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content/book/ewins00_modal.md
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@@ -161,7 +161,7 @@ Indeed, we shall see later how these predictions can be quite detailed, to the p
The main measurement technique studied are those which will permit to make **direct measurements of the various FRF** properties of the test structure.
The type of test best suited to FRF measurement is shown in figure [1](#figure--fig:modal-analysis-schematic).
The type of test best suited to FRF measurement is shown in [Figure 1](#figure--fig:modal-analysis-schematic).
<a id="figure--fig:modal-analysis-schematic"></a>
@@ -233,7 +233,7 @@ Thus there is **no single modal analysis method**, but rater a selection, each b
One of the most widespread and useful approaches is known as the **single-degree-of-freedom curve-fit**, or often as the **circle fit** procedure.
This method uses the fact that **at frequencies close to a natural frequency**, the FRF can often be **approximated to that of a single degree-of-freedom system** plus a constant offset term (which approximately accounts for the existence of other modes).
This assumption allows us to use the circular nature of a modulus/phase polar plot of the frequency response function of a SDOF system (see figure [2](#figure--fig:sdof-modulus-phase)).
This assumption allows us to use the circular nature of a modulus/phase polar plot of the frequency response function of a SDOF system (see [Figure 2](#figure--fig:sdof-modulus-phase)).
This process can be **repeated** for each resonance individually until the whole curve has been analyzed.
At this stage, a theoretical regeneration of the FRF is possible using the set of coefficients extracted.
@@ -272,7 +272,7 @@ Even though the same overall procedure is always followed, there will be a **dif
Theoretical foundations of modal testing are of paramount importance to its successful implementation.
The three phases through a typical theoretical vibration analysis progresses are shown on figure [3](#figure--fig:vibration-analysis-procedure).
The three phases through a typical theoretical vibration analysis progresses are shown on [Figure 3](#figure--fig:vibration-analysis-procedure).
Generally, we start with a description of the structure's physical characteristics (mass, stiffness and damping properties), this is referred to as the **Spatial model**.
<a id="figure--fig:vibration-analysis-procedure"></a>
@@ -295,7 +295,7 @@ Thus our response model will consist of a set of **frequency response functions
<div class="important">
As indicated in figure [3](#figure--fig:vibration-analysis-procedure), it is also possible to do an analysis in the reverse directly: from a description of the response properties (FRFs), we can deduce modal properties and the spatial properties: this is the **experimental route** to vibration analysis.
As indicated in [Figure 3](#figure--fig:vibration-analysis-procedure), it is also possible to do an analysis in the reverse directly: from a description of the response properties (FRFs), we can deduce modal properties and the spatial properties: this is the **experimental route** to vibration analysis.
</div>
@@ -314,7 +314,7 @@ Three classes of system model will be described:
</div>
The basic model for the SDOF system is shown in figure [4](#figure--fig:sdof-model) where \\(f(t)\\) and \\(x(t)\\) are general time-varying force and displacement response quantities.
The basic model for the SDOF system is shown in [Figure 4](#figure--fig:sdof-model) where \\(f(t)\\) and \\(x(t)\\) are general time-varying force and displacement response quantities.
The spatial model consists of a **mass** \\(m\\), a **spring** \\(k\\) and (when damped) either a **viscous dashpot** \\(c\\) or **hysteretic damper** \\(d\\).
<a id="figure--fig:sdof-model"></a>
@@ -392,7 +392,7 @@ which is a single mode of vibration with a complex natural frequency having two
- **An imaginary or oscillatory part**
- **A real or decay part**
The physical significance of these two parts is illustrated in the typical free response plot shown in figure [5](#figure--fig:sdof-response)
The physical significance of these two parts is illustrated in the typical free response plot shown in [Figure 5](#figure--fig:sdof-response)
<a id="figure--fig:sdof-response"></a>
@@ -424,26 +424,26 @@ which is now complex, containing both magnitude and phase information:
All structures exhibit a degree of damping due to the **hysteresis properties** of the material(s) from which they are made.
A typical example of this effect is shown in the force displacement plot in figure [1](#org-target--fig:material_histeresis) in which the **area contained by the loop represents the energy lost in one cycle of vibration** between the extremities shown.
A typical example of this effect is shown in the force displacement plot in [ 1](#org-target--fig-material-histeresis) in which the **area contained by the loop represents the energy lost in one cycle of vibration** between the extremities shown.
The maximum energy stored corresponds to the elastic energy of the structure at the point of maximum deflection.
The damping effect of such a component can conveniently be defined by the ratio of these two:
\\[ \tcmbox{\text{damping capacity} = \frac{\text{energy lost per cycle}}{\text{maximum energy stored}}} \\]
<a id="table--fig:force-deflection-characteristics"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--fig:force-deflection-characteristics">Table 1</a></span>:
<span class="table-number"><a href="#table--fig:force-deflection-characteristics">Table 1</a>:</span>
Force-deflection characteristics
</div>
| ![](/ox-hugo/ewins00_material_histeresis.png) | ![](/ox-hugo/ewins00_dry_friction.png) | ![](/ox-hugo/ewins00_viscous_damper.png) |
|-----------------------------------------------------------------------------------------------|---------------------------------------------------------------------------------|-------------------------------------------------------------------------------------|
| <span class="org-target" id="org-target--fig:material_histeresis"></span> Material hysteresis | <span class="org-target" id="org-target--fig:dry_friction"></span> Dry friction | <span class="org-target" id="org-target--fig:viscous_damper"></span> Viscous damper |
| <span class="org-target" id="org-target--fig-material-histeresis"></span> Material hysteresis | <span class="org-target" id="org-target--fig-dry-friction"></span> Dry friction | <span class="org-target" id="org-target--fig-viscous-damper"></span> Viscous damper |
| height=2cm | height=2cm | height=2cm |
Another common source of energy dissipation in practical structures, is the **friction** which exist in joints between components of the structure.
It may be described very roughly by the simple **dry friction model** shown in figure [1](#org-target--fig:dry_friction).
It may be described very roughly by the simple **dry friction model** shown in [ 1](#org-target--fig-dry-friction).
The mathematical model of the **viscous damper** which we have used can be compared with these more physical effects by plotting the corresponding force-displacement diagram for it, and this is shown in figure [1](#org-target--fig:viscous_damper).
The mathematical model of the **viscous damper** which we have used can be compared with these more physical effects by plotting the corresponding force-displacement diagram for it, and this is shown in [ 1](#org-target--fig-viscous-damper).
Because the relationship is linear between force and velocity, it is necessary to suppose harmonic motion, at frequency \\(\omega\\), in order to construct a force-displacement diagram.
The resulting diagram shows the nature of the approximation provided by the viscous damper model and the concept of the **effective or equivalent viscous damping coefficient** for any of the actual phenomena as being which provides the **same energy loss per cycle** as the real thing.
@@ -503,7 +503,7 @@ Similarly we could use the acceleration parameter so we could define a third FRF
</div>
Table [2](#table--tab:frf-alternatives) gives details of all six of the FRF parameters and of the names used for them.
[Table 2](#table--tab:frf-alternatives) gives details of all six of the FRF parameters and of the names used for them.
**Inverse response** can also be defined. For instance, the **dynamic stiffness** is defined as the force over the displacement.
@@ -521,7 +521,7 @@ It should be noted that that the use of displacement as the response is greatly
<a id="table--tab:frf-alternatives"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--tab:frf-alternatives">Table 2</a></span>:
<span class="table-number"><a href="#table--tab:frf-alternatives">Table 2</a>:</span>
Definition of Frequency Response Functions
</div>
@@ -549,17 +549,17 @@ Any simple plot can only show two of the three quantities and so there are diffe
##### Bode Plot {#bode-plot}
Bode plot are usually displayed using logarithmic scales as shown on figure [3](#table--fig:bode-plots).
