diff --git a/content/article/chen04_decoup_contr_flexur_joint_hexap.md b/content/article/chen04_decoup_contr_flexur_joint_hexap.md index 0f955e5..b3d1286 100644 --- a/content/article/chen04_decoup_contr_flexur_joint_hexap.md +++ b/content/article/chen04_decoup_contr_flexur_joint_hexap.md @@ -5,10 +5,10 @@ draft = false +++ Tags -: [Decoupled Control](decoupled_control.md) +: [Decoupled Control]({{}}) Reference -: ([Chen and McInroy 2004](#org8b1e370)) +: ([Chen and McInroy 2004](#orgbe5d3d7)) Author(s) : Chen, Y., & McInroy, J. @@ -20,4 +20,4 @@ Year ## Bibliography {#bibliography} -Chen, Y., and J.E. McInroy. 2004. “Decoupled Control of Flexure-Jointed Hexapods Using Estimated Joint-Space Mass-Inertia Matrix.” _IEEE Transactions on Control Systems Technology_ 12 (3):413–21. . +Chen, Y., and J.E. McInroy. 2004. “Decoupled Control of Flexure-Jointed Hexapods Using Estimated Joint-Space Mass-Inertia Matrix.” _IEEE Transactions on Control Systems Technology_ 12 (3):413–21. . diff --git a/content/article/li01_simul_vibrat_isolat_point_contr.md b/content/article/li01_simul_vibrat_isolat_point_contr.md deleted file mode 100644 index f501859..0000000 --- a/content/article/li01_simul_vibrat_isolat_point_contr.md +++ /dev/null @@ -1,25 +0,0 @@ -+++ -title = "Simultaneous vibration isolation and pointing control of flexure jointed hexapods" -author = ["Thomas Dehaeze"] -draft = false -+++ - -Tags -: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}}) - -Reference -: ([Li, Hamann, and McInroy 2001](#orgd01725e)) - -Author(s) -: Li, X., Hamann, J. C., & McInroy, J. E. - -Year -: 2001 - -- if the hexapod is designed such that the payload mass/inertia matrix (\\(M\_x\\)) and \\(J^T J\\) are diagonal, the dynamics from \\(u\\) to \\(y\\) are decoupled. - - - -## Bibliography {#bibliography} - -Li, Xiaochun, Jerry C. Hamann, and John E. McInroy. 2001. “Simultaneous Vibration Isolation and Pointing Control of Flexure Jointed Hexapods.” In _Smart Structures and Materials 2001: Smart Structures and Integrated Systems_, nil. . diff --git a/content/article/thayer02_six_axis_vibrat_isolat_system.md b/content/article/thayer02_six_axis_vibrat_isolat_system.md new file mode 100644 index 0000000..75f0ec3 --- /dev/null +++ b/content/article/thayer02_six_axis_vibrat_isolat_system.md @@ -0,0 +1,23 @@ ++++ +title = "Six-axis vibration isolation system using soft actuators and multiple sensors" +author = ["Thomas Dehaeze"] +draft = true ++++ + +Tags +: + + +Reference +: ([Thayer et al. 2002](#org7584b4b)) + +Author(s) +: Thayer, D., Campbell, M., Vagners, J., & Flotow, A. v. + +Year +: 2002 + + +## Bibliography {#bibliography} + +Thayer, Doug, Mark Campbell, Juris Vagners, and Andrew von Flotow. 2002. “Six-Axis Vibration Isolation System Using Soft Actuators and Multiple Sensors.” _Journal of Spacecraft and Rockets_ 39 (2):206–12. . diff --git a/content/phdthesis/li01_simul_fault_vibrat_isolat_point.md b/content/phdthesis/li01_simul_fault_vibrat_isolat_point.md new file mode 100644 index 0000000..dfd170c --- /dev/null +++ b/content/phdthesis/li01_simul_fault_vibrat_isolat_point.md @@ -0,0 +1,293 @@ ++++ +title = "Simultaneous, fault-tolerant vibration isolation and pointing control of flexure jointed hexapods" +author = ["Thomas Dehaeze"] +draft = false ++++ + +Tags +: [Stewart Platforms]({{}}), [Vibration Isolation]({{}}), [Cubic Architecture]({{}}), [Flexible Joints]({{}}), [Multivariable Control]({{}}) + +Reference +: ([Li 2001](#org7277b25)) + +Author(s) +: Li, X. + +Year +: 2001 + + +## Introduction {#introduction} + + +### Flexure Jointed Hexapods {#flexure-jointed-hexapods} + +A general flexible jointed hexapod is shown in Figure [1](#org858f898). + + + +{{< figure src="/ox-hugo/li01_flexure_hexapod_model.png" caption="Figure 1: 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 Figure [2](#orgda07839). + + + +{{< figure src="/ox-hugo/li01_quet_dirty_box.png" caption="Figure 2: (left) Vibration machinery must be isolated from a precision bus. (right) A precision paylaod must be manipulated in the presence of base vibrations and/or exogenous forces." >}} + +Since only small movements are considered in flexure jointed hexapod, the Jacobian matrix, which relates changes in the Cartesian pose to changes in the strut lengths, can be considered constant. +Thus a static kinematic decoupling algorithm can be implemented for both vibration isolation and pointed controls on flexible jointed hexapods. + +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 Figure [3](#orgccc775c)) are: + +- Cubic (mutually orthogonal) +- Flexure Jointed + + + +{{< figure src="/ox-hugo/li01_stewart_platform.png" caption="Figure 3: Flexure jointed Stewart platform used for analysis and control" >}} + +The objectives of the hexapods are: + +- Precise pointing in two axes (sub micro-radians) +- simultaneously, providing both passive and active vibration isolation in six axes + + +### Jacobian matrix, Dynamic model, and decoupling algorithms {#jacobian-matrix-dynamic-model-and-decoupling-algorithms} + + +#### Jacobian Matrix {#jacobian-matrix} + +The Jacobian matrix \\(J\\) relates changes in the cartesian pose \\(\mathcal{X}\\) to changes in the strut lengths \\(l\\): + +\begin{equation} +\delta l = J \delta \mathcal{X} +\end{equation} + +where \\(\mathcal{X}\\) is a 6x1 vector of payload plate translations and rotations + +\begin{equation} +\mathcal{X} = \begin{bmatrix} +p\_x & p\_y & p\_z & \theta\_x & \theta\_y & \theta\_z +\end{bmatrix} +\end{equation} + +\\(J\\) is given by: + +\begin{equation} +J = \begin{bmatrix} +{}^B\hat{u}\_1^T & [({}^B\_PR^P p\_1) \times {}^B\hat{u}\_1]^T \\\\\\ +\vdots & \vdots \\\\\\ +{}^B\hat{u}\_6^T & [({}^B\_PR^P p\_6) \times {}^B\hat{u}\_6]^T +\end{bmatrix} +\end{equation} + +where (see Figure [1](#org858f898)) \\(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. + + +#### Dynamic model of flexure jointed hexapods {#dynamic-model-of-flexure-jointed-hexapods} + +The dynamics of a flexure jointed hexapod can be written in joint space: + +\begin{equation} +\begin{split} +& \left( J^{-T} {}^B\_PR^P M\_x {}^B\_PR^T J^{-1} + M\_s \right) \ddot{l} + B \dot{l} + K (l - l\_r) = \\\\\\ +&\quad f\_m - \left( M\_s + J^{-T} {}^B\_PR^P M\_x {}^U\_PR^T J\_c J\_b^{-1} \right) \ddot{q}\_u + J^{-T} {}^U\_BR\_T(\mathcal{F}\_e + \mathcal{G} + \mathcal{C}) +\end{split} +\end{equation} + + +### Test {#test} + +**Jacobian Analysis**: +\\[ \delta \mathcal{L} = J \delta \mathcal{X} \\] +The origin of \\(\\{P\\}\\) is taken as the center of mass of the payload. + +**Decoupling**: +If we refine the (force) inputs and (displacement) outputs as shown in Figure [4](#org7721136) or in Figure [5](#orgdc42940), we obtain a decoupled plant provided that: + +1. the payload mass/inertia matrix must be diagonal (the CoM is coincident with the origin of frame \\(\\{P\\}\\)) +2. the geometry of the hexapod and the attachment of the payload to the hexapod must be carefully chosen + +> For instance, if the hexapod has a mutually orthogonal geometry (cubic configuration), the payload's center of mass must coincide with the center of the cube formed by the orthogonal struts. + + + +{{< figure src="/ox-hugo/li01_decoupling_conf.png" caption="Figure 4: Decoupling the dynamics of the Stewart Platform using the Jacobians" >}} + + + +{{< figure src="/ox-hugo/li01_decoupling_conf_bis.png" caption="Figure 5: Decoupling the dynamics of the Stewart Platform using the Jacobians" >}} + + +## Simultaneous Vibration Isolation and Pointing Control {#simultaneous-vibration-isolation-and-pointing-control} + +Basic idea: + +- acceleration feedback is used to provide high-frequency vibration isolation +- cartesian pointing feedback can be used to provide low-frequency pointing + +The compensation is divided in frequency because: + +- pointing sensors often have low bandwidth +- acceleration sensors often have a poor low frequency response + +The control bandwidth is divided as follows: + +- low-frequency disturbances as attenuated and tracking is accomplished by feedback from low bandwidth pointing sensors +- mid-frequency disturbances are attenuated by feedback from band-pass sensors like accelerometer or load cells +- high-frequency disturbances are attenuated by passive isolation techniques + + +### Vibration Isolation {#vibration-isolation} + +The system is decoupled into six independent SISO subsystems using the architecture shown in Figure [6](#org0dd19dc). + + + +{{< figure src="/ox-hugo/li01_vibration_isolation_control.png" caption="Figure 6: Figure caption" >}} + +One of the subsystem plant transfer function is shown in Figure [6](#org0dd19dc) + + + +{{< figure src="/ox-hugo/li01_vibration_control_plant.png" caption="Figure 7: Plant transfer function of one of the SISO subsystem for Vibration Control" >}} + +Each compensator is designed using simple loop-shaping techniques. + +The unity control bandwidth of the isolation loop is designed to be from **5Hz to 50Hz**. + +> Despite a reasonably good match between the modeled and the measured transfer functions, the model based decoupling algorithm does not produce the expected decoupling. +> Only about 20 dB separation is achieve between the diagonal and