626 Chapter 18 Mathematics in Engineering
on the beam is represented byW. The boundary conditions tell us what is happening at the bound-
aries of the beam. For Example 1, at supports located atX0 andXL, the deflection of the
beamYis zero. Later, when you take your differential equation class, you will learn how to obtain
the solution for this problem, as given in Table 18.8. The solution shows, for a given loadW,how
the given beam deflects at any locationX. Note that, if you substituteX0 orXLin the solu-
tion, the value ofYis zero. As expected, the solution satisfies the boundary conditions.
The differential equation for Example 2 is derived by applying Newton’s second law to the
given mass. Moreover, for this problem, the initial conditions tell us that, at timet0, we
pulled the mass upward by a distance ofy 0 and then release it without giving the mass any ini-
tial velocity. The solution to Example 2 gives the position of the mass, as denoted by the vari-
abley, with respect to timet. It shows the mass will oscillate according to the given cosinusoidal
function.
In Example 3,Trepresents the temperature of the fin at the location denoted by the
positionX, which varies from zero toL. Note thatXis measured from the base of the fin. The
boundary conditions for this problem tell us that the temperature of the fin at its base isTbase,
and the temperature of the tip of the fin will equal the air temperature, provided that the fin is
very long. The solution then shows how the temperature of the fin varies along the length of
the fin.
Again, please keep in mind that the purpose of this chapter was to focus on important
mathematical models and concepts and to point out why mathematics is so important in your
engineering education. Detailed coverage will be provided later in your math classes.Summary
Now that you have reached this point in the text you should
- be familiar with the examples of math symbols given in this chapter and be prepared to learn
new math symbols as you take additional math and engineering classes. - understand the role of the Greek alphabet in engineering and its importance in terms that
represent angles, dimensions, or a physical variable in a formula. You should also memorize
the Greek alphabet so you can communicate with others effectively. - understand the importance of linear and nonlinear models in describing engineering
problems and their solutions. You should also know the defining characteristics of these mod-
els. - realize that the formulation of many engineering problems leads to a set of linear algebraic
equations that are solved simultaneously. Therefore, a good understanding of matrix algebra
is essential. - know that calculus is divided into two broad areas: differential and integral calculus. You
should also know that differential calculus deals with understanding the rate of change —
how a variable may change with respect to another variable, and that integral calculus is
related to the summation or addition of things. - understand that differential equations contain derivatives of functions and represent the bal-
ance of mass, force, energy, and so on; and boundary conditions provide information about
what is happening physically at the boundaries of a problem. Initial conditions tell us about
the initial condition of a system before a disturbance or a change is introduced.
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).圀圀圀⸀夀䄀娀䐀䄀一倀刀䔀匀匀⸀䌀伀䴀圀圀圀⸀夀䄀娀䐀䄀一倀刀䔀匀匀⸀䌀伀䴀