Bode plot are usually displayed using logarithmic scales as shown on [Table 3](#table--fig:bode-plots).
<a id="table--fig:bode-plots"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--fig:bode-plots">Table 3</a></span>:
<span class="table-number"><a href="#table--fig:bode-plots">Table 3</a>:</span>
FRF plots for undamped SDOF system
</div>
| ![](/ox-hugo/ewins00_bode_receptance.png) | ![](/ox-hugo/ewins00_bode_mobility.png) | ![](/ox-hugo/ewins00_bode_accelerance.png) |
|--------------------------------------------------------------------------------------|----------------------------------------------------------------------------------|----------------------------------------------------------------------------------------|
| <span class="org-target" id="org-target--fig:bode_receptance"></span> Receptance FRF | <span class="org-target" id="org-target--fig:bode_mobility"></span> Mobility FRF | <span class="org-target" id="org-target--fig:bode_accelerance"></span> Accelerance FRF |
| <span class="org-target" id="org-target--fig-bode-receptance"></span> Receptance FRF | <span class="org-target" id="org-target--fig-bode-mobility"></span> Mobility FRF | <span class="org-target" id="org-target--fig-bode-accelerance"></span> Accelerance FRF |
| width=\linewidth | width=\linewidth | width=\linewidth |
Each plot can be divided into three regimes:
@@ -571,18 +571,18 @@ Each plot can be divided into three regimes:
##### Real part and Imaginary part of FRF {#real-part-and-imaginary-part-of-frf}
Real and imaginary part of a receptance FRF of a damped SDOF system is shown on figure [4](#table--fig:plot-receptance-real-imag).
Real and imaginary part of a receptance FRF of a damped SDOF system is shown on [Table 4](#table--fig:plot-receptance-real-imag).
This type of display is not widely used as we cannot use logarithmic axes (as we have to show positive and negative values).
<a id="table--fig:plot-receptance-real-imag"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--fig:plot-receptance-real-imag">Table 4</a></span>:
<span class="table-number"><a href="#table--fig:plot-receptance-real-imag">Table 4</a>:</span>
Plot of real and imaginary part for the receptance of a damped SDOF
</div>
| ![](/ox-hugo/ewins00_plot_receptance_real.png) | ![](/ox-hugo/ewins00_plot_receptance_imag.png) |
|--------------------------------------------------------------------------------------|-------------------------------------------------------------------------------------------|
| <span class="org-target" id="org-target--fig:plot_receptance_real"></span> Real part | <span class="org-target" id="org-target--fig:plot_receptance_imag"></span> Imaginary part |
| <span class="org-target" id="org-target--fig-plot-receptance-real"></span> Real part | <span class="org-target" id="org-target--fig-plot-receptance-imag"></span> Imaginary part |
| width=\linewidth | width=\linewidth |
@@ -590,34 +590,34 @@ This type of display is not widely used as we cannot use logarithmic axes (as we
It can be seen from the expression of the inverse receptance <eq:dynamic_stiffness> that the Real part depends entirely on the mass and stiffness properties while the Imaginary part is a only function of the damping.
Figure [5](#org-target--fig:inverse_frf_mixed) shows an example of a plot of a system with a combination of both viscous and structural damping. The imaginary part is a straight line whose slope is given by the viscous damping rate \\(c\\) and whose intercept at \\(\omega = 0\\) is provided by the structural damping coefficient \\(d\\).
[ 5](#org-target--fig-inverse-frf-mixed) shows an example of a plot of a system with a combination of both viscous and structural damping. The imaginary part is a straight line whose slope is given by the viscous damping rate \\(c\\) and whose intercept at \\(\omega = 0\\) is provided by the structural damping coefficient \\(d\\).
<a id="table--fig:inverse-frf"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--fig:inverse-frf">Table 5</a></span>:
<span class="table-number"><a href="#table--fig:inverse-frf">Table 5</a>:</span>
Inverse FRF plot for the system
</div>
| ![](/ox-hugo/ewins00_inverse_frf_mixed.png) | ![](/ox-hugo/ewins00_inverse_frf_viscous.png) |
|-------------------------------------------------------------------------------|-----------------------------------------------------------------------------------|
| <span class="org-target" id="org-target--fig:inverse_frf_mixed"></span> Mixed | <span class="org-target" id="org-target--fig:inverse_frf_viscous"></span> Viscous |
| <span class="org-target" id="org-target--fig-inverse-frf-mixed"></span> Mixed | <span class="org-target" id="org-target--fig-inverse-frf-viscous"></span> Viscous |
| width=\linewidth | width=\linewidth |
##### Real part vs Imaginary part of FRF {#real-part-vs-imaginary-part-of-frf}
Figure [6](#table--fig:nyquist-receptance) shows Nyquist type FRF plots of a viscously damped SDOF system.
[Table 6](#table--fig:nyquist-receptance) shows Nyquist type FRF plots of a viscously damped SDOF system.
The missing information (in this case, the frequency) must be added by identifying the values of frequency corresponding to particular points on the curve.
<a id="table--fig:nyquist-receptance"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--fig:nyquist-receptance">Table 6</a></span>:
<span class="table-number"><a href="#table--fig:nyquist-receptance">Table 6</a>:</span>
Nyquist FRF plots of the mobility for a SDOF system
</div>
| ![](/ox-hugo/ewins00_nyquist_receptance_viscous.png) | ![](/ox-hugo/ewins00_nyquist_receptance_structural.png) |
|--------------------------------------------------------------------------------------------------|--------------------------------------------------------------------------------------------------------|
| <span class="org-target" id="org-target--fig:nyquist_receptance_viscous"></span> Viscous damping | <span class="org-target" id="org-target--fig:nyquist_receptance_structural"></span> Structural damping |
| <span class="org-target" id="org-target--fig-nyquist-receptance-viscous"></span> Viscous damping | <span class="org-target" id="org-target--fig-nyquist-receptance-structural"></span> Structural damping |
| width=\linewidth | width=\linewidth |
The Nyquist plot has the particularity of distorting the plot so as to focus on the resonance area.
@@ -1110,24 +1110,24 @@ Equally, in a real mode, all parts of the structure pass through their **zero de
</div>
While the real mode has the appearance of a **standing wave**, the complex mode is better described as exhibiting **traveling waves** (illustrated on figure [6](#figure--fig:real-complex-modes)).
While the real mode has the appearance of a **standing wave**, the complex mode is better described as exhibiting **traveling waves** (illustrated on [Figure 6](#figure--fig:real-complex-modes)).
<a id="figure--fig:real-complex-modes"></a>
{{< figure src="/ox-hugo/ewins00_real_complex_modes.png" caption="<span class=\"figure-number\">Figure 6: </span>Real and complex mode shapes displays" >}}
Another method of displaying **modal complexity** is by plotting the elements of the eigenvector on an **Argand diagram**, such as the ones shown in figure [7](#table--fig:argand-diagram).
Another method of displaying **modal complexity** is by plotting the elements of the eigenvector on an **Argand diagram**, such as the ones shown in [Table 7](#table--fig:argand-diagram).
Note that the almost-real mode shape does not necessarily have vector elements with near \\(\SI{0}{\degree}\\) or near \\(\SI{180}{\degree}\\) phase, what matters are the **relative phases** between the different elements.