off-diagonal responses. + + +### Pointing Control {#pointing-control} + +A block diagram of the pointing control system is shown in Figure [8](#orgb338488). + + + +{{< figure src="/ox-hugo/li01_pointing_control.png" caption="Figure 8: Figure caption" >}} + +The plant is decoupled into two independent SISO subsystems. +The compensators are design with inverse-dynamics methods. + +The unity control bandwidth of the pointing loop is designed to be from **0Hz to 20Hz**. + +A feedforward control is added as shown in Figure [9](#orgb372596). + + + +{{< figure src="/ox-hugo/li01_feedforward_control.png" caption="Figure 9: Feedforward control" >}} + + +### Simultaneous Control {#simultaneous-control} + +The simultaneous vibration isolation and pointing control is approached in two ways: + +1. design and implement the vibration isolation control first, identify the pointing plant when the isolation loops are closed, then implement the pointing compensators +2. the reverse design order + +Figure [10](#orgbafcf4b) shows a parallel control structure where \\(G\_1(s)\\) is the dynamics from input force to output strut length. + + + +{{< figure src="/ox-hugo/li01_parallel_control.png" caption="Figure 10: A parallel scheme" >}} + +The transfer function matrix for the pointing loop after the vibration isolation is closed is still decoupled. The same happens when closing the pointing loop first and looking at the transfer function matrix of the vibration isolation. + +The effect of the isolation loop on the pointing loop is large around the natural frequency of the plant as shown in Figure [11](#org2a20ab8). + + + +{{< figure src="/ox-hugo/li01_effect_isolation_loop_closed.png" caption="Figure 11: \\(\theta\_x/\theta\_{x\_d}\\) transfer function with the isolation loop closed (simulation)" >}} + +The effect of pointing control on the isolation plant has not much effect. + +> The interaction between loops may affect the transfer functions of the **first** closed loop, and thus affect its relative stability. + +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 Figures [12](#orgc137ea3) and [13](#orgc06274a)) + +As shown in Figure [12](#orgc137ea3), 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. + + + +{{< figure src="/ox-hugo/li01_closed_loop_pointing.png" caption="Figure 12: 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 (Figure [13](#orgc06274a)). +The first peak resonance of the vibration isolation loop at 15Hz is increased when closing the pointing loop. + + + +{{< figure src="/ox-hugo/li01_closed_loop_vibration.png" caption="Figure 13: Closed-loop transfer functions of the vibration isolation loop before and after the pointing control loop is closed" >}} + +> The isolation loop adds a second resonance peak at its high-frequency crossover in the pointing closed-loop transfer function, which may cause instability. +> Thus, it is recommended to design and implement the isolation control system first, and then identify the pointing plant with the isolation loop closed. + + +### Experimental results {#experimental-results} + +Two hexapods are stacked (Figure [14](#org2a11277)): + +- the bottom hexapod is used to generate disturbances matching candidate applications +- the top hexapod provide simultaneous vibration isolation and pointing control + + + +{{< figure src="/ox-hugo/li01_test_bench.png" caption="Figure 14: Stacked Hexapods" >}} + +Using the vibration isolation control alone, no attenuation is achieved below 1Hz as shown in figure [15](#org5933a45). + + + +{{< figure src="/ox-hugo/li01_vibration_isolation_control_results.png" caption="Figure 15: Vibration isolation control: open-loop (solid) vs. closed-loop (dashed)" >}} + +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 Figure [16](#org996a848) where the bandwidth of the isolation control is expanded to very low frequency. + + + +{{< figure src="/ox-hugo/li01_simultaneous_control_results.png" caption="Figure 16: Simultaneous control: open-loop (solid) vs. closed-loop (dashed)" >}} + + +## Future research areas {#future-research-areas} + +Proposed future research areas include: + +- **Include base dynamics in the control**: + The base dynamics is here neglected since the movements of the base are very small. + The base dynamics could be measured by mounting accelerometers at the bottom of each strut or by using force sensors. + It then could be included in the feedforward path. +- **Robust control and MIMO design** +- **New decoupling method**: + The proposed decoupling algorithm do not produce the expected decoupling, despite a reasonably good match between the modeled and the measured transfer functions. + Incomplete decoupling increases the difficulty in designing the controller. + New decoupling methods are needed. + These methods must be static in order to be implemented practically on precision hexapods +- **Identification**: + Many advanced control methods require a more accurate model or identified plant. + A closed-loop identification method is propose to solve some problems with the current identification methods used. +- **Other possible sensors**: + Many sensors can be used to expand the utility of the Stewart platform: + - **3-axis load cells** to investigate the Coriolis and centripetal terms and new decoupling methods + - **LVDT** to provide differential position of the hexapod payload with respect to the base + - **Geophones** to provide payload and base velocity information + + + +## Bibliography {#bibliography} + +Li, Xiaochun. 2001. “Simultaneous, Fault-Tolerant Vibration Isolation and Pointing Control of Flexure Jointed Hexapods.” University of Wyoming. diff --git a/content/phdthesis/rankers98_machin.md b/content/phdthesis/rankers98_machin.md index 5d7c29e..ce7a120 100644 --- a/content/phdthesis/rankers98_machin.md +++ b/content/phdthesis/rankers98_machin.md @@ -8,7 +8,7 @@ Tags : [Finite Element Model]({{}}) Reference -: ([Rankers 1998](#orgf6a233a)) +: ([Rankers 1998](#org5830c60)) Author(s) : Rankers, A. M. @@ -19,7 +19,9 @@ Year ## Summary {#summary} - +
+
+ 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: @@ -59,7 +61,7 @@ The potential **destabilizing effect** of each of the three typical characterist - 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. +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: @@ -87,7 +89,8 @@ A technique that overcomes these disadvantages is the so-called "**sub-structuri In this approach the system is divided into substructures or components, which are analyzed separately. Then, after application of a reduction technique which preserves the most dominant dynamic properties, the (reduced) models of the components are assemble to form the overall system. By doing so, the size of the final system model is reduced significantly. -
 + + ## Introduction {#introduction} @@ -100,7 +103,7 @@ In the development of servo-controlled positioning devices, it is essential to c - mechanical resonances can endanger the stability of the control loop - vibration of the mechanical system, which are cause by the servo forces during a prescribed motion, can lead to positional errors -To obtain a well-balanced design with respect to the effort in the mechanical design and the control design, one has to adapt a mechatronics approach in which the structural design and the control design are integrated. +To obtain a well-balanced design with respect to the effort in the mechanical design and the control design, one has to adapt a **mechatronics approach** in which the **structural design and the control design are integrated**. Integrated modelling and simulation of structural and control aspects should be part of the product-creation process of any mechatronic positioning device from the very beginning. Such an approach is the only way the enhance the score of success and achieve "first-time-right". @@ -117,9 +120,9 @@ Once this physical model has been derived, the second step consists of translati The following questions are only seldom addressed: -- which analysis must be carried out -- how should the results be interpreted -- what sort of physical model gives a reasonable balance between accuracy and required effort +- Which analysis must be carried out? +- How should the results be interpreted? +- What sort of physical model gives a reasonable balance between accuracy and required effort? There is a huge gap between available theory about modal analysis and engineering practice which is also true for the field of control theory. The integration of machine dynamics and control system design is also limited as the two topics are generally taught by different departments. @@ -137,11 +140,16 @@ The lack of integral knowledge of machine dynamics, control and the interaction This thesis aims at bridging the gap between existing theoretical knowledge in the field of machine dynamics and control, and the practical application of this knowledge during the design of a product or machine. +
+
