarrow A\_r = |\hat{H}| \omega\_r^2 \eta\_r
\end{equation}
It must be noted that the estimates of both damping and modal constant depend heavily on the accuracy of the maximum FRF level \\(|\hat{H}|\\) which is difficult to measure with great accuracy, especially for lightly damped systems.
Only real modal constants and thus real modes can be deduced by this method.
{{< figure src="/ox-hugo/ewins00_peak_amplitude.png" caption="Figure 28: Peak Amplitude method of modal analysis" >}}
Alternatives of this method can be applied using the real part of the receptance FRF instead of the modulus plot.
#### SDOF Modal Analysis II - Circle Fit Method {#sdof-modal-analysis-ii-circle-fit-method}
##### Properties of the modal circle {#properties-of-the-modal-circle}
MDOF systems produce Nyquist plots of FRF data which include **sections of near circular arcs** corresponding to the regions near the natural frequencies.
This characteristic provides the basic of the "**SDOF circle-fit method**".
We here use **structural damping** and we use the **receptance** form of FRF data as this will produces an exact circle in a Nyquist plot.
However, if it is required to use a model incorporating viscous damping, then the mobility version of the FRF data should be used.
In the case of a system assumed to have structural damping, the basic function with which we are dealing is
\begin{equation}
\alpha(\omega) = \frac{1}{\omega\_r^2\left( 1 - \left(\omega/\omega\_r\right)^2 + i\eta\_r \right)}
\end{equation}
since the only effect of including the modal constant \\({}\_rA\_{jk}\\) is to scale the size of the circle by \\(|{}\_rA\_{jk}|\\) and to rotate it by \\(\angle {}\_rA\_{jk}\\).
A plot of the quantity \\(\alpha(\omega)\\) is given in figure [21](#org-target--fig:modal_circle).
|  |  |
|-------------------------------------------------------------------------------|-------------------------------------------------------------------------------------------------------------------|
| Properties | \\(\omega\_b\\) and \\(\omega\_a\\) points |
| width=\linewidth | width=\linewidth |
For any frequency \\(\omega\\), we have the following relationship:
\begin{equation} \label{eq:modal\_circle\_tan}
\begin{aligned}
\tan \gamma &= \frac{\eta\_r}{1 - (\omega/\omega\_r)^2}\\\\
\tan(\SI{90}{\degree}-\gamma) &= \tan\left(\frac{\theta}{2}\right) = \frac{1 - (\omega/\omega\_r)^2}{\eta\_r}
\end{aligned}
\end{equation}
From , 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 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}
\end{equation}
The reciprocal of this quantity is a **measure of the rate at which the locus sweeps around the circular arc**.
It may be seen to reach a maximum value when \\(\omega=\omega\_r\\):
\begin{equation}
\tcmbox{\frac{d}{d\omega} \left(\frac{d\omega^2}{d\theta}\right) = 0 \text{ when } \omega\_r^2 - \omega^2 = 0}
\end{equation}
It may also be seen that an **estimate of the damping** is provided by the sweep rate:
\begin{equation} \label{eq:estimate\_damping\_sweep\_rate}
\tcmbox{\left(\frac{d\theta}{d\omega^2}\right)\_{\omega=\omega\_r} = -\frac{2}{\omega\_r^2 \eta\_r}}
\end{equation}
Suppose now we have two specific points on the circle, one corresponding to a frequency \\(\omega\_b\\) below the natural frequency and the other one \\(\omega\_a\\) above the natural frequency.
Referring to figure [21](#org-target--fig:modal_circle_bis), we can write:
\begin{equation}
\begin{aligned}
\tan\left(\frac{\theta\_b}{2}\right) &= \frac{1 - (\omega\_b/\omega\_r)^2}{\eta\_r}\\\\
\tan\left(\frac{\theta\_a}{2}\right) &= \frac{(\omega\_a/\omega\_r)^2 - 1}{\eta\_r}
\end{aligned}
\end{equation}
From these two equations, we can obtain an expression for the **damping of the mode**:
\begin{equation} \label{eq:estimate\_damping}
\tcmbox{\eta\_r = \frac{\omega\_a^2 - \omega\_b^2}{\omega\_r^2 \left(\tan(\theta\_a/2) + \tan(\theta\_b/2)\right)}}
\end{equation}
which is an exact expression and applies for all levels of damping.
If we take two points for which \\(\theta\_a = \theta\_b = \SI{90}{\degree}\\), we obtain:
\begin{equation}
\begin{aligned}
\eta\_r &= \frac{\omega\_2^2 - \omega\_1^2}{2 \omega\_r^2}\\\\
\eta\_r &= \frac{\omega\_2 - \omega\_1}{\omega\_r} \text{ for light damping}
\end{aligned}
\end{equation}
When scaled by a modal constant \\({}\_rA\_{jk}\\) added in the numerator, the diameter of the circle will be
\\[ {}\_rD\_{jk} = \frac{\left|{}\_rA\_{jk}\right|}{\omega\_r^2 \eta\_r} \\]
and the whole circle will be rotated so that the principal diameter (the one passing through the natural frequency point) is oriented at an angle \\(\arg({}\_rA\_{jk})\\) to the negative Imaginary axis.
For SDOF system with **viscous** damping, rather than structural damping, the **mobility** is
\\[ Y(\omega) = \frac{i\omega}{(k - \omega^2 m) + i \omega c} \\]
And we have
\begin{equation}
\tan\left(\frac{\theta}{2}\right) = \frac{1 - (\omega/\omega\_r)^2}{2 \xi \omega/\omega\_r}
\end{equation}
From points at \\(\omega\_a\\) and \\(\omega\_b\\), we obtain
\begin{equation}
\begin{aligned}
\xi &= \frac{\omega\_a^2 - \omega\_b^2}{2 \omega\_r \left( \omega\_a \tan(\theta\_a/2) + \omega\_b \tan(\theta\_b/2) \right)}\\\\
&= \frac{\omega\_a - \omega\_b}{\omega\_r \left( \tan(\theta\_a/2) + \tan(\theta\_b/2) \right)} \text{ for light damping}
\end{aligned}
\end{equation}
Finally, selecting two points for which \\(\theta\_a = \theta\_b = \SI{90}{\degree}\\):
\begin{equation}
\xi = \frac{\omega\_2 - \omega\_1}{2 \omega\_r}
\end{equation}
##### Circle-fit analysis procedure {#circle-fit-analysis-procedure}
The sequence is:
1. **Select points to be used**.
2. **Fit circle, calculate quality of fit**.
It is generally done by a least-square algorithm.
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 .
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 .
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.
- However, any systematic distortion of the plot is almost certainly caused by some form of error in the data, in the analysis or in the assumed behavior of the system.
5. **Determine modal constant modulus and argument**.
The magnitude and argument of the modal constant is determined from the diameter of the circle and from its orientation relative to the Real and Imaginary axis.
{{< figure src="/ox-hugo/ewins00_circle_fit_natural_frequency.png" caption="Figure 29: Location of natural frequency for a Circle-fit modal analysis" >}}
Then, the theoretically regenerated FRF can be plotted against the original measured data for comparison.
In order to determines the contribution of other modes on the resonance of mode \\(r\\), the distance from the top of the principal diameter to the origin has to be measured and is equal to \\({}\_rB\_{jk}\\).
#### SDOF Modal Analysis III - Inverse or Line-fit method {#sdof-modal-analysis-iii-inverse-or-line-fit-method}
##### Properties of inverse FRF plots {#properties-of-inverse-frf-plots}
The original version of this method uses the fact that a function which generates a circle when plotted in the complex plane will, when plotted as a reciprocal, trace out a **straight line**.
Thus, if we were to plot the reciprocal of receptance of a SDOF system with structural damping, we would find that in the Argand diagram it produces a straight line:
\begin{equation}
\begin{aligned}
\alpha(\omega) &= \frac{(k - \omega^2 m) - i d}{(k - \omega^2 m)^2 + d^2}\\\\
\frac{1}{\alpha(\omega)} &= (k - \omega^2 m) + i d
\end{aligned}
\end{equation}
First, a least squares best-fit straight line is constructed through the data points and an **estimate for the damping parameters** is immediately available from the **intercept of the line with the Imaginary axis**.
Furthermore, an indication of the reliability of that estimate may be gained from the nature of the deviations of the data points from the line itself.
We can here determine whether the damping is structural (imaginary part constant with frequency) or viscous (imaginary part linear with frequency).
Then, a second least squares operation is performed, this time on the deviation between the real part of the measured data points and that of the theoretical model.
Resulting from this, we obtain **estimates for the mass and stiffness parameters**.
It should be noted that this approach is best suited to systems with real modes and to relatively well-separated modes.
##### General inverse analysis method {#general-inverse-analysis-method}
It has been shown that if a purely SDOF system FRF is plotted in this way, then both plots demonstrate straight lines, and separately reveal useful information about the mass, stiffness and damping properties of the measured system.
