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content/phdthesis/jabben07_mechat.md
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@@ -159,7 +159,7 @@ Three factors influence the performance:
The DEB helps identifying which disturbance is the limiting factor, and it should be investigated if the controller can deal with this disturbance before re-designing the plant.
The modelling of disturbance as stochastic variables, is by excellence suitable for the optimal stochastic control framework.
In Figure [1](#figure--fig:jabben07-general-plant), the generalized plant maps the disturbances to the performance channels.
In [Figure 1](#figure--fig:jabben07-general-plant), the generalized plant maps the disturbances to the performance channels.
By minimizing the \\(\mathcal{H}\_2\\) system norm of the generalized plant, the variance of the performance channels is minimized.
<a id="figure--fig:jabben07-general-plant"></a>
@@ -169,11 +169,11 @@ By minimizing the \\(\mathcal{H}\_2\\) system norm of the generalized plant, the
#### Using Weighting Filters for Disturbance Modelling {#using-weighting-filters-for-disturbance-modelling}
Since disturbances are generally not white, the system of Figure [1](#figure--fig:jabben07-general-plant) needs to be augmented with so called **disturbance weighting filters**.
Since disturbances are generally not white, the system of [Figure 1](#figure--fig:jabben07-general-plant) needs to be augmented with so called **disturbance weighting filters**.
A disturbance weighting filter gives the disturbance PSD when white noise as input is applied.
This is illustrated in Figure [2](#figure--fig:jabben07-weighting-functions) where a vector of white noise time signals \\(\underbar{w}(t)\\) is filtered through a weighting filter to obtain the colored physical disturbances \\(w(t)\\) with the desired PSD \\(S\_w\\) .
This is illustrated in [Figure 2](#figure--fig:jabben07-weighting-functions) where a vector of white noise time signals \\(\underbar{w}(t)\\) is filtered through a weighting filter to obtain the colored physical disturbances \\(w(t)\\) with the desired PSD \\(S\_w\\) .
The generalized plant framework also allows to include **weighting filters for the performance channels**.
This is useful for three reasons:
@@ -207,7 +207,7 @@ So, to obtain feasible controllers, the performance channel is a combination of
By choosing suitable weighting filters for \\(y\\) and \\(u\\), the performance can be optimized while keeping the controller effort limited:
\\[ \\|z\\|\_{rms}^2 = \left\\| \begin{bmatrix} y \\\ \alpha u \end{bmatrix} \right\\|\_{rms}^2 = \\|y\\|\_{rms}^2 + \alpha^2 \\|u\\|\_{rms}^2 \\]
By calculation \\(\mathcal{H}\_2\\) optimal controllers for increasing \\(\alpha\\) and plotting the performance \\(\\|y\\|\\) vs the controller effort \\(\\|u\\|\\), the curve as depicted in Figure [3](#figure--fig:jabben07-pareto-curve-H2) is obtained.
By calculation \\(\mathcal{H}\_2\\) optimal controllers for increasing \\(\alpha\\) and plotting the performance \\(\\|y\\|\\) vs the controller effort \\(\\|u\\|\\), the curve as depicted in [Figure 3](#figure--fig:jabben07-pareto-curve-H2) is obtained.
<a id="figure--fig:jabben07-pareto-curve-H2"></a>

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content/phdthesis/li01_simul_fault_vibrat_isolat_point.md
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@@ -24,13 +24,13 @@ Year
### Flexure Jointed Hexapods {#flexure-jointed-hexapods}
A general flexible jointed hexapod is shown in [1](#figure--fig:li01-flexure-hexapod-model).
A general flexible jointed hexapod is shown in [Figure 1](#figure--fig:li01-flexure-hexapod-model).
<a id="figure--fig:li01-flexure-hexapod-model"></a>
{{< figure src="/ox-hugo/li01_flexure_hexapod_model.png" caption="<span class=\"figure-number\">Figure 1: </span>A flexure jointed hexapod. {P} is a cartesian coordinate frame located at, and rigidly attached to the payload's center of mass. {B} is the frame attached to the base, and {U} is a universal inertial frame of reference" >}}
Flexure jointed hexapods have been developed to meet two needs illustrated in [2](#figure--fig:li01-quet-dirty-box).
Flexure jointed hexapods have been developed to meet two needs illustrated in [Figure 2](#figure--fig:li01-quet-dirty-box).
<a id="figure--fig:li01-quet-dirty-box"></a>
@@ -43,7 +43,7 @@ On the other hand, the flexures add some complexity to the hexapod dynamics.
Although the flexure joints do eliminate friction and backlash, they add spring dynamics and severely limit the workspace.
Moreover, base and/or payload vibrations become significant contributors to the motion.
The University of Wyoming hexapods (example in [3](#figure--fig:li01-stewart-platform)) are:
The University of Wyoming hexapods (example in [Figure 3](#figure--fig:li01-stewart-platform)) are:
- Cubic (mutually orthogonal)
- Flexure Jointed
@@ -87,7 +87,7 @@ J = \begin{bmatrix}
\end{bmatrix}
\end{equation}
where (see [1](#figure--fig:li01-flexure-hexapod-model)) \\(p\_i\\) denotes the payload attachment point of strut \\(i\\), the prescripts denote the frame of reference, and \\(\hat{u}\_i\\) denotes a unit vector along strut \\(i\\).
where (see [Figure 1](#figure--fig:li01-flexure-hexapod-model)) \\(p\_i\\) denotes the payload attachment point of strut \\(i\\), the prescripts denote the frame of reference, and \\(\hat{u}\_i\\) denotes a unit vector along strut \\(i\\).
To make the dynamic model as simple as possible, the origin of {P} is located at the payload's center of mass.
Thus all \\({}^Pp\_i\\) are found with respect to the center of mass.
@@ -140,7 +140,7 @@ Equation <eq:hexapod_eq_motion> can be rewritten as:
\end{split}
\end{equation}
If the hexapod is designed such that the payload mass/inertia matrix written in the base frame (\\(^BM\_x = {}^B\_PR \cdot {}^PM\_x \cdot {}^B\_PR\_T\\)) and \\(J^T J\\) are diagonal, the dynamics from \\(u\_1\\) to \\(y\\) are decoupled ([4](#figure--fig:li01-decoupling-conf)).
If the hexapod is designed such that the payload mass/inertia matrix written in the base frame (\\(^BM\_x = {}^B\_PR \cdot {}^PM\_x \cdot {}^B\_PR\_T\\)) and \\(J^T J\\) are diagonal, the dynamics from \\(u\_1\\) to \\(y\\) are decoupled ([Figure 4](#figure--fig:li01-decoupling-conf)).