+ The idea is to show that a basic understanding of machine dynamics suffices to interpret complex mechanical vibrations. Moreover, in combination with basic control theory it is possible to derive the typical patterns that can be observed in an open-loop frequency response of a mechanical servo-system including resonances, and to draw conclusions with respect to the effect of these resonances on the stability of the control loop. Based on the idea that the controlled system must satisfy certain disturbance rejection and bandwidth criteria, design guidelines can be given for the mechanical system such that the chance of realizing the required bandwidth without introduction stability problems is maximized. By using a step-wise modelling approach it is possible to investigate and **predict these phenomena during the design phase**, and to make design decisions which take the dynamics and control aspects into account. +
+ ### Preview {#preview} @@ -164,13 +172,13 @@ 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](#orgd21c65c). -The main task of the system is achieve a desired positional relation between two or more components of the system. +A block diagram representation of a typical servo-system is shown in Figure [1](#orgb58f3ae). +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). 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. - + {{< figure src="/ox-hugo/rankers98_basic_el_mech_servo.png" caption="Figure 1: Basic elements of mechanical servo system" >}} @@ -181,10 +189,10 @@ 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](#orgcea9a7c)). +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](#orge942ce1)). 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)\\). - + {{< figure src="/ox-hugo/rankers98_basic_elastic_struct.png" caption="Figure 2: Basic Elastic Structure" >}} @@ -200,9 +208,9 @@ 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](#orga68790e). +Most modern servo-systems have not only a feedback section, but also a **feedforward** section, as indicated in Figure [3](#orgaea0875). - + {{< figure src="/ox-hugo/rankers98_feedforward_example.png" caption="Figure 3: Mechanical servo system with feedback and feedforward control" >}} @@ -210,7 +218,7 @@ In the feedforward section, control signals are derived from the desired output In practice, a feedforward section gives a significant performance improvement in case of point-to-point and tacking applications, but a feedback section will always be necessary. First, because the model used in the calculation of the feedforward signal is not a perfect representation of the actual system and second because of the presence of unknown disturbances. -Practicing engineers generally accomplish the feedback design and analysis on the basis of the frequency response. +Practicing engineers generally accomplish the feedback design and analysis on the basis of the **frequency response**. One of the major benefits of this approach is the close link to experimental information that can be obtained by exciting the system with sinusoidal inputs and varying frequency and measuring the amplitude and phase of the output. Such frequency response can either be plotted using a Bode diagram of a Nyquist diagram. @@ -223,8 +231,8 @@ In order to quantify the level of stability, two criteria have been introduced: ### Specifications {#specifications} -Specification of a feedback controller is very closely linked to disturbance rejection, especially in modern controllers that incorporate a feedforward section. -The required performance of the feedback section, which is generally expressed in terms of bandwidth, depends very much on the disturbances that act on the system. +Specification of a feedback controller is very closely linked to **disturbance rejection**, especially in modern controllers that incorporate a feedforward section. +The required performance of the feedback section, which is generally expressed in terms of **bandwidth**, depends very much on the disturbances that act on the system. These disturbances can be very different, and vary from application to application: @@ -247,21 +255,21 @@ 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](#org77ec3b1)) 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](#orgb4b4c74)) due to **compliance between the motor and the load**. - + {{< figure src="/ox-hugo/rankers98_actuator_flexibility.png" caption="Figure 4: Actuator Flexibility" >}} #### Guiding System Flexibility {#guiding-system-flexibility} -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). +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](#org420ac3c)). +In the present discussion, a planar actuator with three degrees of freedom will be considered (Figure [5](#org3bdc7e7)). - + {{< figure src="/ox-hugo/rankers98_guiding_flexibility_planar.png" caption="Figure 5: Planar actuator with guiding system flexibility" >}} @@ -270,20 +278,20 @@ The limited support stiffness and the inertia properties of the actuator will re Every actuator as some sort of guiding system in order to suppress certain DoF, and thus possesses guiding modes. However, whether this leads to dynamic problems depends very much on the location of the driving force and the sensor. -By choosing the proper location of the driving force one can avoid excitation of these modes, whereas **the location of the sensor influences the effect of such a mode on the servo stability** where excitation of the mode could not be avoided. +By choosing the **proper location of the driving force one can avoid excitation of these modes**, whereas **the location of the sensor influences the effect of such a mode on the servo stability** where excitation of the mode could not be avoided. In general, it should be attempted to design the actuator (mass distribution and location of driving force) such that it will perform the desired motion even in the absence of the guiding system. #### Limited Mass and Stiffness of Stationary Machine Part {#limited-mass-and-stiffness-of-stationary-machine-part} -The last category of dynamic phenomena results from the limited mass and stiffness of the stationary part of a mechanical servo-system. +The last category of dynamic phenomena results from the **limited mass and stiffness of the stationary part of a mechanical servo-system**. 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](#orgf8032be)). +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](#org01aa9a3)). - + {{< 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 +304,9 @@ 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](#org36b7f2f) 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](#org10e5a34) 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. - + {{< figure src="/ox-hugo/rankers98_1dof_system.png" caption="Figure 7: Elementary dynamic system" >}} @@ -337,9 +345,9 @@ For \\(i=j\\) the result of the multiplication according to equation \eqref{eq:e Because only the direction but not the length of an eigenvector is defined, several scaling methods are used, all based on equation \eqref{eq:modal_mass}: -- \\(|\phi\_i| = 1\\):: Each eigenvector \\(\phi\_i\\) is scaled such that its length is equal to \\(1\\). The modal mass are then calculated from equation \eqref{eq:modal_mass}. -- \\(\max(\phi\_i) = 1\\):: Each eigenvector \\(\phi\_i\\) is scaled such that its largest element is equation to \\(1\\). The modal mass is then calculated from equation \eqref{eq:modal_mass}. -- \\(m\_i = 1\\):: The modal mass \\(\mathit{m}\_i\\) is set to \\(1\\). The scaling of the mode vector \\(\phi\_i\\) follows from equation \eqref{eq:modal_mass}. +- \\(|\phi\_i| = 1\\). Each eigenvector \\(\phi\_i\\) is scaled such that its length is equal to \\(1\\). The modal mass are then calculated from equation \eqref{eq:modal_mass}. +- \\(\max(\phi\_i) = 1\\). Each eigenvector \\(\phi\_i\\) is scaled such that its largest element is equation to \\(1\\). The modal mass is then calculated from equation \eqref{eq:modal_mass}. +- \\(m\_i = 1\\). The modal mass \\(\mathit{m}\_i\\) is set to \\(1\\). The scaling of the mode vector \\(\phi\_i\\) follows from equation \eqref{eq:modal_mass}. The orthogonality properties also apply to the stiffness matrix \\(K\\): @@ -348,7 +356,7 @@ The orthogonality properties also apply to the stiffness matrix \\(K\\): \phi\_i^T K \phi\_i &= \omega\_i^2 \mathit{m}\_i = \mathit{k}\_i \end{align} -Because the \\(n\\) eigenvectors \\(\phi\_i\\) form a **base** in the n-dimensional space, any displacement vector \\(x(t)\\) can be written as a linear combination of the eigenvectors. +Because the \\(n\\) eigenvectors \\(\phi\_i\\) form a **base** in the n-dimensional space, any displacement vector \\(x(t)\\) can be written as a **linear combination of the eigenvectors**. Let \\(q\_i(t)\\) be the response of the decoupled mode \\(i\\), then the resulting displacement vector \\(x(t)\\) will be: \begin{equation} @@ -358,13 +366,18 @@ x(t) = q\_1(t) \phi\_1 + q\_2(t) \phi\_2 + \dots + q\_n(t) \phi\_n For one individual physical DoF \\(x\_k\\): \begin{equation} -x(t) = q\_1(t) \phi\_{1k} + q\_2(t) \phi\_{2k} + \dots + q\_n(t) \phi\_{nk} +x\_k(t) = q\_1(t) \phi\_{1k} + q\_2(t) \phi\_{2k} + \dots + q\_n(t) \phi\_{nk} \end{equation} with \\(\phi\_{ik}\\) being the element of the mode-shape vector \\(\phi\_i\\) that corresponds to the physical DoF \\(x\_k\\). +
+
+ The physical interpretation of the above two equations is that any motion of the system can be regarded as a combination of the contribution of the various modes. +
