Update Content - 2021-04-14
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@ -8,7 +8,7 @@ Tags
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: [Vibration Isolation]({{< relref "vibration_isolation" >}}), [Reference Books]({{< relref "reference_books" >}}), [Stewart Platforms]({{< relref "stewart_platforms" >}}), [HAC-HAC]({{< relref "hac_hac" >}})
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Reference
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: ([Preumont 2018](#orgaa0487d))
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: ([Preumont 2018](#orgd83c544))
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Author(s)
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: Preumont, A.
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@ -61,11 +61,11 @@ There are two radically different approached to disturbance rejection: feedback
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#### Feedback {#feedback}
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<a id="org5636ea9"></a>
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<a id="orgda21dda"></a>
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{{< figure src="/ox-hugo/preumont18_classical_feedback_small.png" caption="Figure 1: Principle of feedback control" >}}
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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.
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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.
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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.
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In the control of lightly damped structures, feedback control is used for two distinct and complementary purposes: **active damping** and **model-based feedback**.
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@ -87,12 +87,12 @@ The objective is to control a variable \\(y\\) to a desired value \\(r\\) in spi
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#### Feedforward {#feedforward}
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<a id="org88ce537"></a>
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<a id="orgf75c047"></a>
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{{< figure src="/ox-hugo/preumont18_feedforward_adaptative.png" caption="Figure 2: Principle of feedforward control" >}}
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The method relies on the availability of a **reference signal correlated to the primary disturbance**.
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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).
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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).
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The filter coefficients are adapted in such a way that the error signal at one or several critical points is minimized.
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@ -123,11 +123,11 @@ The table [1](#table--tab:adv-dis-type-control) summarizes the main features of
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### The Various Steps of the Design {#the-various-steps-of-the-design}
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<a id="org0685157"></a>
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<a id="org1939c0d"></a>
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{{< figure src="/ox-hugo/preumont18_design_steps.png" caption="Figure 3: The various steps of the design" >}}
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The various steps of the design of a controlled structure are shown in figure [3](#org0685157).
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The various steps of the design of a controlled structure are shown in figure [3](#org1939c0d).
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The **starting point** is:
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@ -154,14 +154,14 @@ If the dynamics of the sensors and actuators may significantly affect the behavi
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### Plant Description, Error and Control Budget {#plant-description-error-and-control-budget}
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From the block diagram of the control system (figure [4](#org23c9634)):
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From the block diagram of the control system (figure [4](#orgaf01f6c)):
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\begin{align\*}
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y &= (I - G\_{yu}H)^{-1} G\_{yw} w\\\\\\
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z &= T\_{zw} w = [G\_{zw} + G\_{zu}H(I - G\_{yu}H)^{-1} G\_{yw}] w
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\end{align\*}
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<a id="org23c9634"></a>
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<a id="orgaf01f6c"></a>
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{{< figure src="/ox-hugo/preumont18_general_plant.png" caption="Figure 4: Block diagram of the control System" >}}
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@ -186,12 +186,12 @@ Even more interesting for the design is the **Cumulative Mean Square** response
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It is a monotonously decreasing function of frequency and describes the contribution of all frequencies above \\(\omega\\) to the mean-square value of \\(z\\).
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\\(\sigma\_z(0)\\) is then the global RMS response.
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A typical plot of \\(\sigma\_z(\omega)\\) is shown figure [5](#orgc0a0d3d).
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A typical plot of \\(\sigma\_z(\omega)\\) is shown figure [5](#org7ddcf2a).
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It is useful to **identify the critical modes** in a design, at which the effort should be targeted.
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The diagram can also be used to **assess the control laws** and compare different actuator and sensor configuration.
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<a id="orgc0a0d3d"></a>
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<a id="org7ddcf2a"></a>
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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" >}}
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@ -304,7 +304,7 @@ The mode shapes are orthogonal with respect to the stiffness and mass matrices:
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With \\(\mu\_i\\) the **modal mass** (also called the generalized mass) of mode \\(i\\).
