1
Single Degree of Freedom Systems

1.1 Introduction

In this chapter, the vibration of a single degree of freedom system (SDOF) will be analyzed and reviewed. Analysis, measurement, design and control of SDOF systems are discussed. The concepts developed in this chapter constitute a review of introductory 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 speci-fications of 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 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 (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 this analysis is to lay the groundwork for solving more complex systems discussed in the following chapters.

Figure 1.1 (a) A spring-mass schematic, (b) a free body diagram, and (c) a free body diagram of the static spring mass system.

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

where k is the stiffness of the spring (N/m), xs is the static deflection (m) of the spring under gravity load, g is the acceleration due to gravity (m/s2) and the over dots 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 1.1c), mg = kxs and the above equation of motion becomes

This last expression is the equation of motion of an SDOF 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, xs. The results of this static experiment can be plotted as force (mass times acceleration) versus xs, the slope yielding the value of k for the linear portion of the plot. This is illustrated in Figure 1.3.

Figure 1.2 Measurement of spring constant using static deflection caused by added mass.

Figure 1.3 Determination of the spring constant. The dashed box indicates the linear range of the spring.

1Once 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

where is called the natural frequency in radians per second (rad/s). Here A, the amplitude, and f, the phase shift, 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 in Boyce and DiPrima (2012). Hence, if a solution of the form of Equation (1.2) is guessed and it works, then it is the solution. Fortunately, in this case, the mathematics, physics and observation all agree.

To proceed, if x0 is the specified initial displacement from equilibrium of mass m, and v0 is its specified initial velocity, simple substitution allows the constants of integration A and f to be evaluated. The unique solution is

Alternately, x(t) can be written as

by using a simple trigonometric identity or by direct substitution of the initial conditions (Example 1.2.1).

A purely mathematical approach to the solution of Equation (1.1) is to assume a solution of the form x(t) = Ae?t and solve for ?, i.e.

This implies that (because e?t ? 0 and A ? 0)

or that

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

where A1 and A2 are arbitrary complex conjugate constants of integration to be determined by the initial conditions. Use of Euler's formulas then yields Equations (1.2) and (1.4) (Inman, 2014). 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 (Section 1.10). For now, it suffices to point out that initial conditions and springs should always be checked to make sure that they fall into the linear region, if linear analysis methods are going to be used.

Example 1.2.1

Assume a solution of Equation (1.1) of the form

and calculate the values of the constants of integration A1 and A2 given arbitrary initial conditions x0 and v0, thus verifying Equation (1.4).

Solution: The displacement at time t = 0 is

or A2 = x0. The velocity at time t = 0 is

Solving this last expression for A1 yields A1 = v0/x0, so that Equation (1.4) results in

Example 1.2.2

Compute and plot the time response of a linear spring-mass system to initial conditions of x0 = 0.5 mm and , if the mass is 100 kg and the stiffness is 400 N/m.

Solution: The frequency is

Next compute the amplitude from Equation (1.3):

From Equation (1.3) the phase is

Thus the response has the form

and this is plotted in Figure 1.4.

Figure 1.4 The response of a simple spring-mass system to an initial displacement of x0 = 0.5 mm and an initial velocity of . The period, defined as the time it takes to complete one cycle off oscillation, T = 2p/?n, becomes T = 2p/2 = ps.

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 term, , into Equation (1.1) and examine the equation

This also happens physically with the addition of a dashpot or damper to dissipate energy, as illustrated in Figure 1.5.

Figure 1.5 (a) Schematic of spring-mass-damper system. (b) A free-body diagram of the system in part (a).

Equation (1.6) agrees with summing forces in Figure 1.5 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 Newton-second per meter (Ns/m) or kilograms per second...