The inverse FRF of a MDOF system is not as convenient as SDOF system as:
\begin{align\*}
H\_{jk}^{-1} (\omega) &= \frac{1}{\sum (k - \omega^2 m) + i \omega c}\\\\
&\neq \sum \frac{1}{(k - \omega^2 m) + i \omega c}
\end{align\*}
Thus, in order to determine the modal parameters of a MDOF system using inverse method, some modifications to the basic formulation must be found.
We start with the basic formula for SDOF analysis:
\\[ \alpha\_{jk}(\omega)\_{\omega\simeq\omega\_r} \simeq \frac{{}\_rA\_{jk}}{\omega\_r^2 - \omega^2 + i \eta\_r \omega\_r^2} + {}\_rB\_{jk} \\]
We can note that the presence of the \\({}\_rB\_{jk}\\) term is problematic for the inverse plot.
The trick is to define a **new FRF term** \\(\alpha^\prime\_{ik}(\omega)\\) which is the difference between the actual FRF and the value of the FRF at one fixed frequency \\(\Omega\\) in the range of interest called "**fixing frequency**":
\\[ \alpha^\prime\_{jk}(\omega) = \alpha\_{jk}(\omega) - \alpha\_{jk}(\Omega) \\]
from which the inverse FRF parameter that we shall use for the modal analysis \\(\Delta(\omega)\\), can be defined as:
\begin{align\*}
\Delta(\omega) &= (\omega^2 - \Omega^2)/\alpha^\prime\_{jk}(\omega)\\\\
&= \text{Re}(\Delta) + i \text{Im}(\Delta)
\end{align\*}
It can be seen that
\\[ \text{Re}(\Delta) = m\_R \omega^2 + c\_R; \quad \text{Im}(\Delta) = m\_I \omega^2 + c\_I \\]
and that
\begin{align\*}
m\_R &= a\_R(\Omega^2 - \omega\_r^2) - b\_r (\omega\_r^2 \eta\_r) \\\\
m\_I &= -b\_R(\Omega^2 - \omega\_r^2) - a\_r (\omega\_r^2 \eta\_r) \\\\
{}\_rA\_{jk} &= a\_R + i b\_r
\end{align\*}
The first step of our **analysis procedure** can be made, as follows:
1. Using the FRF data measured in the vicinity of the resonance \\(\omega\_r\\), choose the fixing frequency \\(\Omega\_j\\) and then calculate \\(\Delta(\omega)\\)
2. Plot these values on \\(\text{Re vs } \omega^2\\) and \\(\text{Im vs }\omega^2\\) plots and compute the best fit straight line in order to determine \\(m\_R(\Omega\_j)\\) and \\(m\_I(\Omega\_j)\\)
Now it can be shown that both these straight line slopes \\(m\_R\\) and \\(m\_I\\) are simple functions of \\(\Omega\\), and we can write:
\\[ m\_R = n\_R \Omega^2 + d\_R \text{ and } m\_I = n\_I \Omega^2 + d\_I \\]
where
\begin{equation} \label{eq:four\_parameters}
\begin{aligned}
n\_R &= a\_r; \quad n\_I = -b\_r \\\\
d\_R &= -b\_r(\omega\_r^2 \eta\_r) - a\_r \omega\_r^2; \quad d\_I = b\_r \omega\_r^2 - a\_r\omega\_r^2\eta\_r
\end{aligned}
\end{equation}
Now let \\(p = n\_I/n\_R \text{ and } q = d\_I/d\_R\\), and noting that
\begin{equation} \label{eq:modal\_parameters\_formula}
\begin{aligned}
\eta\_r &= \frac{q - p}{1 + pq}; \quad \omega\_r^2 = \frac{d\_R}{(p\eta\_r - 1)n\_R} \\\\
a\_r &= \frac{\omega\_r^2(p\eta\_r - 1)}{(1 + p^2)d\_R}; \quad b\_r = -a\_r p
\end{aligned}
\end{equation}
we now have sufficient information to extract estimates for the four parameters for the resonance which has been analyzed: \\(\omega\_r, \eta\_r, \text{ and } {}\_rA\_{jk} = a\_r + i b\_r\\).
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 , 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.
#### Residuals {#residuals}
##### Concept of residual terms {#concept-of-residual-terms}
We need to introduce the concept of **residual terms**, necessary in the modal analysis process to take account of those modes which we do not analyze directly but which nevertheless exist and have an influence on the FRF data we use.
The first occasion on which the residual problem is encountered is generally at the end of the analysis of a single FRF curve, such as by the repeated application of an SDOF curve-fit to each of the resonances in turn until all modes visible on the plot have been identified.
At this point, it is often desired to construct a theoretical curve (called "**regenerated**"), based on the modal parameters extracted from the measured data, and to overlay this on the original measured data to assess the success of the curve-fit process.
Then the regenerated curve is compared with the original measurements, the result is often disappointing, as illustrated in figure [22](#org-target--fig:residual_without).
However, by the inclusion of two simple extra terms (the "**residuals**"), the modified regenerated curve is seen to correlate very well with the original experimental data as shown on figure [22](#org-target--fig:residual_with).
Table 22:
Effects of residual terms on FRF regeneration
|  |  |
|-----------------------------------------------------------------------------------------|------------------------------------------------------------------------------------|
| without residual | with residuals |
| width=\linewidth | width=\linewidth |
If we regenerate an FRF curve from the modal parameters we have extracted from the measured data, we shall use a formula of the type
\begin{equation}
H\_{jk}(\omega) = \sum\_{r = m\_1}^{m\_2} \frac{{}\_rA\_{jk}}{\omega\_r^2 - \omega^2 + i \eta\_r \omega\_r^2}
\end{equation}
in which \\(m\_1\\) and \\(m\_2\\) reflects that we do not always start at the first mode (\\(r = 1\\)) and continue to the highest mode (\\(r = N\\)).
However, the equation which most closely represents the measured data is:
\begin{equation}
H\_{jk}(\omega) = \sum\_{r = 1}^{N} \frac{{}\_rA\_{jk}}{\omega\_r^2 - \omega^2 + i \eta\_r \omega\_r^2}
\end{equation}
which may be rewritten as
\begin{equation} \label{eq:sum\_modes}
H\_{jk}(\omega) = \left( \sum\_{r=1}^{m\_1-1} + \sum\_{r=m\_1}^{m\_2} + \sum\_{r = m\_2+1}^{N} \right) \frac{{}\_rA\_{jk}}{\omega\_r^2 - \omega^2 + i \eta\_r \omega\_r^2}
\end{equation}
The three terms corresponds to:
1. the **low frequency modes** not identified
2. the **high frequency modes** not identified
3. the **modes actually identified**
These three terms are illustrated on figure [30](#figure--fig:low-medium-high-modes).
{{< figure src="/ox-hugo/ewins00_low_medium_high_modes.png" caption="Figure 30: Numerical simulation of contribution of low, medium and high frequency modes" >}}
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 :
\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}
\end{equation}
where the quantities \\(M\_{jk}^R\\) and \\(K\_{jk}^R\\) are the **residual mass and stiffness** for that **particular FRF** and **chosen frequency range**.
##### Calculation of residual mass and stiffness terms {#calculation-of-residual-mass-and-stiffness-terms}
First, we compute a few values of the regenerated FRF curve at the lower frequencies covered by the tests, using only the identified modal parameters.
Then, by comparing these values with those from actual measurements, we estimate a mass residual constant which, when added to the regenerated curve, brings this closely into line with the measured data.
Then, the process is repeated at the top end of the frequency range, this time seeking a residual stiffness.
Often, the process is more effective if there is an antiresonance near either end of the frequency range which this is then used as the point of adjustment.
The procedure outlined here may need to be repeated **iteratively** in case the addition of the stiffness residual term then upsets the effectiveness of the mass term.
It should be noted that often there is a **physical significance to the residual terms**.
If the test structure is freely-supported and its rigid body modes are well below the minimum frequency of measurement, then the mass residual term will be a direct reflection of the rigid body mass and inertia properties of the structure.
The high frequency residual can represent the local flexibility at the drive point.
##### Residual and pseudo modes {#residual-and-pseudo-modes}
Sometimes it is convenient to **treat the residual terms as if they were modes**.
Instead of representing each residual effect by a constant, each can be represented by a pseudo mode.
For the low frequency residual effects, this pseudo mode has a natural frequency below the lowest frequency on the measured FRF, and for the high frequency residual effects, that pseudo mode has a natural frequency which is above the highest frequency of the measured FRF.
These pseudo modes can be conveniently included in the list of modes which have been extracted by modal analysis of that FRF.
Using pseudo modes instead of simple residual mass and stiffness terms is a more accurate way of representing the out-of-range modes.
There is one warning, however, and that is to point out that these pseudo modes are **not** genuine modes and that they cannot be used to deduce the corresponding contributions of these same modes for any other FRF curve.
