User:Eml5526.s11.team4.simko/HW1

=Homework 1=

1.1 - Derive EOM and PDE
|Problem 1 - Charles Cook

Problem Statement:
Discuss the specific case where the elastic bar from homework problem 1.1 has a rectangular cross section. The cross section is defined as shown in the figure below, where the height 'h(x)' varies with axial position. The base 'b' is constant for the entire bar.



Solution:
Starting with Equation 1.4 which is in a general form,

the area term 'A(x)' can be modified for the more specific case of a rectangular cross section. The cross sectional area is given by:

Using the equation for the area of a trapezoid, the mass per unit length 'm(x)' can be written as:

where $$\rho$$ is the density. The mid-point of the differential element is used to approximate the density.

Given that:

It should be noted that the normal force terms 'N(x)' and 'N(x+dx)' contain an area dependence as shown previously in Equations 1.7-1.9. With this in mind, Equation 1.9 and Equation 2.2 can be substituted together into Equation 1.4 to produce:

which is the force balance for an elastic bar with a rectangular cross section.

Again taking the limit:

produces the following Equation 2.6. As noted previously in problem 1, dx changes from the finite difference to a differential difference.

Using Equation 1.7 and Equation 1.9, Equation 2.7 is produced.

A simple back substitution of Equation 2.1 into Equation 2.7 yields:

Where $$ A(x)\rho(x)=m(x)$$, Equation 1.1 has again been derived.