International Space Station

One way to think of such a stability problem is to imagine yourself operating a joy stick, where moving the joy stick applies a force at one of the beam's joints. The goal is to apply the appropriate forces to keep the beam from vibrating. For instance, if the beam starts moving to the left, you might move your joystick to the right, applying an opposing force. If we think of this process as a mathematical function, you are supplying the input (the force) that determines the output (the motion of the beam). Your task is then to solve an inverse problem. That is, you know the desired output (stability) and must determine the correct input (force) that produces it. We discuss inverse functions in section 6.2.

Engineers analyzing vibration in beams begin with a differential equation, an equation involving an unknown function and one or more of its derivatives. A solution of a differential equation is a function; in our case, a function describing the motion of the beam. Very often, solutions of differential equations involve exponential functions of the form e^kt . In fact, simple cases of our vibrating beam problem can be solved in terms of such functions. For example, e^kt might represent the displacement of the beam from its resting position at a given point. If k < 0 , then e^kt approaches 0 as t  tends to . This would represent a stable beam. On the other hand, if k > 0 , e^kt tends to as t  tends to . This would be bad news for our space station. In view of this, we would need to make certain that k is negative, to ensure the stability of the structure. We also need to identify cases where the value of k is close to 0 , since a small change in k could then turn a stable system into an unstable one.

In the first four sections of this chapter, we carefully develop and explore the properties and some basic applications of the natural logarithm and exponential functions. Of course, we have already studied exponentials in earlier chapters, but the prominent role that these functions play makes the extra time and effort worthwhile. Many important problems in science and engineering involve differential equations. Accordingly, in section 6.5 we present a particular type of differential equation for which we can find solutions. In section 6.6, we introduce a simple method for numerically approximating solutions of more general differential equations. In the final three sections of the chapter, we introduce several new functions that are important in calculus. The inverse trigonometric functions developed in sections 6.7 and 6.8 apply the notion of inverse functions (from section 6.2) to the trigonometric functions. The hyperbolic functions introduced in section 6.9 are related to both the exponential function and the trigonometric functions.