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### Modal Decomposition {#modal-decomposition}
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### [Modal Decomposition]({{< relref "modal_decomposition" >}}) {#modal-decomposition--modal-decomposition-dot-md}
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#### Structure Without Rigid Body Modes {#structure-without-rigid-body-modes}
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@ -398,11 +398,11 @@ With:
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D\_i(\omega) = \frac{1}{1 - \omega^2/\omega\_i^2 + 2 j \xi\_i \omega/\omega\_i}
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\end{equation}
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<a id="org960ee21"></a>
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<a id="orga618336"></a>
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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\\)" >}}
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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)).
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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)).
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And \\(G(\omega)\\) can be rewritten on terms of the **low frequency modes only**:
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\\[ G(\omega) \approx \sum\_{i=1}^m \frac{\phi\_i \phi\_i^T}{\mu\_i \omega\_i^2} D\_i(\omega) + R \\]
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@ -441,9 +441,9 @@ The open-loop FRF of a collocated system corresponds to a diagonal component of
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If we assumes that the collocated system is undamped and is attached to the DoF \\(k\\), the open-loop FRF is purely real:
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\\[ G\_{kk}(\omega) = \sum\_{i=1}^m \frac{\phi\_i^2(k)}{\mu\_i (\omega\_i^2 - \omega^2)} + R\_{kk} \\]
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\\(G\_{kk}\\) is a monotonously increasing function of \\(\omega\\) (figure [7](#orgcc3baba)).
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\\(G\_{kk}\\) is a monotonously increasing function of \\(\omega\\) (figure [7](#orgecdb253)).
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<a id="orgcc3baba"></a>
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<a id="orgecdb253"></a>
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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" >}}
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@ -457,9 +457,9 @@ For lightly damped structure, the poles and zeros are just moved a little bit in
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</div>
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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).
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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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<a id="org2ea7272"></a>
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{{< figure src="/ox-hugo/preumont18_collocated_zero.png" caption="Figure 8: Structure with collocated actuator and sensor" >}}
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@ -474,9 +474,9 @@ The open-loop poles are independant of the actuator and sensor configuration whi
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</div>
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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.
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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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<a id="orga2c2292"></a>
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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" >}}
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@ -486,7 +486,7 @@ The open-loop transfer function of a lighly damped structure with a collocated a
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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)}
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\end{equation}
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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}\\).
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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}\\).
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In this way, the phase diagram is always contained between \\(\SI{0}{\degree}\\) and \\(\SI{-180}{\degree}\\) as a consequence of the interlacing property.
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@ -508,12 +508,12 @@ Two broad categories of actuators can be distinguish:
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A voice coil transducer is an energy transformer which converts electrical power into mechanical power and vice versa.
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The system consists of (see figure [10](#orgd4ab71d)):
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The system consists of (see figure [10](#org5b9842b)):
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- A permanent magnet which produces a uniform flux density \\(B\\) normal to the gap
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- A coil which is free to move axially
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<a id="orgd4ab71d"></a>
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{{< figure src="/ox-hugo/preumont18_voice_coil_schematic.png" caption="Figure 10: Physical principle of a voice coil transducer" >}}
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@ -551,9 +551,9 @@ Thus, at any time, there is an equilibrium between the electrical power absorbed
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#### Proof-Mass Actuator {#proof-mass-actuator}
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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)).
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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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<a id="org89e3371"></a>
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<a id="org608f53f"></a>
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{{< figure src="/ox-hugo/preumont18_proof_mass_actuator.png" caption="Figure 11: Proof-mass actuator" >}}
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@ -583,9 +583,9 @@ with:
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</div>
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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)).
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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" >}}
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@ -610,7 +610,7 @@ By using the two equations, we obtain:
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Above the corner frequency, the gain of the geophone is equal to the transducer constant \\(T\\).
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<a id="org8e3e3a9"></a>
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{{< figure src="/ox-hugo/preumont18_geophone.png" caption="Figure 13: Model of a geophone based on a voice coil transducer" >}}
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@ -619,9 +619,9 @@ Designing geophones with very low corner frequency is in general difficult. Acti
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### General Electromechanical Transducer {#general-electromechanical-transducer}
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The consitutive behavior of a wide class of electromechanical transducers can be modelled as in figure [14](#org8d49672).
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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" >}}
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@ -646,7 +646,7 @@ With:
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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.
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Thus, if \\(Z\_ei\\) can be measured and substracted from \\(e\\), a signal proportional to the velocity is obtained.