#### Refinement of SDOF modal analysis methods {#refinement-of-sdof-modal-analysis-methods}
In the modal analysis methods discussed above, an assumption is made that near the resonance under analysis, the effect of **all** the other modes could be represented by a constant.
When there are neighboring modes close to the one being analyzed, this assumption may not be valid.
**"Close" modes** is loosely defined as a situation where the separation between the natural frequencies of two adjacent modes is less than the typical damping level, both measured as percentage.
However, we can usually remove that restriction and thereby make a more precise analysis of the data.
We can write the receptance in the frequency range of interest as:
\begin{equation} \label{eq:second\_term\_refinement}
\begin{aligned}
H\_{jk}(\omega) &= \sum\_{s=m\_1}^{m\_2} \left( \frac{{}\_sA\_{jk}}{\omega\_s^2 - \omega^2 + i \eta\_s \omega\_s^2} \right) + \frac{1}{K\_{jk}^R}-\frac{1}{\omega^2 M\_{jk}^R} \\\\
&= \left( \frac{{}\_rA\_{jk}}{\omega\_r^2 - \omega^2 + i\eta\_r \omega\_r^2} \right) \\\\
&+ \left(\sum\_{\substack{s=m\_1 \cr s \neq r}}^{m\_2} \frac{{}\_sA\_{jk}}{\omega\_s^2 - \omega^2 + i\eta\_s \omega\_s^2} + \frac{1}{K\_{jk}^R} - \frac{1}{\omega^2 M\_{jk}^R} \right)
\end{aligned}
\end{equation}
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 , 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**.
### MDOF Modal analysis in the frequency domain (SISO) {#mdof-modal-analysis-in-the-frequency-domain--siso}
#### General Approach {#general-approach}
There are a number of situations in which the SDOF approach to modal analysis is inadequate and for these there exist several alternative methods which may generally be classified as MDOF modal analysis methods.
These situations are generally those with closely coupled modes where the single mode approximation is inappropriate and those with extremely light damping for which measurements at resonance are inaccurate.
Three approach to curve-fit the entire FRF in one step are considered here:
1. a general approach to multi-mode curve-fitting
2. a method based on the rational fraction FRF formulation
3. a method particularly suited to very lightly-damped structures
#### Method I - General Curve Fit approach - Non-linear Least Squares (NLLS) {#method-i-general-curve-fit-approach-non-linear-least-squares--nlls}
We shall denote the individual FRF measured data as:
\\[ H\_{jk}^m(\Omega\_l) = H\_l^m \\]
while the corresponding "theoretical" values are:
\begin{equation}
\begin{aligned}
H\_l &=H\_{jk}(\Omega\_l) \\\\
&= \sum\_{s=m\_1}^{m\_2}\frac{{}\_sA\_{jk}}{\omega\_s^2 - \Omega\_l^2 + i \eta\_s \omega\_s^2} + \frac{1}{K\_{jk}^R} - \frac{1}{\Omega\_l^2 M\_{jk}^R}
\end{aligned}
\end{equation}
where the coefficients \\({}\_1A\_{jk}, {}\_2A\_{jk}, \dots, \omega\_1, \omega\_2, \dots, \eta\_1, \eta\_2, \dots, K\_{jk}^R \text{ and }M\_{jk}^R\\) are all to be **determined**.
We can define an **individual error** as:
\begin{equation}
\epsilon\_l = H\_l^m - H\_l
\end{equation}
and express this as a scalar quantity:
\begin{equation}
E\_l = \left| \epsilon\_l^2 \right|
\end{equation}
If we further increase the generality by attaching a **weighting factor** \\(w\_l\\) to each frequency point of interest, then the curve fit process has to determine the values of the unknown coefficients such that the total error:
\begin{equation} \label{eq:error\_weighted}
E = \sum\_{l = 1}^p w\_l E\_l
\end{equation}
is minimized.
This is achieved by differentiating 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
\end{equation}
Unfortunately, this set of equations are **not linear** in many of the coefficients and thus cannot be solved directly.
It is from this point that the differing algorithms choose their individual procedures: making various simplifications, assumptions or linearizing the expressions.
#### Method II - Rational Fraction Polynomial Method (RFP) {#method-ii-rational-fraction-polynomial-method--rfp}
The method which has emerged as one the **standard** frequency domain modal analysis methods is that known as the **Rational Fraction Polynomial** (**RFP**) method.
This method is a special version of the general curve fitting approach but is based on a different formulation for the theoretical expression used for the FRF.
Rational fraction FRF formulation:
\begin{equation}
H(\omega) = \sum\_{r=1}^N \frac{A\_r}{\omega\_r^2 - \omega^2 + 2 i \omega \omega\_r \xi\_r} \label{eq:frf\_clasic}
\end{equation}
\begin{equation}
H(\omega) = \frac{b\_0 + b\_1(i\omega) + \dots + b\_{2N-1}(i\omega)^{2N-1}}{a\_0 + a\_1(i\omega) + \dots + a\_{2N}(i\omega)^{2N}} \label{eq:frf\_rational}
\end{equation}
In this formulation, we have adopted the **viscous damping model**.
The unknown coefficients \\(a\_0, \dots, a\_{2N}, b\_0, \dots, b\_{2N-1}\\) are not the modal properties but are related to them and are computed in a further stage of processing.
The particular advantage of this approach is the possibility of formulating the curve fitting problem as a **linear set of equations**, thereby making the solution amenable to a direct matrix solution.
We shall denote each of our measured FRF data point by \\(\hat{H}\_k\\), where \\(\hat{H}\_k = \hat{H}(\omega\_k)\\), and define the error between that measured value and the corresponding value derived from the curve-fit expression as
\begin{equation}
e\_k = \frac{b\_0 + b\_1(i\omega\_k) + \dots + b\_{2m-1}(i\omega\_k)^{2m-1}}{a\_0 + a\_1(i\omega\_k) + \dots + a\_{2m}(i\omega\_k)^{2m}} - \hat{H}\_k
\end{equation}
leading to the modified, but more convenient version actually used in the analysis
\begin{equation} \label{eq:rpf\_error}
\begin{aligned}
e\_k^\prime &= \left( b\_0 + b\_1(i\omega\_k) + \dots + b\_{2m-1}(i\omega\_k)^{2m-1} \right)\\\\
&- \hat{H}\_k\left( a\_0 + a\_1(i\omega\_k) + \dots + a\_{2m}(i\omega\_k)^{2m} \right)
\end{aligned}
\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 can be rewritten as follows:
\begin{equation}
\begin{aligned}
e\_k^\prime &= \begin{Bmatrix} 1 & i \omega\_k & \dots & (i\omega\_k)^{2m-1} \end{Bmatrix}
\begin{Bmatrix} b\_0 \\\ \vdots \\\ b\_{2m-1} \end{Bmatrix}\\\\
&- \hat{H}\_k \begin{Bmatrix} 1 & i\omega\_k & \dots & (i\omega\_k)^{2m-1} \end{Bmatrix}
\begin{Bmatrix} a\_0 \\\ \vdots \\\ a\_{2m-1} \end{Bmatrix}\\\\
&- \hat{H}\_k (i\omega\_k)^{2m} a\_{2m}
\end{aligned}
\end{equation}
and the \\(L\\) linear equations corresponding to \\(L\\) individual frequency points can be combined in matrix form:
\begin{equation}
\begin{aligned}
\\{E^\prime\\}\_{L \times 1} &= [P]\_{L\times 2m} \\{b\\}\_{2m\times 1}\\\\
&- [T]\_{L\times(2m+1)} \\{a\\}\_{(2m+1)\times 1}\\\\
&- \\{W\\}\_{L\times 1}
\end{aligned}
\end{equation}
Solution for the unknown coefficients \\(a\_j, \dots, b\_k, \dots\\) is achieved by minimizing the error function
\begin{equation}
J = \\{E^\*\\}^T\\{E\\}
\end{equation}
and this leads to
\begin{equation}
\begin{bmatrix}
[Y] & [X] \\\\
[X]^T & [Z]
\end{bmatrix}\_{L \times (4m+1)}
\begin{Bmatrix}
\\{b\\} \\\ \\{a\\}
\end{Bmatrix}\_{(4m+1) \times 1}
= \begin{Bmatrix}
\\{B\\} \\\ \\{F\\}
\end{Bmatrix}\_{L \times 1}
\end{equation}
where \\([X], [Y], [Z], \\{G\\}\\) and \\(\\{F\\}\\) are known measured quantities:
\begin{equation}
\begin{aligned}
[Y] &= \text{Re}\left( [P^\*]^T[P] \right);\quad [X] = \text{Re}\left( [P^\*]^T[T] \right); \\\\
[Z] &= \text{Re}\left( [T^\*]^T[T] \right); \\\\
\\{G\\} &= \text{Re}\left( [P^\*]\\{W\\} \right);\quad \\{F\\} = \text{Re}\left( [T^\*]\\{W\\} \right);
\end{aligned}
\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 and :
- 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\\)
In order to determine the order, the analysis is repeated using different assumed values for the order \\(m\\) and are compared.
