Then we add to the kinetic energy function a magnetic energy function Wm , x, and add to the potential energy an electric field energy function WeQ, x.. The equations of both the mass an
Trang 1circuits by defining the charge on the capacitor, Q, as another generalized coordinate along with x, i.e.,
in Lagrange’s formulation, q1 = x, q2 = Q Then we add to the kinetic energy function a magnetic energy
function Wm( , x), and add to the potential energy an electric field energy function We(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., Q1 = , Q2 = + 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 d0, 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
-[T+W m]
∂q˙ k
- ∂ T[ +W m]
∂q k
∂q k
cx˙
2
L x ( )Q˙2 1
2
LI2, W e 1
2C x( )
-Q2
F m ∂W m(x, Q˙)
∂x
- 1
2
I2dL x( )
dx
-, F e ∂W e(x, Q)
∂x
2
Q2d dx
- 1
C x( )
W e∗(V, x) 1
2
0V2A d0
d02–x2
-=
Trang 2circuits by defining the charge on the capacitor, Q, as another generalized coordinate along with x, i.e.,
in Lagrange’s formulation, q1 = x, q2 = Q Then we add to the kinetic energy function a magnetic energy
function Wm( , x), and add to the potential energy an electric field energy function We(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., Q1 = , Q2 = + 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 d0, 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
-[T+W m]
∂q˙ k
- ∂ T[ +W m]
∂q k
∂q k
cx˙
2
L x ( )Q˙2 1
2
LI2, W e 1
2C x( )
-Q2
F m ∂W m(x, Q˙)
∂x
- 1
2
I2dL x( )
dx
-, F e ∂W e(x, Q)
∂x
2
Q2d dx
- 1
C x( )
W e∗(V, x) 1
2
0V2A d0
d02–x2
-=
Trang 3
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:
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˙ = ∑Fi
Eniko T Enikov
University of Arizona
0066_Frame_C08 Page 1 Wednesday, January 9, 2002 3:48 PM
Trang 4Modeling 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