Chapter 2 Notes

ME279 CHAPTER 2 NOTES

2.1 SYSTEM MODELING IN THE FREQUENCY DOMAIN

This chapter deals with the mathematical modeling of linear or linearized systems, and the analysis of these models using Laplace transformation. The term "frequency domain" is synonymous to the term Laplace domain. Most of this chapter was covered extensively in ME211, so we will only touch on a few of the highlights.

2.2 CHAPTER OBJECTIVES

1. Be able to apply Laplace Transformation methods to solve ordinary differential equations (ODEs). You must understand the principal Laplace transformation theorems, and be able to apply them.

2. Be able to analyze simple discrete translational and torsional mechanical models, as well as simple electrical circuits. You must be able to solve the resulting differential equations, as well as derive transfer functions relating arbitrary inputs and outputs.

3. Understand the concept of linearity, and be able to linearize simple nonlinear mathematical models. You must understand the limitations inherent in linearizing a nonlinear model.

2.3 LAPLACE TRANSFORMATION

Laplace transformation belongs to a class of analysis methods called integral transformation which are studied in the field of operational calculus. These methods include the Fourier transform, the Mellin transform, etc. In each method, the idea is to transform a difficult problem into an easy problem. For example, taking the Laplace transform of both sides of a linear, ODE results in an algebraic problem. Solving algebraic equations is usually easier than solving differential equations.

The one-sided Laplace transform which we are used to is defined by equation (2.1), and is valid over the interval [0,[[infinity]]). This means that the domain of integration includes its left end point. This is why most authors use the term 0^- to represent the bottom limit of the Laplace integral.

(2.1)

The key thing to note is that Equation (2.1) is not a function of time, but rather a function of the Laplace variable s = [[sigma]] + j[[omega]]. Also, the Laplace transform only transforms functions defined over the interval [0,[[infinity]]), so any part of the function which exists at negative values of t is lost! One of the most useful Laplace transformation theorems is the differentiation theorem.

Theorem 2.1

(2.2)

By repeating the integration by parts, higher derivatives may be similarly transformed. Thus given

, (2.3)

we have by taking the Laplace Transform of both sides of Equation (2.3)

or (2.4)

The difficulty arises when f(t) has either a step function, or a impulse (Dirac delta) function in it. These have the following definitions

Definition 2.1 The unit-step function:

(2.5)

Definition 2.2 The Dirac delta function:

(2.6)

Hence, the unit-step function "turns on" at the right edge (t = 0⁺) of zero, and the Dirac delta function turns on and off at the same place. An additional property of the Dirac delta function is

. (2.7)

Hence, the area under the "curve" defined by the Dirac delta, or impulse function is unity.

The unit-step and the Dirac delta function are derivative and anti-derivative of one another.

(2.8)

Both the unit-step and Dirac delta belong to a class of functions called generalized functions. The term "generalized" stems from the fact that these functions don't satisfy the continuity requirements of regular functions. This fact not withstanding, however, we may define the following relationships:

(2.9)

2.3.1 Additional properties of the Dirac delta function

An interesting property of the Dirac delta function is revealed by taking the Laplace transform of both sides of equation (2.8).

(2.10)

This is the so-called filtering property. In general,

(2.11)

The filtering property may be generalized for an arbitrary "turn-on" time t = b as

(2.12)

for a <= b <= c.

The filtering property may be proved as follows:

Proof 2.1

Equation (2.13) may be proved for a <= b <= c by noting that

(2.13)

Letting [[tau]] = t - b, integrating equation (2.14) by parts, and noting equation (2.8), we have

(2.14)

The last term in the brackets in Equation (2.14) vanishes because U(t) is zero in the interval of integration.

Thus, we have

(2.15)

However, since the limit of U(t) as t approaches zero from the left is zero, and the limit of U(t) as t approaches zero from the right is 1, we have

(2.16)

as claimed. z

The filtering property turns out to be extremely useful in both analytical and experimental vibration analysis, as well as many other areas in applied mathematics.

2.3.2 Use of Dirac delta function in differential equations

In the context of an equation of motion where forces or moments are summed to equal and inertial force or moment ([[Sigma]]F = ma), the Dirac delta function may be thought to carry the units of 1/time, or 1/area, etc. In fact, it may be, and is, thought of as a distribution - albeit a very narrow one. In other words, the force imparted by the delta function is proportional to its integrated area. This means that the Dirac delta function is only useful to describe impulsive forces when integrated over the applicable independent variable. Thus, integral transform techniques, such as the Laplace transform, provide the most natural means to utilize the Dirac delta function.

