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content/book/preumont18_vibrat_contr_activ_struc_fourt_edition.md
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@@ -1,16 +1,16 @@
+++
title = "Vibration Control of Active Structures - Fourth Edition"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
description = "Gives a broad overview of vibration control."
keywords = ["Control", "Vibration"]
draft = false
+++
Tags
: [Vibration Isolation]({{< relref "vibration_isolation" >}}), [Reference Books]({{< relref "reference_books" >}}), [Stewart Platforms]({{< relref "stewart_platforms" >}}), [HAC-HAC]({{< relref "hac_hac" >}})
: [Vibration Isolation]({{< relref "vibration_isolation.md" >}}), [Reference Books]({{< relref "reference_books.md" >}}), [Stewart Platforms]({{< relref "stewart_platforms.md" >}}), [HAC-HAC]({{< relref "hac_hac.md" >}})
Reference
: ([Preumont 2018](#orgf75c814))
: (<a href="#citeproc_bib_item_1">Preumont 2018</a>)
Author(s)
: Preumont, A.
@@ -63,11 +63,11 @@ There are two radically different approached to disturbance rejection: feedback
#### Feedback {#feedback}
<a id="org30e8b62"></a>
<a id="figure--fig:classical-feedback-small"></a>
{{< figure src="/ox-hugo/preumont18_classical_feedback_small.png" caption="Figure 1: Principle of feedback control" >}}
{{< figure src="/ox-hugo/preumont18_classical_feedback_small.png" caption="<span class=\"figure-number\">Figure 1: </span>Principle of feedback control" >}}
The principle of feedback is represented on figure [1](#org30e8b62). The output \\(y\\) of the system is compared to the reference signal \\(r\\), and the error signal \\(\epsilon = r-y\\) is passed into a compensator \\(K(s)\\) and applied to the system \\(G(s)\\), \\(d\\) is the disturbance.
The principle of feedback is represented on figure [1](#figure--fig:classical-feedback-small). The output \\(y\\) of the system is compared to the reference signal \\(r\\), and the error signal \\(\epsilon = r-y\\) is passed into a compensator \\(K(s)\\) and applied to the system \\(G(s)\\), \\(d\\) is the disturbance.
The design problem consists of finding the appropriate compensator \\(K(s)\\) such that the closed-loop system is stable and behaves in the appropriate manner.
In the control of lightly damped structures, feedback control is used for two distinct and complementary purposes: **active damping** and **model-based feedback**.
@@ -89,12 +89,12 @@ The objective is to control a variable \\(y\\) to a desired value \\(r\\) in spi
#### Feedforward {#feedforward}
<a id="org0cb2cac"></a>
<a id="figure--fig:feedforward-adaptative"></a>
{{< figure src="/ox-hugo/preumont18_feedforward_adaptative.png" caption="Figure 2: Principle of feedforward control" >}}
{{< figure src="/ox-hugo/preumont18_feedforward_adaptative.png" caption="<span class=\"figure-number\">Figure 2: </span>Principle of feedforward control" >}}
The method relies on the availability of a **reference signal correlated to the primary disturbance**.
The idea is to produce a second disturbance such that is cancels the effect of the primary disturbance at the location of the sensor error. Its principle is explained in figure [2](#org0cb2cac).
The idea is to produce a second disturbance such that is cancels the effect of the primary disturbance at the location of the sensor error. Its principle is explained in figure [2](#figure--fig:feedforward-adaptative).
The filter coefficients are adapted in such a way that the error signal at one or several critical points is minimized.
@@ -125,11 +125,11 @@ The table [1](#table--tab:adv-dis-type-control) summarizes the main features of
### The Various Steps of the Design {#the-various-steps-of-the-design}
<a id="org5fed023"></a>
<a id="figure--fig:design-steps"></a>
{{< figure src="/ox-hugo/preumont18_design_steps.png" caption="Figure 3: The various steps of the design" >}}
{{< figure src="/ox-hugo/preumont18_design_steps.png" caption="<span class=\"figure-number\">Figure 3: </span>The various steps of the design" >}}
The various steps of the design of a controlled structure are shown in figure [3](#org5fed023).
The various steps of the design of a controlled structure are shown in figure [3](#figure--fig:design-steps).
The **starting point** is:
@@ -156,21 +156,20 @@ If the dynamics of the sensors and actuators may significantly affect the behavi
### Plant Description, Error and Control Budget {#plant-description-error-and-control-budget}
From the block diagram of the control system (figure [4](#orgc558cd1)):
From the block diagram of the control system (figure [4](#figure--fig:general-plant)):
\begin{align\*}
y &= (I - G\_{yu}H)^{-1} G\_{yw} w\\\\\\
y &= (I - G\_{yu}H)^{-1} G\_{yw} w\\\\
z &= T\_{zw} w = [G\_{zw} + G\_{zu}H(I - G\_{yu}H)^{-1} G\_{yw}] w
\end{align\*}
<a id="orgc558cd1"></a>
<a id="figure--fig:general-plant"></a>
{{< figure src="/ox-hugo/preumont18_general_plant.png" caption="Figure 4: Block diagram of the control System" >}}
{{< figure src="/ox-hugo/preumont18_general_plant.png" caption="<span class=\"figure-number\">Figure 4: </span>Block diagram of the control System" >}}
The frequency content of the disturbance \\(w\\) is usually described by its **power spectral density** \\(\Phi\_w (\omega)\\) which describes the frequency distribution of the meas-square value.