+ On can combine the eigenvectors in a matrix \\(\Phi\\) and the coefficients \\(q\_i\\) in a vector \\(q(t)\\) which leads to: \begin{equation} @@ -375,7 +388,12 @@ With: \begin{align} \Phi &= \begin{bmatrix} \phi\_1 & \phi\_2 & \dots & \phi\_n \end{bmatrix} \\\\\\ - q(t) &= \begin{bmatrix} q\_1(t) \\ q\_2(t) \\ \vdots \\ q\_n(t) \end{bmatrix} + q(t) &= \begin{bmatrix} + q\_1(t) \\\\\\ + q\_2(t) \\\\\\ + \vdots \\\\\\ + q\_n(t) +\end{bmatrix} \end{align} Substitution of \\(x(t) = \Phi q(t)\\) into the original equation of motion and premultiplication with \\(\Phi^T\\) results in: @@ -393,16 +411,25 @@ m\_1 & & & \\\\\\ & & \ddots & \\\\\\ & & & m\_n \end{bmatrix} \begin{bmatrix} -\ddot{q}\_1 \\ \ddot{q}\_2 \\ \vdots \\ \ddot{q}\_n +\ddot{q}\_1 \\\\\\ +\ddot{q}\_2 \\\\\\ +\vdots \\\\\\ +\ddot{q}\_n \end{bmatrix} + \begin{bmatrix} k\_1 & & & \\\\\\ & k\_2 & & \\\\\\ & & \ddots & \\\\\\ & & & k\_n \end{bmatrix} \begin{bmatrix} -q\_1 \\ q\_2 \\ \vdots \\ q\_n +q\_1 \\\\\\ +q\_2 \\\\\\ +\vdots \\\\\\ +q\_n \end{bmatrix} = \begin{bmatrix} -\phi\_1^T f \\ \phi\_2^T f \\ \vdots \\ \phi\_n^T f +\phi\_1^T f \\\\\\ +\phi\_2^T f \\\\\\ +\vdots \\\\\\ +\phi\_n^T f \end{bmatrix} \end{equation} @@ -412,10 +439,16 @@ For the i-th modal coordinate \\(q\_i\\) the equation of motion is: m\_i \ddot{q\_i}(t) + k\_i q\_i(t) = \phi\_i^T f(t) \end{equation} -which is a simple second order differential equation similar to that of a single mass spring system. +which is a **simple second order differential equation** similar to that of a single mass spring system. + +
+
+ Using basic formulae that are derived for a simple mass spring system, one is now able to analyze the time and frequency response of all individual modes. Having done that, the total motion of the system can simply be obtained by summing the contributions of all modes. +
+ Characterisation of the dynamics of a mechanical system in terms of frequency response behavior plays a major role in the stability analysis of the control loop of a mechatronic device. In such an analysis one is typically interested in the transfer function between a measured displacement \\(x\_l\\) and a force \\(f\_k\\), which acts at the physical DoF \\(x\_k\\). Applying the principle of modal decomposition, any transfer function can be derived by first calculating the behavior of the individual modes, and then **summing all modal contributions**. @@ -436,7 +469,7 @@ q\_i(s) = f\_k(s) \frac{\phi\_{ik}}{m\_i s^2 + k\_i} Once the modal response \\(q\_i\\) is known, the response of the physical DoF \\(x\_l\\) is found by a simple premultiplication with \\(\phi\_{il}\\), which finally leads to the following expression for the contribution of mode \\(i\\) to the transfer function: \begin{equation} -\left( \frac{x\_l}{f\_k} \right)\_i = \frac{\phi\_{ik}\phi\_{il}}{m\_i s^2 + k\_i} +\boxed{\left( \frac{x\_l}{f\_k} \right)\_i = \frac{\phi\_{ik}\phi\_{il}}{m\_i s^2 + k\_i}} \end{equation}
@@ -456,26 +489,33 @@ 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](#orgdb6f949) for which the three mode shapes are depicted in the traditional graphical representation. +Consider the system in Figure [8](#org494d682) 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. - + {{< 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](#org3b3e340)). +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](#orgbaf5c78)). + +
+
+ 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](#org3b3e340)). +The modal DoF \\(q\_i\\) can be interpreted as the displacement at a distance "1" from the pivot point (Figure [9](#orgbaf5c78)). - + {{< 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](#org48d23d9) (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](#orge97f125) (a). The resulting moment of inertia \\(J\_i\\) of the i-th modal lever then is: \begin{equation} @@ -488,30 +528,30 @@ This result is identical to the modal mass \\(m\_i\\) found with Equation \eqref m\_i = \phi\_j^T M \phi\_j = \sum\_{k=1}^n m\_k \phi\_{ik}^2 \end{equation} -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. +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](#org48d23d9) for the mode 2 of the example system. +The transition from physical masses to modal masses is illustrated in Figure [10](#orge97f125) 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} - + {{< 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](#org9f1d26b), 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](#org3899e03), 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} +\boxed{\left( \frac{x\_l}{f\_k} \right)\_i = \frac{\phi\_{ik}\phi\_{il}}{m\_i s^2 + k\_i}} \end{equation} 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\\). - + {{< 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\\)" >}} @@ -522,38 +562,43 @@ 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](#orge1022d0) for a user DoF \\(x\_u = x\_3 - x\_2\\). +User DoF can be indicated on the modal lever, as illustrated in Figure [12](#org69af888) 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} - + {{< 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\\)" >}} -Even though the dimension mode vector can be very large, only three user DoF are really important for servo-application which define: +Even though the dimension mode vector can be very large, only **three user DoF** are really important for servo-application which define: -- input (how much a mode is excited by the servo force) -- measured output (displacement that is measured by the position sensor) -- actual output (displacement that determines the accuracy of the machine) +- Input: how much a mode is excited by the servo force +- 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](#orgb19de84). +
+
+ +To illustrate this, a servo controlled positioning device is shown in Figure [13](#org8d21910). 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). - + {{< figure src="/ox-hugo/rankers98_servo_system.png" caption="Figure 13: Schematic representation of a servo system" >}} +
+ ### Physical Meaning of Modal Parameters {#physical-meaning-of-modal-parameters} Unfortunately, the mathematical approach of the scaling procedure of mode-shapes and modal parameters sometimes obscures the physical meaning of modal mass and modal stiffness. -The link to the real world can be found bia the **effective modal mass** and the **effective modal stiffness** of a mode as it is "felt" in a certain DoF. +The link to the real world can be found via the **effective modal mass** and the **effective modal stiffness** of a mode as it is "felt" in a certain DoF. These quantities are unique, do not depend on the scaling procedure and have physical meaning and physical units. The effective modal parameters of mode \\(i\\) in physical DoF \\(k\\) can be derived from the modal parameters via the following equations: @@ -568,9 +613,9 @@ These effective modal parameters can be used very effectively in understanding t
-The eigenvalue analysis of the two mass spring system in Figure [14](#org8c59f86) leads to the modal results summarized in Table [1](#table--tab:2dof-example-modal-params) and which are graphically represented in Figure [15](#org3085b08). +The eigenvalue analysis of the two mass spring system in Figure [14](#orgccdbdfe) leads to the modal results summarized in Table [1](#table--tab:2dof-example-modal-params) and which are graphically represented in Figure [15](#org866202d). - + {{< figure src="/ox-hugo/rankers98_example_2dof.png" caption="Figure 14: Two mass spring system" >}} @@ -596,7 +641,7 @@ 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\\) | - + {{< 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 +661,18 @@ 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](#org1398e20). -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](#orgb620f7d)). +The effective modal parameters can then be used in the graphical representation of Figure [16](#orgabaa73a). +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](#org493aaea)). - + {{< 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](#orgb620f7d), 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](#org493aaea), 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. - + {{< figure src="/ox-hugo/rankers98_2dof_example_frf.png" caption="Figure 17: Frequency Response Function \\(x\_1/f\_1\\)" >}} @@ -639,20 +684,26 @@ In the final Bode diagram (Figure [17](#orgb620f7d), below) one can observe an i Sometimes it is required to change the dynamical properties of a system. In such situation it is useful to known how to modify the system so as to bring about the desired change. **Sensitivity analysis**, helps to determine the rate of change of each natural frequency with each of the system parameters. -It typical provides answers to questions such as: + +
+
+ +Sensitivity analysis typical provides answers to questions such as: - Where should one reduce mass in order to achieve the most significant gain in natural frequency? - Between which two points of a structure should one add extra stiffness to increase the natural frequency? +
+ The technique furthermore gives an indication of the amount of frequency shift that can be obtained.