For each analysis, there will be properties found for as many modes as prescribed by the chosen model order.
Some of these will be genuine modes while others will be fictitious modes.
Various strategies may be adopted to separate the fictitious and real modes:
- measuring the difference between the original FRF curve and that regenerated using the modal properties derived
- measuring the consistency of the various modal parameters for different model order choices and eliminating those which vary widely from run to run
In all these checks, interest is concentrated on the **repeatability** of the various modal properties: modes which reappear for all choices of data and model condition are believed to be genuine, while those which vary from run to run are more likely to have computational features due to the curve-fitting requirements as their origins, rather than physical ones.
#### Method III - Lightly Damped Structures {#method-iii-lightly-damped-structures}
It is found that some structures do not provide FRF data which respond very well to the above modal analysis procedures mainly because of the difficulties encountered in acquiring good measurements near resonance.
For such structures, it is often the case that interest is confined to an **undamped model** of the test structure since the damping in a complete structural assembly is provided mostly from the joints and not from the components themselves.
Thus, there is scope for an alternative method of modal analysis which is capable of providing the required modal properties, in this case **natural frequencies** and **real modal constants**, using data measured **away from the resonance regions**.
The requirements for the analysis are as follows:
1. measure the FRF over the frequency range of interest
2. locate the resonances and note the corresponding natural frequencies
3. select individual FRF measurement data points from as many frequencies as there are modes, plus two, confining the selection to points away from resonance
4. using the data thus gathered, compute the modal constants
5. construct a regenerated curve and compare this with the full set of measured data points
### Global modal analysis methods in the frequency domain {#global-modal-analysis-methods-in-the-frequency-domain}
#### General Approach {#general-approach}
More recent curve fitting procedures are capable of performing a multi curve fit instead of just working with individual FRF curves.
They **fit several FRF curves simultaneously**, taking due account of the fact that the **properties of all the individual curves are related** by being from the **same structure**:
all FRF plots on a given testpiece should indicate the same values for natural frequencies and damping factor of each mode.
Such methods have the advantage of producing a **unique and consistent model** as direct output.
A way in which a set of measured FRF curves may be used collectively, rather than singly, is by the construction of a single **Composite Response Function**:
\begin{equation}
\sum\_j\sum\_k H\_{jk}(\omega) = \sum\_j\sum\_k\sum\_{r=1}^N (\dots) = HH(\omega)
\end{equation}
with
\\[ H\_{jk} = \sum\_{r=1}^n \frac{{}\_rA\_{jk}}{\omega\_r^2 - \omega^2 + i \eta\_r \omega\_r^2} \\]
The composite function \\(HH(\omega)\\) can provide a useful means of determining a single (average) value for the natural frequency and damping factor for each mode where the individual functions would each indicate slightly different values.
As an example, a set of mobilities measured are shown individually in figure [23](#org-target--fig:composite_raw) and their summation shown as a single composite curve in figure [23](#org-target--fig:composite_sum).
|  |  |
|---------------------------------------------------------------------------------------|-------------------------------------------------------------------------------------|
| Individual curves | Composite curve |
| width=\linewidth | width=\linewidth |
The global analysis methods have the disadvantages first, that the computation power required is high and second that there may be valid reasons why the various FRF curves exhibit slight differences in their characteristics and it may not always be appropriate to average them.
#### Global Rational Fraction Polynomial Method (GRFP) {#global-rational-fraction-polynomial-method--grfp}
The basic Rational Fraction Polynomial (RFP) method that was described in the context of single FRF curve can be generalized to multi-FRF data.
Indeed, all the FRFs from the same structure will have identical numerator polynomials.
The number of unknown coefficients in a problem where there are \\(n\\) measured FRFs and \\(m\\) modes of vibration is of the order \\((n+1)(2m + 1)\\).
#### Global SVD method {#global-svd-method}
A set of FRFs with a signal reference (such as are contained within a column from the complete FRF matrix) can be referred to the underlying modal model of the structure (assumed to have viscous damping) by the equation:
\begin{align\*}
\\{H(\omega)\\}\_k &= \begin{Bmatrix} H\_{1k}(\omega) \\\ \vdots \\\ H\_{nk}(\omega) \\\ \end{Bmatrix}\_{n\times 1} \\\\
&= [\Phi]\_{n\times N} \\{g\_k(\omega)\\}\_{N\times 1} + \\{R\_k(\omega)\\}\_{n\times 1}
\end{align\*}
where \\(\\{R\_k(\omega)\\}\\) is a vector containing the relevant residual terms and \\(\\{g\_k(\omega)\\}\\) is defined as:
\\[ \\{g\_k(\omega)\\}\_{N\times 1} = [i \omega - s\_r]\_{N\times N}^{-1} \\{\phi\_k\\}\_{N\times 1} \\]
Also
\\[ \\{\dot{H}(\omega)\\}\_k = [\Phi] [s\_r] \\{g\_k(\omega)\\} + \\{\dot{R}\_k(\omega)\\} \\]
Next, we can write the following expressions
\begin{equation}
\begin{aligned}
\\{\Delta H(\omega\_i)\\}\_k &= \\{H(\omega\_i)\\}\_k - \\{H(\omega\_{i+c})\\}\_k \\\\
&\approx [\Phi] \\{\Delta g\_k(\omega\_i)\\}\_{N\times 1} \\\\
\\{\Delta \dot{H}(\omega\_i)\\}\_k &\approx [\Phi] [s\_r] \\{\Delta g\_k(\omega\_i)\\}\_{N\times 1}
\end{aligned}
\end{equation}
If we now consider data at several different frequencies \\(i = 1, 2, \dots, L\\), we can write
\begin{equation}
\begin{aligned}
[\Delta H\_k]\_{n\times L} &= [\Phi] [\Delta g\_k]\_{N\times L} \\\\
[\Delta \dot{H}\_k]\_{n\times L} &= [\Phi] [s\_r] [\Delta g\_k]\_{N\times L}
\end{aligned}
\end{equation}
We can construct an eigenvalue problem:
\\[ \left( [\Delta \dot{H}\_k]^T - s\_r [\Delta H\_k]^T \right) \\{z\\}\_r = \\{0\\} \\]
where
\\[ [z] = [\Phi]^{+T} \\]
If we solve \\([z] = [\Phi]^{+T}\\) using the SVD, we can determine the rank of the FRF matrices and thus the correct number of modes \\(m\\) to be identified, leading to the appropriate eigenvalues \\(s\_r;\ r=1, \dots, m\\).
Then, in order to determine the mode shapes, the modal constants can be recovered from:
\begin{equation}
\begin{aligned}
\begin{Bmatrix} H\_{jk}(\omega\_1) \\\ \vdots \\\ H\_{jk}(\omega\_L) \\\ \end{Bmatrix}\_{L\times 1} = &\begin{bmatrix}
(i\omega\_1 - s\_1)^{-1} & (i\omega\_1 - s\_2)^{-1} & \dots \\\\
(i\omega\_2 - s\_1)^{-1} & (i\omega\_2 - s\_2)^{-1} & \dots \\\\
\vdots & \dots & \dots \\\\
\vdots & \dots & (i\omega\_L - s\_m)^{-1} \\\\
\end{bmatrix}\\\\
&\begin{Bmatrix}
{}\_1A\_{jk}\\\ \vdots \\\ {}\_mA\_{jk}
\end{Bmatrix}\_{m\times 1}
\end{aligned}
\end{equation}
Using this approach, it is possible to extract a consistent set of modal parameters for the model whose FRFs have been supplied.
### Concluding comments {#concluding-comments}
In the task of extracting modal model parameters from measured test data, the analyst must rely on the skill of others who have coded the various analysis algorithms since these are generally complex.
Because of this, the analyst must develop the various skills which enable him to **select the most appropriate analysis procedure** for each case and to make the best interpretation of the output of these analysis methods.
In this chapter, we have first highlighted the **need for accuracy and reliability in the measured data** that is the source of a modal analysis.
If these data are not of high quality, the resulting modal model cannot be expected to be any better.
Thus, attention must be paid at the initial phases to ascertain and to assure the necessary quality of the raw data.
Question as to the **correct order for the model** and the **most appropriate model for damping** are often foremost among these early interpretations.
A hierarchy of different types of modal analysis procedure have been cataloged, from the simple SDOF one-mode-at-a-time for a single response function, through MDOF methods which reveal several modes at a time, to global analysis methods where several modes are extracted simultaneously from several FRFs.
## Derivation of Mathematical Models {#derivation-of-mathematical-models}
### Introduction {#introduction}
We consider now the derivation of a mathematical model to describe the dynamic behavior of the test structure.
Various types of model exists and are suitable in different cases.
