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@@ -1,6 +1,8 @@
+++
title = "Modal testing: theory, practice and application"
title = "Modal Testing: Theory, Practice and Application"
author = ["Thomas Dehaeze"]
description = "Reference book for Modal Testing"
keywords = ["system identification", "modal testing"]
draft = false
+++
@@ -8,7 +10,7 @@ Tags
: [System Identification]({{< relref "system_identification" >}}), [Reference Books]({{< relref "reference_books" >}}), [Modal Analysis]({{< relref "modal_analysis" >}})
Reference
: ([Ewins 2000](#org15876a9))
: ([Ewins 2000](#orgd25baff))
Author(s)
: Ewins, D.
@@ -159,9 +161,9 @@ 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](#org8f0a8f0).
The type of test best suited to FRF measurement is shown in figure [1](#orge16f202).
<a id="org8f0a8f0"></a>
<a id="orge16f202"></a>
{{< figure src="/ox-hugo/ewins00_modal_analysis_schematic.png" caption="Figure 1: Basic components of FRF measurement system" >}}
@@ -231,11 +233,11 @@ 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](#org703f940)).
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](#org5c5d54f)).
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.
<a id="org703f940"></a>
<a id="org5c5d54f"></a>
{{< figure src="/ox-hugo/ewins00_sdof_modulus_phase.png" caption="Figure 2: Curve fit to resonant FRF data" >}}
@@ -270,10 +272,10 @@ 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](#orga0bcee3).
The three phases through a typical theoretical vibration analysis progresses are shown on figure [3](#org2de3899).
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="orga0bcee3"></a>
<a id="org2de3899"></a>
{{< figure src="/ox-hugo/ewins00_vibration_analysis_procedure.png" caption="Figure 3: Theoretical route to vibration analysis" >}}
@@ -295,7 +297,7 @@ Thus our response model will consist of a set of **frequency response functions
<div class="important">
<div></div>
As indicated in figure [3](#orga0bcee3), 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](#org2de3899), 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>
@@ -315,10 +317,10 @@ Three classes of system model will be described:
</div>
The basic model for the SDOF system is shown in figure [4](#org863f8fd) 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](#org2c2a70c) 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="org863f8fd"></a>
<a id="org2c2a70c"></a>
{{< figure src="/ox-hugo/ewins00_sdof_model.png" caption="Figure 4: Single degree-of-freedom system" >}}
@@ -394,9 +396,9 @@ 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](#org777e04b)
The physical significance of these two parts is illustrated in the typical free response plot shown in figure [5](#orgbae45c5)
<a id="org777e04b"></a>
<a id="orgbae45c5"></a>
{{< figure src="/ox-hugo/ewins00_sdof_response.png" caption="Figure 5: Oscillatory and decay part" >}}
@@ -427,7 +429,7 @@ 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](#orgf0a4ea9) 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 figure [1](#orgf870454) 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}}} \\]
@@ -440,13 +442,13 @@ The damping effect of such a component can conveniently be defined by the ratio
| ![](/ox-hugo/ewins00_material_histeresis.png) | ![](/ox-hugo/ewins00_dry_friction.png) | ![](/ox-hugo/ewins00_viscous_damper.png) |
|-----------------------------------------------|----------------------------------------|------------------------------------------|
| <a id="orgf0a4ea9"></a> Material hysteresis | <a id="org2134b3d"></a> Dry friction | <a id="org82f9a69"></a> Viscous damper |
| <a id="orgf870454"></a> Material hysteresis | <a id="orged6e3ed"></a> Dry friction | <a id="org368575f"></a> 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](#org2134b3d).
It may be described very roughly by the simple **dry friction model** shown in figure [1](#orged6e3ed).
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](#org82f9a69).
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](#org368575f).
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.
@@ -567,7 +569,7 @@ Bode plot are usually displayed using logarithmic scales as shown on figure [3](
| ![](/ox-hugo/ewins00_bode_receptance.png) | ![](/ox-hugo/ewins00_bode_mobility.png) | ![](/ox-hugo/ewins00_bode_accelerance.png) |
|-------------------------------------------|-----------------------------------------|--------------------------------------------|
| <a id="orgf6df26a"></a> Receptance FRF | <a id="org58db881"></a> Mobility FRF | <a id="org1c64176"></a> Accelerance FRF |
| <a id="org1eeee38"></a> Receptance FRF | <a id="org2c0cdc1"></a> Mobility FRF | <a id="orgd6f921d"></a> Accelerance FRF |
| width=\linewidth | width=\linewidth | width=\linewidth |
Each plot can be divided into three regimes:
@@ -590,7 +592,7 @@ This type of display is not widely used as we cannot use logarithmic axes (as we
| ![](/ox-hugo/ewins00_plot_receptance_real.png) | ![](/ox-hugo/ewins00_plot_receptance_imag.png) |
|------------------------------------------------|------------------------------------------------|
| <a id="orgb3efe5a"></a> Real part | <a id="org6c0e23c"></a> Imaginary part |
| <a id="orgeaffcf5"></a> Real part | <a id="org7e4b6c1"></a> Imaginary part |
| width=\linewidth | width=\linewidth |
@@ -598,7 +600,7 @@ 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 \eqref{eq:dynamic_stiffness} that the Real part depends entirely on the mass and stiffness properties while the Imaginary part is a only function of the damping.