-Assuming that one is asked to increase the natural frequency of the mode corresponding to Figure [18](#org015ebc2) 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](#org7cab420) 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. - + {{< figure src="/ox-hugo/rankers98_example_3dof_sensitivity.png" caption="Figure 18: Graphical representation of a mod with 3 DoF" >}} @@ -673,13 +724,13 @@ f\_{\text{new},i}(\Delta k) &= \frac{1}{2\pi}\sqrt{\frac{k\_{\text{eff},i} + \De
-Let's use the two mass spring system in Figure [14](#org8c59f86) as an example. +Let's use the two mass spring system in Figure [14](#orgccdbdfe) 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](#orgf24a43b): +This can be graphically done as shown in Figure [19](#org2cdc396): \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 +753,7 @@ 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 | - + {{< 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 +763,9 @@ 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](#orgc27801a), 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](#org1066c2d), the motion of the output \\(y\\) is equals to the sum of the motion of the two inputs \\(x\_1\\) and \\(x\_2\\). - + {{< figure src="/ox-hugo/rankers98_addition_of_motion.png" caption="Figure 20: Addition of motion" >}} @@ -724,9 +775,9 @@ 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](#org67908e9), 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](#org8de0f99), which is a visualization of the transformation between the modal and the physical domains. - + {{< figure src="/ox-hugo/rankers98_conversion_modal_to_physical.png" caption="Figure 21: Conversion between modal DoF to physical DoF" >}} @@ -737,9 +788,9 @@ 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](#org44924f2)). +To illustrate the important of the internal deformation, a very simplified physical model of a precision machine is considered (Figure [22](#org7eacb7b)). - + {{< figure src="/ox-hugo/rankers98_suspension_mode_machine.png" caption="Figure 22: Simplified physical model of a precision machine" >}} @@ -749,7 +800,11 @@ For a proper operation of the machine, the internal deformation \\(\epsilon = x\ Typically, such a machine is designed for high internal stiffness, and it is furthermore very softly supported in order to prevent external (floor) vibrations from entering the machine. Assuming that the natural frequency \\(\omega\_1\\) of the suspension mode \\(\phi\_1\\) is significantly lower than that of the internal mode, one can approximate the frequency of the suspension mode by considering the motion of the entire machine as one rigid body on the stiffness of the suspension system. -However, one should keep in mind that there is always a small amount of internal deformation in case of a non-zero suspension stiffness \\(k\_{20}\\). +However, one should keep in mind that **there is always a small amount of internal deformation** in case of a non-zero suspension stiffness \\(k\_{20}\\). + +
+
+ It can be shown than the internal deformation associated with the suspension mode is: \begin{equation} @@ -757,6 +812,9 @@ It can be shown than the internal deformation associated with the suspension mod \end{equation} with \\(\omega\_{\text{int}} = \sqrt{\frac{k\_{12}}{m\_1}}\\) representing the natural frequency of the manipulator where the base frame is clamped or infinitely heavy. + +
+ This equation shows that the internal deformation associated with the suspension mode depends on the ratio of the natural frequencies of the internal mode compared to the suspension mode.
@@ -778,14 +836,14 @@ which can be a lot for high precision machines. The effect of machine dynamics on the servo control loop stability is discussed in this chapter. -The interaction between the desired (rigid body) motion and the dynamics of one additional mode and its effect on the freuency response function \\(x\_{\text{servo}}/F\_{\text{servo}}\\) is the basis of this chapter. +The interaction between the desired (rigid body) motion and the dynamics of one additional mode and its effect on the frequency response function \\(x\_{\text{servo}}/F\_{\text{servo}}\\) is the basis of this chapter. ### Basic Characteristics of Mechanical FRF {#basic-characteristics-of-mechanical-frf} -Consider the position control loop of Figure [23](#orgcd512cf). +Consider the position control loop of Figure [23](#orga7c3e4d). - + {{< figure src="/ox-hugo/rankers98_mechanical_servo_system.png" caption="Figure 23: Mechanical position servo-system" >}} @@ -795,9 +853,9 @@ 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](#orgb3ba1d2). +The corresponding Bode and Nyquist plots and shown in Figure [24](#org4eddb5f). - + {{< figure src="/ox-hugo/rankers98_ideal_bode_nyquist.png" caption="Figure 24: FRF of an ideal system with no resonances" >}} @@ -809,7 +867,7 @@ In the case of one extra modal contribution, the equation for the mechanical tra The final transfer function and the exact interaction between the two parts depends on the values of the various parameters. -Let's introduce a variable \\(\alpha\\), which relates the high-frequency contribution of the mode to that of the rigid-body motion: +Let's introduce a variable \\(\alpha\\), which **relates the high-frequency contribution of the mode to that of the rigid-body motion**: \begin{equation} \label{eq:alpha} \alpha = \frac{\frac{\phi\_{i,\text{servo}} \phi\_{i,\text{force}}}{m\_i}}{\frac{1}{m}} @@ -818,42 +876,47 @@ Let's introduce a variable \\(\alpha\\), which relates the high-frequency contri which simplifies equation \eqref{eq:effect_one_mode} to: \begin{equation} \label{eq:effect\_one\_mode\_simplified} -\frac{x\_{\text{servo}}}{F\_{\text{servo}}} = \frac{1}{ms^2} + \frac{\alpha}{m s^2 + m \omega\_i^2} +\boxed{\frac{x\_{\text{servo}}}{F\_{\text{servo}}} = \frac{1}{ms^2} + \frac{\alpha}{m s^2 + m \omega\_i^2}} \end{equation} Equation \eqref{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](#orgf995437). +Three different types of intersection pattern can be found in the amplitude plot as shown in Figure [25](#org4d0be94). Depending on the absolute value of \\(\alpha\\) one can observe: - \\(|\alpha| < 1\\): two intersections - \\(|\alpha| = 1\\): one intersection and asymptotic approach at high frequencies - \\(|\alpha| > 1\\): one intersection -The interaction between the rigid body motion and the additional mode will not only depend on \\(|\alpha|\\) but also on the sign of \\(\alpha\\), which determined the phase relation between the two contributions. +The interaction between the rigid body motion and the additional mode will not only depend on \\(|\alpha|\\) but also on the **sign** of \\(\alpha\\), which determined the **phase relation between the two contributions**. - + {{< 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\\)" >}} -The general shape of the overall FRF can be constructed for all cases (Figure [26](#org3659f0b)). +The general shape of the overall FRF can be constructed for all cases (Figure [26](#org7226848)). 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. - + {{< 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\\)" >}} -When analyzing the plots of Figure [26](#org3659f0b), four different types of FRF can be found: +
+
+ +When analyzing the plots of Figure [26](#org7226848), 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\\)) - -2 slope / pole / -4 slope (\\(\alpha = -1\\)) - -2 slope / pole / -2 slope (\\(\alpha < -1\\)) -All cases are shown in Figure [27](#org5f42df0). +
- +All cases are shown in Figure [27](#org8b59092). + + {{< figure src="/ox-hugo/rankers98_interaction_shapes.png" caption="Figure 27: Bode plot of the different types of FRF" >}} @@ -872,49 +935,54 @@ f\_{lp} &= 4 \cdot f\_b \end{align\*} with \\(f\_b\\) the bandwidth frequency. -The asymptotic amplitude plot is shown in Figure [28](#orgc6fa8a4). +The asymptotic amplitude plot is shown in Figure [28](#org57d75c7). - + {{< figure src="/ox-hugo/rankers98_pid_amplitude.png" caption="Figure 28: 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](#orgdf31690). +With these settings, the open loop response of the position loop (controller + mechanics) looks like Figure [29](#orgcc5f9dc). - + {{< figure src="/ox-hugo/rankers98_ideal_frf_pid.png" caption="Figure 29: Ideal open loop FRF of a position servo without mechanical resonances (\\(f\_b = 30\text{ Hz}\\))" >}} +
+
+ 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](#org9480ff1)) and high frequency (Figure [32](#org9b86fef)) regions. - In the medium frequency region (Figure [31](#orga4bae4e)), 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](#org9f4ad31)) and high frequency (Figure [32](#org7659335)) regions. + In the medium frequency region (Figure [31](#org2fc068b)), 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](#orga71718d)) and is harmless in the low (Figure [33](#org9b0e1e4)) and high frequency (Figure [35](#orgc0f181f)) ranges. + It is potentially destabilizing in the medium frequency range (Figure [34](#orgbce6fc9)) and is harmless in the low (Figure [33](#orgb85562f)) and high frequency (Figure [35](#orgc43335a)) 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\\). - +