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To do so, the bridge circuit as shown on figure [15](#org9077cf9) can be used.
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To do so, the bridge circuit as shown on figure [15](#org3b85763) can be used.
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We can show that
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@ -656,7 +656,7 @@ We can show that
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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" >}}
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@ -664,9 +664,9 @@ which is indeed a linear function of the velocity \\(v\\) at the mechanical term
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### Smart Materials {#smart-materials}
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Smart materials have the ability to respond significantly to stimuli of different physical nature.
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Figure [16](#orga08bcd9) lists various effects that are observed in materials in response to various inputs.
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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" >}}
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@ -761,7 +761,7 @@ It measures the efficiency of the conversion of the mechanical energy into elect
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</div>
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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:
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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:
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\begin{equation}
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\begin{bmatrix}Q\\\Delta\end{bmatrix}
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@ -782,7 +782,7 @@ where
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- \\(C = \epsilon^T A n^2/l\\) is the capacitance of the transducer with no external load (\\(f = 0\\))
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- \\(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" >}}
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@ -802,7 +802,7 @@ Equation \eqref{eq:piezo_stack_eq} can be inverted to obtain
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#### Energy Stored in the Piezoelectric Transducer {#energy-stored-in-the-piezoelectric-transducer}
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Let us write the total stored electromechanical energy of a discrete piezoelectric transducer as shown on figure [18](#org62300cf).
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Let us write the total stored electromechanical energy of a discrete piezoelectric transducer as shown on figure [18](#org7c30411).
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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
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@ -810,7 +810,7 @@ The total power delivered to the transducer is the sum of electric power \\(V i\
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dW = V i dt + f \dot{\Delta} dt = V dQ + f d\Delta
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\end{equation}
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{{< figure src="/ox-hugo/preumont18_piezo_discrete.png" caption="Figure 18: Discrete Piezoelectric Transducer" >}}
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@ -844,10 +844,10 @@ The ratio between the remaining stored energy and the initial stored energy is
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#### Admittance of the Piezoelectric Transducer {#admittance-of-the-piezoelectric-transducer}
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Consider the system of figure [19](#orga98ecb7), where the piezoelectric transducer is assumed massless and is connected to a mass \\(M\\).
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Consider the system of figure [19](#org5060008), where the piezoelectric transducer is assumed massless and is connected to a mass \\(M\\).
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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" >}}
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@ -866,9 +866,9 @@ And one can see that
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\frac{z^2 - p^2}{z^2} = k^2
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\end{equation}
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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)).
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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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<a id="orge87e33b"></a>
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{{< figure src="/ox-hugo/preumont18_piezo_admittance_curve.png" caption="Figure 20: Typical admittance FRF of the transducer" >}}
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@ -1566,7 +1566,7 @@ Their design requires a model of the structure, and there is usually a trade-off
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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.
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The HAC/LAC approach consist of combining the two approached in a dual-loop control as shown in Figure [21](#orgeb43e36).
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The HAC/LAC approach consist of combining the two approached in a dual-loop control as shown in Figure [21](#org62e1395).
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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.
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This approach has the following advantages:
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@ -1574,7 +1574,7 @@ This approach has the following advantages:
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- The active damping makes it easier to gain-stabilize the modes outside the bandwidth of the output loop (improved gain margin)
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- 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" >}}
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@ -1818,4 +1818,4 @@ This approach has the following advantages:
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## Bibliography {#bibliography}
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|
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<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>.
|
||||
|
||||
@ -8,7 +8,7 @@ Tags
|
||||
: [Finite Element Model]({{< relref "finite_element_model" >}})
|
||||
|
||||
Reference
|
||||
: ([Rankers 1998](#org1e7e43d))
|
||||
: ([Rankers 1998](#org9a37ad0))
|
||||
|
||||
Author(s)
|
||||
: Rankers, A. M.
|
||||
@ -163,13 +163,13 @@ The basic questions that are addressed in this thesis are:
|
||||
|
||||
### Basic Control Aspects {#basic-control-aspects}
|
||||
|
||||
A block diagram representation of a typical servo-system is shown in Figure [1](#org8d51e97).
|
||||
A block diagram representation of a typical servo-system is shown in Figure [1](#orgf92c352).
|
||||
The main task of the system is achieve a desired positional relation between two or more components of the system.