<a id="figure--fig:li01-decoupling-conf"></a>
@@ -152,7 +152,7 @@ Alternatively, a new set of inputs and outputs can be defined:
u\_2 = J^{-1} f\_m, \quad y = J^{-1} (l - l\_r)
\end{equation}
And another decoupled plant is found ([5](#figure--fig:li01-decoupling-conf-bis)):
And another decoupled plant is found ([Figure 5](#figure--fig:li01-decoupling-conf-bis)):
\begin{equation} \label{eq:hexapod\_eq\_motion\_decoup\_2}
\begin{split}
@@ -200,13 +200,13 @@ The control bandwidth is divided as follows:
### Vibration Isolation {#vibration-isolation}
The system is decoupled into six independent SISO subsystems using the architecture shown in [6](#figure--fig:li01-vibration-isolation-control).
The system is decoupled into six independent SISO subsystems using the architecture shown in [Figure 6](#figure--fig:li01-vibration-isolation-control).
<a id="figure--fig:li01-vibration-isolation-control"></a>
{{< figure src="/ox-hugo/li01_vibration_isolation_control.png" caption="<span class=\"figure-number\">Figure 6: </span>Vibration isolation control strategy" >}}
One of the subsystem plant transfer function is shown in [6](#figure--fig:li01-vibration-isolation-control)
One of the subsystem plant transfer function is shown in [Figure 6](#figure--fig:li01-vibration-isolation-control)
<a id="figure--fig:li01-vibration-isolation-control"></a>
@@ -243,7 +243,7 @@ The reason is not explained.
### Pointing Control Techniques {#pointing-control-techniques}
A block diagram of the pointing control system is shown in [8](#figure--fig:li01-pointing-control).
A block diagram of the pointing control system is shown in [Figure 8](#figure--fig:li01-pointing-control).
<a id="figure--fig:li01-pointing-control"></a>
@@ -252,7 +252,7 @@ A block diagram of the pointing control system is shown in [8](#figure--fig:li01
The plant is decoupled into two independent SISO subsystems.
The decoupling matrix consists of the columns of \\(J\\) corresponding to the pointing DoFs.
[9](#figure--fig:li01-transfer-function-angle) shows the measured transfer function of the \\(\theta\_x\\) axis.
[Figure 9](#figure--fig:li01-transfer-function-angle) shows the measured transfer function of the \\(\theta\_x\\) axis.
<a id="figure--fig:li01-transfer-function-angle"></a>
@@ -268,7 +268,7 @@ A typical compensator consists of the following elements:
The unity control bandwidth of the pointing loop is designed to be from **0Hz to 20Hz**.
A feedforward control is added as shown in [10](#figure--fig:li01-feedforward-control).
A feedforward control is added as shown in [Figure 10](#figure--fig:li01-feedforward-control).
\\(C\_f\\) is the feedforward compensator which is a 2x2 diagonal matrix.
Ideally, the feedforward compensator is an invert of the plant dynamics.
@@ -284,7 +284,7 @@ The simultaneous vibration isolation and pointing control is approached in two w
1. **Closing the vibration isolation loop first**: Design and implement the vibration isolation control first, identify the pointing plant when the isolation loops are closed, then implement the pointing compensators.
2. **Closing the pointing loop first**: Reverse order.
[11](#figure--fig:li01-parallel-control) shows a parallel control structure where \\(G\_1(s)\\) is the dynamics from input force to output strut length.
[Figure 11](#figure--fig:li01-parallel-control) shows a parallel control structure where \\(G\_1(s)\\) is the dynamics from input force to output strut length.
<a id="figure--fig:li01-parallel-control"></a>
@@ -302,16 +302,16 @@ However, the interaction between loops may affect the transfer functions of the
The dynamic interaction effect:
- Only happens in the unity bandwidth of the loop transmission of the first closed loop.
- Affect the closed loop transmission of the loop first closed (see [12](#figure--fig:li01-closed-loop-pointing) and [13](#figure--fig:li01-closed-loop-vibration))
- Affect the closed loop transmission of the loop first closed (see [Figure 12](#figure--fig:li01-closed-loop-pointing) and [Figure 13](#figure--fig:li01-closed-loop-vibration))
As shown in [12](#figure--fig:li01-closed-loop-pointing), the peak resonance of the pointing loop increase after the isolation loop is closed.
As shown in [Figure 12](#figure--fig:li01-closed-loop-pointing), the peak resonance of the pointing loop increase after the isolation loop is closed.
The resonances happen at both crossovers of the isolation loop (15Hz and 50Hz) and they may show of loss of robustness.
<a id="figure--fig:li01-closed-loop-pointing"></a>
{{< figure src="/ox-hugo/li01_closed_loop_pointing.png" caption="<span class=\"figure-number\">Figure 12: </span>Closed-loop transfer functions \\(\theta\_y/\theta\_{y\_d}\\) of the pointing loop before and after the vibration isolation loop is closed" >}}
The same happens when first closing the vibration isolation loop and after the pointing loop ([13](#figure--fig:li01-closed-loop-vibration)).
The same happens when first closing the vibration isolation loop and after the pointing loop ([Figure 13](#figure--fig:li01-closed-loop-vibration)).
The first peak resonance of the vibration isolation loop at 15Hz is increased when closing the pointing loop.
<a id="figure--fig:li01-closed-loop-vibration"></a>
@@ -328,7 +328,7 @@ Thus, it is recommended to design and implement the isolation control system fir
### Experimental results {#experimental-results}
Two hexapods are stacked ([14](#figure--fig:li01-test-bench)):
Two hexapods are stacked ([Figure 14](#figure--fig:li01-test-bench)):
- the bottom hexapod is used to generate disturbances matching candidate applications
- the top hexapod provide simultaneous vibration isolation and pointing control
@@ -338,7 +338,7 @@ Two hexapods are stacked ([14](#figure--fig:li01-test-bench)):
{{< figure src="/ox-hugo/li01_test_bench.png" caption="<span class=\"figure-number\">Figure 14: </span>Stacked Hexapods" >}}
First, the vibration isolation and pointing controls were implemented separately.
Using the vibration isolation control alone, no attenuation is achieved below 1Hz as shown in [15](#figure--fig:li01-vibration-isolation-control-results).
Using the vibration isolation control alone, no attenuation is achieved below 1Hz as shown in [Figure 15](#figure--fig:li01-vibration-isolation-control-results).
<a id="figure--fig:li01-vibration-isolation-control-results"></a>
@@ -349,7 +349,7 @@ The simultaneous control is of dual use:
- it provide simultaneous pointing and isolation control
- it can also be used to expand the bandwidth of the isolation control to low frequencies because the pointing loops suppress pointing errors due to both base vibrations and tracking
The results of simultaneous control is shown in [16](#figure--fig:li01-simultaneous-control-results) where the bandwidth of the isolation control is expanded to very low frequency.
