Vibration With Control (Hardcover)

Chapter One

1.1 INTRODUCTION

In this chapter the vibration of a single-degree-of-freedom system will be analyzed and reviewed. Analysis, measurement, design, and control of a single-degree-of-freedom system (often abbreviated SDOF) is discussed. The concepts developed in this chapter constitute an introductory review of vibrations and serve as an introduction for extending these concepts to more complex systems in later chapters. In addition, basic ideas relating to measurement and control of vibrations are introduced that will later be extended to multiple-degree-of-freedom systems and distributed-parameter systems. This chapter is intended to be a review of vibration basics and an introduction to a more formal and general analysis for more complicated models in the following chapters.

Vibration technology has grown and taken on a more interdisciplinary nature. This has been caused by more demanding performance criteria and design specifications for all types of machines and structures. Hence, in addition to the standard material usually found in introductory chapters of vibration and structural dynamics texts, several topics from control theory and vibration measurement theory are presented. This material is included not to train the reader in control methods (the interested student should study control and system theory texts) but rather to point out some useful connections between vibration and control as related disciplines. In addition, structural control has become an important discipline requiring the coalescence of vibration and control topics. A brief introduction to nonlinear SDOF systems and numerical simulation is also presented.

1.2 SPRING-MASS SYSTEM

Simple harmonic motion, or oscillation, is exhibited by structures that have elastic restoring forces. Such systems can be modeled, in some situations, by a spring-mass schematic, as illustrated in Figure 1.1. This constitutes the most basic vibration model of a structure and can be used successfully to describe a surprising number of devices, machines, and structures. The methods presented here for solving such a simple mathematical model may seem to be more sophisticated than the problem requires. However, the purpose of the analysis is to lay the groundwork for the analysis in the following chapters of more complex systems.

If x = x(t) denotes the displacement (m) of the mass m (kg) from its equilibrium position as a function of time t (s), the equation of motion for this system becomes [upon summing forces in Figure 1.1(b)]

mx + k(x + [x.sub.s]) - mg = 0

where k is the stiffness of the spring (N/m), [x.sub.s] is the static deflection (m) of the spring under gravity load, g is the acceleration due to gravity (m/[s.sup.2]), and the overdots denote differentiation with respect to time. (A discussion of dimensions appears in Appendix A, and it is assumed here that the reader understands the importance of using consistent units.) From summing forces in the free body diagram for the static deflection of the spring [Figure 1.1(c)], mg = [kx.sub.s] and the above equation of motion becomes

mx(t) + kx(t) = 0 (1.1)

This last expression is the equation of motion of a single-degree-of-freedom system and is a linear, second-order, ordinary differential equation with constant coefficients.

Figure 1.2 indicates a simple experiment for determining the spring stiffness by adding known amounts of mass to a spring and measuring the resulting static deflection, [x.sub.s]. The results of this static experiment can be plotted as force (mass times acceleration) versus [x.sub.s], the slope yielding the value of k for the linear portion of the plot. This is illustrated in Figure 1.3.

Once m and k are determined from static experiments, Equation (1.1) can be solved to yield the time history of the position of the mass m, given the initial position and velocity of the mass. The form of the solution of Equation (1.1) is found from substitution of an assumed periodic motion (from experience watching vibrating systems) of the form

x(t) = A sin([omega].sub.n]t + [phi] (1.2)

where [omega].sub.n] = [square root of k/m] is the natural frequency (rad/s). Here, the amplitude, A, and the phase shift, [phi], are constants of integration determined by the initial conditions.

The existence of a unique solution for Equation (1.1) with two specific initial conditions is well known and is given by, for instance, Boyce and DiPrima (2000). Hence, if a solution of the form of Equation (1.2) form is guessed and it works, then it is the solution. Fortunately, in this case the mathematics, physics, and observation all agree.

To proceed, if [x.sub.0] is the specified initial displacement from equilibrium of mass m, and [v.sub.0] is its specified initial velocity, simple substitution allows the constants A and [phi] to be evaluated. The unique solution is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3)

Alternatively, x(t) can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.4)

by using a simple trigonometric identity.