<a id="table--fig:argand-diagram"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--fig:argand-diagram">Table 7</a></span>:
<span class="table-number"><a href="#table--fig:argand-diagram">Table 7</a>:</span>
Complex mode shapes plotted on Argand diagrams
</div>
| ![](/ox-hugo/ewins00_argand_diagram_a.png) | ![](/ox-hugo/ewins00_argand_diagram_b.png) | ![](/ox-hugo/ewins00_argand_diagram_c.png) |
|-----------------------------------------------------------------------------------------|-------------------------------------------------------------------------------------|----------------------------------------------------------------------------------------------|
| <span class="org-target" id="org-target--fig:argand_diagram_a"></span> Almost-real mode | <span class="org-target" id="org-target--fig:argand_diagram_b"></span> Complex Mode | <span class="org-target" id="org-target--fig:argand_diagram_c"></span> Measure of complexity |
| <span class="org-target" id="org-target--fig-argand-diagram-a"></span> Almost-real mode | <span class="org-target" id="org-target--fig-argand-diagram-b"></span> Complex Mode | <span class="org-target" id="org-target--fig-argand-diagram-c"></span> Measure of complexity |
| width=\linewidth | width=\linewidth | width=\linewidth |
@@ -1137,7 +1137,7 @@ There exist few indicators of the modal complexity.
The first one, a simple and crude one, called **MCF1** consists of summing all the phase differences between every combination of two eigenvector elements:
\\[ \text{MCF1} = \sum\_{j=1}^N \sum\_{k=1 \neq j}^N (\theta\_{rj} - \theta\_{rk}) \\]
The second measure is shown on figure [7](#org-target--fig:argand_diagram_c) where a polygon is drawn around the extremities of the individual vectors.
The second measure is shown on [ 7](#org-target--fig-argand-diagram-c) where a polygon is drawn around the extremities of the individual vectors.
The obtained area of this polygon is then compared with the area of the circle which is based on the length of the largest vector element. The resulting ratio is used as an indication of the complexity of the mode, and is defined as **MCF2**.
@@ -1177,11 +1177,11 @@ The second definition comes from the general form of the FRF expression:
Here \\(C\_r\\) may be complex whereas \\(D\_r\\) is real.
\\(\omega\_r\\) is in general different to both \\(\bar{\omega}\_r\\) and \\(\omega\_r^\prime\\).
Table [8](#table--tab:frf-natural-frequencies) summarizes all the different cases.
[Table 8](#table--tab:frf-natural-frequencies) summarizes all the different cases.
<a id="table--tab:frf-natural-frequencies"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--tab:frf-natural-frequencies">Table 8</a></span>:
<span class="table-number"><a href="#table--tab:frf-natural-frequencies">Table 8</a>:</span>
FRF Formulae and Natural Frequencies
</div>
@@ -1233,21 +1233,21 @@ We write \\(\alpha\_{11}\\) the point FRF and \\(\alpha\_{21}\\) the transfer FR
It can be seen that the only difference between the point and transfer receptance is in the sign of the modal constant of the second mode.
Consider the first point mobility (figure [9](#org-target--fig:mobility_frf_mdof_point)), between the two resonances, the two components have opposite signs so that they are substractive rather than additive, and indeed, at the point where they cross, their sum is zero.
Consider the first point mobility ([ 9](#org-target--fig-mobility-frf-mdof-point)), between the two resonances, the two components have opposite signs so that they are substractive rather than additive, and indeed, at the point where they cross, their sum is zero.
On a logarithmic plot, this produces the antiresonance characteristic which reflects that of the resonance.
<a id="table--fig:mobility-frf-mdof"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--fig:mobility-frf-mdof">Table 9</a></span>:
<span class="table-number"><a href="#table--fig:mobility-frf-mdof">Table 9</a>:</span>
Mobility FRF plot for undamped 2DOF system
</div>
| ![](/ox-hugo/ewins00_mobility_frf_mdof_point.png) | ![](/ox-hugo/ewins00_mobility_frf_mdof_transfer.png) |
|-----------------------------------------------------------------------------------------|-----------------------------------------------------------------------------------------------|
| <span class="org-target" id="org-target--fig:mobility_frf_mdof_point"></span> Point FRF | <span class="org-target" id="org-target--fig:mobility_frf_mdof_transfer"></span> Transfer FRF |
| <span class="org-target" id="org-target--fig-mobility-frf-mdof-point"></span> Point FRF | <span class="org-target" id="org-target--fig-mobility-frf-mdof-transfer"></span> Transfer FRF |
| width=\linewidth | width=\linewidth |
For the plot in figure [9](#org-target--fig:mobility_frf_mdof_transfer), between the two resonances, the two components have the same sign and they add up, no antiresonance is present.
For the plot in [ 9](#org-target--fig-mobility-frf-mdof-transfer), between the two resonances, the two components have the same sign and they add up, no antiresonance is present.
##### FRF modulus plots for MDOF systems {#frf-modulus-plots-for-mdof-systems}
@@ -1263,7 +1263,7 @@ If they have apposite signs, there will not be an antiresonance.
##### Bode plots {#bode-plots}
The resonances and antiresonances are blunted by the inclusion of damping, and the phase angles are no longer exactly \\(\SI{0}{\degree}\\) or \\(\SI{180}{\degree}\\), but the general appearance of the plot is a natural extension of that for the system without damping.
Figure [7](#figure--fig:frf-damped-system) shows a plot for the same mobility as appears in figure [9](#org-target--fig:mobility_frf_mdof_point) but here for a system with added damping.
[Figure 7](#figure--fig:frf-damped-system) shows a plot for the same mobility as appears in [ 9](#org-target--fig-mobility-frf-mdof-point) but here for a system with added damping.
Most mobility plots have this general form as long as the modes are relatively well-separated.
@@ -1278,34 +1278,34 @@ This condition is satisfied unless the separation between adjacent natural frequ
Each of the frequency response of a MDOF system in the Nyquist plot is composed of a number of SDOF components.
Figure [10](#org-target--fig:nyquist_point) shows the result of plotting the point receptance \\(\alpha\_{11}\\) for the 2DOF system described above.
[ 10](#org-target--fig-nyquist-point) shows the result of plotting the point receptance \\(\alpha\_{11}\\) for the 2DOF system described above.
The plot for the transfer receptance \\(\alpha\_{21}\\) is presented in figure [10](#org-target--fig:nyquist_transfer) where it may be seen that the opposing signs of the modal constants of the two modes have caused one of the modal circle to be in the upper half of the complex plane.
The plot for the transfer receptance \\(\alpha\_{21}\\) is presented in [ 10](#org-target--fig-nyquist-transfer) where it may be seen that the opposing signs of the modal constants of the two modes have caused one of the modal circle to be in the upper half of the complex plane.
<a id="table--fig:nyquist-frf-plots"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--fig:nyquist-frf-plots">Table 10</a></span>:
<span class="table-number"><a href="#table--fig:nyquist-frf-plots">Table 10</a>:</span>
Nyquist FRF plot for proportionally-damped system
</div>
| ![](/ox-hugo/ewins00_nyquist_point.png) | ![](/ox-hugo/ewins00_nyquist_transfer.png) |
|--------------------------------------------------------------------------------------|--------------------------------------------------------------------------------------------|
| <span class="org-target" id="org-target--fig:nyquist_point"></span> Point receptance | <span class="org-target" id="org-target--fig:nyquist_transfer"></span> Transfer receptance |
| <span class="org-target" id="org-target--fig-nyquist-point"></span> Point receptance | <span class="org-target" id="org-target--fig-nyquist-transfer"></span> Transfer receptance |
| width=\linewidth | width=\linewidth |
In the two figures [11](#org-target--fig:nyquist_nonpropdamp_point) and [11](#org-target--fig:nyquist_nonpropdamp_transfer), we show corresponding data for **non-proportional** damping.
In the two [ 11](#org-target--fig-nyquist-nonpropdamp-point) and [ 11](#org-target--fig-nyquist-nonpropdamp-transfer), we show corresponding data for **non-proportional** damping.
In this case, a relative phase has been introduced between the first and second elements of the eigenvectors: of \\(\SI{30}{\degree}\\) in mode 1 and of \\(\SI{150}{\degree}\\) in mode 2.
