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content/book/ewins00_modal.md
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@@ -141,7 +141,7 @@ The main measurement technique studied are those which will permit to make **dir
The type of test best suited to FRF measurement is shown in figure [fig:modal_analysis_schematic](#fig:modal_analysis_schematic).
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{{< figure src="/ox-hugo/ewins00_modal_analysis_schematic.png" caption="Figure 1: Basic components of FRF measurement system" >}}
@@ -199,7 +199,7 @@ This process itself falls into two stages:
Most of the effort goes into this second stage, which is widely referred to as "modal parameter extraction", or simply as "modal analysis".
We have seen that we can predict the form of the FRF plots for a multi degree-of-freedom system, and that these are directly related to the modal properties of that system.
The great majority of the modal analysis effort involves **curve-fitting** an expression such as equation [eq:frf_modal](#eq:frf_modal) to the measured FRF and thereby finding the appropriate modal parameters.
The great majority of the modal analysis effort involves **curve-fitting** an expression such as equation \eqref{eq:frf_modal} to the measured FRF and thereby finding the appropriate modal parameters.
A completely general curve-fitting approach is possible but generally inefficient.
Mathematically, we can take an equation of the form
@@ -215,7 +215,7 @@ This assumption allows us to use the circular nature of a modulus/phase polar pl
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.
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{{< figure src="/ox-hugo/ewins00_sdof_modulus_phase.png" caption="Figure 2: Curve fit to resonant FRF data" >}}
@@ -253,7 +253,7 @@ Theoretical foundations of modal testing are of paramount importance to its succ
The three phases through a typical theoretical vibration analysis progresses are shown on figure [fig:vibration_analysis_procedure](#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**.
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{{< figure src="/ox-hugo/ewins00_vibration_analysis_procedure.png" caption="Figure 3: Theoretical route to vibration analysis" >}}
@@ -298,7 +298,7 @@ Three classes of system model will be described:
The basic model for the SDOF system is shown in figure [fig:sdof_model](#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\\).
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{{< figure src="/ox-hugo/ewins00_sdof_model.png" caption="Figure 4: Single degree-of-freedom system" >}}
@@ -374,7 +374,7 @@ which is a single mode of vibration with a complex natural frequency having two
The physical significance of these two parts is illustrated in the typical free response plot shown in figure [fig:sdof_response](#fig:sdof_response)
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{{< figure src="/ox-hugo/ewins00_sdof_response.png" caption="Figure 5: Oscillatory and decay part" >}}
@@ -418,7 +418,7 @@ 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="orgd01ea8f"></a> Material hysteresis | <a id="org5fb7a29"></a> Dry friction | <a id="org41cf290"></a> Viscous damper |
| <a id="org54caaf8"></a> Material hysteresis | <a id="org0fc2b44"></a> Dry friction | <a id="org0985c72"></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.
@@ -458,11 +458,11 @@ where \\(\eta\\) is the **structural damping loss factor** and replaces the crit
#### Alternative Forms of FRF {#alternative-forms-of-frf}
So far we have defined our receptance frequency response function \\(\alpha(\omega)\\) as the ratio between a harmonic displacement response and the harmonic force [eq:receptance](#eq:receptance).
So far we have defined our receptance frequency response function \\(\alpha(\omega)\\) as the ratio between a harmonic displacement response and the harmonic force \eqref{eq:receptance}.
This ratio is complex: we can look at its **amplitude** ratio \\(|\alpha(\omega)|\\) and its **phase** angle \\(\theta\_\alpha(\omega)\\).
We could have selected the response velocity \\(v(t)\\) as the output quantity and defined an alternative frequency response function [eq:mobility](#eq:mobility).
Similarly we could use the acceleration parameter so we could define a third FRF parameter [eq:inertance](#eq:inertance).
We could have selected the response velocity \\(v(t)\\) as the output quantity and defined an alternative frequency response function \eqref{eq:mobility}.
Similarly we could use the acceleration parameter so we could define a third FRF parameter \eqref{eq:inertance}.
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@@ -537,7 +537,7 @@ Bode plot are usually displayed using logarithmic scales as shown on figure [fig
| ![](/ox-hugo/ewins00_bode_receptance.png) | ![](/ox-hugo/ewins00_bode_mobility.png) | ![](/ox-hugo/ewins00_bode_accelerance.png) |
|-------------------------------------------|-----------------------------------------|--------------------------------------------|
| <a id="org1478c78"></a> Receptance FRF | <a id="orgd7eb060"></a> Mobility FRF | <a id="orgf95b5b3"></a> Accelerance FRF |
| <a id="orgea747d3"></a> Receptance FRF | <a id="orgc5e3717"></a> Mobility FRF | <a id="orgcf610b2"></a> Accelerance FRF |
| width=\linewidth | width=\linewidth | width=\linewidth |
Each plot can be divided into three regimes:
@@ -560,13 +560,13 @@ 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="org35a1358"></a> Real part | <a id="org69673f5"></a> Imaginary part |
| <a id="org695538e"></a> Real part | <a id="org95c5960"></a> Imaginary part |
| width=\linewidth | width=\linewidth |
##### Real part and Imaginary part of reciprocal FRF {#real-part-and-imaginary-part-of-reciprocal-frf}
It can be seen from the expression of the inverse receptance [eq:dynamic_stiffness](#eq:dynamic_stiffness) that the Real part depends entirely on the mass and stiffness properties while the Imaginary part is a only function of the damping.
It can be seen from the expression of the inverse receptance \eqref{eq:dynamic_stiffness} that the Real part depends entirely on the mass and stiffness properties while the Imaginary part is a only function of the damping.
Figure [fig:inverse_frf_mixed](#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\\).