The most important aspect of the modeling process is to **decide exactly which type of model we should seek** before setting out on the acquisition and processing of experimental data.
Three main categories of model are identified:
- **Spatial model**: mass, stiffness and damping
- **Modal model**: natural frequencies, mode shapes
- **Response model**: frequency response functions
There exist **complete models** of each type and the more realistic **incomplete models** we are obliged to consider in practical cases.
The three types of model are usually derived as \\(\text{Spatial}\Rightarrow\text{Modal}\Rightarrow\text{Response}\\) for theoretical analysis, and conversely, as \\(\text{Response}\Rightarrow\text{Modal}\Rightarrow\text{Spatial}\\) for an experimental study.
We may now view them in a different order, according to the facility with which each may be derived from the test data: Modal, Response and then Spatial.
This reflects the quantity of the completeness of data required in each case.
A **modal model** can be constructed using just one single mode, and including only a handful of degrees of freedom, even though the structure has many modes and many DOFs.
Such a model can be built up by adding data from more modes, but it is not a requirement that **all** the modes should be included nor even that all the modes in the frequency range of interest be taken into account.
Thus such a model may be derived with relatively few, or equally, with many data.
The **response type of model** in the form of a FRF matrix, such as the mobility matrix, also needs only to include information concerning a limited number of point of interest: not all the DOFs need be considered.
However, in this case, it is generally required that the model be valid over a specified frequency range, and here it is necessary that all the modes in that range be included.
Also, some account should be taken of the modes whose natural frequencies lie outside of the range of interest to allow for the residual effects.
Thus, the response type of model demands more data to be collected from the tests.
A representative **spatial model** can only be obtained if we have measured most of the modes of the structure and if we have made measurements at a great many of the DOFs it possesses.
This is generally a very demanding requirement to meet, and as result, the derivation of a spatial model from test data is very difficult to achieve.
This chapter is organized with the following structure:
1. We shall describe what data must be measured in order to construct a suitable model and what checks can be made to access the reliability of the model.
2. We shall discuss a number of techniques for "**refining**" the model which is obtained from the test so that it matches a number of features of the analytical model.
For instance, it is common to extract complex mode shapes from the test data on real structures but the analytical models are usually undamped so that their modes are real.
3. We may wish to expand our experimental model, or, alternatively, reduce the theoretical ones so that the two models which are to be compared are at least of the same order.
4. We shall explore some of the properties of the models which can be derived by the means described here.
### Modal models {#modal-models}
#### Requirements to construct modal model {#requirements-to-construct-modal-model}
A **modal model** of a structure consists of **two matrices**:
- one containing the **natural frequencies** and **damping factors**: the eigenvalues
- one which describes the **shapes of the corresponding modes**: the eigenvectors
Thus, we can construct such a model with just a single mode, and a more complete model is assembled simply by **adding together a set of these single-mode descriptions**.
The basic method of deriving a modal model is as follows.
First, we note that from a single FRF curve, \\(H\_{jk}(\omega)\\), it is possible to extract certain modal properties for the \\(r^\text{th}\\) mode by modal analysis:
\begin{equation} \label{eq:modal\_model\_from\_frf}
H\_{jk}(\omega) \longrightarrow \omega\_r, \eta\_r, {}\_rA\_{jk}; \quad r=1, m
\end{equation}
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 , 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
\end{equation}
If we then measure an associated transfer FRF using the same excitation position, such as \\(H\_{jk}\\), we are able to deduce the mode shape element corresponding to the new response point \\(\phi\_{jr}\\) using the fact that the relevant modal constants may be combined with those from the point measurement:
\begin{equation}
\tcmbox{\phi\_{jr} = \frac{{}\_rA\_{jk}}{\phi\_{kr}}}
\end{equation}
Hence, we find that in order to derive a modal model referred to a particular set of \\(n\\) coordinates, we need to measure and analysis a set of \\(n\\) FRF curves, **all sharing the same excitation point** and thus constituting one point FRF and \\((n-1)\\) transfer FRFs.
In terms of the complete FRF matrix, this corresponds to measure the individual FRF of **one entire column**.
It is also possible to measure one row of the FRF matrix. This corresponds of a set of \\(n\\) FRF curves sharing the same measurement point and varied excitation point.
Often, several additional elements from the FRF matrix would be measured to provide a check, or to replace poor data, and sometimes measurement of a complete **second column** or row might be advised in order to ensure that one or more modes have not been missed by an unfortunate choice of exciter location.
Indeed, if the exciter is placed at a nodal point of one of the modes, then there would be no indications at all of the existence of that mode because every modal constant would be zero for that mode.
It may then require more than one measurement to confirm that we are not exciting the structure at a nodal point.
Once all the selected FRF curves have been measured and individually analyzed, we obtain a set of modal properties containing **more data than needed**:
- we may have determined many separate estimates for the natural frequencies and damping factors as these parameters are extracted from each FRF curve
- in the even we have measured more than one row or one column for the FRF matrix, we also obtain separate estimates for the mode shapes
The simplest procedure is to average all the individual estimates that results in means values \\(\tilde{\omega}\_r\\) and \\(\tilde{\eta}\_r\\).
In practice, not all the estimates should carry equal weight because some would probably be derived from much more satisfactory curve fits than others.
A refined procedure would be to calculate a **weighted mean** of all the estimate using the quality factor obtained from the curve-fit procedure.
If we choose to accept a revised value for \\(\omega\_r\\) and \\(\eta\_r\\) of a particular mode, then the value for the modal constant should also be revised:
\begin{equation}
{}\_r\tilde{A}\_{jk} = {}\_rA\_{jk}\frac{\tilde{\omega}\_r^2 \tilde{\eta}\_r}{\omega\_r^2 \eta\_r}
\end{equation}
The final reduced model obtained consist of the two matrices which constitute a modal model, namely:
\\[ \left[ \omega\_r^2(1 + i\eta\_r) \right]\_{m\times m};\quad \left[ \Phi \right]\_{n\times m} \\]
#### Double modes or repeated roots {#double-modes-or-repeated-roots}
When a structure has two modes that are very close in frequency, it may be impossible to derive a true model for the structure.
All we can define in these circumstances is a single equivalent mode which is, in fact, a combination of the two actual modes that are difficult to identify individually.
However, single equivalent modes can lead to erroneous models and it is very important that we can detect the presence of double modes and that we can identify all the modes which are present.
The only way repeated modes can be detected and identified in a modal test is by using data from more than on reference.
This means that we must measure FRF data from **more than a single row or column** (as many rows/columns as there are repeated roots).
#### Constructing models of NSA structures {#constructing-models-of-nsa-structures}
Structures which are classified as Non-Self-Adjoint (NSA) have **non-symmetric mass**, **stiffness** or **damping matrices**.
This often occurs in structures with rotating components.
As a result, we cannot take advantage of the symmetry of the system matrices and just measuring a single row or column of the FRF matrix.
In the case of NSA structures, we are required to measure and analyze the elements in **both a row and a column of the FRF matrix**.
A mathematical explanation is that this class of system have two types of eigenvectors (left-hand and right-hand) and thus there are twice as many eigenvectors elements to identify.
#### Quality checks for modal models {#quality-checks-for-modal-models}
It is important to check the **reliability** of the obtain results.
There are two such checks that can be recommended for this phase of the process.
First, it is possible to **regenerate FRFs** from the modal model.
These FRFs can be compared with measured data that as been used for the modal analysis.
Furthermore, it is also possible to synthesize FRFs that have not yet been measured (and thus not used for the model), and then to measure the corresponding FRF on the structure and to compare.
This test provides a powerful demonstration of the validity of the modal model.
A second, more demanding but also more convincing, demonstration of the validity of the modal model is to use the model to predict how the dynamic properties of the test structure will change if it is subjected to a small structural modification, such as can be occasioned by adding a small mass at a selected point.
Then such modification can be made and the real structure, measurements done and compare with the modified model.
### Refinement of modal models {#refinement-of-modal-models}
#### Need for model refinement {#need-for-model-refinement}
Several differences exist between most test-derived models and analytical models that make their comparison difficult.
The first difference are on the **mode shapes**:
- **test-derived**: generally complex
- **analytical**: usually real if we use an undamped model
Objective comparison between complex mode shapes and real mode shapes is then not possible and some refinement of one of the two sets are required.
A second incompatibility lies in the difference in the **order of the models**:
- **test-derived**: relatively small order given by the number of measured DOFs \\(n\\)
- **analytical**: generally an order of magnitude greater than \\(n\\)
There is then a desire to **refine** one or other model to **bring them both to the same size** for meaningful comparison.
However, all the refinements involve **approximations** which means that a compromise has been made in order to achieve the greater degree of compatibility which is desired.
#### Complex-to-real conversion {#complex-to-real-conversion}
As we usually don't know the nature, extend and distribution of damping present in the system, the analytical model is chosen to be **undamped**.
We wish here to be able to determine what would be the mode shapes of the tested structure if, by some means, we could remove the damping but leave everything else the same.