+ + {{< figure src="/ox-hugo/rankers98_zero_pole_low_freq.png" caption="Figure 30: Open Loop FRF of type \"-2 slope / zero / pole / -2 slope\" with low frequency resonance" >}} - + {{< figure src="/ox-hugo/rankers98_zero_pole_medium_freq.png" caption="Figure 31: Open Loop FRF of type \"-2 slope / zero / pole / -2 slope\" with medium frequency resonance" >}} - + {{< figure src="/ox-hugo/rankers98_zero_pole_high_freq.png" caption="Figure 32: Open Loop FRF of type \"-2 slope / zero / pole / -2 slope\" with high frequency resonance" >}} - + {{< figure src="/ox-hugo/rankers98_pole_zero_low_freq.png" caption="Figure 33: Open Loop FRF of type \"-2 slope / pole / zero / -2 slope\" with low frequency resonance" >}} - + {{< figure src="/ox-hugo/rankers98_pole_zero_medium_freq.png" caption="Figure 34: Open Loop FRF of type \"-2 slope / pole / zero / -2 slope\" with medium frequency resonance" >}} - + {{< figure src="/ox-hugo/rankers98_pole_zero_high_freq.png" caption="Figure 35: Open Loop FRF of type \"-2 slope / pole / zero / -2 slope\" with high frequency resonance" >}} @@ -924,15 +992,15 @@ These conclusions may differ for different mass ratio \\(\alpha\\). #### Actuator Flexibility {#actuator-flexibility} -Figure [36](#org5c9d040) shows the schematic representation of a system with a certain compliance between the motor and the load. +Figure [36](#orgeddfc41) shows the schematic representation of a system with a certain compliance between the motor and the load. - + {{< figure src="/ox-hugo/rankers98_2dof_actuator_flexibility.png" caption="Figure 36: Servo system with actuator flexibility - Schematic representation" >}} -The corresponding modes are shown in Figure [37](#orgc92a13c). +The corresponding modes are shown in Figure [37](#org28f6c88). - + {{< figure src="/ox-hugo/rankers98_2dof_modes_act_flex.png" caption="Figure 37: Servo System with Actuator Flexibility - Modes" >}} @@ -945,9 +1013,9 @@ 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](#orgb7822b6)). +The resulting FRF exhibit a "-2 slope / zero / pole / -2 slope" (Figure [38](#orgeb05620)). - + {{< figure src="/ox-hugo/rankers98_2dof_act_flex_frf.png" caption="Figure 38: Mechanical FRF of a system with actuator flexibility and position measurement at motor" >}} @@ -955,7 +1023,7 @@ The asymptotes at low and high frequencies are: \begin{align} \left( \frac{x\_1}{F\_{\text{servo}}} \right)\_{s \to 0} &= \frac{1}{(m\_1 + m\_2) s^2} \\\\\\ -\left( \frac{x\_1}{F\_{\text{servo}}} \right)\_{s \to 0} &= \frac{1}{(m\_1 + m\_2) s^2} + \frac{1}{m\_1/m\_2(m\_1 + m\_2) s^2} = \frac{1}{m\_1 s^2} +\left( \frac{x\_1}{F\_{\text{servo}}} \right)\_{s \to \infty} &= \frac{1}{(m\_1 + m\_2) s^2} + \frac{1}{m\_1/m\_2(m\_1 + m\_2) s^2} = \frac{1}{m\_1 s^2} \end{align} which corresponds to the engineering feeling that at very low frequencies the two masses move as one single mass, whereas at very high frequencies the mass \\(m\_2\\) of the load is completely decoupled such that the servo system only "feels and sees" the motion of the motor mass \\(m\_1\\). @@ -966,7 +1034,7 @@ which corresponds to the engineering feeling that at very low frequencies the tw Guideline in presence of actuator flexibility with measurement at the motor position: - The motor inertia should be one to three times to inertial of the load -- The resonance frequency should either be near the bandwidth frequency and much above +- The resonance frequency should either be near the bandwidth frequency or much above
@@ -977,9 +1045,9 @@ 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](#orgbe8ffaf)). +The mass ratio \\(\alpha\\) equal -1, and thus the FRF will be of type "-2 slope / pole / -4 slope" (Figure [39](#org476f71c)). - + {{< figure src="/ox-hugo/rankers98_2dof_act_flex_meas_load_frf.png" caption="Figure 39: FRF \\(k\_p \cdot (x\_{\text{servo}}/F\_{\text{servo}})\\) of a system with actuator flexibility and position measurement at the load" >}} @@ -995,10 +1063,47 @@ 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](#org5654036)) 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. + + + +{{< figure src="/ox-hugo/rankers98_2dof_guiding_flex.png" caption="Figure 40: 2DoF rigid body model of actuator with flexibility of the guiding system" >}} + +Considering the two relevant modes (Figures [41](#org18f041e) and [42](#orgb340007)), 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} +\end{equation} + + + +{{< figure src="/ox-hugo/rankers98_2dof_guiding_flex_x_mode.png" caption="Figure 41: Graphical representation of desired X-motion" >}} + + + +{{< figure src="/ox-hugo/rankers98_2dof_guiding_flex_rock_mode.png" caption="Figure 42: Graphical representation of parasitic rocking mode" >}} + +The two distances \\(a\_F\\) and \\(a\_s\\) have a significant influence on the contribution of the rocking mode on the open loop characteristics. +Note that it is the product of these two distance and not the individual value that are important, so exchanging the position of the sensor and the force does not affect the response. + +By introducing the "gyration radius" \\(\rho\\) such that \\(J = m \cdot \rho^2\\), the following mass ratio is obtained: + +\begin{equation} +\alpha = \frac{a\_s a\_F}{\rho^2} +\end{equation} + +As the location of the sensor and the actuator can be chosen anywhere above or below the center of gravity, \\(a\_s\\) and \\(a\_F\\), and consequently \\(\alpha\\), can become any position or negative value, and the resulting FRF can have any of the discussed characteristics. + +As long as the actuating force and sensor are located on the same side of the center of gravity, \\(\alpha\\) will be positive and the overall FRF will display a "-2 slope / zero / pole / -2 slope" behaviour. +When the force and sensor are located at opposite sides of the center of gravity, one of the other three characteristics shapes will be found, depending on the exact values of \\(a\_s\\) and \\(a\_F\\). +
-Guiding system flexibility: +Guidelines for system with guiding flexibility: 1. Driving force at Center of Mass (Best practice) 2. Locate sensor at Center of Mass (Second best) @@ -1007,53 +1112,341 @@ Guiding system flexibility:
+The best way to avoid any dynamic problems is to drive the system at its center of mass. +By doing so the rocking mode is not excited, which is not only favorable from a stability point of view, but also results in good set-point behaviour. + +When it is impossible to apply the forces at the correct location, one can eliminate the destabilizing effect of the rocking mode by locating the sensor at the height of the center of mass. +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](#org56e126a) 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 src="/ox-hugo/rankers98_frame_dynamics_2dof.png" caption="Figure 43: Model of a servo system including frame dynamics" >}} + +The transfer function \\(x\_{\text{servo}}/F\_{\text{servo}}\\) is: + +\begin{equation} +\frac{x\_2 - x\_1}{F\_{\text{servo}}} = \frac{1}{m\_2 s^2} + \frac{m\_2/m\_1}{m\_2s^2 + c(m\_2/m\_1)} +\end{equation} + +The mass ratio \\(\alpha\\) equal \\(m\_2/m\_1\\) and is therefore always positive. +Consequently, the resulting transfer function is of type "-2 slope / zero / pole / -2 slope". +The asymptotes at low and high frequencies are: + +\begin{align} +\left( \frac{x\_{\text{servo}}}{F\_{\text{servo}}} \right)\_{x \to 0} &= \frac{1}{m\_2 s^2} \\\\\\ +\left( \frac{x\_{\text{servo}}}{F\_{\text{servo}}} \right)\_{x \to \infty} &= \frac{m\_1 + m\_2}{m\_1 m\_2 s^2} +\end{align} + +
+
+ +Guidelines regarding frame motion: + +- frame inertia larger than load inertia in order to limit the increase of high frequency gain. +- frame inertia larger than fifty times the load inertia if additional flexible structures are attached to the frame. + +
+ #### General Guidelines {#general-guidelines} +
+
+ +The amount of contribution of a certain mode (Figure [44](#orge7882a4)) 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. + + + +{{< figure src="/ox-hugo/rankers98_mode_representation_guideline.png" caption="Figure 44: Graphical representation of mode i" >}} + +This observation leads to the idea that an undesired contribution of a mode to the response can be eliminated by one of the following two approaches: + +1. **Mode should not be excited**, which implies that the total moment acting on the modal lever should be equal to zero: + - Locate driving force at a node of the mode + - Modify mode shape such that the location of the driving force becomes a node of the mode + - Apply additional excitation forces such that the overall moment acting on the modal lever is equal to zero +2. **Output of response DoF should be zero**: + - Shift response DoF (sensor) towards node of mode + - Modify structural system such that the sensor location becomes a node of the mode + - Add additional sensors and combine the outputs such that the contribution of the mode to the new response signal is equal to zero. + +It can be shown that these two approaches are closely related to the terms "(un)controllability" and "(un)observability" that are frequently used in modern control theory. +From control theory, it is known that only those modes can be influenced of modified by the feedback loop which are observable and controllable. +If such a modification is not required and the modes are not excited by some other mechanism, it can be very effective to use uncontrollability and unobservability in a positive way in order to eliminate unwanted resonances that could endanger stability. + +