@@ -578,7 +578,7 @@ Figure [fig:inverse_frf_mixed](#fig:inverse_frf_mixed) shows an example of a plo
| ![](/ox-hugo/ewins00_inverse_frf_mixed.png) | ![](/ox-hugo/ewins00_inverse_frf_viscous.png) |
|---------------------------------------------|-----------------------------------------------|
| <a id="org8ba100f"></a> Mixed | <a id="org9bed1e0"></a> Viscous |
| <a id="org9e0909f"></a> Mixed | <a id="orge2690df"></a> Viscous |
| width=\linewidth | width=\linewidth |
@@ -595,7 +595,7 @@ 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="org5cbe7df"></a> Viscous damping | <a id="orgb5e61e4"></a> Structural damping |
| <a id="org86b8a60"></a> Viscous damping | <a id="orgb0d3b09"></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.
@@ -607,7 +607,7 @@ This makes the Nyquist plot very effective for modal testing applications.
#### Free Vibration Solution - The modal Properties {#free-vibration-solution-the-modal-properties}
For an undamped MDOF system, with \\(N\\) degrees of freedom, the governing equations of motion can be written in matrix form [eq:undamped_mdof](#eq:undamped_mdof).
For an undamped MDOF system, with \\(N\\) degrees of freedom, the governing equations of motion can be written in matrix form \eqref{eq:undamped_mdof}.
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@@ -622,7 +622,7 @@ where \\([M]\\) and \\([K]\\) are \\(N\times N\\) mass and stiffness matrices, a
We shall consider first the free vibration solution by taking \\(f(t) = 0\\).
In this case, we assume that a solution exists of the form \\(\\{x(t)\\} = \\{X\\} e^{i \omega t}\\) where \\(\\{X\\}\\) is an \\(N \times 1\\) vector of time-independent amplitudes.
Substitution of this condition into [eq:undamped_mdof](#eq:undamped_mdof) leads to
Substitution of this condition into \eqref{eq:undamped_mdof} leads to
\begin{equation}
\left( [K] - \omega^2 [M] \right) \\{X\\} e^{i\omega t} = \\{0\\}
@@ -632,7 +632,7 @@ for which the non trivial solutions are those which satisfy
\\[ \det \left| [K] - \omega^2 [M] \right| = 0 \\]
from which we can find \\(N\\) values of \\(\omega^2\\) corresponding to the undamped system's **natural frequencies**.
Substituting any of these back into [eq:free_eom_mdof](#eq:free_eom_mdof) yields a corresponding set of relative values for \\(\\{X\\}\\): \\(\\{\psi\\}\_r\\) the so-called **mode shape** corresponding to that natural frequency.
Substituting any of these back into \eqref{eq:free_eom_mdof} yields a corresponding set of relative values for \\(\\{X\\}\\): \\(\\{\psi\\}\_r\\) the so-called **mode shape** corresponding to that natural frequency.
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@@ -774,7 +774,7 @@ An alternative means of deriving the FRF parameters is used which makes use of t
\\[ [K] - \omega^2 [M] = [\alpha(\omega)]^{-1} \\]
Pre-multiply both sides by \\([\Phi]^T\\) and post-multiply both sides by \\([\Phi]\\) to obtain
\\[ [\Phi]^T ([K] - \omega^2 [M]) [\Phi] = [\Phi]^T [\alpha(\omega)]^{-1} [\Phi] \\]
which leads to [eq:receptance_modal](#eq:receptance_modal).
which leads to \eqref{eq:receptance_modal}.
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@@ -783,7 +783,7 @@ which leads to [eq:receptance_modal](#eq:receptance_modal).
[\alpha(\omega)] = [\Phi] \left[ \bar{\omega}\_r^2 - \omega^2 \right]^{-1} [\Phi]^T \label{eq:receptance\_modal}
\end{equation}
Equation [eq:receptance_modal](#eq:receptance_modal) permits us to compute any individual FRF parameters \\(\alpha\_{jk}(\omega)\\) using the following formula
Equation \eqref{eq:receptance_modal} permits us to compute any individual FRF parameters \\(\alpha\_{jk}(\omega)\\) using the following formula
\begin{subequations}
\begin{align}
@@ -800,7 +800,7 @@ where \\({}\_rA\_{jk}\\) is called the **modal constant**.
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It is clear from equation [eq:receptance_modal](#eq:receptance_modal) that the receptance matrix \\([\alpha(\omega)]\\) is **symmetric** and this will be recognized as the **principle of reciprocity**.
It is clear from equation \eqref{eq:receptance_modal} that the receptance matrix \\([\alpha(\omega)]\\) is **symmetric** and this will be recognized as the **principle of reciprocity**.
This principle of reciprocity applies to many structural characteristics.
@@ -938,7 +938,7 @@ From this full matrix equation, we have:
Having derived an expression for the general term in the frequency response function matrix \\(\alpha\_{jk}(\omega)\\), it is appropriate to consider next the analysis of a situation where the system is **excited simultaneously at several points**.
The general behavior for this case is governed by equation [eq:force_response_eom](#eq:force_response_eom) with solution [eq:force_response_eom_solution](#eq:force_response_eom_solution).
The general behavior for this case is governed by equation \eqref{eq:force_response_eom} with solution \eqref{eq:force_response_eom_solution}.
However, a more explicit form of the solution is
\begin{equation}
@@ -962,7 +962,7 @@ The properties of the normal modes of the undamped system are of interest becaus
</div>
We are seeking an excitation vector \\(\\{F\\}\\) such that the **response** \\(\\{X\\}\\) **consists of a single modal component** so that all terms in [eq:ods](#eq:ods) but one is zero.