Now we find that the individual modal circles are no longer "upright" but are **rotated by an amount dictated by the complexity of the modal constants**.
<a id="table--fig:nyquist-frf-nonpropdamp"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--fig:nyquist-frf-nonpropdamp">Table 11</a></span>:
<span class="table-number"><a href="#table--fig:nyquist-frf-nonpropdamp">Table 11</a>:</span>
Nyquist FRF plot for non-proportionally-damped system
</div>
| ![](/ox-hugo/ewins00_nyquist_nonpropdamp_point.png) | ![](/ox-hugo/ewins00_nyquist_nonpropdamp_transfer.png) |
|--------------------------------------------------------------------------------------------------|--------------------------------------------------------------------------------------------------------|
| <span class="org-target" id="org-target--fig:nyquist_nonpropdamp_point"></span> Point receptance | <span class="org-target" id="org-target--fig:nyquist_nonpropdamp_transfer"></span> Transfer receptance |
| <span class="org-target" id="org-target--fig-nyquist-nonpropdamp-point"></span> Point receptance | <span class="org-target" id="org-target--fig-nyquist-nonpropdamp-transfer"></span> Transfer receptance |
| width=\linewidth | width=\linewidth |
@@ -1449,17 +1449,17 @@ The resulting parameter we shall call a **Spectral Density**, in this case the *
The Spectral Density is a real and even function of frequency, and does in fact provides a description of the frequency composition of the original function \\(f(t)\\).
It has units of \\(f^2/\omega\\).
Examples of random signals, autocorrelation function and power spectral density are shown on figure [12](#table--fig:random-signals).
Examples of random signals, autocorrelation function and power spectral density are shown on [Table 12](#table--fig:random-signals).
<a id="table--fig:random-signals"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--fig:random-signals">Table 12</a></span>:
<span class="table-number"><a href="#table--fig:random-signals">Table 12</a>:</span>
Random signals
</div>
| ![](/ox-hugo/ewins00_random_time.png) | ![](/ox-hugo/ewins00_random_autocorrelation.png) | ![](/ox-hugo/ewins00_random_psd.png) |
|--------------------------------------------------------------------------------|-------------------------------------------------------------------------------------------------------|-----------------------------------------------------------------------------------------|
| <span class="org-target" id="org-target--fig:random_time"></span> Time history | <span class="org-target" id="org-target--fig:random_autocorrelation"></span> Autocorrelation Function | <span class="org-target" id="org-target--fig:random_psd"></span> Power Spectral Density |
| <span class="org-target" id="org-target--fig-random-time"></span> Time history | <span class="org-target" id="org-target--fig-random-autocorrelation"></span> Autocorrelation Function | <span class="org-target" id="org-target--fig-random-psd"></span> Power Spectral Density |
| width=\linewidth | width=\linewidth | width=\linewidth |
A similar concept can be applied to a pair of functions such as \\(f(t)\\) and \\(x(t)\\) to produce **cross correlation** and **cross spectral density** functions.
@@ -1540,18 +1540,18 @@ The existence of two equations presents an opportunity to **check the quality**
There are difficulties to implement some of the above formulae in practice because of noise and other limitations concerned with the data acquisition and processing.
One technique involves **three quantities**, rather than two, in the definition of the output/input ratio.
The system considered can best be described with reference to figure [13](#table--fig:frf-determination) which shows first in [13](#org-target--fig:frf_siso_model) the traditional single-input single-output model upon which the previous formulae are based.
Then in [13](#org-target--fig:frf_feedback_model) is given a more detailed and representative model of the system which is used in a modal test.
The system considered can best be described with reference to [Table 13](#table--fig:frf-determination) which shows first in [ 13](#org-target--fig-frf-siso-model) the traditional single-input single-output model upon which the previous formulae are based.
Then in [ 13](#org-target--fig-frf-feedback-model) is given a more detailed and representative model of the system which is used in a modal test.
<a id="table--fig:frf-determination"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--fig:frf-determination">Table 13</a></span>:
<span class="table-number"><a href="#table--fig:frf-determination">Table 13</a>:</span>
System for FRF determination
</div>
| ![](/ox-hugo/ewins00_frf_siso_model.png) | ![](/ox-hugo/ewins00_frf_feedback_model.png) |
|---------------------------------------------------------------------------------------|---------------------------------------------------------------------------------------------------|
| <span class="org-target" id="org-target--fig:frf_siso_model"></span> Basic SISO model | <span class="org-target" id="org-target--fig:frf_feedback_model"></span> SISO model with feedback |
| <span class="org-target" id="org-target--fig-frf-siso-model"></span> Basic SISO model | <span class="org-target" id="org-target--fig-frf-feedback-model"></span> SISO model with feedback |
| width=\linewidth | width=\linewidth |
In this configuration, it can be seen that there are two feedback mechanisms which apply.
@@ -1570,7 +1570,7 @@ where \\(v\\) is a third signal in the system.
##### Derivation of FRF from MIMO data {#derivation-of-frf-from-mimo-data}
A diagram for the general n-input case is shown in figure [8](#figure--fig:frf-mimo).
A diagram for the general n-input case is shown in [Figure 8](#figure--fig:frf-mimo).
We obtain two alternative formulas:
@@ -1849,7 +1849,7 @@ The experimental setup used for mobility measurement contains three major items:
2. **A transduction system**. For the most part, piezoelectric transducer are used, although lasers and strain gauges are convenient because of their minimal interference with the test object. Conditioning amplifiers are used depending of the transducer used
3. **An analyzer**
A typical layout for the measurement system is shown on figure [9](#figure--fig:general-frf-measurement-setup).
A typical layout for the measurement system is shown on [Figure 9](#figure--fig:general-frf-measurement-setup).
<a id="figure--fig:general-frf-measurement-setup"></a>
@@ -1905,7 +1905,7 @@ However, we need a direct measurement of the force applied to the structure (we
The shakers are usually stiff in the orthogonal directions to the excitation.
This can modify the response of the system in those directions.
In order to avoid that, a drive rod which is stiff in one direction and flexible in the other five directions is attached between the shaker and the structure as shown on figure [10](#figure--fig:shaker-rod).
In order to avoid that, a drive rod which is stiff in one direction and flexible in the other five directions is attached between the shaker and the structure as shown on [Figure 10](#figure--fig:shaker-rod).
Typical size for the rod are \\(5\\) to \\(\SI{10}{mm}\\) long and \\(\SI{1}{mm}\\) in diameter, if the rod is longer, it may introduce the effect of its own resonances.
<a id="figure--fig:shaker-rod"></a>
@@ -1914,22 +1914,22 @@ Typical size for the rod are \\(5\\) to \\(\SI{10}{mm}\\) long and \\(\SI{1}{mm}
The support of shaker is also of primary importance.
The setup shown on figure [14](#org-target--fig:shaker_mount_1) presents the most satisfactory arrangement in which the shaker is fixed to ground while the test structure is supported by a soft spring.
The setup shown on [ 14](#org-target--fig-shaker-mount-1) presents the most satisfactory arrangement in which the shaker is fixed to ground while the test structure is supported by a soft spring.
Figure [14](#org-target--fig:shaker_mount_2) shows an alternative configuration in which the shaker itself is supported.
[ 14](#org-target--fig-shaker-mount-2) shows an alternative configuration in which the shaker itself is supported.
It may be necessary to add an additional inertia mass to the shaker in order to generate sufficient excitation forces at low frequencies.
Figure [14](#org-target--fig:shaker_mount_3) shows an unsatisfactory setup. Indeed, the response measured at \\(A\\) would not be due solely to force applied at \\(B\\), but would also be caused by the forces applied at \\(C\\).