<div class="cbox">
<div></div>
\\[\sigma\_w = \sqrt{\int\_0^\infty \Phi\_w(\omega) d\omega}\\]
@@ -179,7 +178,6 @@ The frequency content of the disturbance \\(w\\) is usually described by its **p
Even more interesting for the design is the **Cumulative Mean Square** response defined by the integral of the PSD in the frequency range \\([\omega, \infty[\\).
<div class="cbox">
<div></div>
\\[\sigma\_z^2(\omega) = \int\_\omega^\infty \Phi\_z(\nu) d\nu = \int\_\omega^\infty |T\_{zw}|^2 \Phi\_w(\nu) d\nu \\]
@@ -188,14 +186,14 @@ Even more interesting for the design is the **Cumulative Mean Square** response
It is a monotonously decreasing function of frequency and describes the contribution of all frequencies above \\(\omega\\) to the mean-square value of \\(z\\).
\\(\sigma\_z(0)\\) is then the global RMS response.
A typical plot of \\(\sigma\_z(\omega)\\) is shown figure [5](#orgd0ed9cf).
A typical plot of \\(\sigma\_z(\omega)\\) is shown figure [5](#figure--fig:cas-plot).
It is useful to **identify the critical modes** in a design, at which the effort should be targeted.
The diagram can also be used to **assess the control laws** and compare different actuator and sensor configuration.
<a id="orgd0ed9cf"></a>
<a id="figure--fig:cas-plot"></a>
{{< figure src="/ox-hugo/preumont18_cas_plot.png" caption="Figure 5: Error budget distribution in OL and CL for increasing gains" >}}
{{< figure src="/ox-hugo/preumont18_cas_plot.png" caption="<span class=\"figure-number\">Figure 5: </span>Error budget distribution in OL and CL for increasing gains" >}}
### Pseudo-inverse {#pseudo-inverse}
@@ -254,7 +252,6 @@ This will have usually little impact of the fitting error while reducing conside
The general form of the equation of motion governing the dynamic equilibrium between the external, elastic, inertia and damping forces acting on a discrete, flexible structure with a finite number \\(n\\) of degrees of freedom is
<div class="cbox">
<div></div>
\begin{equation}
M \ddot{x} + C \dot{x} + K x = f
@@ -271,7 +268,6 @@ With:
The damping matrix \\(C\\) represents the various dissipation mechanisms in the structure, which are usually poorly known. One of the popular hypotheses is the Rayleigh damping.
<div class="cbox">
<div></div>
\begin{equation}
C = \alpha M + \beta K
@@ -299,14 +295,14 @@ The number of mode shapes is equal to the number of degrees of freedom \\(n\\).
The mode shapes are orthogonal with respect to the stiffness and mass matrices:
\begin{align}
\phi\_i^T M \phi\_j &= \mu\_i \delta\_{ij} \\\\\\
\phi\_i^T M \phi\_j &= \mu\_i \delta\_{ij} \\\\
\phi\_i^T K \phi\_j &= \mu\_i \omega\_i^2 \delta\_{ij}
\end{align}
With \\(\mu\_i\\) the **modal mass** (also called the generalized mass) of mode \\(i\\).
### [Modal Decomposition]({{< relref "modal_decomposition" >}}) {#modal-decomposition--modal-decomposition-dot-md}
### [Modal Decomposition]({{< relref "modal_decomposition.md" >}}) {#modal-decomposition--modal-decomposition-dot-md}
#### Structure Without Rigid Body Modes {#structure-without-rigid-body-modes}
@@ -314,7 +310,6 @@ With \\(\mu\_i\\) the **modal mass** (also called the generalized mass) of mode
Let perform a change of variable from physical coordinates \\(x\\) to modal coordinates \\(z\\).
<div class="cbox">
<div></div>
\begin{equation}
x = \Phi z
@@ -336,12 +331,11 @@ If we left multiply the equation by \\(\Phi^T\\) and we use the orthogonalily re
If \\(\Phi^T C \Phi\\) is diagonal, the **damping is said classical or normal**. In this case:
\\[ \Phi^T C \Phi = diag(2 \xi\_i \mu\_i \omega\_i) \\]
One can verify that the Rayleigh damping \eqref{eq:rayleigh_damping} complies with this condition with modal damping ratios \\(\xi\_i = \frac{1}{2} ( \frac{\alpha}{\omega\_i} + \beta\omega\_i )\\).
One can verify that the Rayleigh damping <eq:rayleigh_damping> complies with this condition with modal damping ratios \\(\xi\_i = \frac{1}{2} ( \frac{\alpha}{\omega\_i} + \beta\omega\_i )\\).
And we obtain decoupled modal equations \eqref{eq:modal_eom}.
And we obtain decoupled modal equations <eq:modal_eom>.
<div class="cbox">
<div></div>
\begin{equation}
\ddot{z} + 2 \xi \Omega \dot{z} + \Omega^2 z = z^{-1} \Phi^T f
@@ -355,7 +349,7 @@ with:
</div>
Typical values of the modal damping ratio are summarized on table [tab:damping_ratio](#tab:damping_ratio).
