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Tags
: [Finite Element Model]({{< relref "finite_element_model" >}})
: [Finite Element Model]({{<relref "finite_element_model.md#" >}})
Reference
: ([Rankers 1998](#org98ff031))
: ([Rankers 1998](#org2d6d98d))
Author(s)
: Rankers, A. M.
@@ -22,20 +22,20 @@ Year
Despite the fact, that mechanical vibrations in a servo device can be very complex and often involve the motion of many components of the system, there are three fundamental mechanisms that are often observed.
These there basic dynamic phenomena can be indicated by:
- _Actuator flexibility_: the mechanical system does not behave as one rigid body, due to flexibility between the location at which the servo force is applied and the actual point that needs to be positioned
- _Guiding system flexibility_: the device usually rely on the guiding system to suppress motion in an undesired direction
- _Limited mass and stiffness of the stationary machine part_: the reaction force that comes with the driving force will introduce a motion of the "stationary" part of the mechanical system
- **Actuator flexibility**: the mechanical system does not behave as one rigid body, due to flexibility between the location at which the servo force is applied and the actual point that needs to be positioned
- **Guiding system flexibility**: the device usually rely on the guiding system to suppress motion in an undesired direction
- **Limited mass and stiffness of the stationary machine part**: the reaction force that comes with the driving force will introduce a motion of the "stationary" part of the mechanical system
Whereas the first two phenomena mainly affect the stability of the control loop, the last phenomena manifests itself more often as a dynamic positional error in the set-point response.
A tool that can be very useful in understanding the nature of more complex resonance phenomena and the underlying motion of the mechanical system, is "Modal Analysis".
A tool that can be very useful in understanding the nature of more complex resonance phenomena and the underlying motion of the mechanical system, is **Modal Analysis**.
Translating the mathematics of one single decoupled "modal" equation into a graphical representation, which includes all relevant data such as (effective) modal mass and stiffness plus the motion of each physical DoF, facilitates a better understanding of the modal concept.
It enables a very intuitive link between the modal and the physical domain, and thus leads to a more creative use of "modal analysis" without the complications of the mathematical formalism.
Dynamic phenomena of the mechanics in a servo positioning device can lead to stability problems of the control loop.
Therefore it is important to investigate the frequency response (\\(x/F\\)), which characterizes the dynamics of the mechanical system, and especially the influence of mechanical resonances on it.
Once the behavior of one individual mode is fully understood it is not so difficult to construct this frequency response and the interaction between the rigid-body motion of the device, and the dynamics of one additional mode.
This leads to four interaction patterns:
This leads to **four interaction patterns**:
- -2 slope / zero / pole / -2 slope
- -2 slope / pole / zero / -2 slope
@@ -43,14 +43,13 @@ This leads to four interaction patterns:
- -2 slope / pole / -2 slope (non-minimum phase and rarely occurring)
It is not possible to judge the potential destabilizing effect of each of the typical characteristics without considering the frequency of the resonance in relation to the envisaged bandwidth of the control loop.
The phase plot of a typical open loop frequency response of a PID controlled positioning device without mechanical resonances can be divided into three frequency ranges:
The phase plot of a typical open loop frequency response of a PID controlled positioning device without mechanical resonances can be divided into three frequency ranges (supposing the plant model is just a mass line):
- at low frequencies, the phase lies below -180 deg due to integrator action of the controller
- at medium frequency (centered by the bandwidth frequency), the phase lies above -180 deg due to the differential action of the controller, which is necessary in order to achieve a stable position control-loop
- at high frequencies, the phase eventually drops again below -180 deg due to additional low-pass filtering
The potential destabilizing effect of each of the three typical characteristics can be judged in relation to the frequency range.
Whether instability occurs depends very strongly on the resonance amplitude and damping of the additional mode.
The potential **destabilizing effect** of each of the three typical characteristics can be judged in relation to the frequency range:
- A -2 slope / zero / pole / -2 slope characteristics leads to a phase lead and is therefore potentially destabilizing in the low-frequency and high frequency regions.
In the medium frequency region it adds an extra phase leads to the already existing margin, which does not harm the stability.
@@ -58,15 +57,16 @@ Whether instability occurs depends very strongly on the resonance amplitude and
It is potentially destabilizing in the medium-frequency range and is harmless in the low and high frequency ranges.
- The -2 slope / poles / -4 slope behavior always has a devastating effect on the stability of the loop if located in the low or medium frequency range.
