digital-brain/content/phdthesis/rankers98_machin.md

+++
title = "Machine dynamics in mechatronic systems: an engineering approach."
author = ["Thomas Dehaeze"]
draft = false
+++

Tags
:


Reference
: ([Rankers 1998](#org2a286fc))

Author(s)
: Rankers, A. M.

Year
: 1998

**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:
>
> -   _Actuator flexibility_: the mechanical system does not behave as one rigid body, due to flexibility between the location at which the servo force is applied and the actual point that needs to be positioned
> -   _Guiding system flexibility_: the device usually rely on the guiding system to suppress motion in an undesired direction
> -   _Limited mass and stiffness of the stationary machine part_: the reaction force that comes with the driving force will introduce a motion of the "stationary" part of the mechanical system
>
> Whereas the first two phenomena mainly affect the stability of the control loop, the last phenomena manifests itself more often as a dynamic positional error in the set-point response.
>
> A tool that can be very useful in understanding the nature of more complex resonance phenomena and the underlying motion of the mechanical system, is "Modal Analysis".
> Translating the mathematics of one single decoupled "modal" equation into a graphical representation, which includes all relevant data such as (effective) modal mass and stiffness plus the motion of each physical DoF, facilitates a better understanding of the modal concept.
> It enables a very intuitive link between the modal and the physical domain, and thus leads to a more creative use of "modal analysis" without the complications of the mathematical formalism.
>
> Dynamic phenomena of the mechanics in a servo positioning device can lead to stability problems of the control loop.
> Therefore it is important to investigate the frequency response (\\(x/F\\)), which characterizes the dynamics of the mechanical system, and especially the influence of mechanical resonances on it.
> Once the behavior of one individual mode is fully understood it is not so difficult to construct this frequency response and the interaction between the rigid-body motion of the device, and the dynamics of one additional mode.
> This leads to four interaction patterns:
>
> -   -2 slope / zero / pole / -2 slope
> -   -2 slope / pole / zero / -2 slope
> -   -2 slope / pole / -4 slope
> -   -2 slope / pole / -2 slope (non-minimum phase and rarely occurring)
>
> It is not possible to judge the potential destabilizing effect of each of the typical characteristics without considering the frequency of the resonance in relation to the envisaged bandwidth of the control loop.
> The phase plot of a typical open loop frequency response of a PID controlled positioning device without mechanical resonances can be divided into three frequency ranges:
>
> -   at low frequencies, the phase lies below -180 deg due to integrator action of the controller
> -   at medium frequency (centered by the bandwidth frequency), the phase lies above -180 deg due to the differential action of the controller, which is necessary in order to achieve a stable position control-loop
> -   at high frequencies, the phase eventually drops again below -180 deg due to additional low-pass filtering
>
> The potential destabilizing effect of each of the three typical characteristics can be judged in relation to the frequency range.
> Whether instability occurs depends very strongly on the resonance amplitude and damping of the additional mode.
>
> -   A -2 slope / zero / pole / -2 slope characteristics leads to a phase lead and is therefore potentially destabilizing in the low-frequency and high frequency regions.
>     In the medium frequency region it adds an extra phase leads to the already existing margin, which does not harm the stability.
> -   A -2 slope / pole / zero / -2 slope combination has the reverse effect.
>     It is potentially destabilizing in the medium-frequency range and is harmless in the low and high frequency ranges.
> -   The -2 slope / poles / -4 slope behavior always has a devastating effect on the stability of the loop if located in the low or medium frequency range.
>
> 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:
>
> -   usefulness of results
> -   speed
>
> The analysis process has usually a top-down structure.
> Starting with very elementary simulation models to support the selection of the proper concept, these models should become more refined, just like the product or machine under development.
>
> In various project throughout the past years, a three-step modeling approach has evolved, in which the following phases can be distinguished:
>
> -   concept analysis
> -   system analysis
> -   component analysis
>
> In the _concept analysis_ the viability of various concepts is evaluated on the basis of very simple models consisting of a limited number of lumped masses connected by springs.
> Once a concept has been chosen and the first rough 3D sketches become available, a _system analysis_ can be done, based on a limited number of 3D rigid components connected by springs.
> In this phase a lot of important spatial information is added to the model (such as the location of the center of gravity and connecting stiffnessses, plus the location of the driving force and of the sensors).
> Finally, in the _component analysis_ phase critical components are no longer considered rigid, and their internal dynamics are evaluated via FE modeling.
> In cases in which a separate analysis of a critical component is considered insufficient to judge its influence on the overall dynamics, a detailed FE-based description can be used to replace the former rigid description in the system model.
>
> In case many parts of the system need to be modeled in great detail, it is not very practical (error-prone, huge model size, time consuming) to build on, single, huge FE model of the entire system.
> A technique that overcomes these disadvantages is the so-called "sub-structuring technique".
> 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}


### General {#general}


### State of the Art {#state-of-the-art}


### Scope and Purpose {#scope-and-purpose}


### Preview {#preview}


## Mechanical Servo Systems {#mechanical-servo-systems}


### Basic Control Aspects {#basic-control-aspects}


### Specifications {#specifications}


### Interaction Dynamics and Control {#interaction-dynamics-and-control}


### Three Important Dynamic Effects {#three-important-dynamic-effects}


## Modal Decomposition {#modal-decomposition}


### Mathematics of Modal Decomposition {#mathematics-of-modal-decomposition}


### Graphical Representation {#graphical-representation}


### Physical Meaning of Modal Parameters {#physical-meaning-of-modal-parameters}


### A Pragmatic View on Sensitivity Analysis {#a-pragmatic-view-on-sensitivity-analysis}


### Modal Superposition {#modal-superposition}


### Suspension Modes {#suspension-modes}


## Modes and Servo Stability {#modes-and-servo-stability}


### Basic Characteristics of Mechanical FRF {#basic-characteristics-of-mechanical-frf}


### Destabilising Effect of Modes {#destabilising-effect-of-modes}


### Design for Stability {#design-for-stability}


## Predictive Modelling {#predictive-modelling}


### Steps in a Modelling Activity {#steps-in-a-modelling-activity}


### Step-wise Refined Modelling {#step-wise-refined-modelling}


### Practical Modelling Issues {#practical-modelling-issues}


## 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.
> 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.
>
> 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.
>
> 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  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}

<a id="org2a286fc"></a>Rankers, Adrian Mathias. 1998. “Machine Dynamics in Mechatronic Systems: An Engineering Approach.” University of Twente.