Typical values of the modal damping ratio are summarized on table <tab:damping_ratio>.
<a id="table--tab:damping-ratio"></a>
<div class="table-caption">
@@ -372,15 +366,15 @@ Typical values of the modal damping ratio are summarized on table [tab:damping_r
The assumption of classical damping is often justified for light damping, but it is questionable when the damping is large.
If one accepts the assumption of classical damping, the only difference between equation \eqref{eq:general_eom} and \eqref{eq:modal_eom} lies in the change of coordinates.
If one accepts the assumption of classical damping, the only difference between equation <eq:general_eom> and <eq:modal_eom> lies in the change of coordinates.
However, in physical coordinates, the number of degrees of freedom is usually very large.
If a structure is excited in by a band limited excitation, its response is dominated by the modes whose natural frequencies are inside the bandwidth of the excitation and the equation \eqref{eq:modal_eom} can often be restricted to theses modes.
If a structure is excited in by a band limited excitation, its response is dominated by the modes whose natural frequencies are inside the bandwidth of the excitation and the equation <eq:modal_eom> can often be restricted to theses modes.
Therefore, the number of degrees of freedom contribution effectively to the response is **reduced drastically** in modal coordinates.
#### Dynamic Flexibility Matrix {#dynamic-flexibility-matrix}
If we consider the steady-state response of equation \eqref{eq:general_eom} to harmonic excitation \\(f=F e^{j\omega t}\\), the response is also harmonic \\(x = Xe^{j\omega t}\\). The amplitude of \\(F\\) and \\(X\\) is related by:
If we consider the steady-state response of equation <eq:general_eom> to harmonic excitation \\(f=F e^{j\omega t}\\), the response is also harmonic \\(x = Xe^{j\omega t}\\). The amplitude of \\(F\\) and \\(X\\) is related by:
\\[ X = G(\omega) F \\]
Where \\(G(\omega)\\) is called the **Dynamic flexibility Matrix**:
@@ -400,11 +394,11 @@ With:
D\_i(\omega) = \frac{1}{1 - \omega^2/\omega\_i^2 + 2 j \xi\_i \omega/\omega\_i}
\end{equation}
<a id="orgeec9f86"></a>
<a id="figure--fig:neglected-modes"></a>
{{< figure src="/ox-hugo/preumont18_neglected_modes.png" caption="Figure 6: Fourier spectrum of the excitation \\(F\\) and dynamic amplitification \\(D\_i\\) of mode \\(i\\) and \\(k\\) such that \\(\omega\_i < \omega\_b\\) and \\(\omega\_k \gg \omega\_b\\)" >}}
{{< figure src="/ox-hugo/preumont18_neglected_modes.png" caption="<span class=\"figure-number\">Figure 6: </span>Fourier spectrum of the excitation \\(F\\) and dynamic amplitification \\(D\_i\\) of mode \\(i\\) and \\(k\\) such that \\(\omega\_i < \omega\_b\\) and \\(\omega\_k \gg \omega\_b\\)" >}}
If the excitation has a limited bandwidth \\(\omega\_b\\), the contribution of the high frequency modes \\(\omega\_k \gg \omega\_b\\) can be evaluated by assuming \\(D\_k(\omega) \approx 1\\) (as shown on figure [6](#orgeec9f86)).
If the excitation has a limited bandwidth \\(\omega\_b\\), the contribution of the high frequency modes \\(\omega\_k \gg \omega\_b\\) can be evaluated by assuming \\(D\_k(\omega) \approx 1\\) (as shown on figure [6](#figure--fig:neglected-modes)).
And \\(G(\omega)\\) can be rewritten on terms of the **low frequency modes only**:
\\[ G(\omega) \approx \sum\_{i=1}^m \frac{\phi\_i \phi\_i^T}{\mu\_i \omega\_i^2} D\_i(\omega) + R \\]
@@ -418,7 +412,6 @@ The quasi-static correction of the high frequency modes \\(R\\) is called the **
### Collocated Control System {#collocated-control-system}
<div class="cbox">
<div></div>
A **collocated control system** is a control system where:
@@ -443,30 +436,28 @@ The open-loop FRF of a collocated system corresponds to a diagonal component of
If we assumes that the collocated system is undamped and is attached to the DoF \\(k\\), the open-loop FRF is purely real:
\\[ G\_{kk}(\omega) = \sum\_{i=1}^m \frac{\phi\_i^2(k)}{\mu\_i (\omega\_i^2 - \omega^2)} + R\_{kk} \\]
\\(G\_{kk}\\) is a monotonously increasing function of \\(\omega\\) (figure [7](#org2389144)).
\\(G\_{kk}\\) is a monotonously increasing function of \\(\omega\\) (figure [7](#figure--fig:collocated-control-frf)).