Figure [5](#org0339be1) 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\\).
Figure [5](#org8eb6352) 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">
@@ -608,7 +610,7 @@ Figure [5](#org0339be1) shows an example of a plot of a system with a combinatio
| ![](/ox-hugo/ewins00_inverse_frf_mixed.png) | ![](/ox-hugo/ewins00_inverse_frf_viscous.png) |
|---------------------------------------------|-----------------------------------------------|
| <a id="org0339be1"></a> Mixed | <a id="org03893ea"></a> Viscous |
| <a id="org8eb6352"></a> Mixed | <a id="org69817b0"></a> Viscous |
| width=\linewidth | width=\linewidth |
@@ -625,7 +627,7 @@ The missing information (in this case, the frequency) must be added by identifyi
| ![](/ox-hugo/ewins00_nyquist_receptance_viscous.png) | ![](/ox-hugo/ewins00_nyquist_receptance_structural.png) |
|------------------------------------------------------|---------------------------------------------------------|
| <a id="org6baff82"></a> Viscous damping | <a id="orgfadfd34"></a> Structural damping |
| <a id="org8896e35"></a> Viscous damping | <a id="org2b2e557"></a> Structural damping |
| width=\linewidth | width=\linewidth |
The Nyquist plot has the particularity of distorting the plot so as to focus on the resonance area.
@@ -1130,9 +1132,9 @@ 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](#org64f75be)).
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](#org081c1b9)).
<a id="org64f75be"></a>
<a id="org081c1b9"></a>
{{< figure src="/ox-hugo/ewins00_real_complex_modes.png" caption="Figure 6: Real and complex mode shapes displays" >}}
@@ -1147,7 +1149,7 @@ Note that the almost-real mode shape does not necessarily have vector elements w
| ![](/ox-hugo/ewins00_argand_diagram_a.png) | ![](/ox-hugo/ewins00_argand_diagram_b.png) | ![](/ox-hugo/ewins00_argand_diagram_c.png) |
|--------------------------------------------|--------------------------------------------|-----------------------------------------------|
| <a id="orgf1cdf2d"></a> Almost-real mode | <a id="orgf730e4e"></a> Complex Mode | <a id="orgdb80ebd"></a> Measure of complexity |
| <a id="org0bdcc92"></a> Almost-real mode | <a id="orgd1143ca"></a> Complex Mode | <a id="org11d773f"></a> Measure of complexity |
| width=\linewidth | width=\linewidth | width=\linewidth |
@@ -1157,7 +1159,7 @@ 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](#orgdb80ebd) where a polygon is drawn around the extremities of the individual vectors.
The second measure is shown on figure [7](#org11d773f) 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**.
@@ -1253,7 +1255,7 @@ 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](#orgc4a7fb9)), 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 (figure [9](#org0ce9b7d)), 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>
@@ -1264,10 +1266,10 @@ On a logarithmic plot, this produces the antiresonance characteristic which refl
| ![](/ox-hugo/ewins00_mobility_frf_mdof_point.png) | ![](/ox-hugo/ewins00_mobility_frf_mdof_transfer.png) |
|---------------------------------------------------|------------------------------------------------------|
| <a id="orgc4a7fb9"></a> Point FRF | <a id="org55cabb4"></a> Transfer FRF |
| <a id="org0ce9b7d"></a> Point FRF | <a id="org3719638"></a> Transfer FRF |
| width=\linewidth | width=\linewidth |
For the plot in figure [9](#org55cabb4), between the two resonances, the two components have the same sign and they add up, no antiresonance is present.
For the plot in figure [9](#org3719638), 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}
@@ -1283,13 +1285,13 @@ 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](#org6fd9292) shows a plot for the same mobility as appears in figure [9](#orgc4a7fb9) but here for a system with added damping.