We are seeking an excitation vector \\(\\{F\\}\\) such that the **response** \\(\\{X\\}\\) **consists of a single modal component** so that all terms in \eqref{eq:ods} but one is zero.
This can be attained if \\(\\{F\\}\\) is chosen such that
\\[ \\{\phi\_r\\}^T \\{F\\}\_s = 0, \ r \neq s \\]
@@ -1046,7 +1046,7 @@ where \\(\omega\_r\\) is the **natural frequency** and \\(\xi\_r\\) is the **cri
When the modes \\(r\\) and \\(q\\) are a complex conjugate pair:
\\[ s\_r = \omega\_r \left( -\xi\_r - i\sqrt{1 - \xi\_r^2} \right); \quad \\{\psi\\}\_q = \\{\psi\\}\_r^\* \\]
From equations [eq:viscous_damping_orthogonality](#eq:viscous_damping_orthogonality), we can obtain
From equations \eqref{eq:viscous_damping_orthogonality}, we can obtain
\begin{subequations}
\begin{align}
@@ -1103,7 +1103,7 @@ Equally, in a real mode, all parts of the structure pass through their **zero de
While the real mode has the appearance of a **standing wave**, the complex mode is better described as exhibiting **traveling waves** (illustrated on figure [fig:real_complex_modes](#fig:real_complex_modes)).
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{{< figure src="/ox-hugo/ewins00_real_complex_modes.png" caption="Figure 6: Real and complex mode shapes displays" >}}
@@ -1118,7 +1118,7 @@ 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="orgdb142d3"></a> Almost-real mode | <a id="org40e30d6"></a> Complex Mode | <a id="org4c29a71"></a> Measure of complexity |
| <a id="orgd9e3564"></a> Almost-real mode | <a id="orgeedeefa"></a> Complex Mode | <a id="org2d21384"></a> Measure of complexity |
| width=\linewidth | width=\linewidth | width=\linewidth |
@@ -1235,7 +1235,7 @@ 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="org0d0c340"></a> Point FRF | <a id="org13ad8cd"></a> Transfer FRF |
| <a id="org464f787"></a> Point FRF | <a id="orgd21bcd3"></a> Transfer FRF |
| width=\linewidth | width=\linewidth |
For the plot in figure [fig:mobility_frf_mdof_transfer](#fig:mobility_frf_mdof_transfer), between the two resonances, the two components have the same sign and they add up, no antiresonance is present.
@@ -1260,7 +1260,7 @@ Most mobility plots have this general form as long as the modes are relatively w
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.
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{{< figure src="/ox-hugo/ewins00_frf_damped_system.png" caption="Figure 7: Mobility plot of a damped system" >}}
@@ -1281,7 +1281,7 @@ 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="org7bbfae7"></a> Point receptance | <a id="org0f31112"></a> Transfer receptance |
| <a id="org5dbb609"></a> Point receptance | <a id="orgf225939"></a> Transfer receptance |
| width=\linewidth | width=\linewidth |
In the two figures [fig:nyquist_nonpropdamp_point](#fig:nyquist_nonpropdamp_point) and [fig:nyquist_nonpropdamp_transfer](#fig:nyquist_nonpropdamp_transfer), we show corresponding data for **non-proportional** damping.
@@ -1296,7 +1296,7 @@ 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="org460aa35"></a> Point receptance | <a id="org01c8e2c"></a> Transfer receptance |
| <a id="orgae9806e"></a> Point receptance | <a id="orgb532a2f"></a> Transfer receptance |
| width=\linewidth | width=\linewidth |
@@ -1343,7 +1343,7 @@ One these two series are available, the FRF can be defined at the same set of fr
##### Analysis via Fourier transform {#analysis-via-fourier-transform}
For most transient cases, the input function \\(f(t)\\) will satisfy the **Dirichlet condition** and so its Fourier Transform \\(F(\omega)\\) can be computed from [eq:fourier_transform](#eq:fourier_transform).
For most transient cases, the input function \\(f(t)\\) will satisfy the **Dirichlet condition** and so its Fourier Transform \\(F(\omega)\\) can be computed from \eqref{eq:fourier_transform}.
\begin{equation}
F(\omega) = \frac{1}{2 \pi} \int\_{-\infty}^\infty f(t) e^{i\omega t} dt
@@ -1450,7 +1450,7 @@ 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="org3a8665e"></a> Time history | <a id="org5ade2bf"></a> Autocorrelation Function | <a id="org9f69a06"></a> Power Spectral Density |
| <a id="org30bff26"></a> Time history | <a id="org7e07ced"></a> Autocorrelation Function | <a id="orgcb31329"></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.
@@ -1493,10 +1493,10 @@ However, the same equation can be transform to the frequency domain
\tcmbox{ S\_{xx}(\omega) = \left| H(\omega) \right|^2 S\_{ff}(\omega) }
\end{equation}
Although very convenient, equation [eq:psd_input_output](#eq:psd_input_output) does not provide a complete description of the random vibration conditions.
Although very convenient, equation \eqref{eq:psd_input_output} does not provide a complete description of the random vibration conditions.
Further, it is clear that **is could not be used to determine the FRF** from measurement of excitation and response because it **contains only the modulus** of \\(H(\omega)\\), the phase information begin omitted from this formula.
A second equation is required and this may be obtain by a similar analysis, two alternative formulas can be obtained [eq:cross_relation_alternatives](#eq:cross_relation_alternatives).
A second equation is required and this may be obtain by a similar analysis, two alternative formulas can be obtained \eqref{eq:cross_relation_alternatives}.
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@@ -1513,8 +1513,8 @@ A second equation is required and this may be obtain by a similar analysis, two
##### To derive FRF from random vibration signals {#to-derive-frf-from-random-vibration-signals}
The pair of equations [eq:cross_relation_alternatives](#eq:cross_relation_alternatives) provides the basic of determining a system's FRF properties from the measurements and analysis of a random vibration test.