Whether instability occurs depends very strongly on the resonance amplitude and damping of the additional mode.
On the basis of these considerations, it is possible to give design guidelines for servo positioning devices.
The subject of machine dynamics and its interaction with the control system plays a dominant role in fast and accurate positioning devices, so it is vital to consider these issues during the entire design process.
Modeling and simulation can be adequate tools for that purpose; however, two conditions are crucial to the success:
**Modeling and simulation** can be adequate tools for that purpose; however, two conditions are crucial to the success:
- usefulness of results
- speed
The analysis process has usually a top-down structure.
The analysis process has usually a **top-down structure**.
Starting with very elementary simulation models to support the selection of the proper concept, these models should become more refined, just like the product or machine under development.
In various project throughout the past years, a three-step modeling approach has evolved, in which the following phases can be distinguished:
@@ -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](#orgfda2012).
A block diagram representation of a typical servo-system is shown in Figure [1](#orgf3f4585).
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="orgfda2012"></a>
<a id="orgf3f4585"></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](#org2d694a7)).
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](#org7066514)).
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="org2d694a7"></a>
<a id="org7066514"></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](#org0c46a44).
Most modern servo-systems have not only a feedback section, but also a feedforward section, as indicated in Figure [3](#orgd3dd201).
<a id="org0c46a44"></a>
<a id="orgd3dd201"></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](#org765d5f4)) 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](#orgac87759)) due to compliance between the motor and the load.
<a id="org765d5f4"></a>
<a id="orgac87759"></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](#org387cdc7)).
In the present discussion, a planar actuator with three degrees of freedom will be considered (Figure [5](#org7ada0d8)).
<a id="org387cdc7"></a>
<a id="org7ada0d8"></a>
{{< figure src="/ox-hugo/rankers98_guiding_flexibility_planar.png" caption="Figure 5: Planar actuator with guiding system flexibility" >}}
@@ -280,14 +280,14 @@ 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](#orgb3f73d2)).
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](#org60fe278)).
<a id="orgb3f73d2"></a>
<a id="org60fe278"></a>
{{< figure src="/ox-hugo/rankers98_limited_m_k_stationary_machine_part.png" caption="Figure 6: Limited Mass and Stiffness of Stationary Machine Part" >}}
## [Modal Decomposition]({{< relref "modal_decomposition" >}}) {#modal-decomposition--modal-decomposition-dot-md}
## [Modal Decomposition]({{<relref "modal_decomposition.md#" >}}) {#modal-decomposition--modal-decomposition-dot-md}
### Mathematics of Modal Decomposition {#mathematics-of-modal-decomposition}
@@ -451,9 +451,9 @@ The overall transfer function can be found by summation of the individual modal
### Basic Characteristics of Mechanical FRF {#basic-characteristics-of-mechanical-frf}
Consider the position control loop of Figure [7](#orgeee8a5d).
Consider the position control loop of Figure [7](#org96447ac).
<a id="orgeee8a5d"></a>
<a id="org96447ac"></a>
{{< figure src="/ox-hugo/rankers98_mechanical_servo_system.png" caption="Figure 7: Mechanical position servo-system" >}}
@@ -463,7 +463,7 @@ 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="org2ffb1b1"></a>
<a id="orgda68028"></a>
{{< figure src="/ox-hugo/rankers98_ideal_bode_nyquist.png" caption="Figure 8: FRF of an ideal system with no resonances" >}}
@@ -487,11 +487,11 @@ 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="orgc9ecb0c"></a>
<a id="orgdfb8041"></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\\)" >}}
<a id="org5180ee3"></a>
<a id="org080f036"></a>
{{< figure src="/ox-hugo/rankers98_final_frf_alpha.png" caption="Figure 10: Bode diagram of final FRF (\\(x\_{\text{servo}}/F\_{\text{servo}}\\)) for six values of \\(\alpha\\)" >}}
@@ -551,6 +551,7 @@ It has static solution capacity, and the frequency of the highest fixed-interfac
Through the enormous performance drive in mechatronics systems, much has been learned in the past years about the influence of machine dynamics in servo positioning-devices.
## Bibliography {#bibliography}
<a id="org98ff031"></a>Rankers, Adrian Mathias. 1998. “Machine Dynamics in Mechatronic Systems: An Engineering Approach.” University of Twente.
<a id="org2d6d98d"></a>Rankers, Adrian Mathias. 1998. “Machine Dynamics in Mechatronic Systems: An Engineering Approach.” University of Twente.