Then we should be able to compare the modes.
##### Simple method {#simple-method}
This simple method is to convert the mode shape vectors from complex to real by taking the modulus of each element and by assigning a phase to each of \\(\SI{0}{\degree}\\) or \\(\SI{180}{\degree}\\).
Any phase angle of a complex mode shape element which is between \\(\SI{-90}{\degree}\\) and \\(\SI{90}{\degree}\\) is set to \\(\SI{0}{\degree}\\), while those between \\(\SI{90}{\degree}\\) and \\(\SI{270}{\degree}\\) are set to \\(\SI{180}{\degree}\\).
This procedure can become difficult to apply in borderline cases when the phase is poorly defined.
##### Multi point excitation - Asher's method {#multi-point-excitation-asher-s-method}
In this method, the test-derived model based on complex modes is used to synthesize the response that would be produced by the application of several simultaneous harmonic forces in order to establish what those forces would need to be in order to produce a mono-modal response vector.
If the optimum set of excitation forces for a given mode can be found, then they represent the forces that are actually being generated by the damping in the system at resonance of that mode.
We can then deduce the dynamic properties of the structure with these effects removed.
The sequence of steps required to determine this solution is as follows:
1. Compute \\([\alpha(\omega)]\\) from the complex modal model
2. Determine the undamped system natural frequencies \\(\omega\_r\\) by solving the equation \\(\det|\text{Re}[\alpha(\omega)]|=0\\)
3. Calculate the mono-phase vector for each mode of interest using \\(\text{Re}[\alpha(\omega)]\\{\hat{F}\\} = \\{0\\}\\)
4. Calculate the undamped system mode shapes \\(\\{\psi\_u\\}\\) using the just-derived force vector: \\(\\{\psi\_u\\} = \text{Im}[\alpha(\omega)]\\{\hat{F}\\}\\)
##### Matrix transformation {#matrix-transformation}
We here seek a numerical solution to the expression linking the known damped modes and the unknown undamped modes.
The steps are:
1. Assume that \\(\text{Re}[T\_1]\\) is unity and calculate \\(\text{Im}[T\_1]\\) from
\\[ \text{Im}[T\_1] = -[\text{Re}[\phi\_d]]^T [\text{Re}[\phi\_d]]^{-1} [\text{Re}[\phi\_d]]^T \text{Im}[\phi\_d] \\]
2. calculate \\([M\_1]\\) and \\([K\_1]\\) from
\\[ [M\_1] = [T\_1]^T[T\_1]; \quad [K\_1] = [T\_1]^T[\lambda^2][T\_1] \\]
3. Solve the eigen-problem formed by \\([M\_1]\\) and \\([K\_1]\\) leading to
\\[ [\omega\_r^2]; \quad [T\_2] \\]
4. Calculate the real modes using
\\[ [\phi\_u] = [\phi\_d][T\_1][T\_2] \\]
#### Expansion of models {#expansion-of-models}
An important model refinement is called **expansion** and consist of the addition to the actually measured modal data of estimates for selected DOFs which were not measured for one reason or another.
Prior to conducting each modal test, decisions have to be made as to **which of the many DOFs** that exist on the structure **will be measured**.
These decisions are made for various practical reasons:
- Limited test time
- Inaccessibility of some DOFs
- Anticipated low importance of motion in certain DOFs
**Three approaches** to the expansion of measured modes will be mentioned here:
1. **Geometric interpolation** using spline functions
2. Expansion using the analytical model's **spatial properties**
3. Expansion using the analytical model's **modal properties**
In all three approached, we are in effect seeking a transformation matrix \\([T]\\) that allows us to construct a long eigenvector \\(\\{\phi\\}\_{N\times 1}\\) from knowledge of a short (incomplete) one \\(\\{\phi\\}\_{n\times 1}\\):
\\[ \\{\phi\\}\_{N\times 1} = [T]\_{N\times n} \\{\phi\\}\_{n\times 1} \\]
##### Interpolation {#interpolation}
Simple interpolation has a limited range of application and can only be used on structures which have **large regions of relatively homogeneous structure**: those with joints of abrupt changes are must less likely to respond to this form of expansion.
The method is simply geometric interpolation between the measured points themselves, such as by fitting a polynomial function through the measured points.
##### Expansion using theoretical spatial model - Kidder's method {#expansion-using-theoretical-spatial-model-kidder-s-method}
This interpolation uses a **theoretical model's mass and stiffness matrices** in a form of an inverse Guyan reduction procedure.
If we **partition** the eigenvector of interest, \\(\\{\phi\_A\\}\_r\\), into:
- the DOFs to be included: \\(\\{{}\_A\phi\_1\\}\_r\\)
- the DOFs which are not available from the measurements: \\(\\{{}\_A\phi\_2\\}\_r\\)
then we may write:
\begin{equation\*}
\left( \begin{bmatrix}
{}\_AK\_{11} & {}\_AK\_{12} \\\\
{}\_AK\_{21} & {}\_AK\_{22}
\end{bmatrix} - \omega\_r^2 \begin{bmatrix}
{}\_AM\_{11} & {}\_AM\_{12} \\\\
{}\_AM\_{21} & {}\_AM\_{22}
\end{bmatrix} \right) \begin{bmatrix}
{}\_A\phi\_1 \\\\
{}\_A\phi\_2
\end{bmatrix} = \\{0\\}
\end{equation\*}
We can use this relationship between the measured and unmeasured DOFs as the basic for an expansion of the incomplete measured mode shapes:
\\[ \\{{}\_A\phi\_2\\}\_r = [T\_{21}]\\{{}\_A\phi\_1\\}\_r \\]
with
\\[ [T\_{12}] = - \left( [{}\_AK\_{22}] - \omega\_r^2[{}\_AM\_{22}] \right)^{-1} \left( [{}\_AK\_{21}] - \omega\_r^2[{}\_AM\_{21}] \right) \\]
The relation between the incomplete measured vector to the complete expanded vector is then
\begin{equation}
\tcmbox{ \\{\tilde{\phi}\_X\\}\_r = \begin{Bmatrix}
{}\_X\phi\_1 \\\ {}\_X\tilde{\phi}\_2
\end{Bmatrix} = \begin{bmatrix}
[I] \\\ [T\_{21}]
\end{bmatrix} \\{{}\_X\phi\_1\\}\_r }
\end{equation}
##### Expansion using analytical model mode shapes {#expansion-using-analytical-model-mode-shapes}
This method uses the analytical model for the interpolation, but is based on the **mode shapes derived from the analytical modal spatial matrices**, rather than on these matrices themselves.
We may write the following expression which relates the experimental model mode shapes to those of the analytical model:
\begin{equation\*}
\begin{bmatrix}
{}\_X\phi\_1 \\\\
{}\_X\phi\_2
\end{bmatrix} = \begin{bmatrix}
[{}\_A\Phi\_{11}] & [{}\_A\Phi\_{12}] \\\\
[{}\_A\Phi\_{21}] & [{}\_A\Phi\_{22}]
\end{bmatrix} \begin{bmatrix}
\gamma\_1 \\\\
\gamma\_2
\end{bmatrix}\_r
\end{equation\*}
The basic of this method is to assume that the measured mode shape submatrix can be represented exactly by the simple relationship (which assumes that \\(\\{\gamma\_2\\}\_r\\) can be taken to be zero):
\begin{equation}
\\{{}\_X\phi\_1\\}\_r = [{}\_A\Phi\_{11}] \\{\gamma\_1\\}\_r
\end{equation}
so that an estimate can be provided for the unmeasured part of the eigenvector from
\begin{equation}
\begin{aligned}
\\{{}\_X\tilde{\phi}\_2\\} &= [T\_{21}] \\{{}\_X\phi\_1\\}\_r \\\\
&= [{}\_A\Phi\_{21}][{}\_A\Phi\_{11}]^{-1} \\{{}\_X\phi\_1\\}\_r
\end{aligned}
\end{equation}
Thus, we can write the full transformation as:
\begin{equation\*}
\tcmbox{ \\{\tilde{\phi}\_X\\}\_r = \begin{Bmatrix}
{}\_X\phi\_1 \\\ {}\_X\tilde{\phi}\_2
\end{Bmatrix} = \begin{bmatrix}
[{}\_A\Phi\_{11}] \\\ [{}\_A\Phi\_{21}]
\end{bmatrix} [{}\_A\Phi\_{11}]^{-1} \\{{}\_X\phi\_1\\}\_r }
\end{equation\*}
This formula can be generalized to a single expression which covers several measured modes:
\begin{equation\*}
\tcmbox{[\tilde{\Phi}\_X]\_{N\times m\_X} = \underbrace{[\Phi\_A]\_{N\times m\_A} [{}\_A\Phi\_{11}]\_{m\_A \times n}^+}\_{[T]\_{N\times n}} [{}\_A\Phi\_1]\_{n\times m\_X}}
\end{equation\*}
where \\(m\_X\\) and \\(m\_A\\) are the number of experimental and analytical modes used, respectively.
