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@@ -5,10 +5,10 @@ draft = false
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: [System Identification]({{< relref "system_identification" >}}), [Reference Books]({{< relref "reference_books" >}})
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: [System Identification]({{< relref "system_identification" >}}), [Reference Books]({{< relref "reference_books" >}}), [Modal Analysis]({{< relref "modal_analysis" >}})
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Reference
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: ([Ewins 2000](#org84d73f8))
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: ([Ewins 2000](#org57f8bf9))
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Author(s)
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: Ewins, D.
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@@ -141,7 +141,7 @@ The main measurement technique studied are those which will permit to make **dir
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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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<a id="orga193754"></a>
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<a id="org0b82329"></a>
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{{< figure src="/ox-hugo/ewins00_modal_analysis_schematic.png" caption="Figure 1: Basic components of FRF measurement system" >}}
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@@ -215,7 +215,7 @@ This assumption allows us to use the circular nature of a modulus/phase polar pl
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This process can be **repeated** for each resonance individually until the whole curve has been analyzed.
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At this stage, a theoretical regeneration of the FRF is possible using the set of coefficients extracted.
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<a id="org37e66c2"></a>
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<a id="org8ff4e51"></a>
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{{< figure src="/ox-hugo/ewins00_sdof_modulus_phase.png" caption="Figure 2: Curve fit to resonant FRF data" >}}
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@@ -253,7 +253,7 @@ Theoretical foundations of modal testing are of paramount importance to its succ
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The three phases through a typical theoretical vibration analysis progresses are shown on figure [fig:vibration_analysis_procedure](#fig:vibration_analysis_procedure).
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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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<a id="org00d3f58"></a>
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<a id="org4cdbdfc"></a>
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{{< figure src="/ox-hugo/ewins00_vibration_analysis_procedure.png" caption="Figure 3: Theoretical route to vibration analysis" >}}
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@@ -298,7 +298,7 @@ Three classes of system model will be described:
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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.
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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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<a id="org470c5bf"></a>
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<a id="orga199d06"></a>
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{{< figure src="/ox-hugo/ewins00_sdof_model.png" caption="Figure 4: Single degree-of-freedom system" >}}
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@@ -374,7 +374,7 @@ which is a single mode of vibration with a complex natural frequency having two
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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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<a id="org169b90c"></a>
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<a id="org8c327c7"></a>
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{{< figure src="/ox-hugo/ewins00_sdof_response.png" caption="Figure 5: Oscillatory and decay part" >}}
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@@ -418,7 +418,7 @@ The damping effect of such a component can conveniently be defined by the ratio
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|-----------------------------------------------|----------------------------------------|------------------------------------------|
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| <a id="orgb3a7b8e"></a> Material hysteresis | <a id="org68fe7c2"></a> Dry friction | <a id="org03c75ad"></a> Viscous damper |
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| <a id="org30686c0"></a> Material hysteresis | <a id="org151c775"></a> Dry friction | <a id="org45c6c45"></a> Viscous damper |
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| height=2cm | height=2cm | height=2cm |
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Another common source of energy dissipation in practical structures, is the **friction** which exist in joints between components of the structure.