Figure [7](#org0893e81) shows a plot for the same mobility as appears in figure [9](#org0ce9b7d) but here for a system with added damping.
Most mobility plots have this general form as long as the modes are relatively well-separated.
This condition is satisfied unless the separation between adjacent natural frequencies is of the same order as, or less than, the modal damping factors, in which case it becomes difficult to distinguish the individual modes.
<a id="org6fd9292"></a>
<a id="org0893e81"></a>
{{< figure src="/ox-hugo/ewins00_frf_damped_system.png" caption="Figure 7: Mobility plot of a damped system" >}}
@@ -1298,9 +1300,9 @@ 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](#org4caa691) shows the result of plotting the point receptance \\(\alpha\_{11}\\) for the 2DOF system described above.
Figure [10](#org03e786a) 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](#org5060284) 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 figure [10](#orgfec7087) 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">
@@ -1310,10 +1312,10 @@ The plot for the transfer receptance \\(\alpha\_{21}\\) is presented in figure [
| ![](/ox-hugo/ewins00_nyquist_point.png) | ![](/ox-hugo/ewins00_nyquist_transfer.png) |
|------------------------------------------|---------------------------------------------|
| <a id="org4caa691"></a> Point receptance | <a id="org5060284"></a> Transfer receptance |
| <a id="org03e786a"></a> Point receptance | <a id="orgfec7087"></a> Transfer receptance |
| width=\linewidth | width=\linewidth |
In the two figures [11](#org7d25d6c) and [11](#org9e70037), we show corresponding data for **non-proportional** damping.
In the two figures [11](#org8620a5c) and [11](#org75aa0d4), 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**.
@@ -1325,7 +1327,7 @@ Now we find that the individual modal circles are no longer "upright" but are **
| ![](/ox-hugo/ewins00_nyquist_nonpropdamp_point.png) | ![](/ox-hugo/ewins00_nyquist_nonpropdamp_transfer.png) |
|-----------------------------------------------------|--------------------------------------------------------|
| <a id="org7d25d6c"></a> Point receptance | <a id="org9e70037"></a> Transfer receptance |
| <a id="org8620a5c"></a> Point receptance | <a id="org75aa0d4"></a> Transfer receptance |
| width=\linewidth | width=\linewidth |
@@ -1481,7 +1483,7 @@ Examples of random signals, autocorrelation function and power spectral density
| ![](/ox-hugo/ewins00_random_time.png) | ![](/ox-hugo/ewins00_random_autocorrelation.png) | ![](/ox-hugo/ewins00_random_psd.png) |
|---------------------------------------|--------------------------------------------------|------------------------------------------------|
| <a id="org5355634"></a> Time history | <a id="org618edae"></a> Autocorrelation Function | <a id="org363a29a"></a> Power Spectral Density |
| <a id="orgff9751a"></a> Time history | <a id="orga003731"></a> Autocorrelation Function | <a id="org0e8108d"></a> 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.
@@ -1566,8 +1568,8 @@ 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](#org400650f) the traditional single-input single-output model upon which the previous formulae are based.
Then in [13](#org7285276) 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 figure [13](#table--fig:frf-determination) which shows first in [13](#orgd67883e) the traditional single-input single-output model upon which the previous formulae are based.
Then in [13](#orgc7a70ce) 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">
@@ -1577,7 +1579,7 @@ Then in [13](#org7285276) is given a more detailed and representative model of t
| ![](/ox-hugo/ewins00_frf_siso_model.png) | ![](/ox-hugo/ewins00_frf_feedback_model.png) |
|------------------------------------------|--------------------------------------------------|
| <a id="org400650f"></a> Basic SISO model | <a id="org7285276"></a> SISO model with feedback |
| <a id="orgd67883e"></a> Basic SISO model | <a id="orgc7a70ce"></a> SISO model with feedback |
| width=\linewidth | width=\linewidth |
In this configuration, it can be seen that there are two feedback mechanisms which apply.
@@ -1597,7 +1599,7 @@ 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](#org8f4df84).
A diagram for the general n-input case is shown in figure [8](#orga1854d9).
We obtain two alternative formulas:
@@ -1608,7 +1610,7 @@ We obtain two alternative formulas:
In practical application of both of these formulae, care must be taken to ensure the non-singularity of the spectral density matrix which is to be inverted, and it is in this respect that the former version may be found to be more reliable.
<a id="org8f4df84"></a>
<a id="orga1854d9"></a>
{{< figure src="/ox-hugo/ewins00_frf_mimo.png" caption="Figure 8: System for FRF determination via MIMO model" >}}
@@ -1878,9 +1880,9 @@ 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](#org7f3a496).
