0071481133_ar008.pdf

Source: MECHANICAL DESIGN HANDBOOK

●

MECHANICAL SYSTEM

ANALYSIS

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MECHANICAL SYSTEM ANALYSIS

Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com)

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Source: MECHANICAL DESIGN HANDBOOK

CHAPTER 8

SYSTEM DYNAMICS

Sheldon Kaminsky, M.M.E., M.S.E.E.

Consulting Engineer

Weston, Conn.

8.1 INTRODUCTION: PRELIMINARY

CONCEPTS 8.3

8.1.1 Degrees of Freedom 8.5

8.1.2 Coupled and Uncoupled Systems

8.5.4 Linear Time-Invariant Control

System 8.53

8.5.5 Analysis of Control System 8.54

8.5.6 The Problem of Synthesis 8.62

8.5.7 Linear Discontinuous Control: Sampled

Data 8.62

8.5.8 Nonlinear Control Systems

8.6

8.1.3 General System Considerations

8.7

8.2 SYSTEMS OF LINEAR PARTIAL

DIFFERENTIAL EQUATIONS

8.8

8.70

8.2.1 Elastic Systems 8.8

8.2.2 Inelastic Systems 8.15

8.3 SYSTEMS OF ORDINARY DIFFERENTIAL

EQUATIONS 8.17

8.3.1 Fundamentals 8.17

8.3.2 Introduction to Systems of Nonlinear

Differential Equations 8.18

8.4 SYSTEMS OF ORDINARY LINEAR

DIFFERENTIAL EQUATIONS 8.26

8.4.1 Introduction to Matrix Analysis of

Differential Equations 8.29

8.4.2 Fourier-Series Analysis 8.36

8.4.3 Complex Frequency-Domain

Analysis 8.38

8.4.4 Time-Domain Analysis 8.44

8.5 BLOCK DIAGRAMS AND THE TRANSFER

FUNCTION 8.50

8.5.1 General 8.50

8.5.2 Linear Time-Invariant Systems

8.6 SYSTEMS VIEWED FROM STATE

SPACE 8.79

8.6.1 State-Space Characterization 8.79

8.6.2 Transfer Function from State-Space

Representation 8.82

8.6.3 Phase-State Variable-Form Transfer

Function: Canonical (Normal) Form 8.82

8.6.4 Transformation to Normal Form 8.85

8.6.5 System Response from State-Space

Representation 8.86

8.6.6 State Transition matrix for Sampled

Data Systems 8.87

8.6.7 Time-Varying Linear Systems

8.88

8.7 CONTROL THEORY 8.89

8.7.1 Controllability 8.89

8.7.2 Observability 8.89

8.7.3 Introduction to Optimal Control

8.89

8.7.4 Euler-Lagrange Equation 8.90

8.7.5 Multivariable with Constraints and

Independent Variable t

8.50

8.5.3 Feedback Control-System

Dynamics

8.92

8.52

8.7.6 Pontryagin’s Principle

8.95

8.1

INTRODUCTION—PRELIMINARY CONCEPTS

A physical system undergoing a time-varying interchange or dissipation of energy

among or within its elementary storage or dissipative devices is said to be in a

“dynamic state.” The elements are in general inductive, capacitative, or resistive—the

first two being capable of storing energy while the last is dissipative. All are called

“passive,” i.e., they are incapable of generating net energy. A system composed of a

finite number or a denumerable infinity of storage elements is said to be “lumped” or

“discrete,” while a system containing elements which are dense in physical space is

called “continuous.” The mathematical description of the dynamics for the discrete

case is a set of ordinary differential equations, while for the continuous case it is a set

of partial differential equations.