Using either of them, we have a simple formula for determining the FRF from estimates of the relevant spectral densities [eq:frf_estimates_spectral_densities](#eq:frf_estimates_spectral_densities).
The pair of equations \eqref{eq:cross_relation_alternatives} provides the basic of determining a system's FRF properties from the measurements and analysis of a random vibration test.
Using either of them, we have a simple formula for determining the FRF from estimates of the relevant spectral densities \eqref{eq:frf_estimates_spectral_densities}.
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@@ -1547,11 +1547,11 @@ Then in [fig:frf_feedback_model](#fig:frf_feedback_model) is given a more detail
| ![](/ox-hugo/ewins00_frf_siso_model.png) | ![](/ox-hugo/ewins00_frf_feedback_model.png) |
|------------------------------------------|--------------------------------------------------|
| <a id="org5183bee"></a> Basic SISO model | <a id="org7eda16f"></a> SISO model with feedback |
| <a id="orgcf49de0"></a> Basic SISO model | <a id="orgad8dce0"></a> SISO model with feedback |
| width=\linewidth | width=\linewidth |
In this configuration, it can be seen that there are two feedback mechanisms which apply.
We then introduce an alternative formula which is available for the determination of the system FRF from measurements of the input and output quantities [eq:H3](#eq:H3).
We then introduce an alternative formula which is available for the determination of the system FRF from measurements of the input and output quantities \eqref{eq:H3}.
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@@ -1580,7 +1580,7 @@ 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.
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{{< figure src="/ox-hugo/ewins00_frf_mimo.png" caption="Figure 8: System for FRF determination via MIMO model" >}}
@@ -1694,7 +1694,7 @@ First, if we have a **modal incompleteness** (\\(m<N\\) modes included), then we
However, if we have **spatial incompleteness** (only \\(n<N\\) DOFs included), then we cannot express any orthogonality properties at all because the eigenvector matrix is not commutable with the system mass and stiffness matrices.
In both reduced-model cases, it is not possible to use equation [eq:spatial_model_from_modal](#eq:spatial_model_from_modal) to re-construct the system mass and stiffness matrices.
In both reduced-model cases, it is not possible to use equation \eqref{eq:spatial_model_from_modal} to re-construct the system mass and stiffness matrices.
First of all because the eigen matrices are generally singular and even if it is not, the obtained mass and stiffness matrices produced have no physical significance and should not be used.
@@ -1852,7 +1852,7 @@ The experimental setup used for mobility measurement contains three major items:
A typical layout for the measurement system is shown on figure [fig:general_frf_measurement_setup](#fig:general_frf_measurement_setup).
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{{< figure src="/ox-hugo/ewins00_general_frf_measurement_setup.png" caption="Figure 9: General layout of FRF measurement system" >}}
@@ -1909,7 +1909,7 @@ 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 [fig:shaker_rod](#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.
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{{< figure src="/ox-hugo/ewins00_shaker_rod.png" caption="Figure 10: Exciter attachment and drive rod assembly" >}}
@@ -1930,7 +1930,7 @@ Figure [fig:shaker_mount_3](#fig:shaker_mount_3) shows an unsatisfactory setup.
| ![](/ox-hugo/ewins00_shaker_mount_1.png) | ![](/ox-hugo/ewins00_shaker_mount_2.png) | ![](/ox-hugo/ewins00_shaker_mount_3.png) |
|---------------------------------------------|-------------------------------------------------|------------------------------------------|
| <a id="orge0af414"></a> Ideal Configuration | <a id="org21ea9d5"></a> Suspended Configuration | <a id="org272e483"></a> Unsatisfactory |
| <a id="orga9157bf"></a> Ideal Configuration | <a id="org4b90d28"></a> Suspended Configuration | <a id="org3061b55"></a> Unsatisfactory |
| width=\linewidth | width=\linewidth | width=\linewidth |
@@ -1948,7 +1948,7 @@ The frequency range which is effectively excited is controlled by the stiffness
When the hammer tip impacts the test structure, this will experience a force pulse as shown on figure [fig:hammer_impulse](#fig:hammer_impulse).
A pulse of this type (half-sine shape) has a frequency content of the form illustrated on figure [fig:hammer_impulse](#fig:hammer_impulse).
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{{< figure src="/ox-hugo/ewins00_hammer_impulse.png" caption="Figure 11: Typical impact force pulse and spectrum" >}}
@@ -1979,7 +1979,7 @@ By suitable design, such a material may be incorporated into a device which **in
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 [fig:piezo_force_transducer](#fig:piezo_force_transducer)).
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{{< figure src="/ox-hugo/ewins00_piezo_force_transducer.png" caption="Figure 12: Force transducer" >}}
@@ -1992,7 +1992,7 @@ In an accelerometer, transduction is indirect and is achieved using a seismic ma
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="org29aa78f"></a>
<a id="org84766b5"></a>
{{< figure src="/ox-hugo/ewins00_piezo_accelerometer.png" caption="Figure 13: Compression-type of piezoelectric accelerometer" >}}
@@ -2040,7 +2040,7 @@ Shown on figure [fig:transducer_mounting_response](#fig:transducer_mounting_resp
| ![](/ox-hugo/ewins00_transducer_mounting_types.png) | ![](/ox-hugo/ewins00_transducer_mounting_response.png) |
|-----------------------------------------------------|------------------------------------------------------------|
| <a id="org1a66b31"></a> Attachment methods | <a id="org4e23863"></a> Frequency response characteristics |
| <a id="org796e903"></a> Attachment methods | <a id="org308a233"></a> Frequency response characteristics |
| width=\linewidth | width=\linewidth |
@@ -2127,7 +2127,7 @@ Aliasing originates from the discretisation of the originally continuous time hi
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 [fig:aliasing](#fig:aliasing).