<a id="org2389144"></a>
<a id="figure--fig:collocated-control-frf"></a>
{{< figure src="/ox-hugo/preumont18_collocated_control_frf.png" caption="Figure 7: Open-Loop FRF of an undamped structure with collocated actuator/sensor pair" >}}
{{< figure src="/ox-hugo/preumont18_collocated_control_frf.png" caption="<span class=\"figure-number\">Figure 7: </span>Open-Loop FRF of an undamped structure with collocated actuator/sensor pair" >}}
The amplitude of the FRF goes from \\(-\infty\\) at the resonance frequencies \\(\omega\_i\\) to \\(+\infty\\) at the next resonance frequency \\(\omega\_{i+1}\\). Therefore, in every interval, there is a frequency \\(z\_i\\) such that \\(\omega\_i < z\_i < \omega\_{i+1}\\) where the amplitude of the FRF vanishes. The frequencies \\(z\_i\\) are called **anti-resonances**.
<div class="cbox">
<div></div>
Undamped **collocated control systems** have **alternating poles and zeros** on the imaginary axis.
For lightly damped structure, the poles and zeros are just moved a little bit in the left-half plane, but they are still interlacing.
</div>
If the undamped structure is excited harmonically by the actuator at the frequency of the transmission zero \\(z\_i\\), the amplitude of the response of the collocated sensor vanishes. That means that the structure oscillates at the frequency \\(z\_i\\) according to the mode shape shown in dotted line figure [8](#org9a738f7).
If the undamped structure is excited harmonically by the actuator at the frequency of the transmission zero \\(z\_i\\), the amplitude of the response of the collocated sensor vanishes. That means that the structure oscillates at the frequency \\(z\_i\\) according to the mode shape shown in dotted line figure [8](#figure--fig:collocated-zero).
<a id="org9a738f7"></a>
<a id="figure--fig:collocated-zero"></a>
{{< figure src="/ox-hugo/preumont18_collocated_zero.png" caption="Figure 8: Structure with collocated actuator and sensor" >}}
{{< figure src="/ox-hugo/preumont18_collocated_zero.png" caption="<span class=\"figure-number\">Figure 8: </span>Structure with collocated actuator and sensor" >}}
<div class="cbox">
<div></div>
The frequency of the transmission zero \\(z\_i\\) and the mode shape associated are the **natural frequency** and the **mode shape** of the system obtained by **constraining the d.o.f. on which the control systems acts**.
@@ -476,11 +467,11 @@ The open-loop poles are independant of the actuator and sensor configuration whi
</div>
By looking at figure [7](#org2389144), we see that neglecting the residual mode in the modelling amounts to translating the FRF diagram vertically. That produces a shift in the location of the transmission zeros to the right.
By looking at figure [7](#figure--fig:collocated-control-frf), we see that neglecting the residual mode in the modelling amounts to translating the FRF diagram vertically. That produces a shift in the location of the transmission zeros to the right.
<a id="org52c26c5"></a>
<a id="figure--fig:alternating-p-z"></a>
{{< figure src="/ox-hugo/preumont18_alternating_p_z.png" caption="Figure 9: Bode plot of a lighly damped structure with collocated actuator and sensor" >}}
{{< figure src="/ox-hugo/preumont18_alternating_p_z.png" caption="<span class=\"figure-number\">Figure 9: </span>Bode plot of a lighly damped structure with collocated actuator and sensor" >}}
The open-loop transfer function of a lighly damped structure with a collocated actuator/sensor pair can be written:
@@ -488,7 +479,7 @@ The open-loop transfer function of a lighly damped structure with a collocated a
G(s) = G\_0 \frac{\Pi\_i(s^2/z\_i^2 + 2 \xi\_i s/z\_i + 1)}{\Pi\_j(s^2/\omega\_j^2 + 2 \xi\_j s /\omega\_j + 1)}
\end{equation}
The corresponding Bode plot is represented in figure [9](#org52c26c5). Every imaginary pole at \\(\pm j\omega\_i\\) introduces a \\(\SI{180}{\degree}\\) phase lag and every imaginary zero at \\(\pm jz\_i\\) introduces a phase lead of \\(\SI{180}{\degree}\\).
The corresponding Bode plot is represented in figure [9](#figure--fig:alternating-p-z). Every imaginary pole at \\(\pm j\omega\_i\\) introduces a \\(\SI{180}{\degree}\\) phase lag and every imaginary zero at \\(\pm jz\_i\\) introduces a phase lead of \\(\SI{180}{\degree}\\).
In this way, the phase diagram is always contained between \\(\SI{0}{\degree}\\) and \\(\SI{-180}{\degree}\\) as a consequence of the interlacing property.
@@ -510,14 +501,14 @@ Two broad categories of actuators can be distinguish:
A voice coil transducer is an energy transformer which converts electrical power into mechanical power and vice versa.
The system consists of (see figure [10](#orga1a9b67)):
The system consists of (see figure [10](#figure--fig:voice-coil-schematic)):
- A permanent magnet which produces a uniform flux density \\(B\\) normal to the gap
- A coil which is free to move axially
<a id="orga1a9b67"></a>
<a id="figure--fig:voice-coil-schematic"></a>
{{< figure src="/ox-hugo/preumont18_voice_coil_schematic.png" caption="Figure 10: Physical principle of a voice coil transducer" >}}
{{< figure src="/ox-hugo/preumont18_voice_coil_schematic.png" caption="<span class=\"figure-number\">Figure 10: </span>Physical principle of a voice coil transducer" >}}
We note:
@@ -527,7 +518,6 @@ We note:
- \\(i\\) the current into the coil
<div class="cbox">
<div></div>
**Faraday's law**:
@@ -553,11 +543,11 @@ Thus, at any time, there is an equilibrium between the electrical power absorbed
#### Proof-Mass Actuator {#proof-mass-actuator}
A reaction mass \\(m\\) is conected to the support structure by a spring \\(k\\) , and damper \\(c\\) and a force actuator \\(f = T i\\) (figure [11](#orgc439137)).