The results of simultaneous control is shown in [Figure 16](#figure--fig:li01-simultaneous-control-results) where the bandwidth of the isolation control is expanded to very low frequency.
<a id="figure--fig:li01-simultaneous-control-results"></a>

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content/phdthesis/monkhorst04_dynam_error_budget.md
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@@ -106,7 +106,7 @@ Find a controller \\(C\_{\mathcal{H}\_2}\\) which minimizes the \\(\mathcal{H}\_
In order to synthesize an \\(\mathcal{H}\_2\\) controller that will minimize the output error, the total system including disturbances needs to be modeled as a system with zero mean white noise inputs.
This is done by using weighting filter \\(V\_w\\), of which the output signal has a PSD \\(S\_w(f)\\) when the input is zero mean white noise (Figure [1](#figure--fig:monkhorst04-weighting-filter)).
This is done by using weighting filter \\(V\_w\\), of which the output signal has a PSD \\(S\_w(f)\\) when the input is zero mean white noise ([Figure 1](#figure--fig:monkhorst04-weighting-filter)).
<a id="figure--fig:monkhorst04-weighting-filter"></a>
@@ -119,7 +119,7 @@ The PSD \\(S\_w(f)\\) of the weighted signal is:
Given \\(S\_w(f)\\), \\(V\_w(f)\\) can be obtained using a technique called _spectral factorization_.
However, this can be avoided if the modeling of the disturbances is directly done in terms of weighting filters.
Output weighting filters can also be used to scale different outputs relative to each other (Figure [2](#figure--fig:monkhorst04-general-weighted-plant)).
Output weighting filters can also be used to scale different outputs relative to each other ([Figure 2](#figure--fig:monkhorst04-general-weighted-plant)).
<a id="figure--fig:monkhorst04-general-weighted-plant"></a>
@@ -128,7 +128,7 @@ Output weighting filters can also be used to scale different outputs relative to
#### Output scaling and the Pareto curve {#output-scaling-and-the-pareto-curve}
In this research, the outputs of the closed loop system (Figure [3](#figure--fig:monkhorst04-closed-loop-H2)) are:
In this research, the outputs of the closed loop system ([Figure 3](#figure--fig:monkhorst04-closed-loop-H2)) are:
- the performance (error) signal \\(e\\)
- the controller output \\(u\\)

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content/phdthesis/rankers98_machin.md
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@@ -170,7 +170,7 @@ 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](#figure--fig:rankers98-basic-el-mech-servo).
A block diagram representation of a typical servo-system is shown in [Figure 1](#figure--fig:rankers98-basic-el-mech-servo).
The main task of the system is to 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).
@@ -187,7 +187,7 @@ 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](#figure--fig:rankers98-basic-elastic-struct)).
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](#figure--fig:rankers98-basic-elastic-struct)).
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="figure--fig:rankers98-basic-elastic-struct"></a>
@@ -206,7 +206,7 @@ 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](#figure--fig:rankers98-feedforward-example).
Most modern servo-systems have not only a feedback section, but also a **feedforward** section, as indicated in [Figure 3](#figure--fig:rankers98-feedforward-example).
<a id="figure--fig:rankers98-feedforward-example"></a>
@@ -253,7 +253,7 @@ 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](#figure--fig:rankers98-actuator-flexibility)) 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](#figure--fig:rankers98-actuator-flexibility)) due to **compliance between the motor and the load**.
<a id="figure--fig:rankers98-actuator-flexibility"></a>
@@ -265,7 +265,7 @@ 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](#figure--fig:rankers98-guiding-flexibility-planar)).
In the present discussion, a planar actuator with three degrees of freedom will be considered ([Figure 5](#figure--fig:rankers98-guiding-flexibility-planar)).
<a id="figure--fig:rankers98-guiding-flexibility-planar"></a>
@@ -287,7 +287,7 @@ The last category of dynamic phenomena results from the **limited mass and stiff
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](#figure--fig:rankers98-limited-m-k-stationary-machine-part)).
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](#figure--fig:rankers98-limited-m-k-stationary-machine-part)).
<a id="figure--fig:rankers98-limited-m-k-stationary-machine-part"></a>
@@ -302,7 +302,7 @@ The effect of frame vibrations is even worse where the quality of positioning of
To understand and describe the behaviour of a mechanical system in a quantitative way, one usually sets up a model of the system.
The mathematical description of such a model with a finite number of DoF consists of a set of ordinary differential equations.
Although in the case of simple systems, such as illustrated in Figure [7](#figure--fig:rankers98-1dof-system) these equations may be very understandable, in the case of complex systems, the set of differential equations itself gives only limited insight, and mainly serves as a basis for numerical simulations.
Although in the case of simple systems, such as illustrated in [Figure 7](#figure--fig:rankers98-1dof-system) these equations may be very understandable, in the case of complex systems, the set of differential equations itself gives only limited insight, and mainly serves as a basis for numerical simulations.
<a id="figure--fig:rankers98-1dof-system"></a>
@@ -484,14 +484,14 @@ The overall transfer function can be found by summation of the individual modal
Due to the equivalence with the differential equations of a single mass spring system, equation <eq:eoq_modal_i> is often represented by a single mass spring system on which a force \\(f^\prime = \phi\_i^T f\\) acts.
However, this representation implies an important loss of information because it neglects all information about the mode-shape vector.
Consider the system in Figure [8](#figure--fig:rankers98-mode-trad-representation) for which the three mode shapes are depicted in the traditional graphical representation.
Consider the system in [Figure 8](#figure--fig:rankers98-mode-trad-representation) for which the three mode shapes are depicted in the traditional graphical representation.
In this representation, the physical DoF are located at fixed positions and the mode shapes displacement is indicated by the length of an arrow.
<a id="figure--fig:rankers98-mode-trad-representation"></a>
{{< figure src="/ox-hugo/rankers98_mode_trad_representation.png" caption="<span class=\"figure-number\">Figure 8: </span>System and traditional graphical representation of modes" >}}
Alternatively, considering that for each mode the mode shape vector defined a constant relation between the various physical DoF, one could also **represent a mode shape by a lever** (Figure [9](#figure--fig:rankers98-mode-new-representation)).
Alternatively, considering that for each mode the mode shape vector defined a constant relation between the various physical DoF, one could also **represent a mode shape by a lever** ([Figure 9](#figure--fig:rankers98-mode-new-representation)).
<div class="important">
@@ -503,13 +503,13 @@ System with no, very little, or proportional damping exhibit real mode shape vec
Consequently, the respective DoF can only be in phase or in opposite phase.
All DoF on the same side of the rotation point have identical phases, whereas DoF on opposite sides have opposite phases.
The modal DoF \\(q\_i\\) can be interpreted as the displacement at a distance "1" from the pivot point (Figure [9](#figure--fig:rankers98-mode-new-representation)).
