Chapter 9: Modeling of Mechanical Systems for …

1 9 Modeling ofMechanical Systemsfor Mechatronics Applications Int roduction Mechanical system Modelingin Mechatronic Systems Physical Variables and Power Bonds Interconnectionof Components Causality Descriptions of Basic Mechanical Model Components Defining Mechanical Input and Output Model Elements Dissipative Effects in Mechanical Systems Potential Energy Storage Elements Kinetic Energy Storage Coupling Mechanisms Impedance Relationships Physical Laws for Model Formulation. Kinematic and Dynamic Laws Identifying and Representing Motion in a Bond Graph Assigning and Using Causality Developing a Mathematical Model Noteon Some Difficulties in Deriving Equations Energy Methods for Mechanical system Model Formulation Multiport Models Restrictions on Constitutive Relations Deriving Constitutive Relations Checking the Constitutive Relations Rigid Body Multidimensional Dynamics Kinematics of a Rigid Body Dynamic Properties of a Rigid Body Rigid Body Dynamics Lagrange s Equations Classical Approach Dealing with Nonconservative Effects Extensions for Nonholonomic Systems Mechanical Subsystem Models Using Lagrange Methods Methodology for Building Subsystem Model

2 Introduction Mechatronics applications are distinguished by controlled motion of Mechanical Systems coupled toactuators and sensors. Modeling plays a role in understanding how the properties and performance ofmechanical components and Systems affect the overall mechatronic system design. This Chapter reviewsmethods for Modeling Systems of interconnected Mechanical components, initially restricting the Raul G. Longoria The University of Texas at Austin 2002 CRC Press LLC application to basic translational and rotational elements, which characterize a wide class of mechatronicapplications. The underlying basis of Mechanical motion (kinematics) is presumed known and notreviewed here, with more discussion and emphasis placed on a system dynamics perspective.

3 Moreadvanced applications requiring two- or three-dimensional motion is presented in section Systems can be conceptualized as rigid and/or elastic bodies that may move relative to oneanother, depending on how they are interconnected by components such as joints, dampers, and otherpassive devices. This Chapter focuses on those Systems that can be represented using lumped-parameterdescriptions, wherein bodies are treated as rigid and no dependence on spatial extent need be consideredin the elastic effects. The Modeling of Mechanical Systems in general has reached a fairly high level ofmaturity, being based on classical methods rooted in the Newtonian laws of motion. One benefits fromthe extensive and overwhelming knowledge base developed to deal with problems ranging from basicmass-spring Systems to complex multibody Systems .

4 While the underlying physics are well understood,there exist many different means and ways to arrive at an end result. This can be especially true whenthe need arises to model a multibody system , which requires a considerable investment in methods forformulating and solving equations of motion. Those applications are not within the scope of this Chapter ,and the immediate focus is on Modeling basic and moderately complex Systems that may be of primaryinterest to a mechatronic system designer/analyst. Mechanical system Modeling in Mechatronic Systems Initial steps in Modeling any physical system include defining a system boundary, and identifying howbasic components can be partitioned and then put back together. In Mechanical Systems , these analysescan often be facilitated by identifying points in a system that have a distinct velocity.

5 For purposes ofanalysis, active forces and moments are applied at these points, which could represent energetic inter-actions at a system boundary. These forces and moments are typically applied by actuators but mightrepresent other loads applied by the environment. A Mechanical component modeled as a point mass or rigid body is readily identified by its velocity,and depending on the number of bodies and complexity of motion there is a need to introduce acoordinate system to formally describe the kinematics ( , see [12] or [15]). Through a kinematicanalysis, additional (relative) velocities can be identified that indicate the connection with and motionof additional Mechanical components such as springs, dampers, and/or actuators.

6 The interconnectionof Mechanical components can generally have a dependence on geometry. Indeed, it is dependence ofmechanical Systems on geometry that complicates analysis in many cases and requires special consider-ation, especially when handling complex Systems . A preliminary description of a Mechanical system should also account for any constraints on themotional states, which may be functions of time or of the states themselves. The dynamics of mechanicalsystems depends, in many practical cases, on the effect of constraints. Quantifying and accounting forconstraints is of paramount importance, especially in multibody dynamics, and there are different schoolsof thought on how to develop models. Ultimately, the decision on a particular approach depends on theapplication needs as well as on personal preference.

7 It turns out that a fairly large class of Systems can be understood and modeled by first understandingbasic one-dimensional translation and fixed-axis rotation. These Systems can be modeled using methodsconsistent with those used to study other Systems , such as those of an electric or hydraulic type. Fur-thermore, building interconnected mechatronic system models is facilitated, and it is usually easier fora system analyst to conceptualize and analyze these summary, once an understanding of (a) the system components and their interconnections (includ-ing dependence on geometry), (b) applied forces/torques, and (c) the role of constraints, is developed,dynamic equations fundamentally due to Newton can be formulated. The rest of this section introducesthe selection of physical variables consistent with a power flow and energy-based approach to modelingbasic Mechanical translational and rotational Systems .

8 In doing so, a bond graph approach [28,3,17] isintroduced for developing models of Mechanical Systems . This provides a basis for introducing the 2002 CRC Press LLC concept of causality, which captures the input output relationship between power-conveying variablesin a system . The bond graph approach provides a way to understand and mathematically model basic aswell as complex Mechanical Systems that is consistent with other energetic domains (electric, electrome-chanical, thermal, fluid, chemical, etc.). Physical Variables and Power Bonds Power and Energy Basis One way to consistently partition and connect subsystem models is by using power and energy variablesto quantify the system interaction, as illustrated for a Mechanical system in Fig. (a). In this figure,one port is shown at which power flow is given by the product of force and velocity, F V , and anotherfor which power is the product of torque and angular velocity, T.

9 These power-conjugate variables( , those whose product yields power) along with those that would be used for electrical and hydraulicenergy domains are summarized in Table Similar effort ( e ) and flow ( f ) variables can be identifiedfor other energy domains of interest ( , thermal, magnetic, chemical). This basis assures energeticallycorrect models, and provides a consistent way to connect system elements together. In Modeling energetic Systems , energy continuity serves as a basis to classify and to quantify [28] shows how the energy continuity equation, together with a carefully defined port concept, pro-vides a basis for a generalized Modeling framework that eventually leads to a bond graph s reticulated equation of energy continuity,( )concisely identifies the l distinct flows of power, P i , m distinct stores of energy, E j , and the n distinctdissipators of energy, P d.

10 Modeling seeks to refine the descriptions from this point. For example, in asimple mass spring damper system , the mass and spring store energy, a damper dissipates energy, and TABLE Power and Energy Variables for Mechanical Systems Energy Domain Effort, e Flow, f Power, P General efe f [W]TranslationalForce, F [N]Velocity, V [m/sec] F V [N m/sec, W]RotationalTorque, T Angular velocity, T [N m/sec, W]or [N m] [rad/sec] ElectricalVoltage, v [V]Current, i [A] v i [W]HydraulicPressure, P [Pa]Volumetric flowrate, P Q [W] Q [m 3 /sec] FIGURE Basic interconnection of Systems using power dEjdt-------Pd()kk=1n +j=1m =vmwmLmRmJmBmvin(a)(b)FVTwTmiinElectrica lEMMechanical 2002 CRC Press LLC the interconnection of these elements would describe how power flows between them.