|
||||
Therefore, a sensor measures the position which is then compared to the desired value, and the resulting error is used to generate correcting forces.
|
||||
In most systems, the "actual output" (e.g. position of end-effector) cannot be measured directly, and the feedback will therefore be based on a "measured output" (e.g. encoder signal at the motor).
|
||||
It is important to realize that these two outputs can differ, first due to resilience in the mechanical system, and second because of geometrical imperfections in the mechanical transmission between motor and end-effector.
|
||||
|
||||
<a id="org8d51e97"></a>
|
||||
<a id="orgf92c352"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/rankers98_basic_el_mech_servo.png" caption="Figure 1: Basic elements of mechanical servo system" >}}
|
||||
|
||||
@ -180,10 +180,10 @@ The correction force \\(F\\) is defined by:
|
||||
F = k\_p \epsilon + k\_d \dot{\epsilon} + k\_i \int \epsilon dt
|
||||
\end{equation}
|
||||
|
||||
It is illustrative to see that basically the proportional and derivative part of such a position control loop is very similar to a mechanical spring and damper that connect two points (Figure [2](#org7b90722)).
|
||||
It is illustrative to see that basically the proportional and derivative part of such a position control loop is very similar to a mechanical spring and damper that connect two points (Figure [2](#org30b866e)).
|
||||
If \\(c\\) and \\(d\\) represent the constant mechanical stiffness and damping between points \\(A\\) and \\(B\\), and a reference position profile \\(h(t)\\) is applied at \\(A\\), then an opposing force \\(F\\) is generated as soon as the position \\(x\\) and speed \\(\dot{x}\\) of point \\(B\\) does not correspond to \\(h(t)\\) and \\(\dot{h}(t)\\).
|
||||
|
||||
<a id="org7b90722"></a>
|
||||
<a id="org30b866e"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/rankers98_basic_elastic_struct.png" caption="Figure 2: Basic Elastic Structure" >}}
|
||||
|
||||
@ -199,9 +199,9 @@ These properties are very essential since they introduce the issue of **servo st
|
||||
|
||||
An important aspect of a feedback controller is the fact that control forces can only result from an error signal.
|
||||
Thus any desired set-point profile first leads to a position error before the corresponding driving forces are generated.
|
||||
Most modern servo-systems have not only a feedback section, but also a feedforward section, as indicated in Figure [3](#org4adf108).
|
||||
Most modern servo-systems have not only a feedback section, but also a feedforward section, as indicated in Figure [3](#org48cf617).
|
||||
|
||||
<a id="org4adf108"></a>
|
||||
<a id="org48cf617"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/rankers98_feedforward_example.png" caption="Figure 3: Mechanical servo system with feedback and feedforward control" >}}
|
||||
|
||||
@ -246,9 +246,9 @@ Basically, machine dynamics can have two deterioration effects in mechanical ser
|
||||
|
||||
#### Actuator Flexibility {#actuator-flexibility}
|
||||
|
||||
The basic characteristics of what is called "actuator flexibility" is the fact that in the frequency range of interest (usually \\(0-10\times \text{bandwidth}\\)) the driven system no longer behaves as one rigid body (Figure [4](#org72d5adf)) due to compliance between the motor and the load.
|
||||
The basic characteristics of what is called "actuator flexibility" is the fact that in the frequency range of interest (usually \\(0-10\times \text{bandwidth}\\)) the driven system no longer behaves as one rigid body (Figure [4](#org7e67aa4)) due to compliance between the motor and the load.
|
||||
|
||||
<a id="org72d5adf"></a>
|
||||
<a id="org7e67aa4"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/rankers98_actuator_flexibility.png" caption="Figure 4: Actuator Flexibility" >}}
|
||||
|
||||
@ -258,9 +258,9 @@ The basic characteristics of what is called "actuator flexibility" is the fact t
|
||||
The second category of dynamic phenomena results from the limited stiffness of the guiding system in combination with the fact the the device is driven in such a way that it has to rely on the guiding system to suppress motion in an undesired direction (in case of a linear direct drive system this occurs if the driving force is not applied at the center of gravity).
|
||||
|
||||
In general, a rigid actuator possesses six degrees of freedom, five of which need to be suppressed by the guiding system in order to leave one mobile degree of freedom.