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@@ -537,7 +537,7 @@ Bode plot are usually displayed using logarithmic scales as shown on figure [fig
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|  |  |  |
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|-------------------------------------------|-----------------------------------------|--------------------------------------------|
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| <a id="org4673396"></a> Receptance FRF | <a id="org9f41af5"></a> Mobility FRF | <a id="org6696bcf"></a> Accelerance FRF |
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| <a id="org17728b2"></a> Receptance FRF | <a id="org90cee96"></a> Mobility FRF | <a id="orge43a020"></a> Accelerance FRF |
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| width=\linewidth | width=\linewidth | width=\linewidth |
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Each plot can be divided into three regimes:
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@@ -560,7 +560,7 @@ This type of display is not widely used as we cannot use logarithmic axes (as we
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|------------------------------------------------|------------------------------------------------|
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| <a id="org66926ef"></a> Real part | <a id="orgaf2afdd"></a> Imaginary part |
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| <a id="org3aaddc5"></a> Real part | <a id="orgfd7fd7d"></a> Imaginary part |
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| width=\linewidth | width=\linewidth |
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@@ -578,7 +578,7 @@ Figure [fig:inverse_frf_mixed](#fig:inverse_frf_mixed) shows an example of a plo
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|---------------------------------------------|-----------------------------------------------|
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| <a id="org84ad953"></a> Mixed | <a id="orgc18e658"></a> Viscous |
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| <a id="orgeb3ce3c"></a> Mixed | <a id="org8418622"></a> Viscous |
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| width=\linewidth | width=\linewidth |
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@@ -595,7 +595,7 @@ The missing information (in this case, the frequency) must be added by identifyi
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|------------------------------------------------------|---------------------------------------------------------|
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| <a id="orgfee48c0"></a> Viscous damping | <a id="org41c7d29"></a> Structural damping |
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| <a id="orgef2e4cd"></a> Viscous damping | <a id="orgd4187e9"></a> Structural damping |
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| width=\linewidth | width=\linewidth |
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The Nyquist plot has the particularity of distorting the plot so as to focus on the resonance area.
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@@ -1103,7 +1103,7 @@ Equally, in a real mode, all parts of the structure pass through their **zero de
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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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<a id="org05c0f39"></a>
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<a id="org0991bf9"></a>
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{{< figure src="/ox-hugo/ewins00_real_complex_modes.png" caption="Figure 6: Real and complex mode shapes displays" >}}
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@@ -1118,7 +1118,7 @@ Note that the almost-real mode shape does not necessarily have vector elements w
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|--------------------------------------------|--------------------------------------------|-----------------------------------------------|
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| <a id="orgc7a8526"></a> Almost-real mode | <a id="orgcd8be0a"></a> Complex Mode | <a id="orgf34a135"></a> Measure of complexity |
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| <a id="org193debd"></a> Almost-real mode | <a id="org69bb630"></a> Complex Mode | <a id="org3bb718c"></a> Measure of complexity |
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| width=\linewidth | width=\linewidth | width=\linewidth |
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@@ -1235,7 +1235,7 @@ On a logarithmic plot, this produces the antiresonance characteristic which refl
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|---------------------------------------------------|------------------------------------------------------|
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| <a id="org04908dc"></a> Point FRF | <a id="orgc9e36d0"></a> Transfer FRF |
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| <a id="org06f37e2"></a> Point FRF | <a id="orgdc266be"></a> Transfer FRF |
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| width=\linewidth | width=\linewidth |
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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.
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@@ -1260,7 +1260,7 @@ Most mobility plots have this general form as long as the modes are relatively w
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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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<a id="org3342d4f"></a>
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<a id="orgd2ab4ee"></a>
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{{< figure src="/ox-hugo/ewins00_frf_damped_system.png" caption="Figure 7: Mobility plot of a damped system" >}}
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@@ -1281,7 +1281,7 @@ The plot for the transfer receptance \\(\alpha\_{21}\\) is presented in figure [
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|------------------------------------------|---------------------------------------------|
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| <a id="org51d6859"></a> Point receptance | <a id="org49ad44a"></a> Transfer receptance |
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| <a id="org2814a00"></a> Point receptance | <a id="org000b88d"></a> Transfer receptance |
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| width=\linewidth | width=\linewidth |
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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.
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@@ -1296,7 +1296,7 @@ Now we find that the individual modal circles are no longer "upright" but are **
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|-----------------------------------------------------|--------------------------------------------------------|
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| <a id="orgbc84787"></a> Point receptance | <a id="org1fde70c"></a> Transfer receptance |
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| <a id="orge62d92d"></a> Point receptance | <a id="orgaaeb314"></a> Transfer receptance |
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| width=\linewidth | width=\linewidth |
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@@ -1450,7 +1450,7 @@ Examples of random signals, autocorrelation function and power spectral density
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|---------------------------------------|--------------------------------------------------|------------------------------------------------|
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| <a id="org9b223d2"></a> Time history | <a id="orgf89ee65"></a> Autocorrelation Function | <a id="org839a4fd"></a> Power Spectral Density |
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| <a id="org73b51a5"></a> Time history | <a id="org29b2840"></a> Autocorrelation Function | <a id="orgb8db3ea"></a> Power Spectral Density |
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| width=\linewidth | width=\linewidth | width=\linewidth |
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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.