<a id="orgb59b07e"></a>
<a id="orge489af5"></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" >}}
@@ -2144,7 +2144,7 @@ This is illustrated on figure [fig:effect_aliasing](#fig:effect_aliasing).
| ![](/ox-hugo/ewins00_aliasing_no_distortion.png) | ![](/ox-hugo/ewins00_aliasing_distortion.png) |
|--------------------------------------------------|-----------------------------------------------------|
| <a id="org7e6ecc4"></a> True spectrum of signal | <a id="org3c375b8"></a> Indicated spectrum from DFT |
| <a id="org3c7851f"></a> True spectrum of signal | <a id="orgd31d06c"></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.
@@ -2165,7 +2165,7 @@ 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="org3c7efb4"></a> Ideal signal | <a id="orga61ea56"></a> Awkward signal |
| <a id="org62f211a"></a> Ideal signal | <a id="orgd4e0fe1"></a> Awkward signal |
| width=\linewidth | width=\linewidth |
The problem is illustrated on figure [fig:leakage](#fig:leakage).
@@ -2190,7 +2190,7 @@ Windowing involves the imposition of a prescribed profile on the time signal pri
The profiles, or "windows" are generally depicted as a time function \\(w(t)\\) as shown in figure [fig:windowing_examples](#fig:windowing_examples).
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<a id="orge28ad03"></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" >}}
@@ -2211,7 +2211,7 @@ Common filters are: low-pass, high-pass, band-limited, narrow-band, notch.
#### Improving Resolution {#improving-resolution}
<a id="orgf44c4ab"></a>
<a id="org81b4f25"></a>
##### Increasing transform size {#increasing-transform-size}
@@ -2247,10 +2247,10 @@ If we apply a band-pass filter to the signal, as shown on figure [fig:zoom_bandp
| ![](/ox-hugo/ewins00_zoom_range.png) | ![](/ox-hugo/ewins00_zoom_bandpass.png) |
|------------------------------------------------|------------------------------------------|
| <a id="orgafd809b"></a> Spectrum of the signal | <a id="org7d44157"></a> Band-pass filter |
| <a id="org28ed6ec"></a> Spectrum of the signal | <a id="org8a7e75c"></a> Band-pass filter |
| width=\linewidth | width=\linewidth |
<a id="org8e7e54a"></a>
<a id="org60b3e9b"></a>
{{< figure src="/ox-hugo/ewins00_zoom_result.png" caption="Figure 16: Effective frequency translation for zoom" >}}
@@ -2322,7 +2322,7 @@ This is the traditional method of FRF measurement and involves the use of a swee
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 [fig:sweep_distortions](#fig:sweep_distortions).
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<a id="orgeab1f57"></a>
{{< figure src="/ox-hugo/ewins00_sweep_distortions.png" caption="Figure 17: FRF measurements by sine sweep test" >}}
@@ -2440,7 +2440,7 @@ It is known that a low coherence can arise in a measurement where the frequency
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 [fig:coherence_resonance](#fig:coherence_resonance).
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<a id="orgb72faa8"></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" >}}
@@ -2483,7 +2483,7 @@ For the chirp and impulse excitations, each individual sample is collected and p
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 [fig:burst_excitation](#fig:burst_excitation)).
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<a id="org681a980"></a>
{{< figure src="/ox-hugo/ewins00_burst_excitation.png" caption="Figure 19: Example of burst excitation and response signals" >}}
@@ -2502,7 +2502,7 @@ The chirp consist of a short duration signal which has the form shown in figure
The frequency content of the chirp can be precisely chosen by the starting and finishing frequencies of the sweep.
<a id="org56036c2"></a>
<a id="org632f8cc"></a>
{{< figure src="/ox-hugo/ewins00_chirp_excitation.png" caption="Figure 20: Example of chirp excitation and response signals" >}}
@@ -2513,7 +2513,7 @@ The hammer blow produces an input and response as shown in the figure [fig:impul
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="orga792905"></a>
<a id="orgdecf769"></a>
{{< figure src="/ox-hugo/ewins00_impulsive_excitation.png" caption="Figure 21: Example of impulsive excitation and response signals" >}}
@@ -2523,7 +2523,7 @@ However, it should be recorded that in the region below the first cut-off freque
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 [fig:double_hits](#fig:double_hits)).
<a id="org43e765b"></a>
<a id="orgea279f8"></a>
{{< figure src="/ox-hugo/ewins00_double_hits.png" caption="Figure 22: Double hits time domain and frequency content" >}}
@@ -2598,7 +2598,7 @@ Suppose the response parameter is acceleration, then the FRF obtained is inertan
Figure [fig:calibration_setup](#fig:calibration_setup) shows a typical calibration setup.
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<a id="org3a6c052"></a>
{{< figure src="/ox-hugo/ewins00_calibration_setup.png" caption="Figure 23: Mass calibration procedure, measurement setup" >}}
@@ -2613,7 +2613,7 @@ This is because near resonance, the actual applied force becomes very small and
This same argument applies on a lesser scale as we examine the detail around the attachment to the structure, as shown in figure [fig:mass_cancellation](#fig:mass_cancellation).
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<a id="orgf6011aa"></a>
{{< figure src="/ox-hugo/ewins00_mass_cancellation.png" caption="Figure 24: Added mass to be cancelled (crossed area)" >}}
@@ -2670,7 +2670,7 @@ There are two problems to be tackled:
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 [fig:rotational_measurement](#fig:rotational_measurement).