2.3.3 Additional Laplace transform properties

Tables 2.1 and 2.2 in the text contain most of the formulas and properties that you will need in this class, so I will not repeat them here. You should plan on reviewing Laplace transformation during the first couple of weeks of class to get comfortable again with it. Although I will supply the Laplace transforms that you might need on an exam, you should be able to manipulate and interpret the Laplace transforms that you obtain. This includes:

(1) being able to apply all of the theorems in table 2.2,

(2) being able to invert Laplace transforms using any means necessary, including partial fraction expansion, and

(3) being able to solve ordinary differential equations using the above.

Example 2.1

The shifting and filtering properties are useful in specifying the effect of a impulsive force applied to a body which may already be in motion.

Figure 2.1 Impulsively forced mass-spring-damper system

Consider the system shown in Figure 2.1, which consists of a 1 kg mass restrained by a linear spring of stiffness K = 10 N/m, and a damper with damping constant B = 2 N-s/m. The system is forced at time t = 5 seconds by an impulsive force of magnitude 10 N-s. The system has initial conditions , and .

Figure 2.2 Free body diagram of system

The free body diagram of the system is shown in Figure 2.2. Writing the equation of motion, we obtain

(2.17)

Taking the Laplace transform of Equation (2.17), we have

, (2.18)

. (2.19)

Taking the inverse Laplace transform of Equation (2.19), we obtain [you should show this]

. (2.20)

Figure 2.3 Time response of system

The time response of the system is shown in Figure 2.3. As you can see, the displacement of the mass has decayed nearly to zero when the impulsive force is applied - causing additional motion.

2.4 THE TRANSFER FUNCTION

The concept of the transfer function is useful in two principal ways:

1. given the transfer function of a system, we can predict the system response to an arbitrary input, and

2. it allows us to algebraically combine the functions of several subsystems in a natural way.

You should carefully read [[section]] 2.3 in Nise; it explains the essence of transfer functions. Some additional points to consider which often cause confusion are:

1. non-zero initial conditions must be treated as additional inputs,

2. the choice of which input and which output to consider in constructing a transfer function is completely arbitrary (your choice), and

3. a transfer function can only relate one input with one output at a time. Hence, several transfer functions are possible for systems with multiple inputs and outputs.

2.5 TRANSFER FUNCTIONS FOR ELECTRICAL NETWORKS

Read [[section]] 2.4 in Nise, and work through the example problems. Please note, however, that the author's use of symmetry to derive the system equations as in Example 2.9 should only be used as a check. My reasoning is that while I might what these mesh equations are supposed to look like, I will never forget Kirchoff's laws!

2.6 TRANSFER FUNCTIONS FOR TRANSLATIONAL SYSTEMS

Again, you should read through this section, and make sure you understand the examples. One problem I have with the author's presentation of mechanical modeling, however, is his representation of the inertial forces on the free-body diagrams. He is applying what is known as D'Alebert's principle, which while not intrinsically wrong, leads to more confusion, of students than straight-forward application of Newton's laws. Hence, I recommend that you set [[Sigma]] F = mass times acceleration, and then bring all of the dependent variable terms on the left-hand side of the equation. You will make a lot fewer sign errors this way. Also, the author's application of symmetry in determining the equations of motion is "interesting," but not too easy to remember in the long run.

2.7 TRANSFER FUNCTIONS FOR ROTATIONAL SYSTEMS

The same comments mentioned in the previous section apply here as well. Read through [[section]] 2.6. My comments regarding inertial forces are equally valid for inertial torques.

2.8 TRANSFER FUNCTIONS FOR SYSTEMS WITH GEARS

This section describes the reasonably useful concept of impedance transfer in rotational systems with gear trains.

Example 2.2

Figure 2.4 Rotational system with a gear train.

A typical rotational system with a gear train is shown in Figure 2.1, where the mass moments of inertias are denoted by J, the number of teeth on the gears is denoted N, the torsional stiffness is denoted as K, and the torsional viscous damping is denoted B. Lets say we are charged with determining the transfer function relating the input torque Ti(s) with output angle [[theta]]1(s). The conventional way to proceed is to draw free body diagrams of each body, sum the torques equal to the inertial torques, and manipulate the equations of motion to obtain the desired transfer function. Alternatively, we might recognize that our system only has a single degree of freedom because there is no relative motion between the gears (unless they are stripped!). Hence, given two gears of radii r1 and r2, and number of teeth N1, and N2, we see that because there is no slipping between the gears

r1[[theta]]1 = r2 [[theta]]2 (2.21)

and, since the number of teeth is proportional to the gear radius (N = k r), we have

. (2.22)

Assuming no losses between the gears, the work done by torque applied to one of the gears is equal to the work done by the driven gear. Hence,

. (2.23)

Since,

(2.24)

we have that the torque transmitted across the gear train from J2 to J1, is given by , where TR(t) is the reaction torque between the gear train and the J2 body.