A typical layout for the measurement system is shown on figure [9](#org96acfaa).
<a id="org7f3a496"></a>
<a id="org96acfaa"></a>
{{< figure src="/ox-hugo/ewins00_general_frf_measurement_setup.png" caption="Figure 9: General layout of FRF measurement system" >}}
@@ -1934,21 +1936,21 @@ 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](#orge1056cd).
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](#orgf80f52f).
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="orge1056cd"></a>
<a id="orgf80f52f"></a>
{{< figure src="/ox-hugo/ewins00_shaker_rod.png" caption="Figure 10: Exciter attachment and drive rod assembly" >}}
The support of shaker is also of primary importance.
The setup shown on figure [14](#org2ce9b2d) 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 figure [14](#orgad52c50) 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](#orgaf570a9) shows an alternative configuration in which the shaker itself is supported.
Figure [14](#orgfdb8113) 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](#orgc943938) 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\\).
Figure [14](#org2d3f7b0) 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">
@@ -1958,7 +1960,7 @@ Figure [14](#orgc943938) shows an unsatisfactory setup. Indeed, the response mea
| ![](/ox-hugo/ewins00_shaker_mount_1.png) | ![](/ox-hugo/ewins00_shaker_mount_2.png) | ![](/ox-hugo/ewins00_shaker_mount_3.png) |
|---------------------------------------------|-------------------------------------------------|------------------------------------------|
| <a id="org2ce9b2d"></a> Ideal Configuration | <a id="orgaf570a9"></a> Suspended Configuration | <a id="orgc943938"></a> Unsatisfactory |
| <a id="orgad52c50"></a> Ideal Configuration | <a id="orgfdb8113"></a> Suspended Configuration | <a id="org2d3f7b0"></a> Unsatisfactory |
| width=\linewidth | width=\linewidth | width=\linewidth |
@@ -1973,10 +1975,10 @@ 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](#orgb47b9bd).
A pulse of this type (half-sine shape) has a frequency content of the form illustrated on figure [11](#orgb47b9bd).
When the hammer tip impacts the test structure, this will experience a force pulse as shown on figure [11](#org9b1d7fe).
A pulse of this type (half-sine shape) has a frequency content of the form illustrated on figure [11](#org9b1d7fe).
<a id="orgb47b9bd"></a>
<a id="org9b1d7fe"></a>
{{< figure src="/ox-hugo/ewins00_hammer_impulse.png" caption="Figure 11: Typical impact force pulse and spectrum" >}}
@@ -2005,9 +2007,9 @@ 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](#org930ef4e)).
The transmitter force \\(F\\) is applied directly across the crystal, which thus generates a corresponding charge \\(q\\), proportional to \\(F\\) (figure [12](#org493b0fd)).
<a id="org930ef4e"></a>
<a id="org493b0fd"></a>
{{< figure src="/ox-hugo/ewins00_piezo_force_transducer.png" caption="Figure 12: Force transducer" >}}
@@ -2016,11 +2018,11 @@ 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](#orga075bcf)).
In an accelerometer, transduction is indirect and is achieved using a seismic mass (figure [13](#org17d53ed)).
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\\).
<a id="orga075bcf"></a>
<a id="org17d53ed"></a>
{{< figure src="/ox-hugo/ewins00_piezo_accelerometer.png" caption="Figure 13: Compression-type of piezoelectric accelerometer" >}}
@@ -2056,9 +2058,9 @@ 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](#orge053903).
Some of these methods are illustrated in figure [15](#org1468fa8).
Shown on figure [15](#org1b85602) are typical high frequency limits for each type of attachment.
Shown on figure [15](#org7bb2dd9) are typical high frequency limits for each type of attachment.
<a id="table--fig:transducer-mounting"></a>
<div class="table-caption">
@@ -2068,7 +2070,7 @@ Shown on figure [15](#org1b85602) are typical high frequency limits for each typ
| ![](/ox-hugo/ewins00_transducer_mounting_types.png) | ![](/ox-hugo/ewins00_transducer_mounting_response.png) |
|-----------------------------------------------------|------------------------------------------------------------|
| <a id="orge053903"></a> Attachment methods | <a id="org1b85602"></a> Frequency response characteristics |
| <a id="org1468fa8"></a> Attachment methods | <a id="org7bb2dd9"></a> Frequency response characteristics |
| width=\linewidth | width=\linewidth |
@@ -2153,9 +2155,9 @@ 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](#org91dbe3e).