[ 14](#org-target--fig-shaker-mount-3) shows an unsatisfactory setup. Indeed, the response measured at \\(A\\) would not be due solely to force applied at \\(B\\), but would also be caused by the forces applied at \\(C\\).
<a id="table--fig:shaker-mount"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--fig:shaker-mount">Table 14</a></span>:
<span class="table-number"><a href="#table--fig:shaker-mount">Table 14</a>:</span>
Various mounting arrangement of exciter
</div>
| ![](/ox-hugo/ewins00_shaker_mount_1.png) | ![](/ox-hugo/ewins00_shaker_mount_2.png) | ![](/ox-hugo/ewins00_shaker_mount_3.png) |
|------------------------------------------------------------------------------------------|----------------------------------------------------------------------------------------------|-------------------------------------------------------------------------------------|
| <span class="org-target" id="org-target--fig:shaker_mount_1"></span> Ideal Configuration | <span class="org-target" id="org-target--fig:shaker_mount_2"></span> Suspended Configuration | <span class="org-target" id="org-target--fig:shaker_mount_3"></span> Unsatisfactory |
| <span class="org-target" id="org-target--fig-shaker-mount-1"></span> Ideal Configuration | <span class="org-target" id="org-target--fig-shaker-mount-2"></span> Suspended Configuration | <span class="org-target" id="org-target--fig-shaker-mount-3"></span> Unsatisfactory |
| width=\linewidth | width=\linewidth | width=\linewidth |
@@ -1944,8 +1944,8 @@ The magnitude of the impact is determined by the mass of the hammer head and its
The frequency range which is effectively excited is controlled by the stiffness of the contacting surface and the mass of the impactor head: there is a resonance at a frequency given by \\(\sqrt{\frac{\text{contact stiffness}}{\text{impactor mass}}}\\) above which it is difficult to deliver energy into the test structure.
When the hammer tip impacts the test structure, this will experience a force pulse as shown on figure [11](#figure--fig:hammer-impulse).
A pulse of this type (half-sine shape) has a frequency content of the form illustrated on figure [11](#figure--fig:hammer-impulse).
When the hammer tip impacts the test structure, this will experience a force pulse as shown on [Figure 11](#figure--fig:hammer-impulse).
A pulse of this type (half-sine shape) has a frequency content of the form illustrated on [Figure 11](#figure--fig:hammer-impulse).
<a id="figure--fig:hammer-impulse"></a>
@@ -1976,7 +1976,7 @@ By suitable design, such a material may be incorporated into a device which **in
#### Force Transducers {#force-transducers}
The force transducer is the simplest type of piezoelectric transducer.
The transmitter force \\(F\\) is applied directly across the crystal, which thus generates a corresponding charge \\(q\\), proportional to \\(F\\) (figure [12](#figure--fig:piezo-force-transducer)).
The transmitter force \\(F\\) is applied directly across the crystal, which thus generates a corresponding charge \\(q\\), proportional to \\(F\\) ([Figure 12](#figure--fig:piezo-force-transducer)).
<a id="figure--fig:piezo-force-transducer"></a>
@@ -1987,7 +1987,7 @@ There exists an undesirable possibility of a cross sensitivity, i.e. an electric
#### Accelerometers {#accelerometers}
In an accelerometer, transduction is indirect and is achieved using a seismic mass (figure [13](#figure--fig:piezo-accelerometer)).
In an accelerometer, transduction is indirect and is achieved using a seismic mass ([Figure 13](#figure--fig:piezo-accelerometer)).
In this configuration, the force exerted on the crystals is the inertia force of the seismic mass (\\(m\ddot{z}\\)).
Thus, so long as the body and the seismic mass move together, the output of the transducer will be proportional to the acceleration of its body \\(x\\).
@@ -2027,19 +2027,19 @@ However, they cannot be used at such low frequencies as the charge amplifiers an
The correct installation of transducers, especially accelerometers is important.
There are various means of fixing the transducers to the surface of the test structure, some more convenient than others.
Some of these methods are illustrated in figure [15](#org-target--fig:transducer_mounting_types).
Some of these methods are illustrated in [ 15](#org-target--fig-transducer-mounting-types).
Shown on figure [15](#org-target--fig:transducer_mounting_response) are typical high frequency limits for each type of attachment.
Shown on [ 15](#org-target--fig-transducer-mounting-response) are typical high frequency limits for each type of attachment.
<a id="table--fig:transducer-mounting"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--fig:transducer-mounting">Table 15</a></span>:
<span class="table-number"><a href="#table--fig:transducer-mounting">Table 15</a>:</span>
Accelerometer attachment characteristics
</div>
| ![](/ox-hugo/ewins00_transducer_mounting_types.png) | ![](/ox-hugo/ewins00_transducer_mounting_response.png) |
|----------------------------------------------------------------------------------------------------|-----------------------------------------------------------------------------------------------------------------------|
| <span class="org-target" id="org-target--fig:transducer_mounting_types"></span> Attachment methods | <span class="org-target" id="org-target--fig:transducer_mounting_response"></span> Frequency response characteristics |
| <span class="org-target" id="org-target--fig-transducer-mounting-types"></span> Attachment methods | <span class="org-target" id="org-target--fig-transducer-mounting-response"></span> Frequency response characteristics |
| width=\linewidth | width=\linewidth |
@@ -2124,7 +2124,7 @@ That however requires \\(N\\) to be an integral power of \\(2\\).
Aliasing originates from the discretisation of the originally continuous time history.
With this discretisation process, the **existence of very high frequencies in the original signal may well be misinterpreted if the sampling rate is too slow**.
These high frequencies will be **indistinguishable** from genuine low frequency components as shown on figure [14](#figure--fig:aliasing).
These high frequencies will be **indistinguishable** from genuine low frequency components as shown on [Figure 14](#figure--fig:aliasing).
<a id="figure--fig:aliasing"></a>
@@ -2133,17 +2133,17 @@ These high frequencies will be **indistinguishable** from genuine low frequency
A signal of frequency \\(\omega\\) and one of frequency \\(\omega\_s-\omega\\) are indistinguishable and this causes a **distortion of the spectrum** measured via the DFT.
As a result, the part of the signal which has frequency components above \\(\omega\_s/2\\) will appear reflected or **aliased** in the range \\([0, \omega\_s/2]\\).
This is illustrated on figure [16](#table--fig:effect-aliasing).
This is illustrated on [Table 16](#table--fig:effect-aliasing).
<a id="table--fig:effect-aliasing"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--fig:effect-aliasing">Table 16</a></span>:
<span class="table-number"><a href="#table--fig:effect-aliasing">Table 16</a>:</span>
Alias distortion of spectrum by DFT
</div>
| ![](/ox-hugo/ewins00_aliasing_no_distortion.png) | ![](/ox-hugo/ewins00_aliasing_distortion.png) |
|------------------------------------------------------------------------------------------------------|-------------------------------------------------------------------------------------------------------|
| <span class="org-target" id="org-target--fig:aliasing_no_distortion"></span> True spectrum of signal | <span class="org-target" id="org-target--fig:aliasing_distortion"></span> Indicated spectrum from DFT |
| <span class="org-target" id="org-target--fig-aliasing-no-distortion"></span> True spectrum of signal | <span class="org-target" id="org-target--fig-aliasing-distortion"></span> Indicated spectrum from DFT |
| width=\linewidth | width=\linewidth |
The solution of the problem is to use an **anti-aliasing filter** which subjects the original time signal to a low-pass, sharp cut-off filter.