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@@ -1547,7 +1547,7 @@ Then in [fig:frf_feedback_model](#fig:frf_feedback_model) is given a more detail
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|------------------------------------------|--------------------------------------------------|
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| <a id="orgf9a7bf7"></a> Basic SISO model | <a id="org258a6e2"></a> SISO model with feedback |
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| <a id="orgf32d3c7"></a> Basic SISO model | <a id="org56469a4"></a> SISO model with feedback |
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| width=\linewidth | width=\linewidth |
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In this configuration, it can be seen that there are two feedback mechanisms which apply.
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@@ -1580,7 +1580,7 @@ We obtain two alternative formulas:
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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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<a id="org00c19fd"></a>
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<a id="orgf429f5f"></a>
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{{< figure src="/ox-hugo/ewins00_frf_mimo.png" caption="Figure 8: System for FRF determination via MIMO model" >}}
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@@ -1852,7 +1852,7 @@ The experimental setup used for mobility measurement contains three major items:
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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" >}}
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@@ -1909,7 +1909,7 @@ This can modify the response of the system in those directions.
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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).
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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" >}}
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@@ -1930,7 +1930,7 @@ Figure [fig:shaker_mount_3](#fig:shaker_mount_3) shows an unsatisfactory setup.
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|---------------------------------------------|-------------------------------------------------|------------------------------------------|
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| <a id="org5ad1e59"></a> Ideal Configuration | <a id="orge10385d"></a> Suspended Configuration | <a id="orgf027a3a"></a> Unsatisfactory |
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| <a id="orgda674d1"></a> Ideal Configuration | <a id="orgc2ebc7e"></a> Suspended Configuration | <a id="org91922e6"></a> Unsatisfactory |
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| width=\linewidth | width=\linewidth | width=\linewidth |
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@@ -1948,7 +1948,7 @@ The frequency range which is effectively excited is controlled by the stiffness
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When the hammer tip impacts the test structure, this will experience a force pulse as shown on figure [fig:hammer_impulse](#fig:hammer_impulse).
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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" >}}
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@@ -1979,7 +1979,7 @@ By suitable design, such a material may be incorporated into a device which **in
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The force transducer is the simplest type of piezoelectric transducer.
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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" >}}
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@@ -1992,7 +1992,7 @@ In an accelerometer, transduction is indirect and is achieved using a seismic ma
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In this configuration, the force exerted on the crystals is the inertia force of the seismic mass (\\(m\ddot{z}\\)).
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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\\).
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{{< figure src="/ox-hugo/ewins00_piezo_accelerometer.png" caption="Figure 13: Compression-type of piezoelectric accelerometer" >}}
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@@ -2040,7 +2040,7 @@ Shown on figure [fig:transducer_mounting_response](#fig:transducer_mounting_resp
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|-----------------------------------------------------|------------------------------------------------------------|
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| <a id="org7c446c6"></a> Attachment methods | <a id="org9920b7a"></a> Frequency response characteristics |
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| <a id="org4f8e7e0"></a> Attachment methods | <a id="orga55d5b4"></a> Frequency response characteristics |
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| width=\linewidth | width=\linewidth |
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@@ -2127,7 +2127,7 @@ Aliasing originates from the discretisation of the originally continuous time hi
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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**.
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These high frequencies will be **indistinguishable** from genuine low frequency components as shown on figure [fig:aliasing](#fig:aliasing).
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{{< 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" >}}
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@@ -2144,7 +2144,7 @@ This is illustrated on figure [fig:effect_aliasing](#fig:effect_aliasing).