<a id="org92fa8a1"></a>
<a id="org6c1a993"></a>
{{< figure src="/ox-hugo/ewins00_rotational_measurement.png" caption="Figure 25: Measurement of rotational response" >}}
@@ -2691,7 +2691,7 @@ First, a single applied excitation force \\(F\_1\\) corresponds to a simultaneou
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\\).
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<a id="org19d9418"></a>
{{< figure src="/ox-hugo/ewins00_rotational_excitation.png" caption="Figure 26: Application of moment excitation" >}}
@@ -2700,7 +2700,7 @@ Then, the full \\(6 \times 6\\) mobility matrix can be measured, however this pr
Other methods for measuring rotational effects include specially developed rotational accelerometers and shakers.
However, there is a major problem that is encountered when measuring rotational FRF: the translational components of the structure's movement tends to overshadow those due to the rotational motions.
For example, the magnitude of the difference in equation [eq:rotational_diff](#eq:rotational_diff) is often of the order of \\(\SI{1}{\%}\\) of the two individual values which is similar to the transverse sensitivity of the accelerometers: potential errors in rotations are thus enormous.
For example, the magnitude of the difference in equation \eqref{eq:rotational_diff} is often of the order of \\(\SI{1}{\%}\\) of the two individual values which is similar to the transverse sensitivity of the accelerometers: potential errors in rotations are thus enormous.
### Multi-point excitation methods {#multi-point-excitation-methods}
@@ -2739,7 +2739,7 @@ The two vectors are related by the system's FRF properties as:
\\{X\\}\_{n\times 1} = [H(\omega)]\_{n\times p} \\{F\\}\_{p\times 1}
\end{equation}
However, it is not possible to derive the FRF matrix from the single equation [eq:mpss_equation](#eq:mpss_equation), because there will be insufficient data in the two vectors (one of length \\(p\\), the other of length \\(n\\)) to define completely the \\(n\times p\\) FRF matrix.
However, it is not possible to derive the FRF matrix from the single equation \eqref{eq:mpss_equation}, because there will be insufficient data in the two vectors (one of length \\(p\\), the other of length \\(n\\)) to define completely the \\(n\times p\\) FRF matrix.
What is required is to make a series of \\(p^\prime\\) measurements of the same basic type using different excitation vectors \\(\\{F\\}\_i\\) that should be chosen such that the forcing matrix \\([F]\_{p\times p^\prime} = [\\{F\\}\_1, \dots, \\{F\\}\_p]\\) is non-singular.
This can be assured if:
@@ -3031,7 +3031,7 @@ 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="org3c82345"></a> FRF | <a id="orga931e29"></a> Singular Values | <a id="orgf4a6ae7"></a> PRF |
| <a id="org911bfc8"></a> FRF | <a id="org60f84fb"></a> Singular Values | <a id="orgdf8522b"></a> PRF |
| width=\linewidth | width=\linewidth | width=\linewidth |
<a id="table--fig:PRF-measured"></a>
@@ -3042,7 +3042,7 @@ 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="orge590e19"></a> FRF | <a id="org176a55f"></a> Singular Values | <a id="orge859b81"></a> PRF |
| <a id="org3d1c696"></a> FRF | <a id="orgeb81dac"></a> Singular Values | <a id="orgc25aeb3"></a> PRF |
| width=\linewidth | width=\linewidth | width=\linewidth |
@@ -3084,7 +3084,7 @@ 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**
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{{< figure src="/ox-hugo/ewins00_mifs.png" caption="Figure 27: Complex Mode Indicator Function (CMIF)" >}}
@@ -3179,7 +3179,7 @@ The peak-picking method is applied as follows (illustrated on figure [fig:peak_a
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="org12489c1"></a>
<a id="org0d4b46a"></a>
{{< figure src="/ox-hugo/ewins00_peak_amplitude.png" caption="Figure 28: Peak Amplitude method of modal analysis" >}}
@@ -3214,7 +3214,7 @@ A plot of the quantity \\(\alpha(\omega)\\) is given in figure [fig:modal_circle
| ![](/ox-hugo/ewins00_modal_circle.png) | ![](/ox-hugo/ewins00_modal_circle_bis.png) |
|----------------------------------------|--------------------------------------------------------------------|
| <a id="orge6f933c"></a> Properties | <a id="org852e910"></a> \\(\omega\_b\\) and \\(\omega\_a\\) points |
| <a id="org187efdc"></a> Properties | <a id="org0e24b72"></a> \\(\omega\_b\\) and \\(\omega\_a\\) points |
| width=\linewidth | width=\linewidth |
For any frequency \\(\omega\\), we have the following relationship:
@@ -3226,13 +3226,13 @@ For any frequency \\(\omega\\), we have the following relationship:
\end{align}
\end{subequations}
From [eq:modal_circle_tan](#eq:modal_circle_tan), we obtain:
From \eqref{eq:modal_circle_tan}, we obtain:
\begin{equation}
\omega^2 = \omega\_r^2 \left(1 - \eta\_r \tan\left(\frac{\theta}{2}\right) \right)
\end{equation}
If we differentiate [eq:modal_circle_omega](#eq:modal_circle_omega) with respect to \\(\theta\\), we obtain:
If we differentiate \eqref{eq:modal_circle_omega} with respect to \\(\theta\\), we obtain:
\begin{equation}
\frac{d\omega^2}{d\theta} = \frac{-\omega\_r^2 \eta\_r}{2} \frac{\left(1 - (\omega/\omega\_r)^2\right)^2}{\eta\_r^2}
@@ -3317,10 +3317,10 @@ The sequence is:
Then we obtain the **center** and **radius** of the circle and the **quality factor** is the mean square deviation of the chosen points from the circle.