A reaction mass \\(m\\) is conected to the support structure by a spring \\(k\\) , and damper \\(c\\) and a force actuator \\(f = T i\\) (figure [11](#figure--fig:proof-mass-actuator)).
<a id="orgc439137"></a>
<a id="figure--fig:proof-mass-actuator"></a>
{{< figure src="/ox-hugo/preumont18_proof_mass_actuator.png" caption="Figure 11: Proof-mass actuator" >}}
{{< figure src="/ox-hugo/preumont18_proof_mass_actuator.png" caption="<span class=\"figure-number\">Figure 11: </span>Proof-mass actuator" >}}
If we apply the second law of Newton on the mass:
\\[ m\ddot{x} + c\dot{x} + kx = f = Ti \\]
@@ -571,7 +561,6 @@ The total force applied on the support is:
The transfer function between the total force and the current \\(i\\) applied to the coil is :
<div class="cbox">
<div></div>
\begin{equation}
\frac{F}{i} = \frac{-s^2 T}{s^2 + 2\xi\_p \omega\_p s + \omega\_p^2}
@@ -585,11 +574,11 @@ with:
</div>
Above some critical frequency \\(\omega\_c \approx 2\omega\_p\\), **the proof-mass actuator can be regarded as an ideal force generator** (figure [12](#org3b93a8e)).
Above some critical frequency \\(\omega\_c \approx 2\omega\_p\\), **the proof-mass actuator can be regarded as an ideal force generator** (figure [12](#figure--fig:proof-mass-tf)).
<a id="org3b93a8e"></a>
<a id="figure--fig:proof-mass-tf"></a>
{{< figure src="/ox-hugo/preumont18_proof_mass_tf.png" caption="Figure 12: Bode plot \\(F/i\\) of the proof-mass actuator" >}}
{{< figure src="/ox-hugo/preumont18_proof_mass_tf.png" caption="<span class=\"figure-number\">Figure 12: </span>Bode plot \\(F/i\\) of the proof-mass actuator" >}}
#### Geophone {#geophone}
@@ -600,7 +589,7 @@ The voltage \\(e\\) of the coil is used as the sensor output.
If \\(x\_0\\) is the displacement of the support and if the voice coil is open (\\(i=0\\)), the governing equations are:
\begin{align\*}
m\ddot{x} + c(\dot{x}-\dot{x\_0}) + k(x-x\_0) &= 0\\\\\\
m\ddot{x} + c(\dot{x}-\dot{x\_0}) + k(x-x\_0) &= 0\\\\
T(\dot{x}-\dot{x\_0}) &= e
\end{align\*}
@@ -612,25 +601,25 @@ By using the two equations, we obtain:
Above the corner frequency, the gain of the geophone is equal to the transducer constant \\(T\\).
<a id="org7ded49f"></a>
<a id="figure--fig:geophone"></a>
{{< figure src="/ox-hugo/preumont18_geophone.png" caption="Figure 13: Model of a geophone based on a voice coil transducer" >}}
{{< figure src="/ox-hugo/preumont18_geophone.png" caption="<span class=\"figure-number\">Figure 13: </span>Model of a geophone based on a voice coil transducer" >}}
Designing geophones with very low corner frequency is in general difficult. Active geophones where the frequency is lowered electronically may constitute a good alternative option.
### General Electromechanical Transducer {#general-electromechanical-transducer}
The consitutive behavior of a wide class of electromechanical transducers can be modelled as in figure [14](#org82c090c).
The consitutive behavior of a wide class of electromechanical transducers can be modelled as in figure [14](#figure--fig:electro-mechanical-transducer).
<a id="org82c090c"></a>
<a id="figure--fig:electro-mechanical-transducer"></a>
{{< figure src="/ox-hugo/preumont18_electro_mechanical_transducer.png" caption="Figure 14: Electrical analog representation of an electromechanical transducer" >}}
{{< figure src="/ox-hugo/preumont18_electro_mechanical_transducer.png" caption="<span class=\"figure-number\">Figure 14: </span>Electrical analog representation of an electromechanical transducer" >}}
In Laplace form the constitutive equations read:
\begin{align}
e & = Z\_e i + T\_{em} v \label{eq:gen\_trans\_e} \\\\\\
e & = Z\_e i + T\_{em} v \label{eq:gen\_trans\_e} \\\\
f & = T\_{em} i + Z\_m v \label{eq:gen\_trans\_f}
\end{align}
@@ -645,10 +634,10 @@ With:
- \\(T\_{me}\\) is the transduction coefficient representing the force acting on the mechanical terminals to balance the electromagnetic force induced per unit current input (in \\(\si{\newton\per\ampere}\\))
- \\(Z\_m\\) is the mechanical impedance measured when \\(i=0\\)
Equation \eqref{eq:gen_trans_e} shows that the voltage across the electrical terminals of any electromechanical transducer is the sum of a contribution proportional to the current applied and a contribution proportional to the velocity of the mechanical terminals.
