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@ -8,7 +8,7 @@ Tags
: [Finite Element Model]({{<relref "finite_element_model.md#" >}})
Reference
: ([Rankers 1998](#orgb3cf37f))
: ([Rankers 1998](#org6436602))
Author(s)
: Rankers, A. M.
@ -164,13 +164,13 @@ 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](#org93f8ab7).
A block diagram representation of a typical servo-system is shown in Figure [1](#orga7794da).
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.
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{{< figure src="/ox-hugo/rankers98_basic_el_mech_servo.png" caption="Figure 1: Basic elements of mechanical servo system" >}}
@ -181,10 +181,10 @@ 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](#org926a1c1)).
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](#org3f53580)).
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)\\).
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{{< figure src="/ox-hugo/rankers98_basic_elastic_struct.png" caption="Figure 2: Basic Elastic Structure" >}}
@ -200,9 +200,9 @@ 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](#org982a751).
Most modern servo-systems have not only a feedback section, but also a **feedforward** section, as indicated in Figure [3](#org4a1de51).
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<a id="org4a1de51"></a>
{{< figure src="/ox-hugo/rankers98_feedforward_example.png" caption="Figure 3: Mechanical servo system with feedback and feedforward control" >}}
@ -247,9 +247,9 @@ 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](#orgc72d1ed)) 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](#orgd374c3a)) due to compliance between the motor and the load.
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<a id="orgd374c3a"></a>
{{< figure src="/ox-hugo/rankers98_actuator_flexibility.png" caption="Figure 4: Actuator Flexibility" >}}
@ -259,9 +259,9 @@ 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](#org6fb625d)).
In the present discussion, a planar actuator with three degrees of freedom will be considered (Figure [5](#orgc0e4433)).
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<a id="orgc0e4433"></a>
{{< figure src="/ox-hugo/rankers98_guiding_flexibility_planar.png" caption="Figure 5: Planar actuator with guiding system flexibility" >}}
@ -281,9 +281,9 @@ 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](#orgcc64485)).
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](#org22ae9b4)).
<a id="orgcc64485"></a>
<a id="org22ae9b4"></a>
{{< figure src="/ox-hugo/rankers98_limited_m_k_stationary_machine_part.png" caption="Figure 6: Limited Mass and Stiffness of Stationary Machine Part" >}}
@ -296,9 +296,9 @@ 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](#orgd59a93b) 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](#orgb3fe7a3) 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="orgd59a93b"></a>
<a id="orgb3fe7a3"></a>
{{< figure src="/ox-hugo/rankers98_1dof_system.png" caption="Figure 7: Elementary dynamic system" >}}
@ -456,26 +456,26 @@ 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 \eqref{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](#orgc8410d5) for which the three mode shapes are depicted in the traditional graphical representation.
Consider the system in Figure [8](#org04e36cc) 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.
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<a id="org04e36cc"></a>
{{< figure src="/ox-hugo/rankers98_mode_trad_representation.png" caption="Figure 8: 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](#org121fe62)).
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](#org17d6c66)).
For each individual mode \\(i\\), each physical DoF \\(x\_k\\) is indicated on the lever at a position with respect to the point of rotation that corresponds to the amplitude and sign of that DoF in the mode shape vector (\\(\phi\_{ik}\\)).
System with no, very little, or proportional damping exhibit real mode shape vectors, and thus the various DoF each their maximum values at the same moment of the cycle.
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](#org121fe62)).
The modal DoF \\(q\_i\\) can be interpreted as the displacement at a distance "1" from the pivot point (Figure [9](#org17d6c66)).
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<a id="org17d6c66"></a>
{{< figure src="/ox-hugo/rankers98_mode_new_representation.png" caption="Figure 9: 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](#org8a13d61) (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](#org0de4b98) (a).
The resulting moment of inertia \\(J\_i\\) of the i-th modal lever then is:
\begin{equation}
@ -490,20 +490,20 @@ 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](#org8a13d61) for the mode 2 of the example system.
The transition from physical masses to modal masses is illustrated in Figure [10](#org0de4b98) 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}
k\_i = \omega\_i^2 m\_i
\end{equation}
<a id="org8a13d61"></a>
<a id="org0de4b98"></a>
{{< figure src="/ox-hugo/rankers98_mode_2_lumped_masses.png" caption="Figure 10: Graphical representation of mode 2 with (a.) lumped masses and (b.) modal mass and stiffness" >}}
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](#org45f9908), 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](#orgef7eb39), it is not difficult to understand the contribution of mode i to the transfer function \\(x\_l/f\_k\\):
\begin{equation}
\left( \frac{x\_l}{f\_k} \right)\_i = \frac{\phi\_{ik}\phi\_{il}}{m\_i s^2 + k\_i}
@ -511,7 +511,7 @@ Based on the graphical representation in Figure [11](#org45f9908), it is not dif
Hence, the force \\(f\_k\\) must be multiplied by the distance \\(\phi\_{ik}\\) in order to find the equivalent excitation force at the location of \\(q\_i\\) on the lever, whereas the resulting modal displacement \\(q\_i\\) must be multiplied by the distance \\(\phi\_{il}\\) in order to obtain the displacement of the physical DoF \\(x\_l\\).