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|--------------------------------------------------|-----------------------------------------------------|
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| <a id="org6412686"></a> True spectrum of signal | <a id="orgd099bc4"></a> Indicated spectrum from DFT |
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| <a id="orga51438f"></a> True spectrum of signal | <a id="org2e9d2ef"></a> Indicated spectrum from DFT |
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| width=\linewidth | width=\linewidth |
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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.
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@@ -2165,7 +2165,7 @@ Leakage is a problem which is a direct **consequence of the need to take only a
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|--------------------------------------|----------------------------------------|
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| <a id="org18c664c"></a> Ideal signal | <a id="org71abe57"></a> Awkward signal |
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| <a id="org86baf6e"></a> Ideal signal | <a id="orge49931c"></a> Awkward signal |
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| width=\linewidth | width=\linewidth |
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The problem is illustrated on figure [fig:leakage](#fig:leakage).
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@@ -2190,7 +2190,7 @@ Windowing involves the imposition of a prescribed profile on the time signal pri
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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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{{< figure src="/ox-hugo/ewins00_windowing_examples.png" caption="Figure 15: Different types of window. (a) Boxcar, (b) Hanning, (c) Cosine-taper, (d) Exponential" >}}
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@@ -2211,7 +2211,7 @@ Common filters are: low-pass, high-pass, band-limited, narrow-band, notch.
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#### Improving Resolution {#improving-resolution}
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##### Increasing transform size {#increasing-transform-size}
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@@ -2247,10 +2247,10 @@ If we apply a band-pass filter to the signal, as shown on figure [fig:zoom_bandp
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|------------------------------------------------|------------------------------------------|
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| <a id="org78b0c83"></a> Spectrum of the signal | <a id="orge62379a"></a> Band-pass filter |
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| <a id="org7059865"></a> Spectrum of the signal | <a id="org833d09d"></a> Band-pass filter |
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| width=\linewidth | width=\linewidth |
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{{< figure src="/ox-hugo/ewins00_zoom_result.png" caption="Figure 16: Effective frequency translation for zoom" >}}
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@@ -2322,7 +2322,7 @@ This is the traditional method of FRF measurement and involves the use of a swee
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It is necessary to check that progress through the frequency range is sufficiently slow to check that steady-state response conditions are attained.
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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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{{< figure src="/ox-hugo/ewins00_sweep_distortions.png" caption="Figure 17: FRF measurements by sine sweep test" >}}
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@@ -2440,7 +2440,7 @@ It is known that a low coherence can arise in a measurement where the frequency
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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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{{< 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" >}}
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@@ -2483,7 +2483,7 @@ For the chirp and impulse excitations, each individual sample is collected and p
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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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{{< figure src="/ox-hugo/ewins00_burst_excitation.png" caption="Figure 19: Example of burst excitation and response signals" >}}
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@@ -2502,7 +2502,7 @@ The chirp consist of a short duration signal which has the form shown in figure
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The frequency content of the chirp can be precisely chosen by the starting and finishing frequencies of the sweep.
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<a id="org9c55941"></a>
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{{< figure src="/ox-hugo/ewins00_chirp_excitation.png" caption="Figure 20: Example of chirp excitation and response signals" >}}
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@@ -2513,7 +2513,7 @@ The hammer blow produces an input and response as shown in the figure [fig:impul
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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.
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<a id="org0ed8171"></a>
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{{< figure src="/ox-hugo/ewins00_impulsive_excitation.png" caption="Figure 21: Example of impulsive excitation and response signals" >}}
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@@ -2523,7 +2523,7 @@ However, it should be recorded that in the region below the first cut-off freque
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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.
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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)).
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<a id="org6bd77b6"></a>
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{{< figure src="/ox-hugo/ewins00_double_hits.png" caption="Figure 22: Double hits time domain and frequency content" >}}
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@@ -2598,7 +2598,7 @@ Suppose the response parameter is acceleration, then the FRF obtained is inertan
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Figure [fig:calibration_setup](#fig:calibration_setup) shows a typical calibration setup.