Figure 2.5 Free-body diagram of system.

Hence, summing up the torques applied to the first body, we have

, (2.25)

Similarly, for the second body, we obtain

(2.26)

Solving Equation (2.26) for TR, and substiting into Equation (2.25) we obtain, after collecting terms,

. (2.27)

From Equation (2.27), we see that the impedances from the second body may be "reflected" to the first body by multiplying them by the squared ratio of the destination gear (N1) over the source gear (N2) and adding them to the first body impedances. The applied torques from the second body may be reflected to the first by same method, but with the ratio unsquared. Using equation (2.27), we may now determine the input torque (T1) required to excite a desired response [[theta]]1(t). In the time domain, we have

, (2.28)

so, for example, if we want a harmonic response of the form [[theta]]1 = 10 sin ([[omega]] t) radians per second, with an applied torque T2(t) = 5 cos ([[omega]] t) Newton-meters, we must apply a torque

. (2.28)

Thus, the ability to reflect mechanical impedances across gear trains is quite handy in determining the required control torque to generate a prescribed motion. The reflecting method, is limited to gear trains, or other mechanisms which multiply torque or motion with a single degree of freedom. In other words, there can be no relative motion anywhere between the input gear, and the output gear. [read [[section]]2.7 in Nise]

2.9 TRANSFER FUNCTIONS OF ELECTROMECHANICAL SYSTEMS

The principal electromechanical system which we will study in ME279 is the DC motor. DC motors are quite commonly used as servos to position mechanisms. Servomotors typically have relatively high torque, but low no-load speed. The DC motor with a fixed magnetic field (due to a permanent magnet) is well described in [[section]] 2.8 in Nise. This section will form the basis for much of our future modeling.

2.10 NONLINEARITIES

The concept of linearity is one of the most important ideas in engineering, physics, and applied mathematics. The difference between a linear and nonlinear system is often the difference between finding a solution or not. Recently, in the past 20 years or so, engineers have been made aware of the existance of chaos in nonlinear systems. This awareness has spawned a whole new branch of applied mathematics - the study of chaotic systems.

Unfortunately, about 99.99999% of the availible mathematical techniques developed in the last 300 years are only useful for linear systems! This includes all of the classical control methods which we will learn in ME279! What are we going to do about this limitation? Well, we're going to linearize the nonlinear terms in our mathematical models as we encounter them. But first, let us carefully define the term linear.

Definition 2.3

A function, or operation f(x) is said to be linear if and only if for f(x1) = y1, and for f(x2) = y2, we have f(x1+x2) = y1 + y2. In addition, we must have that f(kx) = kf(x) for any constant k.

Definition 2.3 is somewhat more restrictive than the definition of a line [Why?] What do we mean by the term linear operation? The most common linear operations that you will encounter in this course are integration, including Laplace transformation, and differentiation. What are the consequences of linearizing a nonlinear function or operation?

Example 2.3

Figure 2.6 ____ ___ ___ x, __ __ __ x - , ______ sin(x).

A commonly encountered nonlinear term is sin(x). For x "near" zero, sin(s) ~ x according to Taylor's series. If we need more accuracy we can keep the first two terms in the Taylor series expansion, but then we have a nonlinear system again. A comparison of sin(x) and its one and two term Taylor series expansions is shown in Figure 2.6. Clearly, the linear approximation of sin(x) rapidly becomes inaccurate for large magnitudes of x.

2.10.1 Linearizing nonlinear differential equations

As mentioned earlier, it may be impossible to analytically solve a nonlinear ODE, and it is impossible to control a nonlinear system using the methods covered in this course. However, we can linearize the nonlinear terms in a nonlinear system, and obtain an approximate solution which will be valid for small excursions of the dependent variable about an operating point. In Figure 2.6, the operating point is also a point of equilibrium - sin(0) = 0.

Definition 2.4

A differential equation is said to be nonlinear if any of its terms are nonlinear in the dependent variable, or any of the derivatives of the dependent variable with respect to the independent variable.

In most of the systems which we will study in this class, the dependent variable will be displacement, and the independent variable will be time.

Definition 2.5

A system is said to be in a state of equilibrium if and only if its dependent variable is invariant with respect to the independent variable.