The mathematical formulation depends upon the constraints (e.g., kinematic or

geometric) and the physical laws governing the behavior of the system. For example,

8.3

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SYSTEM DYNAMICS

8.4

MECHANICAL SYSTEM ANALYSIS

the motion of a single point mass obeys F m ( d v / dt ) in accordance with Newton’s

second law of motion. Analogously, the voltage drop across a perfect coil of self-

inductance L is V L ( di / dt ), a consequence of Faraday’s law. In the first case the

energy-storage element is the mass, which stores mv 2 /2 units of kinetic energy while

the inductance L stores Li 2 /2 units of energy in the second case. A spring-mass system

and its electrical analog, an inductive-capacitive series circuit, represent higher-order

discrete systems. The unbalanced force acting on the mass is F kx . Thus

m $ m , k

(8.1)

Analogously for the electrical case,

L $

q / c

L , c

following Kirchhoff’s voltage-drop law (i.e., the sum of voltage drop around a closed

loop is zero). To show that Eq. (8.1) expresses the dynamic exchange of energy, multiply

Eq. (8.1) by

x # dt

(which is equal to dx ) and integrate:

Fx . dt

mx . $ dt

k . xdt

Fdx

x x 0

mx . 2

mx . 0

d t

kx 2

d t

kx 2

kx 0

Work input

which is a statement of the law of conservation of energy. This illustrates that work

input is divided into two parts, one part increasing the kinetic energy, the remainder

increasing the potential energy. The actual partition between the two energy sources at

any instant is time-varying, depending on the solution to Eq. (8.1).

If a viscous damping element is added to the system the force equation becomes

(see Fig. 8.1 a )

m $

cx .

F c

and performing the same operation of multiplying by

dt ,( dx ) and integrating we

obtain

m $ x . dt

cx . 2 dt

kxx . dt

Fdx

(8.2 a )

x 0

m . 2

d t

kx 2

cx . 2 dt

Fdx

(8.2 b )

x 0

FIG. 8.1

Second-order systems. ( a ) Mechanical system. ( b ) Electrical analog.

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SYSTEM DYNAMICS

8.5

cx . 2

again expressing the energy-conservation law. Note that the integrand and that

the integral in Eq. (8.2 b ) is thus a monotonically increasing function of time. This

condition assures that, for F 0, the free (homogeneous) system must eventually

come to rest since under this condition Eq. (8.2 b ) becomes

mx . 2

mx . 0

kx 2

kx 0

cx . 2 dt

const

(8.3)

which again is an expression of the law of energy conservation. The first two terms are

positive since they contain the squared factors and x 2 , while the third term, as noted

above, increases with time. It follows that the sum of the first two must decrease

monotonically in order to satisfy Eq. (8.3); moreover, neither term can be greater than

the sum. It follows that, as , and .

Formulation of the foregoing simple problems was based upon fundamental physi-

cal laws. The derivation by Lagrange equations, which in this simple case offers little

advantage, is (Chap. 1)

x . 2

t S `

x S 0

2 mx . 2

kx 2 /2

V T

For conservative systems (e.g., spring-mass),

dt ' L

' L

m $

' x

' x .

For nonconservative systems with dissipation function

cx . 2 /2

dt '

' x .

' x 2 ' f

' x .

m $

2 cx .

m $

cx .

Precisely the same form is deducible from a Lagrange statement of the electrical

equivalent (Fig. 8.1 b ).

8.1.1

Degrees of Freedom

Thus far it has been observed that one independent variable x was employed to

describe the system dynamics. In general, however, several variables x 1 , x 2 , . . ., x n are

necessary to describe the motion of a complex system. The minimum number of coor-

dinates that are so required is defined as the number of degrees of freedom of the sys-

tem. Simple examples of two-degree-of-freedom systems are shown in Fig. 8.2. The

respective equations of motion are

m 1 $ 1

Mechanical:

k 1 s x 1

x 2 d

(8.4 a )

m 2 $ 2

k 2 x 2

k 1 s x 2

x 1 d

L 1 $ 1

Electrical:

s q 1

q 2 d / c 1

(8.4 b )

L 2 $ 2

q 2 / c 2

s q 2

q 1 d / c 1

derivable from force and loop voltage-drop considerations.

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