<a id="org45f9908"></a>
<a id="orgef7eb39"></a>
{{< figure src="/ox-hugo/rankers98_lever_representation_with_force.png" caption="Figure 11: Graphical representation of mode \\(i\\), including the proper location of a force component \\(f\_k\\) that acts on physical DoF \\(x\_k\\)" >}}
@ -522,14 +522,14 @@ 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](#org9ad55d4) for a user DoF \\(x\_u = x\_3 - x\_2\\).
User DoF can be indicated on the modal lever, as illustrated in Figure [12](#orga82e374) 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}
\phi\_{iu} = b^T \phi\_i
\end{equation}
<a id="org9ad55d4"></a>
<a id="orga82e374"></a>
{{< figure src="/ox-hugo/rankers98_representation_user_dof.png" caption="Figure 12: Graphical representation of mode including user DoF \\(x\_u = x\_3 - x\_2\\)" >}}
@ -539,13 +539,13 @@ Even though the dimension mode vector can be very large, only three user DoF are
- measured output (displacement that is measured by the position sensor)
- actual output (displacement that determines the accuracy of the machine)
To illustrate this, a servo controlled positioning device is shown in Figure [13](#org17236f2).
To illustrate this, a servo controlled positioning device is shown in Figure [13](#orgae9e430).
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).
The slide is driven by a linear motor which transforms the output of the controller into a force on the slide and a reaction force on the stator (input).
<a id="org17236f2"></a>
<a id="orgae9e430"></a>
{{< figure src="/ox-hugo/rankers98_servo_system.png" caption="Figure 13: Schematic representation of a servo system" >}}
@ -568,9 +568,9 @@ These effective modal parameters can be used very effectively in understanding t
<div class="exampl">
<div></div>
The eigenvalue analysis of the two mass spring system in Figure [14](#org046aea2) leads to the modal results summarized in Table [1](#table--tab:2dof-example-modal-params) and which are graphically represented in Figure [15](#orgef2a3f3).
The eigenvalue analysis of the two mass spring system in Figure [14](#orgbf0f71b) leads to the modal results summarized in Table [1](#table--tab:2dof-example-modal-params) and which are graphically represented in Figure [15](#org9a6d0a0).
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{{< figure src="/ox-hugo/rankers98_example_2dof.png" caption="Figure 14: Two mass spring system" >}}
@ -596,7 +596,7 @@ whereas the modal stiffnesses follow from \\(k\_i = \omega\_i^2 m\_i\\).
| Modal Mass [kg] | \\(m\_1 = 50.8\\) | \\(m\_2 = 11.1\\) |
| Modal Stiff [N/m] | \\(k\_1 = 0.46\cdot 10^7\\) | \\(k\_2 = 1.23\cdot 10^7\\) |
<a id="orgef2a3f3"></a>
<a id="org9a6d0a0"></a>
{{< figure src="/ox-hugo/rankers98_example_2dof_modal.png" caption="Figure 15: Graphical representation of modes and modal parameters of the two mass spring system" >}}
@ -616,18 +616,18 @@ 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](#orgd22c3b7).
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](#org66f501f)).
The effective modal parameters can then be used in the graphical representation of Figure [16](#orgad02a5c).
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](#org5c3b059)).
<a id="orgd22c3b7"></a>
<a id="orgad02a5c"></a>
{{< figure src="/ox-hugo/rankers98_example_2dof_effective_modal.png" caption="Figure 16: Alternative graphical representation of modes of two mass spring system based on the effective modal mass and stiffnesses in DoF \\(x\_1\\)" >}}
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](#org66f501f), 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](#org5c3b059), 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.
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{{< figure src="/ox-hugo/rankers98_2dof_example_frf.png" caption="Figure 17: Frequency Response Function \\(x\_1/f\_1\\)" >}}
@ -649,10 +649,10 @@ The technique furthermore gives an indication of the amount of frequency shift t
<div class="exampl">
<div></div>
Assuming that one is asked to increase the natural frequency of the mode corresponding to Figure [18](#org6b192b5) 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](#org78df8fe) 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="org6b192b5"></a>
<a id="org78df8fe"></a>
{{< figure src="/ox-hugo/rankers98_example_3dof_sensitivity.png" caption="Figure 18: Graphical representation of a mod with 3 DoF" >}}
@ -673,13 +673,13 @@ f\_{\text{new},i}(\Delta k) &= \frac{1}{2\pi}\sqrt{\frac{k\_{\text{eff},i} + \De
<div class="exampl">
<div></div>
Let's use the two mass spring system in Figure [14](#org046aea2) as an example.