The modal DoF \\(q\_i\\) can be interpreted as the displacement at a distance "1" from the pivot point ([Figure 9](#figure--fig:rankers98-mode-new-representation)).
<a id="figure--fig:rankers98-mode-new-representation"></a>
{{< figure src="/ox-hugo/rankers98_mode_new_representation.png" caption="<span class=\"figure-number\">Figure 9: </span>System and new graphical representation of mode-shape" >}}
In the case of a lumped mass model, as in the previous example, it is possible to indicate at each physical DoF on the modal lever the corresponding physical mass, as shown in Figure [10](#figure--fig:rankers98-mode-2-lumped-masses) (a).
In the case of a lumped mass model, as in the previous example, it is possible to indicate at each physical DoF on the modal lever the corresponding physical mass, as shown in [Figure 10](#figure--fig:rankers98-mode-2-lumped-masses) (a).
The resulting moment of inertia \\(J\_i\\) of the i-th modal lever then is:
\begin{equation}
@@ -524,7 +524,7 @@ m\_i = \phi\_j^T M \phi\_j = \sum\_{k=1}^n m\_k \phi\_{ik}^2
As a result of this, the **modal mass** \\(m\_i\\) could be interpreted as the resulting mass moment of inertia of the modal lever, or alternatively as a **mass located at a distance "1" from the pivot point**.
The transition from physical masses to modal masses is illustrated in Figure [10](#figure--fig:rankers98-mode-2-lumped-masses) for the mode 2 of the example system.
The transition from physical masses to modal masses is illustrated in [Figure 10](#figure--fig:rankers98-mode-2-lumped-masses) for the mode 2 of the example system.
The modal stiffness \\(k\_2\\) is simply calculated via the relation between natural frequency, mass and stiffness:
\begin{equation}
@@ -537,7 +537,7 @@ k\_i = \omega\_i^2 m\_i
Let's now consider the effect of excitation forces that act on the physical DoF.
The scalar product \\(\phi\_{ik}f\_k\\) of each force component with the corresponding element of the mode shape vector can be seen as the moment that acts on the modal level, or as an equivalent force that acts at the location of \\(q\_i\\) on the lever.
Based on the graphical representation in Figure [11](#figure--fig:rankers98-lever-representation-with-force), it is not difficult to understand the contribution of mode i to the transfer function \\(x\_l/f\_k\\):
Based on the graphical representation in [Figure 11](#figure--fig:rankers98-lever-representation-with-force), it is not difficult to understand the contribution of mode i to the transfer function \\(x\_l/f\_k\\):
\begin{equation}
\boxed{\left( \frac{x\_l}{f\_k} \right)\_i = \frac{\phi\_{ik}\phi\_{il}}{m\_i s^2 + k\_i}}
@@ -556,7 +556,7 @@ This linear combination of physical DoF, which will be called "User DoF" can be
x\_u = b\_1 x\_1 + \dots + b\_n x\_n = b^T x
\end{equation}
User DoF can be indicated on the modal lever, as illustrated in Figure [12](#figure--fig:rankers98-representation-user-dof) for a user DoF \\(x\_u = x\_3 - x\_2\\).
User DoF can be indicated on the modal lever, as illustrated in [Figure 12](#figure--fig:rankers98-representation-user-dof) for a user DoF \\(x\_u = x\_3 - x\_2\\).
The location of this user DoF \\(x\_u\\) with respect to the pivot point of modal lever \\(i\\) is defined by \\(\phi\_{iu}\\):
\begin{equation}
@@ -575,7 +575,7 @@ Even though the dimension mode vector can be very large, only **three user DoF**
<div class="exampl">
To illustrate this, a servo controlled positioning device is shown in Figure [13](#figure--fig:rankers98-servo-system).
To illustrate this, a servo controlled positioning device is shown in [Figure 13](#figure--fig:rankers98-servo-system).
The task of the device is to position the payload with respect to a tool that is mounted to the machine frame.
The actual accuracy of the machine is determined by the relative motion of these two components (actual output).
However, direct measurement of the distance between the tool and the payload is not possible and therefore the control action is based on the measured distance between a sensor and the slide on which the payload is mounted (measured output).
@@ -605,7 +605,7 @@ These effective modal parameters can be used very effectively in understanding t
<div class="exampl">
The eigenvalue analysis of the two mass spring system in Figure [14](#figure--fig:rankers98-example-2dof) leads to the modal results summarized in Table [1](#table--tab:2dof-example-modal-params) and which are graphically represented in Figure [15](#figure--fig:rankers98-example-2dof-modal).
The eigenvalue analysis of the two mass spring system in [Figure 14](#figure--fig:rankers98-example-2dof) leads to the modal results summarized in [Table 1](#table--tab:2dof-example-modal-params) and which are graphically represented in [Figure 15](#figure--fig:rankers98-example-2dof-modal).
<a id="figure--fig:rankers98-example-2dof"></a>
@@ -622,7 +622,7 @@ whereas the modal stiffnesses follow from \\(k\_i = \omega\_i^2 m\_i\\).
<a id="table--tab:2dof-example-modal-params"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--tab:2dof-example-modal-params">Table 1</a></span>:
<span class="table-number"><a href="#table--tab:2dof-example-modal-params">Table 1</a>:</span>
Modal results for the two mass spring system
</div>
@@ -638,11 +638,11 @@ whereas the modal stiffnesses follow from \\(k\_i = \omega\_i^2 m\_i\\).
{{< figure src="/ox-hugo/rankers98_example_2dof_modal.png" caption="<span class=\"figure-number\">Figure 15: </span>Graphical representation of modes and modal parameters of the two mass spring system" >}}
From these results, the effective modal parameters for each mode, and for each individual DoF can be defined using equations <eq:m_modal_eff> and <eq:k_modal_eff>.
The results are summarized in Table [2](#table--tab:2dof-example-modal-params-eff).
The results are summarized in [Table 2](#table--tab:2dof-example-modal-params-eff).
<a id="table--tab:2dof-example-modal-params-eff"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--tab:2dof-example-modal-params-eff">Table 2</a></span>:
<span class="table-number"><a href="#table--tab:2dof-example-modal-params-eff">Table 2</a>:</span>
Effective modal parameters for the two mass spring system
</div>
@@ -653,8 +653,8 @@ The results are summarized in Table [2](#table--tab:2dof-example-modal-params-ef
| Effective stiff - DoF 1 | \\(k\_{\text{eff},11} = 1.02 \cdot 10^7 N/m\\) | \\(k\_{\text{eff},21} = 1.02 \cdot 10^9 N/m\\) |
| Effective stiff - DoF 2 | \\(k\_{\text{eff},12} = 0.84 \cdot 10^7 N/m\\) | \\(k\_{\text{eff},22} = 1.25 \cdot 10^7 N/m\\) |
The effective modal parameters can then be used in the graphical representation of Figure [16](#figure--fig:rankers98-example-2dof-effective-modal).