These high frequencies will be **indistinguishable** from genuine low frequency components as shown on figure [14](#orgeaeb967).
<a id="org91dbe3e"></a>
<a id="orgeaeb967"></a>
{{< figure src="/ox-hugo/ewins00_aliasing.png" caption="Figure 14: The phenomenon of aliasing. On top: Low-frequency signal, On the bottom: High frequency signal" >}}
@@ -2172,7 +2174,7 @@ This is illustrated on figure [16](#table--fig:effect-aliasing).
| ![](/ox-hugo/ewins00_aliasing_no_distortion.png) | ![](/ox-hugo/ewins00_aliasing_distortion.png) |
|--------------------------------------------------|-----------------------------------------------------|
| <a id="orgb4560b8"></a> True spectrum of signal | <a id="orgd413cee"></a> Indicated spectrum from DFT |
| <a id="org3e3d162"></a> True spectrum of signal | <a id="org65765c2"></a> 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.
@@ -2193,12 +2195,12 @@ Leakage is a problem which is a direct **consequence of the need to take only a
| ![](/ox-hugo/ewins00_leakage_ok.png) | ![](/ox-hugo/ewins00_leakage_nok.png) |
|--------------------------------------|----------------------------------------|
| <a id="orgd54be6b"></a> Ideal signal | <a id="org95b6cdc"></a> Awkward signal |
| <a id="org7c07e86"></a> Ideal signal | <a id="orgdd714e0"></a> Awkward signal |
| width=\linewidth | width=\linewidth |
The problem is illustrated on figure [17](#table--fig:leakage).
In the first case (figure [17](#orgd54be6b)), 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](#org95b6cdc)), the periodicity assumption is not strictly valid as there is a discontinuity at each end of the sample.
In the first case (figure [17](#org7c07e86)), 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](#orgdd714e0)), 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.
@@ -2216,14 +2218,14 @@ 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](#org105c7d0).
The profiles, or "windows" are generally depicted as a time function \\(w(t)\\) as shown in figure [15](#orga7ce3a7).
<a id="org105c7d0"></a>
<a id="orga7ce3a7"></a>
{{< figure src="/ox-hugo/ewins00_windowing_examples.png" caption="Figure 15: 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](#org105c7d0).
The result of using a window is seen in the third column of figure [15](#orga7ce3a7).
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.
@@ -2239,7 +2241,7 @@ Common filters are: low-pass, high-pass, band-limited, narrow-band, notch.
#### Improving Resolution {#improving-resolution}
<a id="org4b52cde"></a>
<a id="org009a00a"></a>
##### Increasing transform size {#increasing-transform-size}
@@ -2263,9 +2265,9 @@ 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](#orgfeb63a7), 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 figure [18](#orgfbb1177), 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](#org94b4dd9), 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](#org0cfcb53)).
If we apply a band-pass filter to the signal, as shown on figure [18](#orgf8735d6), 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](#orga5b098f)).
<a id="table--fig:frequency-zoom"></a>
<div class="table-caption">
@@ -2275,10 +2277,10 @@ If we apply a band-pass filter to the signal, as shown on figure [18](#org94b4dd
| ![](/ox-hugo/ewins00_zoom_range.png) | ![](/ox-hugo/ewins00_zoom_bandpass.png) |
|------------------------------------------------|------------------------------------------|
| <a id="orgfeb63a7"></a> Spectrum of the signal | <a id="org94b4dd9"></a> Band-pass filter |
| <a id="orgfbb1177"></a> Spectrum of the signal | <a id="orgf8735d6"></a> Band-pass filter |
| width=\linewidth | width=\linewidth |
<a id="org0cfcb53"></a>
<a id="orga5b098f"></a>
{{< figure src="/ox-hugo/ewins00_zoom_result.png" caption="Figure 16: Effective frequency translation for zoom" >}}
@@ -2348,9 +2350,9 @@ 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](#orgd1e88bf).
If excessive sweep rate is used, then distortions of the FRF plot are introduced as shown on figure [17](#org3a0fa7e).
<a id="orgd1e88bf"></a>
<a id="org3a0fa7e"></a>
{{< figure src="/ox-hugo/ewins00_sweep_distortions.png" caption="Figure 17: FRF measurements by sine sweep test" >}}
@@ -2466,9 +2468,9 @@ 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](#org2d9ba99).
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](#orga2df003).
<a id="org2d9ba99"></a>
<a id="orga2df003"></a>
{{< figure src="/ox-hugo/ewins00_coherence_resonance.png" caption="Figure 18: Coherence \\(\gamma^2\\) and FRF estimate \\(H\_1(\omega)\\) for a lightly damped structure" >}}
@@ -2509,9 +2511,9 @@ 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](#org63d0501)).