+ ## Predictive Modelling {#predictive-modelling} ### Steps in a Modelling Activity {#steps-in-a-modelling-activity} +One can distinguish at least four steps in any modelling activity (Figure [45](#org60f12a2)). + + + +{{< figure src="/ox-hugo/rankers98_steps_modelling.png" caption="Figure 45: Steps in a modelling activity" >}} + +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](#orgf77c197). +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](#orgf77c197)). +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. + On the basis of this comparison, design decisions are taken. + It is important to realize that the design decisions taken in this step are the actual outcome of the modelling process. + + + +{{< figure src="/ox-hugo/rankers98_illustration_first_two_steps.png" caption="Figure 46: Illustration of the first two steps in the modelling process" >}} + +Modelling and simulation can only have an impact on the design process when the last step is properly done. +Often, a lot of time and energy is wasted because extended modelling and simulations is done with great enthusiasm only to find out at the end that nobody is capable of interpreting the results and to take design decisions on the basis of the obtained results. +**It is therefore recommended to start any modelling process by specifying the criteria that will be used in the interpretation and evaluation phase.** + ### Step-wise Refined Modelling {#step-wise-refined-modelling} +Modelling and simulations can have three main applications: + +- **Decision support** +- **Design optimization** +- **Trouble shooting** (help to better understand unexpected problems and help to find solutions) + +Two aspects are crucial for the success of modelling and simulation as a tool in the product creation process, mainly the usefulness of results and the speed. + +The analysis program must be capable of providing **useful results**, that is to say the answers to the proper questions. +Simply stating that the first resonance frequency of a machine lies at 150 Hz does not satisfy the needs of the control engineer who wants to know whether the machine dynamics could endanger the stability of the servo system. +The results of the simulations could be presented as Bode or Nyquist diagrams. + +The second critical success factor is the **speed** with which is simulation results are obtained. +The decision making process can only be affected if the analysis results are available on time. + +
+
+ +A **three step modelling approach** is proposed: + +1. **Concept evaluation**: one checks whether a concept would work in uni-axial on the basis of a limited number of lumped masses connected by springs. +2. **System evaluation**: one checks whether it still works in 3D, again assuming rigid components connected by springs. +3. **Component evaluation**: one checks the deformation of the individual components, and how this affects the overall behaviour. + +
+ +
+
+ +Opponents of computer simulation often doubt the predictive value of these simulations, especially is the models are very elementary, and therefore do not carry out these simulations. +One must agree that successful simulations based on a 2DoF lumped mass model of a compact disc player are no guarantee that the final product will work according to specifications. +However, if these simulations, based on an elementary model of the product, show that the specifications are not met, then chances are extremely small that the final product will perform according to specifications. +Therefore, computer simulations should be regarded as a means to guide the design process by supporting the design choices and to detect unfit design concepts at a very early state in the design process. + +
+ +
+
+ +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](#orgc7e396f)). + + + +{{< figure src="/ox-hugo/rankers98_pattern_generator.png" caption="Figure 47: The basic elements of the pattern generator" >}} + +The basic elements of this machine are the work-piece and the optical unit. +The relative motion of these two elements in X and Y direction enables the generation of any pattern on the work-piece. +Based on the required throughput of the machine, an acceleration level of \\(1m/s^2\\) is required, whereas the positioning accuracy is \\(1\mu m\\) or better. + +
+ + +#### Specifications {#specifications} + +
+
+ +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](#orgf322acd)). + +
+ + + +{{< figure src="/ox-hugo/rankers98_system_performance_spec.png" caption="Figure 48: System performance specifications need to be translated into criteria on the basis of which simulation results can be judged" >}} + +In a first step one needs to make some initial **estimation about the required bandwidth** of the controlled system, because this is a prerequisite for evaluating the influence of the dynamics of the mechanical system on servo stability. + +
+
+ +Based on some analysis, disturbance (mainly friction) forces are foreseen to be in the order of 10N. +With the wanted accuracy is \\(1 \mu m\\), the initial estimate of the required servo stiffness \\(k\_p\\) is: + +\begin{equation} +k\_p = \frac{10 N}{10^{-6} m} = 10^7 N/m +\end{equation} + +Neglecting the effect of the derivative action of the controller, one can obtain a first estimate of the required bandwidth \\(f\_b\\): + +\begin{equation} +f\_b \approx \frac{1}{2\pi}\sqrt{\frac{k\_p}{m}} \approx 50Hz +\end{equation} + +with \\(m = 100kg\\) is the total moving mass. + +
+ +Having derived this estimate of the required bandwidth on the basis of the necessary disturbance rejection, one has to consider whether this bandwidth can be achieved without introducing stability problems and what the consequences are for the mechanical design. +Dynamic properties of the various designs can be now be evaluated. + + +#### Concept evaluation {#concept-evaluation} + +In the initial stage of the development a number of different concepts will be considered. +The designer will generally use his experience and engineering judgment to select one of these concepts. +In this stage, the designer only has a rough idea about the outlines of the machine, and the feasibility of this idea can be judged on the basis of very elementary calculations. + +
+
+ +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](#orgda306b4)). +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. + + + +{{< figure src="/ox-hugo/rankers98_pattern_generator_concept.png" caption="Figure 49: 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](#orgdefe8fe)). + + + +{{< figure src="/ox-hugo/rankers98_concept_1dof_evaluation.png" caption="Figure 50: Simple 1D model for the analysis of the dynamic behaviour in the X direction" >}} + +By using this fast and simple method of analysis, potential risks associated with the different concepts can be evaluated. + +
+ + +#### System Evaluation {#system-evaluation} + +Once the concept of the machine has been chosen, first rough three dimensional sketches become available and one can add extra spatial information to the simulations such as: + +- mass and mass moment of inertia of the different components +- location of the center of mass +- location of connecting stiffness +- location of driving forces +- location of sensors + +Typically, such a model contains 5-10 rigid bodies connected by suitable connectors that incorporate flexibility, whereas damping is in most cases added in the form of modal damping (1% relative damping is in most cases a good first estimate). + +
+
+ +Figure [51](#orgd35f0b3) shows such a 3D model of a different concept for the pattern generator. + + + +{{< figure src="/ox-hugo/rankers98_pattern_generator_rigid_body.png" caption="Figure 51: Rigid body model of a concept based on a movement of the work-piece in X direction, and a movement of the optical unit in Y direction" >}} + +
+ + +#### Component Evaluation {#component-evaluation} + +On the basis of previous analyses, experimental evaluation of previous designs, or engineering judgment, it is generally possible to identify **critical components** in the design. +These components will then need to be analyzed in more detail using FEM. + +Sometimes it is possible to judge the influence of the internal dynamics of such a component on the performance of the total system, based on a separate analysis of the component. +However, this approach requires serious consideration of the boundary conditions and is not always feasible. + +When a separate analysis of a component is not feasible, the detailed FEM description of the component can be used to replace for former rigid body description that has been used in the "system evaluation". +Such a step normally required the use of so-called "**sub-structuring**" techniques. + +
+
+ +In the patter generator it is very important that the connection between the linear motor module and the work piece is sufficiently stiff. +The reason lies in the fact that due to accuracy specifications the position is measured at the work piece and not at the motor. +Consequently, one has to ensure that the internal stiffness of the actuator is high enough to avoid stability problems. +FE model of this part can be used for such purpose. + +