Based on this representation, it is now very easy to construct the individual modal contributions to the frequency response function \\(x\_1/F\_1\\) of the example system (Figure [17](#figure--fig:rankers98-2dof-example-frf)).
The effective modal parameters can then be used in the graphical representation of [Figure 16](#figure--fig:rankers98-example-2dof-effective-modal).
Based on this representation, it is now very easy to construct the individual modal contributions to the frequency response function \\(x\_1/F\_1\\) of the example system ([Figure 17](#figure--fig:rankers98-2dof-example-frf)).
<a id="figure--fig:rankers98-example-2dof-effective-modal"></a>
@@ -662,7 +662,7 @@ Based on this representation, it is now very easy to construct the individual mo
One can observe that the low frequency part of each modal contribution corresponds to the inverse of the calculated effective modal mass stiffness at DoF \\(x\_1\\) whereas the high frequency contribution is defined by the effective modal mass.
In the final Bode diagram (Figure [17](#figure--fig:rankers98-2dof-example-frf), below) one can observe an interference of the two modal contributions in the frequency range of the second natural frequency, which in this example leads to a combination of an anti-resonance an a resonance.
In the final Bode diagram ([Figure 17](#figure--fig:rankers98-2dof-example-frf), below) one can observe an interference of the two modal contributions in the frequency range of the second natural frequency, which in this example leads to a combination of an anti-resonance an a resonance.
<a id="figure--fig:rankers98-2dof-example-frf"></a>
@@ -690,7 +690,7 @@ The technique furthermore gives an indication of the amount of frequency shift t
<div class="exampl">
Assuming that one is asked to increase the natural frequency of the mode corresponding to Figure [18](#figure--fig:rankers98-example-3dof-sensitivity) by attaching a linear spring \\(k\\) between two of the three represented DoF.
Assuming that one is asked to increase the natural frequency of the mode corresponding to [Figure 18](#figure--fig:rankers98-example-3dof-sensitivity) by attaching a linear spring \\(k\\) between two of the three represented DoF.
As the relative motion between \\(x\_A\\) and \\(x\_B\\) is the largest of all possible combinations, this is the choice that will maximize the natural frequency of the mode.
<a id="figure--fig:rankers98-example-3dof-sensitivity"></a>
@@ -713,13 +713,13 @@ f\_{\text{new},i}(\Delta k) &= \frac{1}{2\pi}\sqrt{\frac{k\_{\text{eff},i} + \De
<div class="exampl">
Let's use the two mass spring system in Figure [14](#figure--fig:rankers98-example-2dof) as an example.
Let's use the two mass spring system in [Figure 14](#figure--fig:rankers98-example-2dof) as an example.
In order to analyze the effect of an extra mass at \\(x\_2\\), the effective modal mass at that DoF needs to be known for both modes (see Table [2](#table--tab:2dof-example-modal-params-eff)).
In order to analyze the effect of an extra mass at \\(x\_2\\), the effective modal mass at that DoF needs to be known for both modes (see [Table 2](#table--tab:2dof-example-modal-params-eff)).
Then using equation <eq:sensitivity_add_m>, one can estimate the effect of an extra mass \\(\Delta m = 1 kg\\) added to \\(m\_2\\).
To estimate the influence of extra stiffness between the two DoF, one needs to calculate the effective modal stiffness that corresponds to the relative motion between \\(x\_2\\) and \\(x\_1\\).
This can be graphically done as shown in Figure [19](#figure--fig:rankers98-example-sensitivity-2dof):
This can be graphically done as shown in [Figure 19](#figure--fig:rankers98-example-sensitivity-2dof):
\begin{align}
k\_{\text{eff},1,(2-1)} &= 0.46 \cdot 10^7 / 0.07^2 = 93.9 \cdot 10^7 N/m \\\\
@@ -728,11 +728,11 @@ k\_{\text{eff},2,(2-1)} &= 1.23 \cdot 10^7 / 1.1^2 = 1.0 \cdot 10^7 N/m
And using equation <eq:sensitivity_add_m>, the effect of additional stiffness on the frequency of the two modes can be computed.
The results are summarized in Table [3](#table--tab:example-sensitivity-2dof-results).
The results are summarized in [Table 3](#table--tab:example-sensitivity-2dof-results).
<a id="table--tab:example-sensitivity-2dof-results"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--tab:example-sensitivity-2dof-results">Table 3</a></span>:
<span class="table-number"><a href="#table--tab:example-sensitivity-2dof-results">Table 3</a>:</span>
Sensitivity analysis results
</div>
@@ -752,7 +752,7 @@ The results are summarized in Table [3](#table--tab:example-sensitivity-2dof-res
### Modal Superposition {#modal-superposition}
Previously, the lever representation was used only to represent the individual mode shapes.
In the mechanism shown in Figure [20](#figure--fig:rankers98-addition-of-motion), the motion of the output \\(y\\) is equals to the sum of the motion of the two inputs \\(x\_1\\) and \\(x\_2\\).
In the mechanism shown in [Figure 20](#figure--fig:rankers98-addition-of-motion), the motion of the output \\(y\\) is equals to the sum of the motion of the two inputs \\(x\_1\\) and \\(x\_2\\).
<a id="figure--fig:rankers98-addition-of-motion"></a>
@@ -764,7 +764,7 @@ This approach can be applied to the concept of modal superposition, which expres
x\_k(t) = \sum\_{i=1}^n \phi\_{ik} q\_i(t) = \sum\_{i=1}^n x\_{ki}(t)
\end{equation}
Combining the concept of summation of modal contribution with the lever representation of mode shapes leads to Figure [21](#figure--fig:rankers98-conversion-modal-to-physical), which is a visualization of the transformation between the modal and the physical domains.
Combining the concept of summation of modal contribution with the lever representation of mode shapes leads to [Figure 21](#figure--fig:rankers98-conversion-modal-to-physical), which is a visualization of the transformation between the modal and the physical domains.
<a id="figure--fig:rankers98-conversion-modal-to-physical"></a>
@@ -777,7 +777,7 @@ The "rigid body modes" usually refer to the lower natural frequencies of a machi
This is misleading at it suggests that the structure exhibits no internal deformation.
A better term for such a mode would be **suspension mode**.
To illustrate the important of the internal deformation, a very simplified physical model of a precision machine is considered (Figure [22](#figure--fig:rankers98-suspension-mode-machine)).
To illustrate the important of the internal deformation, a very simplified physical model of a precision machine is considered ([Figure 22](#figure--fig:rankers98-suspension-mode-machine)).