In other words, in the systems encountered in this course, all terms which are explicit functions of time, and all terms which are time-derivatives of the dependent variable must vanish. Thus, to locate the equilibrium point(s) of a system, set all time-derivatives of the dependent variable to zero, and ignore all explicit functions of time. Then, solve the resulting nonlinear equation for the equilibrium value of the dependent variable.

Once the equilibrium point has been determined, the nonlinear terms may then be linearized about this point, and substituted back into the original equation of motion. If the equation of motion is linearized about an equilibrium point, any constant forcing terms will cancel out. If the equation of motion is linearized about any other point, there will still be some constant forcing terms in the equation. This makes sense if you realize that added force is required to hold a mechanism in any position other than an equilibrium position.

Example 2.4

A mass of 1 kg is suspended by a nonlinear spring and damper as shown above. The tension force resisting the downward pull of gravity in the spring and damper is given by Newtons, where x is measured from the unstretched length of the spring. The mass is subjected to an impulsive force of 10d(t) Newtons as shown.

Find: (a) the nonlinear equation of motion, (b) the equilibrium point , (c) the linearized equation of motion in terms of the incremental deviation, , from the equilibrium point, and (d) the solution to the linearized equation of motion if .

Solution:

The following steps are applicable to all linearization problems, and you should make sure you understand all of them.

FBD:

Nonlinear equation of motion: (2.29)

Equilibrium equation: (2.30)

Equilibrium point: (2.31)

Linearizing:

, (2.32)

since .

Letting :

(2.33)

Linearized equation of motion: (2.34)

Laplace transform of solution: (2.35)

Solution in time-domain: (2.36)

2.11 USING MAPLE(TM) TO SOLVE ODE's IN THE LAPLACE DOMAIN

The symbolic mathematics program Maple(TM) is available on the UNIX server (Saucer, for example) and the Macintosh server. It should also soon be available on the PC servers as well. Maple is an extremely powerful means to perform computer algebra as well as numerical solutions in a manner similar to Mathematica. There are some notable differences between Maple and Mathematica, however. Maple has an extensive on-line help facility which may be accessed at the command prompt by entering ?subject. You are encouraged to explore the capabilities of Maple beyond what is required in ME279, but for now, we shall focus on a relatively small subset of Maple's capabilities. Specifically, we will use Maple to invert Laplace transforms, and plot the resulting time-domain solution. The following example will illustrate this process, but you will have to do some browsing through Maple's functions via the on-line help before you become comfortable with Maple's syntax.

Example 2.5

Determine the unit step response for the system defined by .

Solution:

The following commands are entered at the command line within Maple. First, Maple is executed (on UNIX systems) by entering "xmaple" at the prompt. Comments (non executed lines) are denoted by %.

% Suppose we forgot what the inverse Laplace command does, or that it requires us first to issue % the readlib(laplace) command. ?invlaplace tells us what we need. The following series of commands, each of which is followed by a semicolon except the ? command, will be echoed to the screen (not shown here) by Maple. The end result is the plot shown below.

>?invlaplace

> readlib(laplace);

> T:=10*(s+10)/(s^2+6*s+25);

> R:=1/s;

> c:=invlaplace(R*T,s,t);

> plot(c,t=0..10);

There are several key commands with which one must become familiar to navigate Maple. Issue the following commands to learn about some of these functions: ?with, ?readlib, ?allvalues, ?solve, ?fsolve, ?dsolve, ?int, ?diff, ?simplify, ?expand, ?plot, ?odeplot, ?laplace, ?invlaplace.

For example, issuing the ?Heaviside command yields the following output:

FUNCTION: Dirac - the Dirac delta function

FUNCTION: Heaviside - the Heaviside step function

CALLING SEQUENCE:

Dirac(t)

Dirac(n, t)

Heaviside(t)

PARAMETERS:

t - algebraic expression

n - nonnegative integer

SYNOPSIS:

- The Dirac(t) delta function is defined as zero everywhere except at t = 0

where it has a singularity. It has an additional property, specifically:

Int(Dirac(t),t = -infinity..infinity) = 1

- Derivatives of the Dirac function are denoted by the two-argument Dirac func-

tion. The first argument denotes the order of the derivative. For example,

diff(Dirac(t), t$n) will be automatically simplified to Dirac(n, t) for any

integer n.

- The Heaviside(t) unit step function is defined as zero for t < 0, 1 for

t >= 0. It is related to the Dirac function by

diff(Heaviside(t),t) = Dirac(t).

- These functions are typically used in the context of integral transforms such

as laplace(), mellin() or fourier().