3. **Locate natural frequency, obtain damping estimate**.
The rate of sweep through the region is estimated numerically and the frequency at which it reaches the maximum is deduced.
At the same time, an estimate of the damping is derived using [eq:estimate_damping_sweep_rate](#eq:estimate_damping_sweep_rate).
At the same time, an estimate of the damping is derived using \eqref{eq:estimate_damping_sweep_rate}.
A typical example is shown on figure [fig:circle_fit_natural_frequency](#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](#eq:estimate_damping).
A set of damping estimates using all possible combination of the selected data points are computed using \eqref{eq:estimate_damping}.
Then, we can choose the damping estimate to be the mean value.
We also look at the distribution of the obtained damping estimates as is permits a useful diagnostic of the quality of the entire analysis:
- Good measured data should lead to a smooth plot of these damping estimates, any roughness of the surface can be explained in terms of noise in the original data.
@@ -3328,7 +3328,7 @@ 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.
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<a id="org379e1a2"></a>
{{< figure src="/ox-hugo/ewins00_circle_fit_natural_frequency.png" caption="Figure 29: Location of natural frequency for a Circle-fit modal analysis" >}}
@@ -3427,7 +3427,7 @@ we now have sufficient information to extract estimates for the four parameters
<ol class="org-ol">
<li value="3">Plot graphs of \\(m\_R(\Omega)\\) vs \\(\Omega^2\\) and of \\(m\_I(\Omega)\\) vs \\(\Omega^2\\) using the results from step 1., each time using a different measurement points as the fixing frequency \\(\Omega\_j\\)</li>
<li>Determine the slopes of the best fit straight lines through these two plots, \\(n\_R\\) and \\(n\_I\\), and their intercepts with the vertical axis \\(d\_R\\) and \\(d\_I\\)</li>
<li>Use these four quantities, and equation [eq:modal_parameters_formula](#eq:modal_parameters_formula), to determine the **four modal parameters** required for that mode</li>
<li>Use these four quantities, and equation \eqref{eq:modal_parameters_formula}, to determine the **four modal parameters** required for that mode</li>
</ol>
This procedure which places more weight to points slightly away from the resonance region is likely to be less sensitive to measurement difficulties of measuring the resonance region.
@@ -3453,7 +3453,7 @@ 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="org4fd3d88"></a> without residual | <a id="orgdb69a63"></a> with residuals |
| <a id="orge96a388"></a> without residual | <a id="org92a8b32"></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
@@ -3484,7 +3484,7 @@ The three terms corresponds to:
These three terms are illustrated on figure [fig:low_medium_high_modes](#fig:low_medium_high_modes).
<a id="org85845f3"></a>
<a id="org745f0a4"></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" >}}
@@ -3493,7 +3493,7 @@ From the sketch, it may be seen that within the frequency range of interest:
- the first term tends to approximate to a **mass-like behavior**
- the third term approximates to a **stiffness effect**
Thus, we have a basis for the residual terms and shall rewrite equation [eq:sum_modes](#eq:sum_modes):
Thus, we have a basis for the residual terms and shall rewrite equation \eqref{eq:sum_modes}:
\begin{equation}
H\_{jk}(\omega) \simeq -\frac{1}{\omega^2 M\_{jk}^R} + \sum\_{r=m\_1}^{m\_2} \left( \frac{{}\_rA\_{jk}}{\omega\_r^2 - \omega^2 + i \eta\_r \omega\_r^2} \right) + \frac{1}{K\_{jk}^R}
@@ -3554,7 +3554,7 @@ We can write the receptance in the frequency range of interest as:
In the previous methods, the second term was assumed to be a constant in the curve-fit procedure for mode \\(r\\).
However, if we have good **estimates** for the coefficients which constitutes the second term, for example by having already completed an SDOF analysis, we may remove the restriction on the analysis.
Indeed, suppose we take a set of measured data points around the resonance at \\(\omega\_r\\), and that we can compute the magnitude of the second term in [eq:second_term_refinement](#eq:second_term_refinement), we then subtract this from the measurement and we obtain adjusted data points that are conform to a true SDOF behavior and we can use the same technique as before to obtain **improved estimated** to the modal parameters of more \\(r\\).
Indeed, suppose we take a set of measured data points around the resonance at \\(\omega\_r\\), and that we can compute the magnitude of the second term in \eqref{eq:second_term_refinement}, we then subtract this from the measurement and we obtain adjusted data points that are conform to a true SDOF behavior and we can use the same technique as before to obtain **improved estimated** to the modal parameters of more \\(r\\).
This procedure can be repeated iteratively for all the modes in the range of interest and it can significantly enhance the quality of found modal parameters for system with **strong coupling**.
@@ -3614,7 +3614,7 @@ If we further increase the generality by attaching a **weighting factor** \\(w\_
is minimized.
This is achieved by differentiating [eq:error_weighted](#eq:error_weighted) with respect to each unknown in turn, thus generating a set of as many equations as there are unknown:
This is achieved by differentiating \eqref{eq:error_weighted} with respect to each unknown in turn, thus generating a set of as many equations as there are unknown:
\begin{equation}
\frac{d E}{d q} = 0; \quad q = {}\_1A\_{jk}, {}\_2A\_{jk}, \dots
@@ -3663,7 +3663,7 @@ leading to the modified, but more convenient version actually used in the analys
\end{equation}
In these expressions, only \\(m\\) modes are included in the theoretical FRF formula: the true number of modes, \\(N\\), is actually one of the **unknowns** to be determined during the analysis.