<a id="figure--fig:rankers98-suspension-mode-machine"></a>
@@ -828,7 +828,7 @@ The interaction between the desired (rigid body) motion and the dynamics of one
### Basic Characteristics of Mechanical FRF {#basic-characteristics-of-mechanical-frf}
Consider the position control loop of Figure [23](#figure--fig:rankers98-mechanical-servo-system).
Consider the position control loop of [Figure 23](#figure--fig:rankers98-mechanical-servo-system).
<a id="figure--fig:rankers98-mechanical-servo-system"></a>
@@ -840,7 +840,7 @@ In the ideal situation the mechanical system behaves as one rigid body with mass
\frac{x\_{\text{servo}}}{F\_{\text{servo}}} = \frac{1}{m s^2}
\end{equation}
The corresponding Bode and Nyquist plots and shown in Figure [24](#figure--fig:rankers98-ideal-bode-nyquist).
The corresponding Bode and Nyquist plots and shown in [Figure 24](#figure--fig:rankers98-ideal-bode-nyquist).
<a id="figure--fig:rankers98-ideal-bode-nyquist"></a>
@@ -868,7 +868,7 @@ which simplifies equation <eq:effect_one_mode> to:
Equation <eq:effect_one_mode_simplified> will be the basis for the discussion of the various patterns that can be observe in the frequency response functions and the effect of resonances on servo stability.
Three different types of intersection pattern can be found in the amplitude plot as shown in Figure [25](#figure--fig:rankers98-frf-effect-alpha).
Three different types of intersection pattern can be found in the amplitude plot as shown in [Figure 25](#figure--fig:rankers98-frf-effect-alpha).
Depending on the absolute value of \\(\alpha\\) one can observe:
- \\(|\alpha| < 1\\): two intersections
@@ -881,7 +881,7 @@ The interaction between the rigid body motion and the additional mode will not o
{{< figure src="/ox-hugo/rankers98_frf_effect_alpha.png" caption="<span class=\"figure-number\">Figure 25: </span>Contribution of rigid-body motion and modal dynamics to the amplitude and phase of FRF for various values of \\(\alpha\\)" >}}
The general shape of the overall FRF can be constructed for all cases (Figure [26](#figure--fig:rankers98-final-frf-alpha)).
The general shape of the overall FRF can be constructed for all cases ([Figure 26](#figure--fig:rankers98-final-frf-alpha)).
Interesting points are the interaction of the two parts at the frequency that corresponds to an intersection in the amplitude plot.
At this frequency the magnitudes are equal, so it depends on the phase of the two contributions whether they cancel each other, thus leading to a zero, or just add up.
@@ -891,7 +891,7 @@ At this frequency the magnitudes are equal, so it depends on the phase of the tw
<div class="important">
When analyzing the plots of Figure [26](#figure--fig:rankers98-final-frf-alpha), four different types of FRF can be found:
When analyzing the plots of [Figure 26](#figure--fig:rankers98-final-frf-alpha), four different types of FRF can be found:
- -2 slope / zero / pole / -2 slope (\\(\alpha > 0\\))
- -2 slope / pole / zero / -2 slope (\\(-1 < \alpha < 0\\))
@@ -900,7 +900,7 @@ When analyzing the plots of Figure [26](#figure--fig:rankers98-final-frf-alpha),
</div>
All cases are shown in Figure [27](#figure--fig:rankers98-interaction-shapes).
All cases are shown in [Figure 27](#figure--fig:rankers98-interaction-shapes).
<a id="figure--fig:rankers98-interaction-shapes"></a>
@@ -921,13 +921,13 @@ f\_{lp} &= 4 \cdot f\_b
\end{align\*}
with \\(f\_b\\) the bandwidth frequency.
The asymptotic amplitude plot is shown in Figure [28](#figure--fig:rankers98-pid-amplitude).
The asymptotic amplitude plot is shown in [Figure 28](#figure--fig:rankers98-pid-amplitude).
<a id="figure--fig:rankers98-pid-amplitude"></a>
{{< figure src="/ox-hugo/rankers98_pid_amplitude.png" caption="<span class=\"figure-number\">Figure 28: </span>Typical crossover frequencies of a PID controller with 2nd order low pass filtering" >}}
With these settings, the open loop response of the position loop (controller + mechanics) looks like Figure [29](#figure--fig:rankers98-ideal-frf-pid).
With these settings, the open loop response of the position loop (controller + mechanics) looks like [Figure 29](#figure--fig:rankers98-ideal-frf-pid).
<a id="figure--fig:rankers98-ideal-frf-pid"></a>
@@ -937,10 +937,10 @@ With these settings, the open loop response of the position loop (controller + m
Conclusions are:
- A "-2 slope / zero / pole / -2 slope" characteristic leads to a phase lead, and is therefore potentially destabilizing in the low frequency (Figure [30](#figure--fig:rankers98-zero-pole-low-freq)) and high frequency (Figure [32](#figure--fig:rankers98-zero-pole-high-freq)) regions.
In the medium frequency region (Figure [31](#figure--fig:rankers98-zero-pole-medium-freq)), it adds an extra phase lead to the already existing margin, which does not harm the stability.
- A "-2 slope / zero / pole / -2 slope" characteristic leads to a phase lead, and is therefore potentially destabilizing in the low frequency ([Figure 30](#figure--fig:rankers98-zero-pole-low-freq)) and high frequency ([Figure 32](#figure--fig:rankers98-zero-pole-high-freq)) regions.
In the medium frequency region ([Figure 31](#figure--fig:rankers98-zero-pole-medium-freq)), it adds an extra phase lead to the already existing margin, which does not harm the stability.
- A "-2 slope / pole / zero / -2 slope" combination has the reverse effect.
It is potentially destabilizing in the medium frequency range (Figure [34](#figure--fig:rankers98-pole-zero-medium-freq)) and is harmless in the low (Figure [33](#figure--fig:rankers98-pole-zero-low-freq)) and high frequency (Figure [35](#figure--fig:rankers98-pole-zero-high-freq)) ranges.
It is potentially destabilizing in the medium frequency range ([Figure 34](#figure--fig:rankers98-pole-zero-medium-freq)) and is harmless in the low ([Figure 33](#figure--fig:rankers98-pole-zero-low-freq)) and high frequency ([Figure 35](#figure--fig:rankers98-pole-zero-high-freq)) ranges.
- The "-2 slope / pole / -4 slope" behavior always has a devastating effect on the stability of the loop if located in the low of medium frequency ranges.
These conclusions may differ for different mass ratio \\(\alpha\\).
@@ -977,13 +977,13 @@ These conclusions may differ for different mass ratio \\(\alpha\\).
#### Actuator Flexibility {#actuator-flexibility}
Figure [36](#figure--fig:rankers98-2dof-actuator-flexibility) shows the schematic representation of a system with a certain compliance between the motor and the load.