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](#orgb1cdd01)).
<a id="org63d0501"></a>
<a id="orgb1cdd01"></a>
{{< figure src="/ox-hugo/ewins00_burst_excitation.png" caption="Figure 19: Example of burst excitation and response signals" >}}
@@ -2526,22 +2528,22 @@ 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](#org3d7182f).
The chirp consist of a short duration signal which has the form shown in figure [20](#org3e96514).
The frequency content of the chirp can be precisely chosen by the starting and finishing frequencies of the sweep.
<a id="org3d7182f"></a>
<a id="org3e96514"></a>
{{< figure src="/ox-hugo/ewins00_chirp_excitation.png" caption="Figure 20: Example of chirp excitation and response signals" >}}
##### Impulsive excitation {#impulsive-excitation}
The hammer blow produces an input and response as shown in the figure [21](#orgee86d4a).
The hammer blow produces an input and response as shown in the figure [21](#org7d16186).
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.
<a id="orgee86d4a"></a>
<a id="org7d16186"></a>
{{< figure src="/ox-hugo/ewins00_impulsive_excitation.png" caption="Figure 21: Example of impulsive excitation and response signals" >}}
@@ -2549,9 +2551,9 @@ 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](#org2914aa8)).
In such cases, the spectrum of the excitation is seen to have "holes" in it at certain frequencies (figure [22](#org465da50)).
<a id="org2914aa8"></a>
<a id="org465da50"></a>
{{< figure src="/ox-hugo/ewins00_double_hits.png" caption="Figure 22: Double hits time domain and frequency content" >}}
@@ -2624,9 +2626,9 @@ 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](#org1f3d9fc) shows a typical calibration setup.
Figure [23](#org3793510) shows a typical calibration setup.
<a id="org1f3d9fc"></a>
<a id="org3793510"></a>
{{< figure src="/ox-hugo/ewins00_calibration_setup.png" caption="Figure 23: Mass calibration procedure, measurement setup" >}}
@@ -2639,9 +2641,9 @@ 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](#org5a54bfb).
This same argument applies on a lesser scale as we examine the detail around the attachment to the structure, as shown in figure [24](#org8b94b73).
<a id="org5a54bfb"></a>
<a id="org8b94b73"></a>
{{< figure src="/ox-hugo/ewins00_mass_cancellation.png" caption="Figure 24: Added mass to be cancelled (crossed area)" >}}
@@ -2696,9 +2698,9 @@ 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](#org7c44a7c).
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](#org9d4e788).
<a id="org7c44a7c"></a>
<a id="org9d4e788"></a>
{{< figure src="/ox-hugo/ewins00_rotational_measurement.png" caption="Figure 25: Measurement of rotational response" >}}
@@ -2714,12 +2716,12 @@ 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](#org69e6665) shows a device to simulate a moment excitation.
Figure [26](#org65dc42a) 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\\).
<a id="org69e6665"></a>
<a id="org65dc42a"></a>
{{< figure src="/ox-hugo/ewins00_rotational_excitation.png" caption="Figure 26: Application of moment excitation" >}}
@@ -3043,8 +3045,8 @@ Then, each PRF is, simply, a particular combination of the original FRFs, and th
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](#org1966197) 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](#orgf7309a1) shows the genuine modes distinct from the computational modes.
The second plot [19](#org21a5511) 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](#org57a90c0) shows the genuine modes distinct from the computational modes.
<div class="important">
<div></div>
@@ -3071,7 +3073,7 @@ The two groups are usually separated by a clear gap (depending of the noise pres
| ![](/ox-hugo/ewins00_PRF_numerical_FRF.png) | ![](/ox-hugo/ewins00_PRF_numerical_svd.png) | ![](/ox-hugo/ewins00_PRF_numerical_PRF.png) |
|---------------------------------------------|---------------------------------------------|---------------------------------------------|
| <a id="org85391fe"></a> FRF | <a id="org1966197"></a> Singular Values | <a id="orgf7309a1"></a> PRF |
| <a id="org866353d"></a> FRF | <a id="org21a5511"></a> Singular Values | <a id="org57a90c0"></a> PRF |
| width=\linewidth | width=\linewidth | width=\linewidth |
<a id="table--fig:PRF-measured"></a>
@@ -3082,7 +3084,7 @@ The two groups are usually separated by a clear gap (depending of the noise pres
| ![](/ox-hugo/ewins00_PRF_measured_FRF.png) | ![](/ox-hugo/ewins00_PRF_measured_svd.png) | ![](/ox-hugo/ewins00_PRF_measured_PRF.png) |
|--------------------------------------------|--------------------------------------------|--------------------------------------------|
| <a id="org410e27f"></a> FRF | <a id="org6b7d854"></a> Singular Values | <a id="orgc6a23d0"></a> PRF |
| <a id="org071dcb1"></a> FRF | <a id="org61b6872"></a> Singular Values | <a id="org760fea0"></a> PRF |
| width=\linewidth | width=\linewidth | width=\linewidth |
@@ -3114,7 +3116,7 @@ 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](#org08ac181):
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](#org405ffa2):
- **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**.