Equation [eq:rpf_error](#eq:rpf_error) can be rewritten as follows:
Equation \eqref{eq:rpf_error} can be rewritten as follows:
\begin{equation}
\begin{aligned}
@@ -3717,7 +3717,7 @@ where \\([X], [Y], [Z], \\{G\\}\\) and \\(\\{F\\}\\) are known measured quantiti
\end{equation}
Once the solution has been obtained for the coefficients \\(a\_k, \dots , b\_k, \dots\\) then the second stage of the modal analysis can be performed in which the required **modal parameters are derived**.
This is usually done by solving the two polynomial expressions which form the numerator and denominator of equations [eq:frf_clasic](#eq:frf_clasic) and [eq:frf_rational](#eq:frf_rational):
This is usually done by solving the two polynomial expressions which form the numerator and denominator of equations \eqref{eq:frf_clasic} and \eqref{eq:frf_rational}:
- the denominator is used to obtain the natural frequencies \\(\omega\_r\\) and damping factors \\(\xi\_r\\)
- the numerator is used to determine the complex modal constants \\(A\_r\\)
@@ -3785,7 +3785,7 @@ As an example, a set of mobilities measured are shown individually in figure [fi
| ![](/ox-hugo/ewins00_composite_raw.png) | ![](/ox-hugo/ewins00_composite_sum.png) |
|-------------------------------------------|-----------------------------------------|
| <a id="org9ce04ed"></a> Individual curves | <a id="orgf0fa0fe"></a> Composite curve |
| <a id="org3f9a0d6"></a> Individual curves | <a id="org9ebc973"></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.
@@ -3949,7 +3949,7 @@ First, we note that from a single FRF curve, \\(H\_{jk}(\omega)\\), it is possib
Now, although this gives us the natural frequency and damping properties directly, it does not explicitly yield the mode shape: only a modal constant \\({}\_rA\_{jk}\\) which is formed from the mode shape data.
In order to extract the individual elements \\(\phi\_{jr}\\) of the mode shape matrix \\([\Phi]\\), it is necessary to make a series of measurements of specific FRFs including, especially, the point FRF at the excitation position.
If we measure \\(H\_{kk}\\), then by using [eq:modal_model_from_frf](#eq:modal_model_from_frf), we also obtain the specific elements in the mode shape matrix corresponding to the excitation point:
If we measure \\(H\_{kk}\\), then by using \eqref{eq:modal_model_from_frf}, we also obtain the specific elements in the mode shape matrix corresponding to the excitation point:
\begin{equation}
H\_{kk}(\omega) \longrightarrow \omega\_r, \eta\_r, {}\_rA\_{jk} \longrightarrow \phi\_{kr}; \quad r=1, m
@@ -4332,7 +4332,7 @@ Measured coordinates of the test structure are first linked as shown on figure [
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 [fig:static_display](#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="orgbbe2809"></a>
<a id="orge0d2fb3"></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" >}}
@@ -4377,7 +4377,7 @@ If we consider the first six modes of the beam, whose mode shapes are sketched i
All the higher modes will be indistinguishable from these first few.
This is a well known problem of **spatial aliasing**.
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<a id="org5c16ec7"></a>
{{< figure src="/ox-hugo/ewins00_beam_modes.png" caption="Figure 32: Misinterpretation of mode shapes by spatial aliasing" >}}
@@ -4415,7 +4415,7 @@ In this respect, the demands of the response model are more stringent that those
#### Synthesis of FRF curves {#synthesis-of-frf-curves}
One of the implications of equation [eq:regenerate_full_frf_matrix](#eq:regenerate_full_frf_matrix) is that **it is possible to synthesize the FRF curves which were not measured**.
One of the implications of equation \eqref{eq:regenerate_full_frf_matrix} is that **it is possible to synthesize the FRF curves which were not measured**.
This arises because if we measured three individual FRF such as \\(H\_{ik}(\omega)\\), \\(H\_{jk}(\omega)\\) and \\(K\_{kk}(\omega)\\), then modal analysis of these yields the modal parameters from which it is possible to generate the FRF \\(H\_{ij}(\omega)\\), \\(H\_{jj}(\omega)\\), etc.
However, it must be noted that there is an important **limitation to this procedure** which is highlighted in the example below.
@@ -4440,7 +4440,7 @@ 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="org0111dfe"></a> Using measured modal data only | <a id="org2abbdff"></a> After inclusion of residual terms |
| <a id="orgee3fc43"></a> Using measured modal data only | <a id="org959e2d5"></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
@@ -4495,7 +4495,7 @@ If the transmissibility is measured during a modal test which has a single excit
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 [fig:transmissibility_plots](#fig:transmissibility_plots).
This may explain why transmissibilities are not widely used in modal analysis.
<a id="org6f53493"></a>
<a id="orgb69dd65"></a>
{{< figure src="/ox-hugo/ewins00_transmissibility_plots.png" caption="Figure 33: Transmissibility plots" >}}
@@ -4516,7 +4516,7 @@ 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="orgfcd5181"></a> Conventional modal test setup | <a id="org544b696"></a> Base excitation setup |
| <a id="orgfb8d62b"></a> Conventional modal test setup | <a id="orgb803ff7"></a> Base excitation setup |
| height=4cm | height=4cm |
@@ -4541,7 +4541,7 @@ from which is would appear that we can write
\end{aligned}
\end{equation}
However, equation [eq:m_k_from_modes](#eq:m_k_from_modes) is **only applicable when we have available the complete \\(N \times N\\) modal model**.
However, equation \eqref{eq:m_k_from_modes} is **only applicable when we have available the complete \\(N \times N\\) modal model**.
It is much more usual to have an incomplete model in which the eigenvector matrix is rectangle and, as such, is non-invertible.
One step which can be made using the incomplete data is the construction of "pseudo" flexibility and inverse-mass matrices.