[Figure 36](#figure--fig:rankers98-2dof-actuator-flexibility) shows the schematic representation of a system with a certain compliance between the motor and the load.
<a id="figure--fig:rankers98-2dof-actuator-flexibility"></a>
{{< figure src="/ox-hugo/rankers98_2dof_actuator_flexibility.png" caption="<span class=\"figure-number\">Figure 36: </span>Servo system with actuator flexibility - Schematic representation" >}}
The corresponding modes are shown in Figure [37](#figure--fig:rankers98-2dof-modes-act-flex).
The corresponding modes are shown in [Figure 37](#figure--fig:rankers98-2dof-modes-act-flex).
<a id="figure--fig:rankers98-2dof-modes-act-flex"></a>
@@ -998,7 +998,7 @@ The following transfer function must be considered:
\end{align}
with \\(\alpha = m\_2/m\_1\\) (mass ratio) relates the "mass" of the additional modal contribution to the mass of the rigid body motion.
The resulting FRF exhibit a "-2 slope / zero / pole / -2 slope" (Figure [38](#figure--fig:rankers98-2dof-act-flex-frf)).
The resulting FRF exhibit a "-2 slope / zero / pole / -2 slope" ([Figure 38](#figure--fig:rankers98-2dof-act-flex-frf)).
<a id="figure--fig:rankers98-2dof-act-flex-frf"></a>
@@ -1029,7 +1029,7 @@ Now we are interested by the following transfer function:
\frac{x\_2}{F\_{\text{servo}}} = = \frac{1}{m\_1 + m\_2} \left( \frac{1}{s^2} - \frac{1}{s^2 + \omega\_2^2} \right)
\end{equation}
The mass ratio \\(\alpha\\) equal -1, and thus the FRF will be of type "-2 slope / pole / -4 slope" (Figure [39](#figure--fig:rankers98-2dof-act-flex-meas-load-frf)).
The mass ratio \\(\alpha\\) equal -1, and thus the FRF will be of type "-2 slope / pole / -4 slope" ([Figure 39](#figure--fig:rankers98-2dof-act-flex-meas-load-frf)).
<a id="figure--fig:rankers98-2dof-act-flex-meas-load-frf"></a>
@@ -1046,7 +1046,7 @@ Guideline in presence of actuator flexibility with measurement at the load posit
#### Guiding System Flexibility {#guiding-system-flexibility}
Here, the influence of a limited guiding stiffness (Figure [40](#figure--fig:rankers98-2dof-guiding-flex)) on the FRF of such an actuator system will be analyzed.
Here, the influence of a limited guiding stiffness ([Figure 40](#figure--fig:rankers98-2dof-guiding-flex)) on the FRF of such an actuator system will be analyzed.
The servo force \\(F\_{\text{servo}}\\) acts at a certain distance \\(a\_F\\) with respect to the center of gravity, and the servo position is measured at a distance \\(a\_s\\) with respect to the center of gravity.
Due to the symmetry of the system, the Y motion is decoupled from the X and \\(\phi\\) motions and can therefore be omitted in this analysis.
@@ -1055,7 +1055,7 @@ Due to the symmetry of the system, the Y motion is decoupled from the X and \\(\
{{< figure src="/ox-hugo/rankers98_2dof_guiding_flex.png" caption="<span class=\"figure-number\">Figure 40: </span>2DoF rigid body model of actuator with flexibility of the guiding system" >}}
Considering the two relevant modes (Figures [41](#figure--fig:rankers98-2dof-guiding-flex-x-mode) and [42](#figure--fig:rankers98-2dof-guiding-flex-rock-mode)), the resulting transfer function \\(x\_{\text{servo}}/F\_{\text{servo}}\\) can be constructed from the contributions of the individual modes:
Considering the two relevant modes ([Figure 41](#figure--fig:rankers98-2dof-guiding-flex-x-mode) and [Figure 42](#figure--fig:rankers98-2dof-guiding-flex-rock-mode)), the resulting transfer function \\(x\_{\text{servo}}/F\_{\text{servo}}\\) can be constructed from the contributions of the individual modes:
\begin{equation}
\frac{x\_{\text{servo}}}{F\_{\text{servo}}} = \frac{1}{ms^2} + \frac{a\_s a\_F}{Js^2 + 2cb^2}
@@ -1103,7 +1103,7 @@ As this point, the resonance will not be present in the FRF.
#### Limited Mass and Stiffness of Stationary Machine Part {#limited-mass-and-stiffness-of-stationary-machine-part}
Figure [43](#figure--fig:rankers98-frame-dynamics-2dof) shows a simple model of a translational direct drive motion on a frame with limited mass and stiffness, and in which the control system operates on the measured position \\(x\_{\text{servo}} = x\_2 - x\_1\\).
[Figure 43](#figure--fig:rankers98-frame-dynamics-2dof) shows a simple model of a translational direct drive motion on a frame with limited mass and stiffness, and in which the control system operates on the measured position \\(x\_{\text{servo}} = x\_2 - x\_1\\).
<a id="figure--fig:rankers98-frame-dynamics-2dof"></a>
@@ -1138,7 +1138,7 @@ Guidelines regarding frame motion:
<div class="sum">
The amount of contribution of a certain mode (Figure [44](#figure--fig:rankers98-mode-representation-guideline)) to the open loop response and its interaction with the desired motion is determined by the modal mass and stiffness, but also by the location of the driving force and the location of the response DoF.
The amount of contribution of a certain mode ([Figure 44](#figure--fig:rankers98-mode-representation-guideline)) to the open loop response and its interaction with the desired motion is determined by the modal mass and stiffness, but also by the location of the driving force and the location of the response DoF.
<a id="figure--fig:rankers98-mode-representation-guideline"></a>
@@ -1167,7 +1167,7 @@ If such a modification is not required and the modes are not excited by some oth
### Steps in a Modelling Activity {#steps-in-a-modelling-activity}
One can distinguish at least four steps in any modelling activity (Figure [45](#figure--fig:rankers98-steps-modelling)).
One can distinguish at least four steps in any modelling activity ([Figure 45](#figure--fig:rankers98-steps-modelling)).
<a id="figure--fig:rankers98-steps-modelling"></a>
@@ -1176,10 +1176,10 @@ One can distinguish at least four steps in any modelling activity (Figure [45](#
1. The first step consists of a translation of the real structure or initial design drawing of a structure in a **physical model**.
Such a physical model is a simplification of the reality, but contains all relations that are considered to be important to describe the investigated phenomenon.
This step requires experience and engineering judgment in order to determine which simplifications are valid.
See for example Figure [46](#figure--fig:rankers98-illustration-first-two-steps).