+ + +#### Final remarks {#final-remarks} + +For the industrial application of "predictive modelling" it is essential that the amount of detail in a simulation model corresponds to the current phase in the design process. +A design team profits from the application of simulation tools only if a proper balance is found between detail and accuracy on one hand, and the total throughput time of the analysis on the other hand. + ### Practical Modelling Issues {#practical-modelling-issues} +In the "component evaluation" stage, detailed FE models of critical components need to be created and analyzed. +Sometimes, components can be evaluated individually against component specifications, which are often defined in terms of lower internal natural frequencies. +In other cases, such a separate analysis of a component is not sufficient to judge the impact of its dynamics on the overall system, and one is forced to combine these detailed component-models into a detailed model of the entire system. + +Due to the complexity of the structures it is normally not very practical to build one, single, huge, FE model of the entire device: + +- Building one huge model of a machine tends to be very error-prone +- It is not feasible to work on one huge model with a group of people +- 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](#org475956c), 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. + + + +{{< figure src="/ox-hugo/rankers98_substructuring_technique.png" caption="Figure 52: Steps in the creation of an overall system model based on detailed FE models of the components" >}} + +The process involves the following steps: + +- In the first step, the entire system is sub-divided into components +- In the second step, a detailed FE model of each component is generated, resulting in a component mass and stiffness matrix +- In the step three, a reduced model of the component is generated by applying a "component reduction" technique to the original model. + The intention of this step is to reduce the size of the matrices that describe the behaviour of the component, yet retain its main dynamic characteristics. +- Finally, the reduced models are assembled into one overall system + ## Conclusions {#conclusions} - +
+
+ Machine dynamics, and the interaction with the control system, plays a dominant role in the performance of fast and accurate servo-controlled positioning devices such as compact disc, wafer-steppers, and component-mounters. -"Modal analysis" is a numerical and experimental tool that can be very profitable in understanding the nature of complicated mechanical resonances. +**Modal analysis** is a numerical and experimental tool that can be very profitable in understanding the nature of complicated mechanical resonances. The mathematics of a single decoupled "modal" equation of motion can be translated into a graphical representation including all relevant data, which simplifies the understanding and creative use of the modal concept. -The introduction of the terms "effective" modal mass and stiffnesses enables a unique link between the modal and the physical domain. +The introduction of the terms "effective" modal mass and stiffness enables a unique link between the modal and the physical domain. From a servo stability point of view it is essential to investigate the mechanical FRF (\\(x/F\\)) which characterizes the dynamic properties of the mechanical system. Once the dynamics of the one individual mode is fully understood it is straightforward to construct this FRF and the interaction between the desired rigid body motion and the contribution of one additional mode. A closer investigation of this interaction reveals that only four interaction patterns exists. The destabilizing effect of a mechanical resonance depends not only on the resulting typical interaction pattern in the FRF, but also on its frequency in relation to the intended bandwidth frequency of the control loop. -On the basis of these stability considerations, design guidelines for the mechanics of a servo positioning devices are derived, so as to minimize the effect of mechanical vibrations on the stability of the controlled system. +On the basis of these stability considerations, **design guidelines** for the mechanics of a servo positioning devices are derived, so as to minimize the effect of mechanical vibrations on the stability of the controlled system. -In view of its importance to the overall performance, the effect of machine dynamics should be monitored during the entire design process through the use of modelling and simulation techniques. +In view of its importance to the overall performance, the effect of machine dynamics should be monitored during the entire design process through the use of **modelling and simulation** techniques. However, it is vital for the success of modelling and simulation as a tool to support design decisions, that analysis data are translated into useful information, and that this information is available on time. This requires a proper balance between accuracy and speed that can best be achieved by a top-down analysis process, which is closely linked to the phases in the design process, and in which the simulation models are step-wise refined. -When many parts of the mechanical system need to be modelled in great detail it is not advisable to build one, single, huge FE model but rather to apply a so-called "sub-structuring" techniques. +When many parts of the mechanical system need to be modelled in great detail it is not advisable to build one, single, huge FE model but rather to apply a so-called "**sub-structuring**" techniques. The Craig-Bampton approach, which is a component mode technique based on a combination of all boundary constraint modes plus a limited number of fixed interface normal modes, was found to be favorable. It has static solution capacity, and the frequency of the highest fixed-interface normal mode gives a good indication of the frequency range up to which the overall system results are valid. -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} -Rankers, Adrian Mathias. 1998. “Machine Dynamics in Mechatronic Systems: An Engineering Approach.” University of Twente. +Rankers, Adrian Mathias. 1998. “Machine Dynamics in Mechatronic Systems: An Engineering Approach.” University of Twente. diff --git a/static/ox-hugo/li01_flexure_hexapod_model.png b/static/ox-hugo/li01_flexure_hexapod_model.png new file mode 100644 index 0000000..9577fa0 Binary files /dev/null and b/static/ox-hugo/li01_flexure_hexapod_model.png differ diff --git a/static/ox-hugo/li01_quet_dirty_box.png b/static/ox-hugo/li01_quet_dirty_box.png new file mode 100644 index 0000000..8ac8d9b Binary files /dev/null and b/static/ox-hugo/li01_quet_dirty_box.png differ diff --git a/static/ox-hugo/rankers98_2dof_guiding_flex.png b/static/ox-hugo/rankers98_2dof_guiding_flex.png new file mode 100644 index 0000000..bf01314 Binary files /dev/null and b/static/ox-hugo/rankers98_2dof_guiding_flex.png differ diff --git a/static/ox-hugo/rankers98_2dof_guiding_flex_rock_mode.png b/static/ox-hugo/rankers98_2dof_guiding_flex_rock_mode.png new file mode 100644 index 0000000..91a4462 Binary files /dev/null and b/static/ox-hugo/rankers98_2dof_guiding_flex_rock_mode.png differ diff --git a/static/ox-hugo/rankers98_2dof_guiding_flex_x_mode.png b/static/ox-hugo/rankers98_2dof_guiding_flex_x_mode.png new file mode 100644 index 0000000..85f4e3c Binary files /dev/null and b/static/ox-hugo/rankers98_2dof_guiding_flex_x_mode.png differ diff --git a/static/ox-hugo/rankers98_concept_1dof_evaluation.png b/static/ox-hugo/rankers98_concept_1dof_evaluation.png new file mode 100644 index 0000000..94de9b5 Binary files /dev/null and b/static/ox-hugo/rankers98_concept_1dof_evaluation.png differ diff --git a/static/ox-hugo/rankers98_frame_dynamics_2dof.png b/static/ox-hugo/rankers98_frame_dynamics_2dof.png new file mode 100644 index 0000000..5c9a057 Binary files /dev/null and b/static/ox-hugo/rankers98_frame_dynamics_2dof.png differ diff --git a/static/ox-hugo/rankers98_illustration_first_two_steps.png b/static/ox-hugo/rankers98_illustration_first_two_steps.png new file mode 100644 index 0000000..0ca70e0 Binary files /dev/null and b/static/ox-hugo/rankers98_illustration_first_two_steps.png differ diff --git a/static/ox-hugo/rankers98_mode_representation_guideline.png b/static/ox-hugo/rankers98_mode_representation_guideline.png new file mode 100644 index 0000000..8774d31 Binary files /dev/null and b/static/ox-hugo/rankers98_mode_representation_guideline.png differ diff --git a/static/ox-hugo/rankers98_pattern_generator.png b/static/ox-hugo/rankers98_pattern_generator.png new file mode 100644 index 0000000..5319e9a Binary files /dev/null and b/static/ox-hugo/rankers98_pattern_generator.png differ diff --git a/static/ox-hugo/rankers98_pattern_generator_concept.png b/static/ox-hugo/rankers98_pattern_generator_concept.png new file mode 100644 index 0000000..c0e1f5f Binary files /dev/null and b/static/ox-hugo/rankers98_pattern_generator_concept.png differ diff --git a/static/ox-hugo/rankers98_pattern_generator_rigid_body.png b/static/ox-hugo/rankers98_pattern_generator_rigid_body.png new file mode 100644 index 0000000..f380d89 Binary files /dev/null and b/static/ox-hugo/rankers98_pattern_generator_rigid_body.png differ diff --git a/static/ox-hugo/rankers98_steps_modelling.png b/static/ox-hugo/rankers98_steps_modelling.png new file mode 100644 index 0000000..747a4b9 Binary files /dev/null and b/static/ox-hugo/rankers98_steps_modelling.png differ diff --git a/static/ox-hugo/rankers98_substructuring_technique.png b/static/ox-hugo/rankers98_substructuring_technique.png new file mode 100644 index 0000000..b82c5af Binary files /dev/null and b/static/ox-hugo/rankers98_substructuring_technique.png differ diff --git a/static/ox-hugo/rankers98_system_performance_spec.png b/static/ox-hugo/rankers98_system_performance_spec.png new file mode 100644 index 0000000..98e7876 Binary files /dev/null and b/static/ox-hugo/rankers98_system_performance_spec.png differ