Equation <eq:gen_trans_e> shows that the voltage across the electrical terminals of any electromechanical transducer is the sum of a contribution proportional to the current applied and a contribution proportional to the velocity of the mechanical terminals.
Thus, if \\(Z\_ei\\) can be measured and substracted from \\(e\\), a signal proportional to the velocity is obtained.
To do so, the bridge circuit as shown on figure [15](#org8e1c5fb) can be used.
To do so, the bridge circuit as shown on figure [15](#figure--fig:bridge-circuit) can be used.
We can show that
@@ -658,19 +647,19 @@ We can show that
which is indeed a linear function of the velocity \\(v\\) at the mechanical terminals.
<a id="org8e1c5fb"></a>
<a id="figure--fig:bridge-circuit"></a>
{{< figure src="/ox-hugo/preumont18_bridge_circuit.png" caption="Figure 15: Bridge circuit for self-sensing actuation" >}}
{{< figure src="/ox-hugo/preumont18_bridge_circuit.png" caption="<span class=\"figure-number\">Figure 15: </span>Bridge circuit for self-sensing actuation" >}}
### Smart Materials {#smart-materials}
Smart materials have the ability to respond significantly to stimuli of different physical nature.
Figure [16](#org29efe87) lists various effects that are observed in materials in response to various inputs.
Figure [16](#figure--fig:smart-materials) lists various effects that are observed in materials in response to various inputs.
<a id="org29efe87"></a>
<a id="figure--fig:smart-materials"></a>
{{< figure src="/ox-hugo/preumont18_smart_materials.png" caption="Figure 16: Stimulus response relations indicating various effects in materials. The smart materials corresponds to the non-diagonal cells" >}}
{{< figure src="/ox-hugo/preumont18_smart_materials.png" caption="<span class=\"figure-number\">Figure 16: </span>Stimulus response relations indicating various effects in materials. The smart materials corresponds to the non-diagonal cells" >}}
### Piezoelectric Transducer {#piezoelectric-transducer}
@@ -678,14 +667,12 @@ Figure [16](#org29efe87) lists various effects that are observed in materials in
Piezoelectric materials exhibits two effects described below.
<div class="cbox">
<div></div>
Ability to generate an electrical charge in proportion to an external applied force.
</div>
<div class="cbox">
<div></div>
An electric filed parallel to the direction of polarization induces an expansion of the material.
@@ -696,11 +683,10 @@ The most popular piezoelectric materials are Lead-Zirconate-Titanate (PZT) which
We here consider a transducer made of one-dimensional piezoelectric material.
<div class="cbox">
<div></div>
\begin{subequations}
\begin{align}
D & = \epsilon^T E + d\_{33} T\\\\\\
D & = \epsilon^T E + d\_{33} T\\\\
S & = d\_{33} E + s^E T
\end{align}
\end{subequations}
@@ -720,16 +706,16 @@ With:
#### Constitutive Relations of a Discrete Transducer {#constitutive-relations-of-a-discrete-transducer}
The set of equations \eqref{eq:piezo_eq} can be written in a matrix form:
The set of equations <eq:piezo_eq> can be written in a matrix form:
\begin{equation}
\begin{bmatrix}D\\S\end{bmatrix}
\begin{bmatrix}D\\\S\end{bmatrix}
=
\begin{bmatrix}
\epsilon^T & d\_{33}\\\\\\
\epsilon^T & d\_{33}\\\\
d\_{33} & s^E
\end{bmatrix}
\begin{bmatrix}E\\T\end{bmatrix}
\begin{bmatrix}E\\\T\end{bmatrix}
\end{equation}
Where \\((E, T)\\) are the independent variables and \\((D, S)\\) are the dependent variable.