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<a id="org5e0d830"></a>
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{{< figure src="/ox-hugo/ewins00_calibration_setup.png" caption="Figure 23: Mass calibration procedure, measurement setup" >}}
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@@ -2613,7 +2613,7 @@ This is because near resonance, the actual applied force becomes very small and
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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="org3d2d464"></a>
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{{< figure src="/ox-hugo/ewins00_mass_cancellation.png" caption="Figure 24: Added mass to be cancelled (crossed area)" >}}
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@@ -2670,7 +2670,7 @@ There are two problems to be tackled:
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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).
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<a id="org8a3adca"></a>
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{{< figure src="/ox-hugo/ewins00_rotational_measurement.png" caption="Figure 25: Measurement of rotational response" >}}
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@@ -2691,7 +2691,7 @@ First, a single applied excitation force \\(F\_1\\) corresponds to a simultaneou
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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\\).
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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="orgd9d3238"></a>
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{{< figure src="/ox-hugo/ewins00_rotational_excitation.png" caption="Figure 26: Application of moment excitation" >}}
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@@ -3031,7 +3031,7 @@ The two groups are usually separated by a clear gap (depending of the noise pres
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|  |  |  |
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|---------------------------------------------|---------------------------------------------|---------------------------------------------|
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| <a id="org27a7bd2"></a> FRF | <a id="org0725348"></a> Singular Values | <a id="orgcc8943d"></a> PRF |
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| <a id="org729f249"></a> FRF | <a id="org5ffbfd5"></a> Singular Values | <a id="org61f8f16"></a> PRF |
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| width=\linewidth | width=\linewidth | width=\linewidth |
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<a id="table--fig:PRF-measured"></a>
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@@ -3042,7 +3042,7 @@ The two groups are usually separated by a clear gap (depending of the noise pres
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|--------------------------------------------|--------------------------------------------|--------------------------------------------|
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| <a id="orgad6d59c"></a> FRF | <a id="orged00ce0"></a> Singular Values | <a id="orga025551"></a> PRF |
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| <a id="org830b4d0"></a> FRF | <a id="orgac004ea"></a> Singular Values | <a id="org4f86091"></a> PRF |
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| width=\linewidth | width=\linewidth | width=\linewidth |
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@@ -3084,7 +3084,7 @@ Associated with the CMIF values at each natural frequency \\(\omega\_r\\) are tw
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- the left singular vector \\(\\{U(\omega\_r)\\}\_1\\) which approximates the **mode shape** of that mode
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- 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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<a id="org80fd4e8"></a>
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{{< figure src="/ox-hugo/ewins00_mifs.png" caption="Figure 27: Complex Mode Indicator Function (CMIF)" >}}
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@@ -3179,7 +3179,7 @@ The peak-picking method is applied as follows (illustrated on figure [fig:peak_a
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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.
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Only real modal constants and thus real modes can be deduced by this method.
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<a id="org7d69374"></a>
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{{< figure src="/ox-hugo/ewins00_peak_amplitude.png" caption="Figure 28: Peak Amplitude method of modal analysis" >}}
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@@ -3214,7 +3214,7 @@ A plot of the quantity \\(\alpha(\omega)\\) is given in figure [fig:modal_circle
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|----------------------------------------|--------------------------------------------------------------------|
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| <a id="org290c571"></a> Properties | <a id="orgc059e31"></a> \\(\omega\_b\\) and \\(\omega\_a\\) points |
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| <a id="org1b60ce7"></a> Properties | <a id="orgb28d972"></a> \\(\omega\_b\\) and \\(\omega\_a\\) points |
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| width=\linewidth | width=\linewidth |
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For any frequency \\(\omega\\), we have the following relationship:
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@@ -3328,7 +3328,7 @@ The sequence is:
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5. **Determine modal constant modulus and argument**.