@@ -3124,7 +3126,7 @@ Associated with the CMIF values at each natural frequency \\(\omega\_r\\) are tw
- 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**
<a id="org08ac181"></a>
<a id="org405ffa2"></a>
{{< figure src="/ox-hugo/ewins00_mifs.png" caption="Figure 27: Complex Mode Indicator Function (CMIF)" >}}
@@ -3197,7 +3199,7 @@ 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](#orgd1dacfd)):
The peak-picking method is applied as follows (illustrated on figure [28](#org017fa0f)):
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.
@@ -3219,7 +3221,7 @@ The peak-picking method is applied as follows (illustrated on figure [28](#orgd1
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.
<a id="orgd1dacfd"></a>
<a id="org017fa0f"></a>
{{< figure src="/ox-hugo/ewins00_peak_amplitude.png" caption="Figure 28: Peak Amplitude method of modal analysis" >}}
@@ -3244,7 +3246,7 @@ 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](#org0c46692).
A plot of the quantity \\(\alpha(\omega)\\) is given in figure [21](#org6e0c9b9).
<a id="table--fig:modal-circle-figures"></a>
<div class="table-caption">
@@ -3254,7 +3256,7 @@ A plot of the quantity \\(\alpha(\omega)\\) is given in figure [21](#org0c46692)
| ![](/ox-hugo/ewins00_modal_circle.png) | ![](/ox-hugo/ewins00_modal_circle_bis.png) |
|----------------------------------------|--------------------------------------------------------------------|
| <a id="org0c46692"></a> Properties | <a id="orga918af0"></a> \\(\omega\_b\\) and \\(\omega\_a\\) points |
| <a id="org6e0c9b9"></a> Properties | <a id="orgbc7eafe"></a> \\(\omega\_b\\) and \\(\omega\_a\\) points |
| width=\linewidth | width=\linewidth |
For any frequency \\(\omega\\), we have the following relationship:
@@ -3292,7 +3294,7 @@ 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](#orga918af0), we can write:
Referring to figure [21](#orgbc7eafe), we can write:
\begin{equation}
\begin{aligned}
@@ -3358,7 +3360,7 @@ 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 \eqref{eq:estimate_damping_sweep_rate}.
A typical example is shown on figure [29](#org96a13a2).
A typical example is shown on figure [29](#orgb1f1c40).
4. **Calculate multiple damping estimates, and scatter**.
A set of damping estimates using all possible combination of the selected data points are computed using \eqref{eq:estimate_damping}.
Then, we can choose the damping estimate to be the mean value.
@@ -3368,7 +3370,7 @@ The sequence is:
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.
<a id="org96a13a2"></a>
<a id="orgb1f1c40"></a>
{{< figure src="/ox-hugo/ewins00_circle_fit_natural_frequency.png" caption="Figure 29: Location of natural frequency for a Circle-fit modal analysis" >}}
@@ -3482,8 +3484,8 @@ 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](#org398d4d8).
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](#org8ee9d90).
Then the regenerated curve is compared with the original measurements, the result is often disappointing, as illustrated in figure [22](#org3cda8ae).
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](#org92dafed).
<a id="table--fig:residual-modes"></a>
<div class="table-caption">
@@ -3493,7 +3495,7 @@ However, by the inclusion of two simple extra terms (the "**residuals**"), the m
| ![](/ox-hugo/ewins00_residual_without.png) | ![](/ox-hugo/ewins00_residual_with.png) |
|--------------------------------------------|-----------------------------------------|
| <a id="org398d4d8"></a> without residual | <a id="org8ee9d90"></a> with residuals |
| <a id="org3cda8ae"></a> without residual | <a id="org92dafed"></a> 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
@@ -3522,9 +3524,9 @@ 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](#org473ef14).
These three terms are illustrated on figure [30](#org8849a18).