Other formulations for \\([T]\\) are possible, they involve various combinations of the available experimental mode shape data and those from the analytical model:
\begin{equation}
\begin{aligned}
[T\_{(1)}] &= [\Phi\_A][{}\_A\Phi\_1]^+ & \text{A model - based} \\\\
[T\_{(2)}] &= [\Phi\_A][{}\_X\Phi\_1]^+ & \text{X model - based} \\\\
[T\_{(3)}] &= \begin{bmatrix}
{}\_X\Phi\_1 \\\\
{}\_A\Phi\_2
\end{bmatrix}[{}\_A\Phi\_1]^+ & \text{Mixed/A - based} \\\\
[T\_{(4)}] &= \begin{bmatrix}
{}\_X\Phi\_1 \\\\
{}\_A\Phi\_2
\end{bmatrix}[{}\_X\Phi\_1]^+ & \text{Mixed/X - based}
\end{aligned}
\end{equation}
It must be pointed out that all the above formula are approximate because of the initial assumption that the higher modes are not required to be included in the process (that \\(\\{\gamma\_2\\}\\) is zero).
#### Reduction of models {#reduction-of-models}
The model reduction, which is the inverse of the expansion process, is used when it is decided to obtain compatibility between two otherwise disparate models by reducing the size of the larger of the two models (almost always, the analytical model).
Model reduction has less importance nowadays as computing power is widely available and because such reduction introduces approximations.
There are basically **two different types of model reduction**, both of which are applied to the spatial model (as opposed to the modal model as is the case in model expansion), and both achieve the same end result of yielding a smaller order model, with system matrices which are \\(n\times n\\) instead of \\(N\times N\\):
1. a **condensed model** which seeks to represent the entire structure completely at a smaller number of DOFs. This type of model introduces approximation.
2. a **reduced model** which has removed information related to the DOFs that are eliminated from the model, and which is thus an incomplete model. However, for the retained DOFs, no information is lost.
Let's summarize the basic feature of model reduction by **condensation**.
The basic equation of motion for the original model can be expressed as:
\\[ [M] \ddot{x} + [K]\\{x\\} = \\{f\\} \\]
and this can be partitioned into the **kept DOFs** \\(\\{x\_1\\}\\) and the **eliminated DOFs** \\(\\{x\_2\\}\\) (which by definition cannot have any excitation forces applied to them):
\begin{equation\*}
\begin{bmatrix}
M\_{11} & M\_{12} \\\\
M\_{21} & M\_{22}
\end{bmatrix} \begin{Bmatrix}
\ddot{x}\_1 \\\ \ddot{x}\_2
\end{Bmatrix} + \begin{bmatrix}
K\_{11} & K\_{12} \\\\
K\_{21} & K\_{22}
\end{bmatrix} \begin{Bmatrix}
x\_1 \\\ x\_2
\end{Bmatrix} = \begin{Bmatrix}
f\_1 \\\ 0
\end{Bmatrix}
\end{equation\*}
A relationship between the kept and eliminated DOFs can be written in the form:
\begin{equation}
\begin{Bmatrix}
x\_1 \\\ x\_2
\end{Bmatrix}\_{N\times 1} = \begin{bmatrix}
[I] \\\ [T]
\end{bmatrix}\_{N\times n} \\{x\_1\\}\_{n\times 1}
\end{equation}
where the transformation matrix \\([T]\\) can be defined by
\begin{equation\*}
[T] = (1 - \beta)\left(-[K\_{22}]^{-1}[K\_{21}]\right) + \beta\left(-[M\_{22}]^{-1}[M\_{21}]\right)
\end{equation\*}
in which \\(\beta\\) is a reduction coefficient whose limiting values are \\(\beta = 0\\) for **static reduction** and \\(\beta = 1\\) for **dynamic reduction**.
The **reduced mass and stiffness matrices** which are produced by this process are:
\begin{align\*}
\left[M^R\right]\_{n\times n} &= \begin{bmatrix}[I] & [T]^T\end{bmatrix}\_{n\times N} \begin{bmatrix}
M\_{22} & M\_{21} \\\ M\_{12} & M\_{22}
\end{bmatrix}\_{N\times N} \begin{bmatrix}
[I] \\\ [T]
\end{bmatrix}\_{N\times n}\\\\
\left[K^R\right]\_{n\times n} &= \begin{bmatrix}[I] & [T]^T\end{bmatrix}\_{n\times N} \begin{bmatrix}
K\_{22} & K\_{21} \\\ K\_{12} & K\_{22}
\end{bmatrix}\_{N\times N} \begin{bmatrix}
[I] \\\ [T]
\end{bmatrix}\_{N\times n}
\end{align\*}
The two limiting cases of static and dynamic reduction are of particular interest.
In each case, one of the two system matrices is unchanged and the other one is:
\begin{align\*}
\beta = 1:\ &[M^{R\text{static}}] = [M\_{12}] \left(-[M\_{22}]^{-1}[M\_{21}]\right)^{-1} + [M\_{11}]\\\\
&[K^{R\text{static}}] = [K]\\\\
\beta = 0:\ &[M^{R\text{dynamic}}] = [M]\\\\
&[K^{R\text{dynamic}}] = [K\_{12}] \left(-[K\_{22}]^{-1}[K\_{21}]\right)^{-1} + [K\_{11}]
\end{align\*}
These reduction procedure can provide useful approximate models of the structure if an optimum choice of which DOFs to retain and which can be eliminated is made.
However, a reduced theoretical model of this type does not correspond to a similarly low-order model which is obtained from experiments since that is formed simply by ignoring the eliminated DOFs.
The measured data for the included DOFs are the same no matter how many DOFs are eliminated.
Thus, there are inherent difficulties involved in using this mixture of condensed (but complete) theoretical models and reduced (but incomplete) experimental models.
### Display of modal models {#display-of-modal-models}
One of the attraction of the modal model is possibility of obtaining a **graphic display** of its form by plotting the mode shapes.
There are basically two choices for the graphical display of a modal model:
- a **static plot**
- a **dynamic** (animated) **display**
#### Static Displays {#static-displays}
##### Deflected shapes {#deflected-shapes}
A static display is often adequate for depicting relatively simple mode shapes.
Measured coordinates of the test structure are first linked as shown on figure [31](#figure--fig:static-display) (a).
Then, the grid of measured coordinate points is redrawn on the same plot but this time displaced by an amount proportional to the corresponding element in the mode shape vector as shown on figure [31](#figure--fig:static-display) (b).
The elements in the vector are scaled according the normalization process used (usually mass-normalized), and their absolute magnitudes have no particular significance.
{{< figure src="/ox-hugo/ewins00_static_display.png" caption="Figure 31: Static display of modes shapes. (a) basic grid (b) single-frame deflection pattern (c) multiple-frame deflection pattern (d) complex mode (e) Argand diagram - quasi-real mode (f) Argand diagram - complex mode" >}}
It is customary to select the largest eigenvector element and to scale the whole vector by an amount that makes that displacement on the plot a viable amount.
##### Multiple frames {#multiple-frames}
If a series of deflection patterns that has been computed for a different instant of time are superimposed, we obtain a result as shown on figure [31](#figure--fig:static-display) (c).
Some indication of the motion of the structure can be obtained, and the points of zero motion (nodes) can be clearly identified.
It is also possible, in this format, to give some indication of the essence of complex modes, as shown in figure [31](#figure--fig:static-display) (d).
Complex modes do not, in general, exhibit fixed nodal points.
##### Argand diagram plots {#argand-diagram-plots}
Another form of representation which is useful for complex modes is the representation of the individual complex elements of the eigenvectors on a polar plot, as shown in the examples of figure [31](#figure--fig:static-display) (e) and (f).
Although there is no attempt to show the physical deformation of the actual structure in this format, the complexity of the mode shape is graphically displayed.
#### Dynamic Display {#dynamic-display}
The coordinates for the basic picture are computed and stored for multiple fractions of a cycle.
Then, 10 to 20 frames are stored and displayed with an update rate which is suitable to give a clear picture of the distortion of the structure during vibration.
The dynamic character of animation is the only really effective way to view modal complexity and is very useful to display complex modes.
#### Interpretation of mode shape displays {#interpretation-of-mode-shape-displays}
There are a number of features associated with mode shape displays that warrant a mention in the context of ensuring that the **correct interpretation** is made from viewing these displays.
The first concerns the consequences of viewing an **incomplete model**.
In that case, there are no mode shape data from some of the points which comprise the grid which outlines the structure, and the indicated result is zero motion of those DOFs and this can be very **misleading**.
For instance, if we measure the displacement of grid points in only one direction, \\(x\\) for instance, then the shape display will show significant x-direction motion of those points with no motion in the other transverse directions.