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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="orga4f6a8d"></a>
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{{< figure src="/ox-hugo/ewins00_circle_fit_natural_frequency.png" caption="Figure 29: Location of natural frequency for a Circle-fit modal analysis" >}}
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@@ -3453,7 +3453,7 @@ However, by the inclusion of two simple extra terms (the "**residuals**"), the m
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|--------------------------------------------|-----------------------------------------|
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| <a id="orgb0a10e7"></a> without residual | <a id="org7168563"></a> with residuals |
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| <a id="org441a50e"></a> without residual | <a id="org8c87686"></a> with residuals |
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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
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@@ -3484,7 +3484,7 @@ The three terms corresponds to:
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These three terms are illustrated on figure [fig:low_medium_high_modes](#fig:low_medium_high_modes).
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<a id="org3ba03ab"></a>
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{{< figure src="/ox-hugo/ewins00_low_medium_high_modes.png" caption="Figure 30: Numerical simulation of contribution of low, medium and high frequency modes" >}}
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@@ -3785,7 +3785,7 @@ As an example, a set of mobilities measured are shown individually in figure [fi
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|-------------------------------------------|-----------------------------------------|
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| <a id="orgf1eae63"></a> Individual curves | <a id="org156012b"></a> Composite curve |
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| <a id="org3564b55"></a> Individual curves | <a id="org7736a33"></a> Composite curve |
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| width=\linewidth | width=\linewidth |
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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.
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@@ -4332,7 +4332,7 @@ Measured coordinates of the test structure are first linked as shown on figure [
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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).
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The elements in the vector are scaled according the normalization process used (usually mass-normalized), and their absolute magnitudes have no particular significance.
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<a id="orgaffacf3"></a>
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{{< 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" >}}
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@@ -4377,7 +4377,7 @@ If we consider the first six modes of the beam, whose mode shapes are sketched i
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All the higher modes will be indistinguishable from these first few.
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This is a well known problem of **spatial aliasing**.
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<a id="org1952587"></a>
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{{< figure src="/ox-hugo/ewins00_beam_modes.png" caption="Figure 32: Misinterpretation of mode shapes by spatial aliasing" >}}
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@@ -4440,7 +4440,7 @@ The inclusion of these two additional terms (obtained here only after measuring
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|--------------------------------------------------------|-----------------------------------------------------------|
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| <a id="org7d9a13a"></a> Using measured modal data only | <a id="orgae3b985"></a> After inclusion of residual terms |
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| <a id="org376d498"></a> Using measured modal data only | <a id="orgb025b02"></a> After inclusion of residual terms |
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The appropriate expression for a "correct" response model, derived via a set of modal properties is thus
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@@ -4495,7 +4495,7 @@ If the transmissibility is measured during a modal test which has a single excit
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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).
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This may explain why transmissibilities are not widely used in modal analysis.
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<a id="orgd4fb092"></a>
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<a id="org5d97d3b"></a>
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{{< figure src="/ox-hugo/ewins00_transmissibility_plots.png" caption="Figure 33: Transmissibility plots" >}}
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@@ -4516,7 +4516,7 @@ The fact that the excitation force is not measured is responsible for the lack o
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|---------------------------------------------------------|-------------------------------------------------------|
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| <a id="org1dc5bf9"></a> Conventional modal test setup | <a id="orge8f2893"></a> Base excitation setup |
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| <a id="org5b39165"></a> Conventional modal test setup | <a id="org6815dc0"></a> Base excitation setup |
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| height=4cm | height=4cm |
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@@ -4559,4 +4559,4 @@ Because the rank of each pseudo matrix is less than its order, it cannot be inve
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## Bibliography {#bibliography}
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<a id="org84d73f8"></a>Ewins, DJ. 2000. _Modal Testing: Theory, Practice and Application_. _Research Studies Pre, 2nd Ed., ISBN-13_. Baldock, Hertfordshire, England Philadelphia, PA: Wiley-Blackwell.
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<a id="org57f8bf9"></a>Ewins, DJ. 2000. _Modal Testing: Theory, Practice and Application_. _Research Studies Pre, 2nd Ed., ISBN-13_. Baldock, Hertfordshire, England Philadelphia, PA: Wiley-Blackwell.
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