A purely mathematical approach to the solution of Equation (1.1) is to assume a solution of the form [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and solve for [lambda], i.e.,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This implies that (because [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and A [not equal to] 0)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

or that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where j = [(-1).sup.1/2]. Then the general solution becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.5)

where [A.sub.1] and [A.sub.2] are arbitrary complex conjugate constants of integration to be determined by the initial conditions. Use of Euler''s formulae then yields Equations (1.2) and (1.4) (see, for instance, Inman, 2001). For more complicated systems, the exponential approach is often more appropriate than first guessing the form (sinusoid) of the solution from watching the motion.

Another mathematical comment is in order. Equation (1.1) and its solution are valid only as long as the spring is linear. If the spring is stretched too far, or too much force is applied to it, the curve in Figure 1.3 will no longer be linear. Then Equation (1.1) will be nonlinear (see Section 1.8). For now, it suffices to point out that initial conditions and springs should always be checked to make sure that they fall in the linear region if linear analysis methods are going to be used.

1.3 SPRING-MASS-DAMPER SYSTEM

Most systems will not oscillate indefinitely when disturbed, as indicated by the solution in Equation (1.3). Typically, the periodic motion dies down after some time. The easiest way to treat this mathematically is to introduce a velocity based force term, cx, into Equation (1.1) and examine the equation

mx + cx + kx = 0 (1.6)

This also happens physically with the addition of a dashpot or damper to dissipate energy, as illustrated in Figure 1.4.

Equation (1.6) agrees with summing forces in Figure 1.4 if the dashpot exerts a dissipative force proportional to velocity on the mass m. Unfortunately, the constant of proportionality, c, cannot be measured by static methods as m and k are. In addition, many structures dissipate energy in forms not proportional to velocity. The constant of proportionality c is given in N s/m or kg/s in terms of fundamental units.

Again, the unique solution of Equation (1.6) can be found for specified initial conditions by assuming that x(t) is of the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and substituting this into Equation (1.6) to yield

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.7)

Since a trivial solution is not desired, A [not equal to] 0, and since [e.sub.[lambda]t] is never zero, Equation (1.7) yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.8)

Equation (1.8) is called the characteristic equation of Equation (1.6). Using simple algebra, the two solutions for [lambda] are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.9)

The quantity under the radical is called the discriminant and, together with the sign of m, c, and k, determines whether or not the roots are complex or real. Physically, m, c, and k are all positive in this case, so the value of the discriminant determines the nature of the roots of Equation (1.8).

It is convenient to define the dimensionless damping ratio, [zeta], as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In addition, let the damped natural frequency, [[omega].sub.d], be defined (for 0 < [zeta] < 1) by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.10)

Then, Equation (1.6) becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.11)

and Equation (1.9) becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.12)

Clearly, the value of the damping ratio, [zeta], determines the nature of the solution of Equation (1.6). There are three cases of interest. The derivation of each case is left as a problem and can be found in almost any introductory text on vibrations (see, for instance, Meirovitch, 1986 or Inman, 2001).

Underdamped. This case occurs if the parameters of the system are such that

0 < [zeta] < 1

so that the discriminant in Equation (1.12) is negative and the roots form a complex conjugate pair of values. The solution of Equation (1.11) then becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.13)

or

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where A, B, C, and [phi] are constants determined by the specified initial velocity, [v.sub.0], and position, [x.sub.0]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.14)

The underdamped response has the form given in Figure 1.5 and consists of a decaying oscillation of frequency [[omega].sub.d].

Overdamped. This case occurs if the parameters of the system are such that

[zeta] > 1

so that the discriminant in Equation (1.12) is positive and the roots are a pair of negative real numbers. The solution of Equation (1.11) then becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.15)

where A and B are again constants determined by [v.sub.0] and [x.sub.0]. They are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The overdamped response has the form given in Figure 1.6. An overdamped system does not oscillate, but rather returns to its rest position exponentially.

Critically damped. This case occurs if the parameters of the system are such that

[zeta] = 1

so that the discriminant in Equation (1.12) is zero and the roots are a pair of negative real repeated numbers. The solution of Equation (1.11) then becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.16)

The critically damped response is plotted in Figure 1.7 for various values of the initial conditions [v.sub.0] and [x.sub.0].