We then tend to interpret this as a motion which is purely in the x-direction which may be clearly not true.
The second problem arises when the **grid of measurement points** that is chosen to display the mode shapes is **too coarse in relation to the complexity of the deformation patterns** that are to be displayed.
This can be illustrated using a very simple example: suppose that our test structure is a straight beam, and that we decide to use just three response measurements points.
If we consider the first six modes of the beam, whose mode shapes are sketched in figure [32](#figure--fig:beam-modes), then we see that with this few measurement points, modes 1 and 5 look the same as do modes 2, 4 and 6.
All the higher modes will be indistinguishable from these first few.
This is a well known problem of **spatial aliasing**.
{{< figure src="/ox-hugo/ewins00_beam_modes.png" caption="Figure 32: Misinterpretation of mode shapes by spatial aliasing" >}}
### Response models {#response-models}
There are two main requirements demanded for a response model:
- the capability of regeneration "theoretical" curves for the FRFs actually measured and analyzed
- synthesizing the other response functions which were not measured
In general, the form of response model with which we are concerned is an **FRF matrix** whose order is dictated by the number of coordinates \\(n\\) included in the test.
Also, as explained, it is normal in practice to measured and to analyze just a **subset of the FRF matrix** but rather to measure the full FRF matrix.
Usually **one column** or **one row** with a few additional elements are measured.
Thus, if we are to construct an acceptable response model, it will be necessary to synthesize those elements which have not been directly measured.
However, in principle, this need present no major problem as it is possible to compute the full FRF matrix from a modal model using:
\begin{equation} \label{eq:regenerate\_full\_frf\_matrix}
\tcmbox{[H]\_{n\times n} = [\Phi]\_{n\times m} [\lambda\_r^2 - \omega^2]\_{m\times m}^{-1} [\Phi]\_{m\times n}^T}
\end{equation}
#### Regenerated FRF curves {#regenerated-frf-curves}
It is usual practice to regenerate an FRF curve using the results from the modal analysis as a mean of **checking the success of that analysis**.
It should be noted that in order to construct an acceptable response model, it is essential that all the modes in the frequency range of interest be included, and that suitable residual terms are added to take account of out-of-range modes.
In this respect, the demands of the response model are more stringent that those of the modal model.
#### Synthesis of FRF curves {#synthesis-of-frf-curves}
One of the implications of equation 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.
As an example, suppose that FRF data \\(H\_{11}\\) and \\(H\_{21}\\) are measured and analyzed in order to synthesize the FRF \\(H\_{22}\\) initially unmeasured.
The predict curve is compared with the measurements on figure [24](#org-target--fig:H22_without_residual).
Clearly, the agreement is poor and would tend to indicate that the measurement/analysis process had not been successful.
However, the synthesized curve contained only those terms relating to the modes which had actually been studied from \\(H\_{11}\\) and \\(H\_{21}\\) and this set of modes did not include **all** the modes of the structure.
Thus, \\(H\_{22}\\) **omitted the influence of out-of-range modes**.
The inclusion of these two additional terms (obtained here only after measuring and analyzing \\(H\_{22}\\) itself) resulted in the greatly improved predicted vs measured comparison shown in figure [24](#org-target--fig:H22_with_residual).
|  |  |
|-----------------------------------------------------------------------------------------------------------|-----------------------------------------------------------------------------------------------------------|
| Using measured modal data only | After inclusion of residual terms |
| width=\linewidth | width=\linewidth |
The appropriate expression for a "correct" response model, derived via a set of modal properties is thus
\begin{equation}
[H] = [\Phi] [\lambda\_r^2 - \omega^2]^{-1} [\Phi]^T + [\text{Res}]
\end{equation}
In order to obtain all the data necessary to form such a model, we must first derive the modal model on which it is based and then find some means of **determining the elements in the residual matrix** \\([\text{Res}]\\).
This latter task may be done in several ways:
- It may be most accurately achieved by **measuring all** (or at least over half) **of the elements in the FRF matrix**, but this would increase a lot the quantity of data to be measured.
- **Extend the frequency range of the modal test** beyond that over which the model is eventually required.
In this way, much of the content of the residual terms is included in separate modes and their actual magnitudes can be reduced to relatively unimportant dimensions.
- Try to access **which of the many FRF elements are liable to need large residual terms** and to make sure that these are included in the list of those which are measured and analyzed.
We noted earlier that it is the point mobilities which are expected to have the highest-valued residuals and the remote transfers which will have the smallest.
Thus, the significant terms in the \\([\text{Res}]\\) matrix will generally be grouped close to the leading diagonal, and this suggests **making measurements of most of the point mobility parameters**.
#### Direct measurement {#direct-measurement}
It should be noted that it is quite possible to develop a response model by measuring and analyzing all the elements in one half of the FRF matrix (this being symmetric) and by storing the results of this process without constructing a modal model.
This procedure clearly solves the residual problem discussed above, but it will introduce **inconsistencies** into to model which renders it unsatisfactory.
#### Transmissibilities {#transmissibilities}
One vibration parameter which has not been mentioned so far is that of **transmissibility**.
This is a quantity which is quite widely used in vibration engineering practice to indicate the relative vibration levels between two points.
In general, transmissibility is considered to be a frequency dependent response function \\(T\_{jk}(\omega)\\) which defines the ratio between the response levels at two DOFs \\(j\\) and \\(k\\).
Simply defined, we can write:
\begin{equation}
T\_{jk} (\omega) = \frac{X\_j e^{i\omega t}}{X\_k e^{i\omega t}}
\end{equation}
but, in fact, we need also to **specify the excitation conditions that give rise to the two responses** in question and these are missing from the above definition which is thus not rigorous.
It does not give us enough information to be able to reproduce the conditions which have been used to measured \\(T\_{jk}(\omega)\\).
If the **transmissibility** is measured during a modal test which has a single excitation, say at DOF \\(i\\), then we can define the transmissibility thus obtained more precisely:
\begin{equation}
{}\_iT\_{jk}(\omega) = \frac{H\_{ji}(\omega)}{H\_{ki}(\omega)}
\end{equation}
In general, the transmissibility **depends significantly on the excitation point** (\\({}\_iT\_{jk}(\omega) \neq {}\_qT\_{jk}(\omega)\\) where \\(q\\) is a different DOF than \\(i\\)) and it is shown on figure [33](#figure--fig:transmissibility-plots).
This may explain why transmissibilities are not widely used in modal analysis.
{{< figure src="/ox-hugo/ewins00_transmissibility_plots.png" caption="Figure 33: Transmissibility plots" >}}
#### Base excitation {#base-excitation}
The one application area where transmissibilities can be used as part of modal testing is in the case of **base excitation**.
Base excitation is a type of test where the input is measured as a response at the drive point \\(x\_0(t)\\), instead of as a force \\(f\_1(t)\\), as illustrated in figure [25](#table--fig:base-excitation-configuration).
We can show that it is possible to determine, from measurements of \\(x\_i\\) and \\(x\_0\\), modal properties of natural frequency, damping factor and **unscaled** mode shape for each of the modes that are visible in the frequency range of measurement.
The fact that the excitation force is not measured is responsible for the lack of formal scaling of the mode shapes.
|  |  |
|-------------------------------------------------------------------------------------------------------------------|---------------------------------------------------------------------------------------------------------|
| Conventional modal test setup | Base excitation setup |
| height=4cm | height=4cm |
### Spatial models {#spatial-models}
It would appear from the basic orthogonality properties of the modal model that there exists a simple means of constructing a spatial model from the modal model, thus this is not so.
We have that:
\begin{equation}
\begin{aligned}
[\Phi]^T[M][\Phi] &= [I]\\\\
[\Phi]^T[K][\Phi] &= [\lambda\_r^2]
\end{aligned}
\end{equation}
from which is would appear that we can write
\begin{equation} \label{eq:m\_k\_from\_modes}
\begin{aligned}
[M] &= [\Phi]^{-T} [I] [\Phi]^{-1}\\\\
[K] &= [\Phi]^{-T} [\lambda\_r^2] [\Phi]^{-1}
\end{aligned}
\end{equation}
However, equation 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.
This is accomplished using the above equation in the form:
\begin{equation}
\begin{aligned}
[K]\_{n\times n}^{-1} &= [\Phi]\_{n\times m} [\lambda\_r^2]\_{m\times m}^{-1} [\Phi]\_{m\times n}^T\\\\
[M]\_{n\times n}^{-1} &= [\Phi]\_{n\times m} [\Phi]\_{m\times n}^T
\end{aligned}
\end{equation}
Because the rank of each pseudo matrix is less than its order, it cannot be inverted and so we are unable to construct stiffness or mass matrix from this approach.
## Bibliography {#bibliography}
Ewins, DJ. 2000.
Modal Testing: Theory, Practice and Application.
Research Studies Pre, 2nd Ed., Isbn-13. Baldock, Hertfordshire, England Philadelphia, PA: Wiley-Blackwell.