@@ -2158,18 +2158,18 @@ Leakage is a problem which is a direct **consequence of the need to take only a
<a id="table--fig:leakage"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--fig:leakage">Table 17</a></span>:
<span class="table-number"><a href="#table--fig:leakage">Table 17</a>:</span>
Sample length and leakage of spectrum
</div>
| ![](/ox-hugo/ewins00_leakage_ok.png) | ![](/ox-hugo/ewins00_leakage_nok.png) |
|-------------------------------------------------------------------------------|----------------------------------------------------------------------------------|
| <span class="org-target" id="org-target--fig:leakage_ok"></span> Ideal signal | <span class="org-target" id="org-target--fig:leakage_nok"></span> Awkward signal |
| <span class="org-target" id="org-target--fig-leakage-ok"></span> Ideal signal | <span class="org-target" id="org-target--fig-leakage-nok"></span> Awkward signal |
| width=\linewidth | width=\linewidth |
The problem is illustrated on figure [17](#table--fig:leakage).
In the first case (figure [17](#org-target--fig:leakage_ok)), the signal is perfectly periodic and the resulting spectrum is just a single line at the frequency of the sine wave.
In the second case (figure [17](#org-target--fig:leakage_nok)), the periodicity assumption is not strictly valid as there is a discontinuity at each end of the sample.
The problem is illustrated on [Table 17](#table--fig:leakage).
In the first case ([ 17](#org-target--fig-leakage-ok)), the signal is perfectly periodic and the resulting spectrum is just a single line at the frequency of the sine wave.
In the second case ([ 17](#org-target--fig-leakage-nok)), the periodicity assumption is not strictly valid as there is a discontinuity at each end of the sample.
As a result, the spectrum produced for this case does not indicate the single frequency which the original time signal possessed.
Energy has "leaked" into a number of the spectral lines close to the true frequency and the spectrum is spread over several lines.
@@ -2187,14 +2187,14 @@ Leakage is a serious problem in many applications, **ways of avoiding its effect
Windowing involves the imposition of a prescribed profile on the time signal prior to performing the Fourier transform.
The profiles, or "windows" are generally depicted as a time function \\(w(t)\\) as shown in figure [15](#figure--fig:windowing-examples).
The profiles, or "windows" are generally depicted as a time function \\(w(t)\\) as shown in [Figure 15](#figure--fig:windowing-examples).
<a id="figure--fig:windowing-examples"></a>
{{< figure src="/ox-hugo/ewins00_windowing_examples.png" caption="<span class=\"figure-number\">Figure 15: </span>Different types of window. (a) Boxcar, (b) Hanning, (c) Cosine-taper, (d) Exponential" >}}
The analyzed signal is then \\(x^\prime(t) = x(t) w(t)\\).
The result of using a window is seen in the third column of figure [15](#figure--fig:windowing-examples).
The result of using a window is seen in the third column of [Figure 15](#figure--fig:windowing-examples).
The **Hanning and Cosine Taper windows are typically used for continuous signals**, such as are produced by steady periodic or random vibration, while the **Exponential window is used for transient vibration** applications where much of the important information is concentrated in the initial part of the time record.
@@ -2210,7 +2210,7 @@ Common filters are: low-pass, high-pass, band-limited, narrow-band, notch.
#### Improving Resolution {#improving-resolution}
<span class="org-target" id="org-target--sec:improving_resolution"></span>
<span class="org-target" id="org-target--sec-improving-resolution"></span>
##### Increasing transform size {#increasing-transform-size}
@@ -2234,19 +2234,19 @@ The common solution to the need for finer frequency resolution is to zoom on the
There are various ways of achieving this result.
The easiest way is to use a frequency shifting process coupled with a controlled aliasing device.
Suppose the signal to be analyzed \\(x(t)\\) has a spectrum \\(X(\omega)\\) has shown on figure [18](#org-target--fig:zoom_range), and that we are interested in a detailed analysis between \\(\omega\_1\\) and \\(\omega\_2\\).
Suppose the signal to be analyzed \\(x(t)\\) has a spectrum \\(X(\omega)\\) has shown on [ 18](#org-target--fig-zoom-range), and that we are interested in a detailed analysis between \\(\omega\_1\\) and \\(\omega\_2\\).
If we apply a band-pass filter to the signal, as shown on figure [18](#org-target--fig:zoom_bandpass), and perform a DFT between \\(0\\) and \\((\omega\_2 - \omega\_1)\\), then because of the aliasing phenomenon described earlier, the frequency components between \\(\omega\_1\\) and \\(\omega\_2\\) will appear between \\(0\\) and \\((\omega\_2 - \omega\_1)\\) with the advantage of a finer resolution (see figure [16](#figure--fig:zoom-result)).
If we apply a band-pass filter to the signal, as shown on [ 18](#org-target--fig-zoom-bandpass), and perform a DFT between \\(0\\) and \\((\omega\_2 - \omega\_1)\\), then because of the aliasing phenomenon described earlier, the frequency components between \\(\omega\_1\\) and \\(\omega\_2\\) will appear between \\(0\\) and \\((\omega\_2 - \omega\_1)\\) with the advantage of a finer resolution (see [Figure 16](#figure--fig:zoom-result)).
<a id="table--fig:frequency-zoom"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--fig:frequency-zoom">Table 18</a></span>:
<span class="table-number"><a href="#table--fig:frequency-zoom">Table 18</a>:</span>
Controlled aliasing for frequency zoom
</div>
| ![](/ox-hugo/ewins00_zoom_range.png) | ![](/ox-hugo/ewins00_zoom_bandpass.png) |
|-----------------------------------------------------------------------------------------|--------------------------------------------------------------------------------------|
| <span class="org-target" id="org-target--fig:zoom_range"></span> Spectrum of the signal | <span class="org-target" id="org-target--fig:zoom_bandpass"></span> Band-pass filter |
| <span class="org-target" id="org-target--fig-zoom-range"></span> Spectrum of the signal | <span class="org-target" id="org-target--fig-zoom-bandpass"></span> Band-pass filter |
| width=\linewidth | width=\linewidth |
<a id="figure--fig:zoom-result"></a>
@@ -2319,7 +2319,7 @@ For instance, the typical FRF curve has large region of relatively slow changes
This is the traditional method of FRF measurement and involves the use of a sweep oscillator to provide a sinusoidal command signal with a frequency that varies slowly in the range of interest.
It is necessary to check that progress through the frequency range is sufficiently slow to check that steady-state response conditions are attained.
If excessive sweep rate is used, then distortions of the FRF plot are introduced as shown on figure [17](#figure--fig:sweep-distortions).
If excessive sweep rate is used, then distortions of the FRF plot are introduced as shown on [Figure 17](#figure--fig:sweep-distortions).
<a id="figure--fig:sweep-distortions"></a>
@@ -2437,7 +2437,7 @@ where \\(v(t)\\) is a third signal in the system, such as the voltage supplied t
It is known that a low coherence can arise in a measurement where the frequency resolution of the analyzer is not fine enough to describe adequately the very rapidly changing functions such as are encountered near resonance and anti-resonance on lightly-damped structures.
This is known as a **bias** error and leakage is often the most likely source of low coherence on lightly-damped structures as shown on figure [18](#figure--fig:coherence-resonance).
This is known as a **bias** error and leakage is often the most likely source of low coherence on lightly-damped structures as shown on [Figure 18](#figure--fig:coherence-resonance).
<a id="figure--fig:coherence-resonance"></a>
@@ -2480,7 +2480,7 @@ For the chirp and impulse excitations, each individual sample is collected and p
##### Burst excitation signals {#burst-excitation-signals}
Burst excitation signals consist of short sections of an underlying continuous signal (which may be a sine wave, a sine sweep or a random signal), followed by a period of zero output, resulting in a response which shows a transient build-up followed by a decay (see figure [19](#figure--fig:burst-excitation)).
Burst excitation signals consist of short sections of an underlying continuous signal (which may be a sine wave, a sine sweep or a random signal), followed by a period of zero output, resulting in a response which shows a transient build-up followed by a decay (see [Figure 19](#figure--fig:burst-excitation)).