|
||||
In the present discussion, a planar actuator with three degrees of freedom will be considered (Figure [5](#orgc8208b1)).
|
||||
In the present discussion, a planar actuator with three degrees of freedom will be considered (Figure [5](#org8db4207)).
|
||||
|
||||
<a id="orgc8208b1"></a>
|
||||
<a id="org8db4207"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/rankers98_guiding_flexibility_planar.png" caption="Figure 5: Planar actuator with guiding system flexibility" >}}
|
||||
|
||||
@ -280,9 +280,9 @@ The last category of dynamic phenomena results from the limited mass and stiffne
|
||||
In contrast to many textbooks on mechanics and machine dynamics, it is good practice always to look at the combination of driving force on the moving part, and reaction force on the stationary part, of a positioning device.
|
||||
When doing so, one has to consider what the effect of the reaction force on the systems performance will be.
|
||||
In the discussion of the previous two dynamic phenomena, the stationary part of the machine was assumed to be infinitely stiff and heavy, and therefore the effect of the reaction force was negligible.
|
||||
However, in general the stationary part is neither infinitely heavy, nor is it connected to its environment with infinite stiffness, so the stationary part will exhibit a resonance that is excited by the reaction forces (Figure [6](#org1252ea7)).
|
||||
However, in general the stationary part is neither infinitely heavy, nor is it connected to its environment with infinite stiffness, so the stationary part will exhibit a resonance that is excited by the reaction forces (Figure [6](#org235784d)).
|
||||
|
||||
<a id="org1252ea7"></a>
|
||||
<a id="org235784d"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/rankers98_limited_m_k_stationary_machine_part.png" caption="Figure 6: Limited Mass and Stiffness of Stationary Machine Part" >}}
|
||||
|
||||
@ -393,7 +393,7 @@ q\_1 \\ q\_2 \\ \vdots \\ q\_n
|
||||
|
||||
For the i-th modal coordinate \\(q\_i\\) the equation of motion is:
|
||||
|
||||
\begin{equation}
|
||||
\begin{equation} \label{eq:eoq\_modal\_i}
|
||||
m\_i \ddot{q\_i}(t) + k\_i q\_i(t) = \phi\_i^T f(t)
|
||||
\end{equation}
|
||||
|
||||
@ -401,6 +401,35 @@ which is a simple second order differential equation similar to that of a single
|
||||
Using basic formulae that are derived for a simple mass spring system, one is now able to analyse the time and frequency response of all individual modes.
|
||||
Having done that, the total motion of the system can simply be obtained by summing the contributions of all modes.
|
||||
|
||||
Characterisation of the dynamics of a mechanical system in terms of frequency response behavior plays a major role in the stability analysis of the control loop of a mechatronic device.
|
||||
In such an analysis one is typically interested in the transfer function between a measured displacement \\(x\_l\\) and a force \\(f\_k\\), which acts at the physical DoF \\(x\_k\\).
|
||||
Applying the principle of modal decomposition, any transfer function can be derived by first calculating the behavior of the individual modes, and then summing all modal contributions.
|
||||
|
||||
The contribution of one single mode \\(i\\) to the transfer function \\(x\_l/f\_k\\) can be derived by first considering the response of the modal DoF \\(q\_i\\) to a force vector \\(f\\) with only one non-zero component \\(f\_k\\).