@@ -737,13 +723,13 @@ Where \\((E, T)\\) are the independent variables and \\((D, S)\\) are the depend
If \\((E, S)\\) are taken as independant variables:
\begin{equation}
\begin{bmatrix}D\\T\end{bmatrix}
\begin{bmatrix}D\\\T\end{bmatrix}
=
\begin{bmatrix}
\epsilon^T(1-k^2) & e\_{33}\\\\\\
\epsilon^T(1-k^2) & e\_{33}\\\\
-e\_{33} & c^E
\end{bmatrix}
\begin{bmatrix}E\\S\end{bmatrix}
\begin{bmatrix}E\\\S\end{bmatrix}
\end{equation}
With:
@@ -752,7 +738,6 @@ With:
- \\(e\_{33} = \frac{d\_{33}}{s^E}\\) is the constant relating the electric displacement to the strain for short-circuited electrodes \\([C/m^2]\\)
<div class="cbox">
<div></div>
\begin{equation}
k^2 = \frac{{d\_{33}}^2}{s^E \epsilon^T} = \frac{{e\_{33}}^2}{c^E \epsilon^T}
@@ -763,16 +748,16 @@ It measures the efficiency of the conversion of the mechanical energy into elect
</div>
If one assumes that all the electrical and mechanical quantities are uniformly distributed in a linear transducer formed by a **stack** (see figure [17](#org226015b)) of \\(n\\) disks of thickness \\(t\\) and cross section \\(A\\), the global constitutive equations of the transducer are obtained by integrating \eqref{eq:piezo_eq_matrix_bis} over the volume of the transducer:
If one assumes that all the electrical and mechanical quantities are uniformly distributed in a linear transducer formed by a **stack** (see figure [17](#figure--fig:piezo-stack)) of \\(n\\) disks of thickness \\(t\\) and cross section \\(A\\), the global constitutive equations of the transducer are obtained by integrating <eq:piezo_eq_matrix_bis> over the volume of the transducer:
\begin{equation}
\begin{bmatrix}Q\\\Delta\end{bmatrix}
\begin{bmatrix}Q\\\\Delta\end{bmatrix}
=
\begin{bmatrix}
C & nd\_{33}\\\\\\
C & nd\_{33}\\\\
nd\_{33} & 1/K\_a
\end{bmatrix}
\begin{bmatrix}V\\f\end{bmatrix}
\begin{bmatrix}V\\\f\end{bmatrix}
\end{equation}
where
@@ -784,27 +769,27 @@ where
- \\(C = \epsilon^T A n^2/l\\) is the capacitance of the transducer with no external load (\\(f = 0\\))
- \\(K\_a = A/s^El\\) is the stiffness with short-circuited electrodes (\\(V = 0\\))
<a id="org226015b"></a>
<a id="figure--fig:piezo-stack"></a>
{{< figure src="/ox-hugo/preumont18_piezo_stack.png" caption="Figure 17: Piezoelectric linear transducer" >}}
{{< figure src="/ox-hugo/preumont18_piezo_stack.png" caption="<span class=\"figure-number\">Figure 17: </span>Piezoelectric linear transducer" >}}
Equation \eqref{eq:piezo_stack_eq} can be inverted to obtain
Equation <eq:piezo_stack_eq> can be inverted to obtain
\begin{equation}
\begin{bmatrix}V\\f\end{bmatrix}
\begin{bmatrix}V\\\f\end{bmatrix}
=
\frac{K\_a}{C(1-k^2)}
\begin{bmatrix}
1/K\_a & -nd\_{33}\\\\\\
1/K\_a & -nd\_{33}\\\\
-nd\_{33} & C
\end{bmatrix}
\begin{bmatrix}Q\\\Delta\end{bmatrix}
\begin{bmatrix}Q\\\\Delta\end{bmatrix}
\end{equation}
#### Energy Stored in the Piezoelectric Transducer {#energy-stored-in-the-piezoelectric-transducer}
Let us write the total stored electromechanical energy of a discrete piezoelectric transducer as shown on figure [18](#org4316115).
Let us write the total stored electromechanical energy of a discrete piezoelectric transducer as shown on figure [18](#figure--fig:piezo-discrete).
The total power delivered to the transducer is the sum of electric power \\(V i\\) and the mechanical power \\(f \dot{\Delta}\\). The net work of the transducer is
@@ -812,11 +797,11 @@ The total power delivered to the transducer is the sum of electric power \\(V i\
dW = V i dt + f \dot{\Delta} dt = V dQ + f d\Delta
\end{equation}
<a id="org4316115"></a>
<a id="figure--fig:piezo-discrete"></a>
{{< figure src="/ox-hugo/preumont18_piezo_discrete.png" caption="Figure 18: Discrete Piezoelectric Transducer" >}}
{{< figure src="/ox-hugo/preumont18_piezo_discrete.png" caption="<span class=\"figure-number\">Figure 18: </span>Discrete Piezoelectric Transducer" >}}
By integrating equation \eqref{eq:piezo_work} and using the constitutive equations \eqref{eq:piezo_stack_eq_inv}, we obtain the analytical expression of the stored electromechanical energy for the discrete transducer:
By integrating equation <eq:piezo_work> and using the constitutive equations <eq:piezo_stack_eq_inv>, we obtain the analytical expression of the stored electromechanical energy for the discrete transducer:
\begin{equation}
W\_e(\Delta, Q) = \frac{Q^2}{2 C (1 - k^2)} - \frac{n d\_{33} K\_a}{C(1-k^2)} Q\Delta + \frac{K\_a}{1-k^2}\frac{\Delta^2}{2}
@@ -830,7 +815,7 @@ The constitutive equations can be recovered by differentiate the stored energy:
\\[ f = \frac{\partial W\_e}{\partial \Delta}, \quad V = \frac{\partial W\_e}{\partial Q} \\]
#### Interpretation of \\(k^2\\) {#interpretation-of--k-2}
#### Interpretation of \\(k^2\\) {#interpretation-of-k-2}
Consider a piezoelectric transducer subjected to the following mechanical cycle: first, it is loaded with a force \\(F\\) with short-circuited electrodes; the resulting extension is \\(\Delta\_1 = F/K\_a\\) where \\(K\_a = A/(s^El)\\) is the stiffness with short-circuited electrodes.
The energy stored in the system is:
@@ -846,12 +831,12 @@ The ratio between the remaining stored energy and the initial stored energy is
#### Admittance of the Piezoelectric Transducer {#admittance-of-the-piezoelectric-transducer}
Consider the system of figure [19](#orgcdbb831), where the piezoelectric transducer is assumed massless and is connected to a mass \\(M\\).