It should be noted that critically damped systems can be thought of in several ways. First, they represent systems with the minimum value of damping rate that yields a nonoscillating system (Problem 1.5). Critical damping can also be thought of as the case that separates nonoscillation from oscillation.

1.4 FORCED RESPONSE

The preceding analysis considers the vibration of a device or structure as a result of some initial disturbance (i.e., [v.sub.0] and [x.sub.0]). In this section, the vibration of a spring-mass-damper system subjected to an external force is considered. In particular, the response to harmonic excitations, impulses, and step forcing functions is examined.

In many environments, rotating machinery, motors, and so on, cause periodic motions of structures to induce vibrations into other mechanical devices and structures nearby. It is common to approximate the driving forces, F(t), as periodic of the form

F(t)=[F.sub.0] sin [[omega].sub.t]

where [F.sub.0] represents the amplitude of the applied force and [omega] denotes the frequency of the applied force, or the driving frequency (rad/s). On summing the forces, the equation for the forced vibration of the system in Figure 1.8 becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.17)

Recall from the discipline of differential equations (Boyce and DiPrima, 2000), that the solution of Equation (1.17) consists of the sum of the homogeneous solution in Equation (1.5) and a particular solution. These are usually referred to as the transient response and the steady state response respectively. Physically, there is motivation to assume that the steady state response will follow the forcing function. Hence, it is tempting to assume that the particular solution has the form

[x.sub.p](t) = X sin([[omega].sub.t] - [theta] (1.18)

where X is the steady state amplitude and [theta] is the phase shift at steady state. Mathematically, the method is referred to as the method of undetermined coefficients. Substitution of Equation (1.18) into Equation (1.17) yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

or

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.19)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.20)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] as before. Since the system is linear, the sum of two solutions is a solution, and the total time response for the system of Figure 1.8 for the case 0 < [zeta] < 1 becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.21)

Here, A and B are constants of integration determined by the initial conditions and the forcing function (and in general will be different from the values of A and B determined for the free response).

Examining Equation (1.21), two features are important and immediately obvious. First, as t becomes larger, the transient response (the first term) becomes very small, and hence the term steady state response is assigned to the particular solution (the second term). The second observation is that the coefficient of the steady state response, or particular solution, becomes large when the excitation frequency is close to the undamped natural frequency, i.e., [omega] [approximately equals] [[omega].sub.n]. This phenomenon is known as resonance and is extremely important in design, vibration analysis, and testing.

Example 1.4.1

Compute the response of the following system (assuming consistent units):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

First, solve for the particular solution by using the more convenient form of

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

rather than the magnitude and phase form, where [X.sub.1] and [X.sub.2] are the constants to be determined. Differentiating [x.sub.p] yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Substitution of [x.sub.p] and its derivatives into the equation of motion and collecting like terms yield

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since the sine and cosine are independent, the two coefficients in parentheses must vanish, resulting in two equations in the two unknowns [X.sub.1] and [X.sub.2]. This solution yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Next, consider adding the free response to this. From the problem statement

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus, the system is underdamped, and the total solution is of the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Applying the initial conditions requires the following derivative

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The initial conditions yield the constants A and B:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus the total solution is

x(t)=[-e.sup.-0.2t (0.008 sin 1.99t +2.089 cos 1.99t)-0.134 sin 3t -0.032 cos 3t

Resonance is generally to be avoided in designing structures, since it means large-amplitude vibrations, which can cause fatigue failure, discomfort, loud noises, and so on. Occasionally, the effects of resonance are catastrophic. However, the concept of resonance is also very useful in testing structures. In fact, the process of modal testing (see Chapter 8) is based on resonance. Figure 1.9 illustrates how [[omega].sub.n] and [zeta] affect the amplitude at resonance. The dimensionless quantity Xk/[F.sub.0] is called the magnification factor and Figure 1.9 is magnification curve or magnitude plot. The maximum value at resonance, called the peak resonance, and denoted by [M.sub.p], can be shown (see, for instance, Inman, 2001) to be related to the damping ratio by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.22)

(Continues...)

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Excerpted from Vibration with Controlby D. J. Inman Copyright © 2006 by John Wiley & Sons, Ltd. Excerpted by permission.
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