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content/book/preumont18_vibrat_contr_activ_struc_fourt_edition.md
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@@ -8,7 +8,7 @@ Tags
: [Vibration Isolation]({{< relref "vibration_isolation" >}}), [Reference Books]({{< relref "reference_books" >}}), [Stewart Platforms]({{< relref "stewart_platforms" >}}), [HAC-HAC]({{< relref "hac_hac" >}})
Reference
: ([Preumont 2018](#orgaa0487d))
: ([Preumont 2018](#orgd83c544))
Author(s)
: Preumont, A.
@@ -61,11 +61,11 @@ There are two radically different approached to disturbance rejection: feedback
#### Feedback {#feedback}
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{{< figure src="/ox-hugo/preumont18_classical_feedback_small.png" caption="Figure 1: Principle of feedback control" >}}
The principle of feedback is represented on figure [1](#org5636ea9). 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](#orgda21dda). 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**.
@@ -87,12 +87,12 @@ The objective is to control a variable \\(y\\) to a desired value \\(r\\) in spi
#### Feedforward {#feedforward}
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{{< figure src="/ox-hugo/preumont18_feedforward_adaptative.png" caption="Figure 2: 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](#org88ce537).
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](#orgf75c047).
The filter coefficients are adapted in such a way that the error signal at one or several critical points is minimized.
@@ -123,11 +123,11 @@ 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}
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{{< figure src="/ox-hugo/preumont18_design_steps.png" caption="Figure 3: The various steps of the design" >}}
The various steps of the design of a controlled structure are shown in figure [3](#org0685157).
The various steps of the design of a controlled structure are shown in figure [3](#org1939c0d).
The **starting point** is:
@@ -154,14 +154,14 @@ 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](#org23c9634)):
From the block diagram of the control system (figure [4](#orgaf01f6c)):
\begin{align\*}
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\*}
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{{< figure src="/ox-hugo/preumont18_general_plant.png" caption="Figure 4: Block diagram of the control System" >}}
@@ -186,12 +186,12 @@ 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](#orgc0a0d3d).
A typical plot of \\(\sigma\_z(\omega)\\) is shown figure [5](#org7ddcf2a).
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.
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{{< figure src="/ox-hugo/preumont18_cas_plot.png" caption="Figure 5: Error budget distribution in OL and CL for increasing gains" >}}
@@ -304,7 +304,7 @@ The mode shapes are orthogonal with respect to the stiffness and mass matrices:
With \\(\mu\_i\\) the **modal mass** (also called the generalized mass) of mode \\(i\\).
### Modal Decomposition {#modal-decomposition}
### [Modal Decomposition]({{< relref "modal_decomposition" >}}) {#modal-decomposition--modal-decomposition-dot-md}
#### Structure Without Rigid Body Modes {#structure-without-rigid-body-modes}
@@ -398,11 +398,11 @@ With:
D\_i(\omega) = \frac{1}{1 - \omega^2/\omega\_i^2 + 2 j \xi\_i \omega/\omega\_i}
\end{equation}
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{{< 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\\)" >}}
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](#org960ee21)).
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](#orga618336)).
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 \\]
@@ -441,9 +441,9 @@ 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](#orgcc3baba)).
\\(G\_{kk}\\) is a monotonously increasing function of \\(\omega\\) (figure [7](#orgecdb253)).
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{{< figure src="/ox-hugo/preumont18_collocated_control_frf.png" caption="Figure 7: Open-Loop FRF of an undamped structure with collocated actuator/sensor pair" >}}
@@ -457,9 +457,9 @@ For lightly damped structure, the poles and zeros are just moved a little bit in
</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](#org2ea7272).
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](#org2e6ee6b).
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{{< figure src="/ox-hugo/preumont18_collocated_zero.png" caption="Figure 8: Structure with collocated actuator and sensor" >}}
@@ -474,9 +474,9 @@ The open-loop poles are independant of the actuator and sensor configuration whi
</div>
By looking at figure [7](#orgcc3baba), 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](#orgecdb253), 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.
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{{< figure src="/ox-hugo/preumont18_alternating_p_z.png" caption="Figure 9: Bode plot of a lighly damped structure with collocated actuator and sensor" >}}
@@ -486,7 +486,7 @@ 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](#orga2c2292). 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](#org8e5acfb). 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.
@@ -508,12 +508,12 @@ 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](#orgd4ab71d)):
The system consists of (see figure [10](#org5b9842b)):
- A permanent magnet which produces a uniform flux density \\(B\\) normal to the gap
- A coil which is free to move axially
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{{< figure src="/ox-hugo/preumont18_voice_coil_schematic.png" caption="Figure 10: Physical principle of a voice coil transducer" >}}
@@ -551,9 +551,9 @@ 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](#org89e3371)).