Let's use the two mass spring system in Figure [14](#orgbf0f71b) 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)).
Then using equation \eqref{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](#orgcd2e26e):
This can be graphically done as shown in Figure [19](#org4893595):
\begin{align}
k\_{\text{eff},1,(2-1)} &= 0.46 \cdot 10^7 / 0.07^2 = 93.9 \cdot 10^7 \, N/m \\\\\\
@ -702,7 +702,7 @@ The results are summarized in Table [3](#table--tab:example-sensitivity-2dof-res
| \\(\Delta m = 1\,kg\\) added to \\(m\_2\\) | 47.5 | 160.7 |
| \\(\Delta k = 10^7\, N/m\\) added between \\(x\_2\\) and \\(x\_1\\) | 48.1 | 237.2 |
<a id="orgcd2e26e"></a>
<a id="org4893595"></a>
{{< figure src="/ox-hugo/rankers98_example_sensitivity_2dof.png" caption="Figure 19: Graphical representation of modes and modal parameters of two mass spring system" >}}
@ -712,9 +712,9 @@ 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](#orgbd67800), 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](#org3df11f4), 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="orgbd67800"></a>
<a id="org3df11f4"></a>
{{< figure src="/ox-hugo/rankers98_addition_of_motion.png" caption="Figure 20: Addition of motion" >}}
@ -724,9 +724,9 @@ 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](#orga9463e6), 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](#orgb8f94d2), which is a visualization of the transformation between the modal and the physical domains.
<a id="orga9463e6"></a>
<a id="orgb8f94d2"></a>
{{< figure src="/ox-hugo/rankers98_conversion_modal_to_physical.png" caption="Figure 21: Conversion between modal DoF to physical DoF" >}}
@ -737,9 +737,9 @@ 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](#org4de80a4)).
To illustrate the important of the internal deformation, a very simplified physical model of a precision machine is considered (Figure [22](#orgc5ca877)).
<a id="org4de80a4"></a>
<a id="orgc5ca877"></a>
{{< figure src="/ox-hugo/rankers98_suspension_mode_machine.png" caption="Figure 22: Simplified physical model of a precision machine" >}}
@ -776,12 +776,14 @@ which can be a lot for high precision machines.
## Modes and Servo Stability {#modes-and-servo-stability}
One of the two limiting effects
### Basic Characteristics of Mechanical FRF {#basic-characteristics-of-mechanical-frf}
Consider the position control loop of Figure [23](#orgd683f0f).
Consider the position control loop of Figure [23](#org18af44d).
<a id="orgd683f0f"></a>
<a id="org18af44d"></a>
{{< figure src="/ox-hugo/rankers98_mechanical_servo_system.png" caption="Figure 23: Mechanical position servo-system" >}}
@ -791,7 +793,7 @@ 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}
<a id="org715f673"></a>
<a id="orgac4be89"></a>
{{< figure src="/ox-hugo/rankers98_ideal_bode_nyquist.png" caption="Figure 24: FRF of an ideal system with no resonances" >}}
@ -815,11 +817,11 @@ which simplifies equation \eqref{eq:effect_one_mode} to:
\frac{x\_{\text{servo}}}{F\_{\text{servo}}} = \frac{1}{ms^2} + \frac{\alpha}{m s^2 + m \omega\_i^2}
\end{equation}
<a id="org8ee0e38"></a>
<a id="orgda4ef86"></a>
{{< figure src="/ox-hugo/rankers98_frf_effect_alpha.png" caption="Figure 25: Contribution of rigid-body motion and modal dynamics to the amplitude and phase of FRF for various values of \\(\alpha\\)" >}}
<a id="org80c3467"></a>
<a id="org6c0aa47"></a>
{{< figure src="/ox-hugo/rankers98_final_frf_alpha.png" caption="Figure 26: Bode diagram of final FRF (\\(x\_{\text{servo}}/F\_{\text{servo}}\\)) for six values of \\(\alpha\\)" >}}
@ -905,4 +907,4 @@ Through the enormous performance drive in mechatronics systems, much has been le
## Bibliography {#bibliography}
<a id="orgb3cf37f"></a>Rankers, Adrian Mathias. 1998. “Machine Dynamics in Mechatronic Systems: An Engineering Approach.” University of Twente.
<a id="org6436602"></a>Rankers, Adrian Mathias. 1998. “Machine Dynamics in Mechatronic Systems: An Engineering Approach.” University of Twente.