|
||||
In that case, equation \eqref{eq:eoq_modal_i} is reduced to:
|
||||
|
||||
\begin{equation}
|
||||
m\_i \ddot{q}\_i(t) + k\_i q\_i(t) = \phi\_{ik} f\_k(t)
|
||||
\end{equation}
|
||||
|
||||
After a Laplace transformation and some rearrangement:
|
||||
|
||||
\begin{equation}
|
||||
q\_i(s) = f\_k(s) \frac{\phi\_{ik}}{m\_i s^2 + k\_i}
|
||||
\end{equation}
|
||||
|
||||
Once the modal response \\(q\_i\\) is known, the response of the physical DoF \\(x\_l\\) is found by a simple premultiplication with \\(\phi\_{il}\\), which finally leads to the following expression for the contribution of mode \\(i\\) to the transfer function:
|
||||
|
||||
\begin{equation}
|
||||
\left( \frac{x\_l}{f\_k} \right)\_i = \frac{\phi\_{ik}\phi\_{il}}{m\_i s^2 + k\_i}
|
||||
\end{equation}
|
||||
|
||||
The overall transfer function can be found by summation of the individual modal contributions, which all have the same structure:
|
||||
|
||||
\begin{equation}
|
||||
\left( \frac{x\_l}{f\_k} \right) = \sum\_{i = 1}^n \left( \frac{x\_l}{f\_k} \right)\_i = \sum\_{i = 1}^n \frac{\phi\_{ik} \phi\_{il}}{m\_i s^2 + k\_i}
|
||||
\end{equation}
|
||||
|
||||
|
||||
### Graphical Representation {#graphical-representation}
|
||||
|
||||
@ -422,6 +451,46 @@ Having done that, the total motion of the system can simply be obtained by summi
|
||||
|
||||
### Basic Characteristics of Mechanical FRF {#basic-characteristics-of-mechanical-frf}
|
||||
|
||||
Consider the position control loop of Figure [7](#orgd7fce0d).
|
||||
|
||||
<a id="orgd7fce0d"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/rankers98_mechanical_servo_system.png" caption="Figure 7: Mechanical position servo-system" >}}
|
||||
|
||||
In the ideal situation the mechanical system behaves as one rigid body with mass \\(m\\), so the mechanical transfer function can be written as:
|
||||
|
||||
\begin{equation}
|
||||
\frac{x\_{\text{servo}}}{F\_{\text{servo}}} = \frac{1}{m s^2}
|
||||
\end{equation}
|
||||
|
||||
<a id="org6bb431c"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/rankers98_ideal_bode_nyquist.png" caption="Figure 8: FRF of an ideal system with no resonances" >}}
|
||||
|
||||
In the case of one extra modal contribution, the equation for the mechanical transfer function needs to be extended with one extra term:
|
||||
|
||||
\begin{equation} \label{eq:effect\_one\_mode}
|
||||
\frac{x\_{\text{servo}}}{F\_{\text{servo}}} = \frac{1}{m s^2} + \frac{\phi\_{i,\text{servo}} \phi\_{i,\text{force}}}{m\_i s^2 + k\_i} = \frac{1}{m s^2} + \frac{\phi\_{i,\text{servo}} \phi\_{i,\text{force}}}{m\_i s^2 + m\_i \omega\_i^2}
|
||||
\end{equation}
|
||||
|
||||
The final transfer function and the exact interaction between the two parts depends on the values of the various parameters.
|
||||
|
||||
Let's introduce a variable \\(\alpha\\), which relates the high-frequency contribution of the mode to that of the rigid-body motion:
|
||||
|
||||
\begin{equation} \label{eq:alpha}
|
||||
\alpha = \frac{\frac{\phi\_{i,\text{servo}} \phi\_{i,\text{force}}}{m\_i}}{\frac{1}{m}}
|
||||
\end{equation}
|
||||
|
||||
which simplifies equation \eqref{eq:effect_one_mode} to:
|
||||
|
||||
\begin{equation}
|
||||
\frac{x\_{\text{servo}}}{F\_{\text{servo}}} = \frac{1}{ms^2} + \frac{\alpha}{m s^2 + m \omega\_i^2}
|
||||
\end{equation}
|
||||
|
||||
<a id="org1d0aa47"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/rankers98_frf_effect_alpha.png" caption="Figure 9: Contribution of rigid-body motion and modal dynamics to the amplitude and phase of FRF for various values of \\(\alpha\\)" >}}
|
||||
|
||||
|
||||
### Destabilising Effect of Modes {#destabilising-effect-of-modes}
|
||||
|
||||
@ -468,4 +537,4 @@ Through the enormous performance drive in mechatronics systems, much has been le
|
||||
|
||||
## Bibliography {#bibliography}
|
||||
|
||||
<a id="org1e7e43d"></a>Rankers, Adrian Mathias. 1998. “Machine Dynamics in Mechatronic Systems: An Engineering Approach.” University of Twente.
|
||||
<a id="org9a37ad0"></a>Rankers, Adrian Mathias. 1998. “Machine Dynamics in Mechatronic Systems: An Engineering Approach.” University of Twente.
|
||||
|
||||
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Reference in New Issue
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