<a id="org473ef14"></a>
<a id="org8849a18"></a>
{{< figure src="/ox-hugo/ewins00_low_medium_high_modes.png" caption="Figure 30: Numerical simulation of contribution of low, medium and high frequency modes" >}}
@@ -3818,7 +3820,7 @@ 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](#org1ee7063) and their summation shown as a single composite curve in figure [23](#orgfb3f6a3).
As an example, a set of mobilities measured are shown individually in figure [23](#org433946d) and their summation shown as a single composite curve in figure [23](#orgd3aaebe).
<a id="table--fig:composite"></a>
<div class="table-caption">
@@ -3828,7 +3830,7 @@ As an example, a set of mobilities measured are shown individually in figure [23
| ![](/ox-hugo/ewins00_composite_raw.png) | ![](/ox-hugo/ewins00_composite_sum.png) |
|-------------------------------------------|-----------------------------------------|
| <a id="org1ee7063"></a> Individual curves | <a id="orgfb3f6a3"></a> Composite curve |
| <a id="org433946d"></a> Individual curves | <a id="orgd3aaebe"></a> 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.
@@ -4382,11 +4384,11 @@ 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](#org0dcf72a) (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](#org0dcf72a) (b).
Measured coordinates of the test structure are first linked as shown on figure [31](#org873fbad) (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](#org873fbad) (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="org0dcf72a"></a>
<a id="org873fbad"></a>
{{< 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" >}}
@@ -4395,16 +4397,16 @@ 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](#org0dcf72a) (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](#org873fbad) (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](#org0dcf72a) (d).
It is also possible, in this format, to give some indication of the essence of complex modes, as shown in figure [31](#org873fbad) (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](#org0dcf72a) (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](#org873fbad) (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.
@@ -4427,11 +4429,11 @@ 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](#org843940c), 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](#orgd0cec90), 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**.
<a id="org843940c"></a>
<a id="orgd0cec90"></a>
{{< figure src="/ox-hugo/ewins00_beam_modes.png" caption="Figure 32: Misinterpretation of mode shapes by spatial aliasing" >}}
@@ -4478,11 +4480,11 @@ However, it must be noted that there is an important **limitation to this proced
<div></div>
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](#orga9d477f).
The predict curve is compared with the measurements on figure [24](#org257d04e).
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](#orgc3d79ab).
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](#org82dd447).
</div>
@@ -4494,7 +4496,7 @@ The inclusion of these two additional terms (obtained here only after measuring
| ![](/ox-hugo/ewins00_H22_without_residual.png) | ![](/ox-hugo/ewins00_H22_with_residual.png) |
|--------------------------------------------------------|-----------------------------------------------------------|
| <a id="orga9d477f"></a> Using measured modal data only | <a id="orgc3d79ab"></a> After inclusion of residual terms |
| <a id="org257d04e"></a> Using measured modal data only | <a id="org82dd447"></a> 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
@@ -4546,10 +4548,10 @@ 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](#orgf71911f).
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](#orgcaddc1e).
This may explain why transmissibilities are not widely used in modal analysis.
<a id="orgf71911f"></a>
<a id="orgcaddc1e"></a>
{{< figure src="/ox-hugo/ewins00_transmissibility_plots.png" caption="Figure 33: Transmissibility plots" >}}
@@ -4570,7 +4572,7 @@ The fact that the excitation force is not measured is responsible for the lack o
| ![](/ox-hugo/ewins00_conventional_modal_test_setup.png) | ![](/ox-hugo/ewins00_base_excitation_modal_setup.png) |
|---------------------------------------------------------|-------------------------------------------------------|
| <a id="org8c4c8e9"></a> Conventional modal test setup | <a id="orge759ea9"></a> Base excitation setup |
| <a id="orge8ea85c"></a> Conventional modal test setup | <a id="org5a6dcef"></a> Base excitation setup |
| height=4cm | height=4cm |
@@ -4614,4 +4616,4 @@ Because the rank of each pseudo matrix is less than its order, it cannot be inve
## Bibliography {#bibliography}
<a id="org15876a9"></a>Ewins, DJ. 2000. _Modal Testing: Theory, Practice and Application_. _Research Studies Pre, 2nd Ed., ISBN-13_. Baldock, Hertfordshire, England Philadelphia, PA: Wiley-Blackwell.
<a id="orgd25baff"></a>Ewins, DJ. 2000. _Modal Testing: Theory, Practice and Application_. _Research Studies Pre, 2nd Ed., ISBN-13_. Baldock, Hertfordshire, England Philadelphia, PA: Wiley-Blackwell.