Consider the system of figure [19](#figure--fig:piezo-stack-admittance), where the piezoelectric transducer is assumed massless and is connected to a mass \\(M\\).
The force acting on the mass is negative of that acting on the transducer, \\(f = -M \ddot{x}\\).
<a id="orgcdbb831"></a>
<a id="figure--fig:piezo-stack-admittance"></a>
{{< figure src="/ox-hugo/preumont18_piezo_stack_admittance.png" caption="Figure 19: Elementary dynamical model of the piezoelectric transducer" >}}
{{< figure src="/ox-hugo/preumont18_piezo_stack_admittance.png" caption="<span class=\"figure-number\">Figure 19: </span>Elementary dynamical model of the piezoelectric transducer" >}}
From the constitutive equations, one finds
@@ -868,11 +853,11 @@ And one can see that
\frac{z^2 - p^2}{z^2} = k^2
\end{equation}
Equation \eqref{eq:distance_p_z} constitutes a practical way to determine the electromechanical coupling factor from the poles and zeros of the admittance measurement (figure [20](#org15dd7b6)).
Equation <eq:distance_p_z> constitutes a practical way to determine the electromechanical coupling factor from the poles and zeros of the admittance measurement (figure [20](#figure--fig:piezo-admittance-curve)).
<a id="org15dd7b6"></a>
<a id="figure--fig:piezo-admittance-curve"></a>
{{< figure src="/ox-hugo/preumont18_piezo_admittance_curve.png" caption="Figure 20: Typical admittance FRF of the transducer" >}}
{{< figure src="/ox-hugo/preumont18_piezo_admittance_curve.png" caption="<span class=\"figure-number\">Figure 20: </span>Typical admittance FRF of the transducer" >}}
## Piezoelectric Beam, Plate and Truss {#piezoelectric-beam-plate-and-truss}
@@ -1004,13 +989,12 @@ Equation \eqref{eq:distance_p_z} constitutes a practical way to determine the el
#### Equivalent Damping Ratio {#equivalent-damping-ratio}
## Collocated Versus Non-collocated Control {#collocated-versus-non-collocated-control}
## BKMK Collocated Versus Non-collocated Control {#bkmk-collocated-versus-non-collocated-control}
### Pole-Zero Flipping {#pole-zero-flipping}
<div class="cbox">
<div></div>
The Root Locus shows, in a graphical form, the evolution of the poles of the closed-loop system as a function of the scalar gain \\(g\\) applied to the compensator.
The Root Locus is the locus of the solution \\(s\\) of the closed loop characteristic equation \\(1 + gG(s)H(s) = 0\\) when \\(g\\) goes from zero to infinity.
@@ -1380,7 +1364,7 @@ Weakness of LQG:
- use frequency independant cost function
- use noise statistics with uniform distribution
To overcome the weakness => frequency shaping either by:
To overcome the weakness =&gt; frequency shaping either by:
- considering a frequency dependant cost function
- using colored noise statistics
@@ -1568,7 +1552,7 @@ Their design requires a model of the structure, and there is usually a trade-off
When collocated actuator/sensor pairs can be used, stability can be achieved using positivity concepts, but in many situations, collocated pairs are not feasible for HAC.
The HAC/LAC approach consist of combining the two approached in a dual-loop control as shown in Figure [21](#org0c9fed0).
The HAC/LAC approach consist of combining the two approached in a dual-loop control as shown in Figure [21](#figure--fig:hac-lac-control).
The inner loop uses a set of collocated actuator/sensor pairs for decentralized active damping with guaranteed stability ; the outer loop consists of a non-collocated HAC based on a model of the actively damped structure.
This approach has the following advantages:
@@ -1576,9 +1560,9 @@ This approach has the following advantages:
- The active damping makes it easier to gain-stabilize the modes outside the bandwidth of the output loop (improved gain margin)
- The larger damping of the modes within the controller bandwidth makes them more robust to the parmetric uncertainty (improved phase margin)
<a id="org0c9fed0"></a>
<a id="figure--fig:hac-lac-control"></a>
{{< figure src="/ox-hugo/preumont18_hac_lac_control.png" caption="Figure 21: Principle of the dual-loop HAC/LAC control" >}}
{{< figure src="/ox-hugo/preumont18_hac_lac_control.png" caption="<span class=\"figure-number\">Figure 21: </span>Principle of the dual-loop HAC/LAC control" >}}
#### Wide-Band Position Control {#wide-band-position-control}
@@ -1818,7 +1802,8 @@ This approach has the following advantages:
### Problems {#problems}
## Bibliography {#bibliography}
<a id="orgf75c814"></a>Preumont, Andre. 2018. _Vibration Control of Active Structures - Fourth Edition_. Solid Mechanics and Its Applications. Springer International Publishing. <https://doi.org/10.1007/978-3-319-72296-2>.
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Preumont, Andre. 2018. <i>Vibration Control of Active Structures - Fourth Edition</i>. Solid Mechanics and Its Applications. Springer International Publishing. doi:<a href="https://doi.org/10.1007/978-3-319-72296-2">10.1007/978-3-319-72296-2</a>.</div>
</div>