See for example [Figure 46](#figure--fig:rankers98-illustration-first-two-steps).
2. Once a physical model has been derived, the second step consists of translating this physical model into a **mathematical model**.
The real world is now represented by a set of differential equations.
This step is fairly straightforward, because it is based and existing approaches and rules (Example in Figure [46](#figure--fig:rankers98-illustration-first-two-steps)).
This step is fairly straightforward, because it is based and existing approaches and rules (Example in [Figure 46](#figure--fig:rankers98-illustration-first-two-steps)).
3. The third step consists of actuator **simulation run**, the outcome of which is the value of some quantity (for instance stress of some part, resonance frequency, FRF, etc.).
4. The final step is the **interpretation** of results.
Here, the calculated results and previously defined specifications are compared.
@@ -1233,7 +1233,7 @@ Therefore, computer simulations should be regarded as a means to guide the desig
<div class="exampl">
This three step modelling approach is now illustrated by the example of a fast and accurate pattern generator in which an optical unit has to move in X and Y directions with respect to a work piece (Figure [47](#figure--fig:rankers98-pattern-generator)).
This three step modelling approach is now illustrated by the example of a fast and accurate pattern generator in which an optical unit has to move in X and Y directions with respect to a work piece ([Figure 47](#figure--fig:rankers98-pattern-generator)).
<a id="figure--fig:rankers98-pattern-generator"></a>
@@ -1251,7 +1251,7 @@ Based on the required throughput of the machine, an acceleration level of \\(1m/
<div class="important">
One of the most crucial step in the modelling process is the **definition of proper criteria on the basis of which the simulation results can be judged**.
In most cases, this implies that functional system-specifications in combination with **assumed imperfections and disturbances** need to be translated into **dynamics and control specifications** (Figure [48](#figure--fig:rankers98-system-performance-spec)).
In most cases, this implies that functional system-specifications in combination with **assumed imperfections and disturbances** need to be translated into **dynamics and control specifications** ([Figure 48](#figure--fig:rankers98-system-performance-spec)).
</div>
@@ -1292,14 +1292,14 @@ In this stage, the designer only has a rough idea about the outlines of the mach
<div class="exampl">
One of the potential concepts for this machine consists of a stationary work piece with an optical unit that moves in both the X and Y directions (Figure [49](#figure--fig:rankers98-pattern-generator-concept)).
One of the potential concepts for this machine consists of a stationary work piece with an optical unit that moves in both the X and Y directions ([Figure 49](#figure--fig:rankers98-pattern-generator-concept)).
In the X direction, two driving forces are applied to the slides, whereas the position is measured by two linear encoders mounted between the slide and the granite frame.
<a id="figure--fig:rankers98-pattern-generator-concept"></a>
{{< figure src="/ox-hugo/rankers98_pattern_generator_concept.png" caption="<span class=\"figure-number\">Figure 49: </span>One of the possible concepts of the pattern generator" >}}
In this stage of the design, a simple model of the dynamic effects in the X direction could consist of the base, the slides, the guiding rail, the optical housing and intermediate flexibility (Figure [50](#figure--fig:rankers98-concept-1dof-evaluation)).
In this stage of the design, a simple model of the dynamic effects in the X direction could consist of the base, the slides, the guiding rail, the optical housing and intermediate flexibility ([Figure 50](#figure--fig:rankers98-concept-1dof-evaluation)).
<a id="figure--fig:rankers98-concept-1dof-evaluation"></a>
@@ -1324,7 +1324,7 @@ Typically, such a model contains 5-10 rigid bodies connected by suitable connect
<div class="exampl">
Figure [51](#figure--fig:rankers98-pattern-generator-rigid-body) shows such a 3D model of a different concept for the pattern generator.
[Figure 51](#figure--fig:rankers98-pattern-generator-rigid-body) shows such a 3D model of a different concept for the pattern generator.
<a id="figure--fig:rankers98-pattern-generator-rigid-body"></a>
@@ -1373,7 +1373,7 @@ Due to the complexity of the structures it is normally not very practical to bui
- The resulting mass and stiffness matrices can easily have many thousands degrees of freedom, which puts high demands on the required computing capacity.
A technique which overcomes these disadvantages is the co-called **sub-structuring technique**.
In this approach, illustrated in Figure [52](#figure--fig:rankers98-substructuring-technique), the system is divided into substructures or components, which are analyzed separately.
In this approach, illustrated in [Figure 52](#figure--fig:rankers98-substructuring-technique), the system is divided into substructures or components, which are analyzed separately.
Then, the (reduced) models of the components are assembled to form the overall system.
By doing so, the size of the final system model is significantly reduced.

9
content/phdthesis/zuo04_elemen_system_desig_activ_passiv_vibrat_isolat.md
View File

@@ -1,5 +1,6 @@
+++
title = "Element and system design for active and passive vibration isolation"
author = ["Dehaeze Thomas"]
draft = false
ref_author = "Zuo, L."
ref_year = 2004
@@ -55,19 +56,19 @@ Year
<a id="figure--fig:zuo04-piezo-spring-series"></a>
{{< figure src="/ox-hugo/zuo04_piezo_spring_series.png" caption="<span class=\"figure-number\">Figure 1: </span>PZT actuator and spring in series" >}}
{{< figure src="/ox-hugo/zuo04_piezo_spring_series.png" caption="<span class=\"figure-number\">Figure 3: </span>PZT actuator and spring in series" >}}
<a id="figure--fig:zuo04-voice-coil-spring-parallel"></a>
{{< figure src="/ox-hugo/zuo04_voice_coil_spring_parallel.png" caption="<span class=\"figure-number\">Figure 1: </span>Voice coil actuator and spring in parallel" >}}
{{< figure src="/ox-hugo/zuo04_voice_coil_spring_parallel.png" caption="<span class=\"figure-number\">Figure 4: </span>Voice coil actuator and spring in parallel" >}}
<a id="figure--fig:zuo04-piezo-plant"></a>
{{< figure src="/ox-hugo/zuo04_piezo_plant.png" caption="<span class=\"figure-number\">Figure 1: </span>Transmission from PZT voltage to geophone output" >}}
{{< figure src="/ox-hugo/zuo04_piezo_plant.png" caption="<span class=\"figure-number\">Figure 5: </span>Transmission from PZT voltage to geophone output" >}}
<a id="figure--fig:zuo04-voice-coil-plant"></a>
{{< figure src="/ox-hugo/zuo04_voice_coil_plant.png" caption="<span class=\"figure-number\">Figure 1: </span>Transmission from voice coil voltage to geophone output" >}}
{{< figure src="/ox-hugo/zuo04_voice_coil_plant.png" caption="<span class=\"figure-number\">Figure 6: </span>Transmission from voice coil voltage to geophone output" >}}
## Bibliography {#bibliography}