circuits by defining the charge on the capacitor, Q, as another generalized coordinate along with x, i.e., in Lagrange’s formulation, q 1 = x, q 2 = Q. Then we add to the kinetic energy function a magnetic energy function W m (,x), and add to the potential energy an electric field energy function W e (Q, x). The equations of both the mass and the circuit can then be derived from (7.44) The generalized force must also be modified to account for the energy dissipation in the resistor and the energy input of the applied voltage V(t), i.e., Q 1 = , Q 2 = + V(t). In this example the magnetic energy is proportional to the inductance L(x), and the electric energy function is inversely proportional to the capacitance C(x). Applying Lagrange’s equations automatically results in expressions for the magnetic and electric forces as derivatives of the magnetic and electric energy functions, respectively, i.e., (7.45) (7.46) These remarkable formulii are very useful in that one can calculate the electromagnetic forces by just knowing the dependence of the inductance and capacitance on the displacement x. These functions can often be found from electrical measurements of L and C. Example: Electric Force on a Comb-Drive MEMS Actuator Consider the motion of an elastically constrained plate between two grounded fixed plates as in a MEMS comb-drive actuator in Fig. 7.15. When the moveable plate has a voltage V applied, there is stored electric field energy in the two gaps given by (7.47) In this expression the electric energy function is written in terms of the voltage V instead of the charge on the plates Q as in Eqs. (7.45) and (7.46). Also the initial gap is d 0 , and the area of the plate is A. FIGURE 7.14 Coupled lumped parameter electromechanical system with single degree of freedom mechanical motion x(t). Q ˙ d∂ dt TW m +[] ∂q ˙ k ∂ TW m +[] ∂q k – ∂ WW e +[] ∂q k + Q k = cx ˙ – RQ ˙ – W m 1 2 Lx()Q ˙ 2 1 2 LI 2 , W e 1 2Cx() Q 2 == = F m ∂W m x, Q ˙ () ∂x 1 2 I 2 dL x() dx , F e ∂W e x, Q() ∂x – − 1 2 Q 2 d dx 1 Cx() == = = W e ∗ V, x() 1 2 e 0 V 2 A d 0 d 0 2 x 2 – = 0066_Frame_C07 Page 16 Wednesday, January 9, 2002 3:39 PM ©2002 CRC Press LLC circuits by defining the charge on the capacitor, Q, as another generalized coordinate along with x, i.e., in Lagrange’s formulation, q 1 = x, q 2 = Q. Then we add to the kinetic energy function a magnetic energy function W m (,x), and add to the potential energy an electric field energy function W e (Q, x). The equations of both the mass and the circuit can then be derived from (7.44) The generalized force must also be modified to account for the energy dissipation in the resistor and the energy input of the applied voltage V(t), i.e., Q 1 = , Q 2 = + V(t). In this example the magnetic energy is proportional to the inductance L(x), and the electric energy function is inversely proportional to the capacitance C(x). Applying Lagrange’s equations automatically results in expressions for the magnetic and electric forces as derivatives of the magnetic and electric energy functions, respectively, i.e., (7.45) (7.46) These remarkable formulii are very useful in that one can calculate the electromagnetic forces by just knowing the dependence of the inductance and capacitance on the displacement x. These functions can often be found from electrical measurements of L and C. Example: Electric Force on a Comb-Drive MEMS Actuator Consider the motion of an elastically constrained plate between two grounded fixed plates as in a MEMS comb-drive actuator in Fig. 7.15. When the moveable plate has a voltage V applied, there is stored electric field energy in the two gaps given by (7.47) In this expression the electric energy function is written in terms of the voltage V instead of the charge on the plates Q as in Eqs. (7.45) and (7.46). Also the initial gap is d 0 , and the area of the plate is A. FIGURE 7.14 Coupled lumped parameter electromechanical system with single degree of freedom mechanical motion x(t). Q ˙ d∂ dt TW m +[] ∂q ˙ k ∂ TW m +[] ∂q k – ∂ WW e +[] ∂q k + Q k = cx ˙ – RQ ˙ – W m 1 2 Lx()Q ˙ 2 1 2 LI 2 , W e 1 2Cx() Q 2 == = F m ∂W m x, Q ˙ () ∂x 1 2 I 2 dL x() dx , F e ∂W e x, Q() ∂x – − 1 2 Q 2 d dx 1 Cx() == = = W e ∗ V, x() 1 2 e 0 V 2 A d 0 d 0 2 x 2 – = 0066_Frame_C07 Page 16 Wednesday, January 9, 2002 3:39 PM ©2002 CRC Press LLC 8 Structures and Materials 8.1 Fundamental Laws of Mechanics Statics and Dynamics of Mechatronic Systems • Equations of Motion of Deformable Bodies • Electric Phenomena 8.2 Common Structures in Mechatronic Systems Beams • Torsional Springs • Thin Plates 8.3 Vibration and Modal Analysis 8.4 Buckling Analysis . 