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](#org608f53f)).
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{{< figure src="/ox-hugo/preumont18_proof_mass_actuator.png" caption="Figure 11: Proof-mass actuator" >}}
@@ -583,9 +583,9 @@ 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](#org1b03971)).
Above some critical frequency \\(\omega\_c \approx 2\omega\_p\\), **the proof-mass actuator can be regarded as an ideal force generator** (figure [12](#org21ce10b)).
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{{< figure src="/ox-hugo/preumont18_proof_mass_tf.png" caption="Figure 12: Bode plot \\(F/i\\) of the proof-mass actuator" >}}
@@ -610,7 +610,7 @@ By using the two equations, we obtain:
Above the corner frequency, the gain of the geophone is equal to the transducer constant \\(T\\).
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{{< figure src="/ox-hugo/preumont18_geophone.png" caption="Figure 13: Model of a geophone based on a voice coil transducer" >}}
@@ -619,9 +619,9 @@ Designing geophones with very low corner frequency is in general difficult. Acti
### General Electromechanical Transducer {#general-electromechanical-transducer}
The consitutive behavior of a wide class of electromechanical transducers can be modelled as in figure [14](#org8d49672).
The consitutive behavior of a wide class of electromechanical transducers can be modelled as in figure [14](#org98492c9).
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{{< figure src="/ox-hugo/preumont18_electro_mechanical_transducer.png" caption="Figure 14: Electrical analog representation of an electromechanical transducer" >}}
@@ -646,7 +646,7 @@ With:
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.
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](#org9077cf9) can be used.
To do so, the bridge circuit as shown on figure [15](#org3b85763) can be used.
We can show that
@@ -656,7 +656,7 @@ We can show that
which is indeed a linear function of the velocity \\(v\\) at the mechanical terminals.
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{{< figure src="/ox-hugo/preumont18_bridge_circuit.png" caption="Figure 15: Bridge circuit for self-sensing actuation" >}}
@@ -664,9 +664,9 @@ which is indeed a linear function of the velocity \\(v\\) at the mechanical term
### Smart Materials {#smart-materials}
Smart materials have the ability to respond significantly to stimuli of different physical nature.
Figure [16](#orga08bcd9) lists various effects that are observed in materials in response to various inputs.
Figure [16](#org6279c77) lists various effects that are observed in materials in response to various inputs.
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{{< 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" >}}
@@ -761,7 +761,7 @@ 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](#orge7aeb11)) 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](#org8006b4a)) 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:
\begin{equation}
\begin{bmatrix}Q\\\Delta\end{bmatrix}
@@ -782,7 +782,7 @@ 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\\))
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{{< figure src="/ox-hugo/preumont18_piezo_stack.png" caption="Figure 17: Piezoelectric linear transducer" >}}
@@ -802,7 +802,7 @@ Equation \eqref{eq:piezo_stack_eq} can be inverted to obtain
#### 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](#org62300cf).
Let us write the total stored electromechanical energy of a discrete piezoelectric transducer as shown on figure [18](#org7c30411).
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
@@ -810,7 +810,7 @@ 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}
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{{< figure src="/ox-hugo/preumont18_piezo_discrete.png" caption="Figure 18: Discrete Piezoelectric Transducer" >}}
@@ -844,10 +844,10 @@ 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](#orga98ecb7), where the piezoelectric transducer is assumed massless and is connected to a mass \\(M\\).
Consider the system of figure [19](#org5060008), 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}\\).
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{{< figure src="/ox-hugo/preumont18_piezo_stack_admittance.png" caption="Figure 19: Elementary dynamical model of the piezoelectric transducer" >}}
@@ -866,9 +866,9 @@ 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](#orge87e33b)).
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](#org7f3b3bf)).
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{{< figure src="/ox-hugo/preumont18_piezo_admittance_curve.png" caption="Figure 20: Typical admittance FRF of the transducer" >}}
@@ -1566,7 +1566,7 @@ 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](#orgeb43e36).
The HAC/LAC approach consist of combining the two approached in a dual-loop control as shown in Figure [21](#org62e1395).
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:
@@ -1574,7 +1574,7 @@ 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)
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{{< figure src="/ox-hugo/preumont18_hac_lac_control.png" caption="Figure 21: Principle of the dual-loop HAC/LAC control" >}}
@@ -1818,4 +1818,4 @@ This approach has the following advantages:
## Bibliography {#bibliography}
<a id="orgaa0487d"></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>.
<a id="orgd83c544"></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>.