<a id="figure--fig:burst-excitation"></a>
@@ -2497,7 +2497,7 @@ In the case of burst random, however, each individual burst will be different to
##### Chirp excitation {#chirp-excitation}
The chirp consist of a short duration signal which has the form shown in figure [20](#figure--fig:chirp-excitation).
The chirp consist of a short duration signal which has the form shown in [Figure 20](#figure--fig:chirp-excitation).
The frequency content of the chirp can be precisely chosen by the starting and finishing frequencies of the sweep.
@@ -2508,7 +2508,7 @@ The frequency content of the chirp can be precisely chosen by the starting and f
##### Impulsive excitation {#impulsive-excitation}
The hammer blow produces an input and response as shown in the figure [21](#figure--fig:impulsive-excitation).
The hammer blow produces an input and response as shown in the [Figure 21](#figure--fig:impulsive-excitation).
This and the chirp excitation are very similar in the analysis point of view, the main difference is that the chirp offers the possibility of greater control of both amplitude and frequency content of the input and also permits the input of a greater amount of vibration energy.
@@ -2520,7 +2520,7 @@ The frequency content of the hammer blow is dictated by the **materials** involv
However, it should be recorded that in the region below the first cut-off frequency induced by the elasticity of the hammer tip structure contact, the spectrum of the force signal tends to be **very flat**.
On some structures, the movement of the structure in response to the hammer blow can be such that it returns and **rebounds** on the hammer tip before the user has had time to move that out of the way.
In such cases, the spectrum of the excitation is seen to have "holes" in it at certain frequencies (figure [22](#figure--fig:double-hits)).
In such cases, the spectrum of the excitation is seen to have "holes" in it at certain frequencies ([Figure 22](#figure--fig:double-hits)).
<a id="figure--fig:double-hits"></a>
@@ -2595,7 +2595,7 @@ and so **what is required is the ratio of the two sensitivities**:
The overall sensitivity can be more readily obtained by a calibration process because we can easily make an independent measurement of the quantity now being measured: the ratio of response to force.
Suppose the response parameter is acceleration, then the FRF obtained is inertance which has the units of \\(1/\text{mass}\\), a quantity which can readily be independently measured by other means.
Figure [23](#figure--fig:calibration-setup) shows a typical calibration setup.
[Figure 23](#figure--fig:calibration-setup) shows a typical calibration setup.
<a id="figure--fig:calibration-setup"></a>
@@ -2610,7 +2610,7 @@ Thus, frequent checks on the overall calibration factors are strongly recommende
It is very important the ensure that the force is measured directly at the point at which it is applied to the structure, rather than deducing its magnitude from the current flowing in the shaker coil or other similar **indirect** processes.
This is because near resonance, the actual applied force becomes very small and is thus very prone to inaccuracy.
This same argument applies on a lesser scale as we examine the detail around the attachment to the structure, as shown in figure [24](#figure--fig:mass-cancellation).
This same argument applies on a lesser scale as we examine the detail around the attachment to the structure, as shown in [Figure 24](#figure--fig:mass-cancellation).
<a id="figure--fig:mass-cancellation"></a>
@@ -2667,7 +2667,7 @@ There are two problems to be tackled:
1. measurement of rotational responses
2. generation of measurement of rotation excitation
The first of these is less difficult and techniques usually use a pair a matched conventional accelerometers placed at a short distance apart on the structure to be measured as shown on figure [25](#figure--fig:rotational-measurement).
The first of these is less difficult and techniques usually use a pair a matched conventional accelerometers placed at a short distance apart on the structure to be measured as shown on [Figure 25](#figure--fig:rotational-measurement).
<a id="figure--fig:rotational-measurement"></a>
@@ -2685,7 +2685,7 @@ The principle of operation is that by measuring both accelerometer signals, the
This approach permits us to measure half of the possible FRFs: all those which are of the \\(X/F\\) and \\(\Theta/F\\) type.
The others can only be measured directly by applying a moment excitation.
Figure [26](#figure--fig:rotational-excitation) shows a device to simulate a moment excitation.
[Figure 26](#figure--fig:rotational-excitation) shows a device to simulate a moment excitation.
First, a single applied excitation force \\(F\_1\\) corresponds to a simultaneous force \\(F\_0 = F\_1\\) and a moment \\(M\_0 = -F\_1 l\_1\\).
Then, the same excitation force is applied at the second position that gives a force \\(F\_0 = F\_2\\) and moment \\(M\_0 = F\_2 l\_2\\).
By adding and subtracting the responses produced by these two separate excitations conditions, we can deduce the translational and rotational responses to the translational force and the rotational moment separately, thus enabling the measurement of all four types of FRF: \\(X/F\\), \\(\Theta/F\\), \\(X/M\\) and \\(\Theta/M\\).
@@ -3005,10 +3005,10 @@ Then, each PRF is, simply, a particular combination of the original FRFs, and th
</div>
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.
On example of this form of pre-processing is shown on [Table 19](#table--fig:PRF-numerical) for a numerically-simulation test data, and another in [Table 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.
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.
<div class="important">
@@ -3028,24 +3028,24 @@ The two groups are usually separated by a clear gap (depending of the noise pres
<a id="table--fig:PRF-numerical"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--fig:PRF-numerical">Table 19</a></span>:
<span class="table-number"><a href="#table--fig:PRF-numerical">Table 19</a>:</span>
FRF and PRF characteristics for numerical model
</div>
| ![](/ox-hugo/ewins00_PRF_numerical_FRF.png) | ![](/ox-hugo/ewins00_PRF_numerical_svd.png) | ![](/ox-hugo/ewins00_PRF_numerical_PRF.png) |
|-----------------------------------------------------------------------------|-----------------------------------------------------------------------------------------|-----------------------------------------------------------------------------|
| <span class="org-target" id="org-target--fig:PRF_numerical_FRF"></span> FRF | <span class="org-target" id="org-target--fig:PRF_numerical_svd"></span> Singular Values | <span class="org-target" id="org-target--fig:PRF_numerical_PRF"></span> PRF |
| <span class="org-target" id="org-target--fig-PRF-numerical-FRF"></span> FRF | <span class="org-target" id="org-target--fig-PRF-numerical-svd"></span> Singular Values | <span class="org-target" id="org-target--fig-PRF-numerical-PRF"></span> PRF |
| width=\linewidth | width=\linewidth | width=\linewidth |
<a id="table--fig:PRF-measured"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--fig:PRF-measured">Table 20</a></span>:
<span class="table-number"><a href="#table--fig:PRF-measured">Table 20</a>:</span>
FRF and PRF characteristics for measured model
</div>
| ![](/ox-hugo/ewins00_PRF_measured_FRF.png) | ![](/ox-hugo/ewins00_PRF_measured_svd.png) | ![](/ox-hugo/ewins00_PRF_measured_PRF.png) |
|----------------------------------------------------------------------------|----------------------------------------------------------------------------------------|----------------------------------------------------------------------------|
| <span class="org-target" id="org-target--fig:PRF_measured_FRF"></span> FRF | <span class="org-target" id="org-target--fig:PRF_measured_svd"></span> Singular Values | <span class="org-target" id="org-target--fig:PRF_measured_PRF"></span> PRF |
| <span class="org-target" id="org-target--fig-PRF-measured-FRF"></span> FRF | <span class="org-target" id="org-target--fig-PRF-measured-svd"></span> Singular Values | <span class="org-target" id="org-target--fig-PRF-measured-PRF"></span> PRF |
| width=\linewidth | width=\linewidth | width=\linewidth |
@@ -3076,7 +3076,7 @@ The **Complex mode indicator function** (CMIF) is defined as
</div>
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):
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**.
@@ -3157,7 +3157,7 @@ In this method, it is assumed that close to one local mode, any effects due to t
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)):
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.