8.5 Transducers Electrostatic Transducers • Electromagnetic Transducers • Thermal Actuators • Electroactive Polymer Actuators 8.6 Future Trends The term mechatronics was first used by Japanese engineers to define a mechanical system with embedded electronics, capable of providing intelligence and control functions. Since then, the continued progress in integration has led to the development of microelectromechanical systems (MEMS) in which the mechanical structures themselves are part of the electrical subsystem. The development and design of such mechatronic systems requires interdisciplinary knowledge in several disciplines—electronics, mechanics, materials, and chemistry. This section contains an overview of the main mechanical struc- tures, the materials they are built from, and the governing laws describing the interaction between electrical and mechanical processes. It is intended for use in the initial stage of the design, when quick estimates are necessary to validate or reject a particular concept. Special attention is devoted to the newly emerging smart materials—electroactive polymer actuators. Several tables of material constants are also provided for reference. 8.1 Fundamental Laws of Mechanics Statics and Dynamics of Mechatronic Systems The fundamental laws of mechanics are the balance of linear and angular momentum. For an idealized system consisting of a point mass m moving with velocity v , the linear momentum is defined as the product of the mass and the velocity: L = m v (8.1) The conservation of linear momentum for a single particle postulates that the rate of change of linear momentum is equal to the sum of all forces acting on the particle (8.2)L ˙ mv ˙ F i ∑ == Eniko T. Enikov University of Arizona 0066_Frame_C08 Page 1 Wednesday, January 9, 2002 3:48 PM ©2002 CRC Press LLC 9 Modeling of Mechanical Systems for Mechatronics Applications 9.1 Introduction 9.2 Mechanical System Modeling in Mechatronic Systems Physical Variables and Power Bonds • Interconnection of Components • Causality 9.3 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 9.4 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 • Note on Some Difficulties in Deriving Equations 9.5 Energy Methods for Mechanical System Model Formulation Multiport Models • Restrictions on Constitutive Relations • Deriving Constitutive Relations • Checking the Constitutive Relations 9.6 Rigid Body Multidimensional Dynamics Kinematics of a Rigid Body • Dynamic Properties of a Rigid Body • Rigid Body Dynamics 9.7 Lagrange’s Equations Classical Approach • Dealing with Nonconservative Effects • Extensions for Nonholonomic Systems • Mechanical Subsystem Models Using Lagrange Methods • Methodology for Building Subsystem Model 9.1 Introduction Mechatronics applications are distinguished by controlled motion of mechanical systems coupled to actuators and sensors. Modeling plays a role in understanding how the properties and performance of mechanical components and systems affect the overall mechatronic system design. This chapter reviews methods for modeling systems of interconnected mechanical components, initially restricting the Raul G. Longoria The University of Texas at Austin ©2002 CRC Press LLC . Arizona 0066_Frame_C08 Page 1 Wednesday, January 9, 2002 3 :48 PM 2002 CRC Press LLC 9 Modeling of Mechanical Systems for Mechatronics Applications 9 .1 Introduction 9.2 Mechanical. RQ ˙ – W m 1 2 Lx()Q ˙ 2 1 2 LI 2 , W e 1 2Cx() Q 2 == = F m ∂W m x, Q ˙ () ∂x 1 2 I 2 dL x() dx , F e ∂W e x, Q() ∂x – − 1 2 Q 2 d dx 1 Cx() == = = W e ∗ V, x() 1 2 e 0 V 2 A d 0 d 0 2 x 2 – . RQ ˙ – W m 1 2 Lx()Q ˙ 2 1 2 LI 2 , W e 1 2Cx() Q 2 == = F m ∂W m x, Q ˙ () ∂x 1 2 I 2 dL x() dx , F e ∂W e x, Q() ∂x – − 1 2 Q 2 d dx 1 Cx() == = = W e ∗ V, x() 1 2 e 0 V 2 A d 0 d 0 2 x 2 –