@@ -3204,17 +3204,17 @@ In the case of a system assumed to have structural damping, the basic function w
\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).
A plot of the quantity \\(\alpha(\omega)\\) is given in [ 21](#org-target--fig-modal-circle).
<a id="table--fig:modal-circle-figures"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--fig:modal-circle-figures">Table 21</a></span>:
<span class="table-number"><a href="#table--fig:modal-circle-figures">Table 21</a>:</span>
Modal Circle
</div>
| ![](/ox-hugo/ewins00_modal_circle.png) | ![](/ox-hugo/ewins00_modal_circle_bis.png) |
|-------------------------------------------------------------------------------|-------------------------------------------------------------------------------------------------------------------|
| <span class="org-target" id="org-target--fig:modal_circle"></span> Properties | <span class="org-target" id="org-target--fig:modal_circle_bis"></span> \\(\omega\_b\\) and \\(\omega\_a\\) points |
| <span class="org-target" id="org-target--fig-modal-circle"></span> Properties | <span class="org-target" id="org-target--fig-modal-circle-bis"></span> \\(\omega\_b\\) and \\(\omega\_a\\) points |
| width=\linewidth | width=\linewidth |
For any frequency \\(\omega\\), we have the following relationship:
@@ -3252,7 +3252,7 @@ It may also be seen that an **estimate of the damping** is provided by the sweep
\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:
Referring to [ 21](#org-target--fig-modal-circle-bis), we can write:
\begin{equation}
\begin{aligned}
@@ -3318,7 +3318,7 @@ The sequence is:
3. **Locate natural frequency, obtain damping estimate**.
The rate of sweep through the region is estimated numerically and the frequency at which it reaches the maximum is deduced.
At the same time, an estimate of the damping is derived using <eq:estimate_damping_sweep_rate>.
A typical example is shown on figure [29](#figure--fig:circle-fit-natural-frequency).
A typical example is shown on [Figure 29](#figure--fig:circle-fit-natural-frequency).
4. **Calculate multiple damping estimates, and scatter**.
A set of damping estimates using all possible combination of the selected data points are computed using <eq:estimate_damping>.
Then, we can choose the damping estimate to be the mean value.
@@ -3440,18 +3440,18 @@ We need to introduce the concept of **residual terms**, necessary in the modal a
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).
Then the regenerated curve is compared with the original measurements, the result is often disappointing, as illustrated in [ 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 [ 22](#org-target--fig-residual-with).
<a id="table--fig:residual-modes"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--fig:residual-modes">Table 22</a></span>:
<span class="table-number"><a href="#table--fig:residual-modes">Table 22</a>:</span>
Effects of residual terms on FRF regeneration
</div>
| ![](/ox-hugo/ewins00_residual_without.png) | ![](/ox-hugo/ewins00_residual_with.png) |
|-----------------------------------------------------------------------------------------|------------------------------------------------------------------------------------|
| <span class="org-target" id="org-target--fig:residual_without"></span> without residual | <span class="org-target" id="org-target--fig:residual_with"></span> with residuals |
| <span class="org-target" id="org-target--fig-residual-without"></span> without residual | <span class="org-target" id="org-target--fig-residual-with"></span> 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
@@ -3480,7 +3480,7 @@ The three terms corresponds to:
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).
These three terms are illustrated on [Figure 30](#figure--fig:low-medium-high-modes).
<a id="figure--fig:low-medium-high-modes"></a>
@@ -3772,17 +3772,17 @@ with
</div>
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).
As an example, a set of mobilities measured are shown individually in [ 23](#org-target--fig-composite-raw) and their summation shown as a single composite curve in [ 23](#org-target--fig-composite-sum).
<a id="table--fig:composite"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--fig:composite">Table 23</a></span>:
<span class="table-number"><a href="#table--fig:composite">Table 23</a>:</span>
Set of measured FRF
</div>
| ![](/ox-hugo/ewins00_composite_raw.png) | ![](/ox-hugo/ewins00_composite_sum.png) |
|---------------------------------------------------------------------------------------|-------------------------------------------------------------------------------------|
| <span class="org-target" id="org-target--fig:composite_raw"></span> Individual curves | <span class="org-target" id="org-target--fig:composite_sum"></span> Composite curve |
| <span class="org-target" id="org-target--fig-composite-raw"></span> Individual curves | <span class="org-target" id="org-target--fig-composite-sum"></span> 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.
@@ -4331,8 +4331,8 @@ There are basically two choices for the graphical display of a modal model:
##### 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).
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.
<a id="figure--fig:static-display"></a>
@@ -4344,16 +4344,16 @@ It is customary to select the largest eigenvector element and to scale the whole
##### 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).
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).
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).
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.
@@ -4376,7 +4376,7 @@ We then tend to interpret this as a motion which is purely in the x-direction wh
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.
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**.
@@ -4425,23 +4425,23 @@ However, it must be noted that there is an important **limitation to this proced
<div class="exampl">
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).
The predict curve is compared with the measurements on [ 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).
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 [ 24](#org-target--fig-H22-with-residual).
</div>
<a id="table--fig:H22"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--fig:H22">Table 24</a></span>:
<span class="table-number"><a href="#table--fig:H22">Table 24</a>:</span>
Synthesized FRF plot
</div>
| ![](/ox-hugo/ewins00_H22_without_residual.png) | ![](/ox-hugo/ewins00_H22_with_residual.png) |
|-----------------------------------------------------------------------------------------------------------|-----------------------------------------------------------------------------------------------------------|
| <span class="org-target" id="org-target--fig:H22_without_residual"></span> Using measured modal data only | <span class="org-target" id="org-target--fig:H22_with_residual"></span> After inclusion of residual terms |
| <span class="org-target" id="org-target--fig-H22-without-residual"></span> Using measured modal data only | <span class="org-target" id="org-target--fig-H22-with-residual"></span> 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
@@ -4492,7 +4492,7 @@ If the **transmissibility** is measured during a modal test which has a single e
</div>
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).
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.
<a id="figure--fig:transmissibility-plots"></a>
@@ -4503,20 +4503,20 @@ This may explain why transmissibilities are not widely used in modal analysis.
#### 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).
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 [Table 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.
<a id="table--fig:base-excitation-configuration"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--fig:base-excitation-configuration">Table 25</a></span>:
<span class="table-number"><a href="#table--fig:base-excitation-configuration">Table 25</a>:</span>
Base excitation configuration
</div>
| ![](/ox-hugo/ewins00_conventional_modal_test_setup.png) | ![](/ox-hugo/ewins00_base_excitation_modal_setup.png) |
|-------------------------------------------------------------------------------------------------------------------|---------------------------------------------------------------------------------------------------------|
| <span class="org-target" id="org-target--fig:conventional_modal_test_setup"></span> Conventional modal test setup | <span class="org-target" id="org-target--fig:base_excitation_modal_setup"></span> Base excitation setup |
| <span class="org-target" id="org-target--fig-conventional-modal-test-setup"></span> Conventional modal test setup | <span class="org-target" id="org-target--fig-base-excitation-modal-setup"></span> Base excitation setup |
| height=4cm | height=4cm |
@@ -4560,5 +4560,5 @@ Because the rank of each pseudo matrix is less than its order, it cannot be inve
## Bibliography {#bibliography}
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Ewins, DJ. 2000. <i>Modal Testing: Theory, Practice and Application</i>. <i>Research Studies Pre, 2nd Ed., Isbn-13</i>. Baldock, Hertfordshire, England Philadelphia, PA: Wiley-Blackwell.</div>
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Ewins, DJ. 2000. <i>Modal Testing: Theory, Practice and Application</i>. <i>Research Studies Pre, 2nd Ed., ISBN-13</i>. Baldock, Hertfordshire, England Philadelphia, PA: Wiley-Blackwell.</div>
</div>