User:Eml5526.s11.team4.rhee/HW1

Problem Statement
Derive the dynamic case (1D PDE)

The PDE is

Solution
For regular static case, summing the forces in the x-direction gives

where


 * $$N(x)$$ || is the normal force at $$x$$
 * $$f(x)$$ || is the body force normal force (in the units: force/length)
 * }
 * $$f(x)$$ || is the body force normal force (in the units: force/length)
 * }

Dividing Eq(2) by dx, we obtain

If we take the limit as $$dx->0 $$, it becomes

Now, let's define several terms. The stress is

The strain is

Taking the limit as $$dx->0$$

In our case, $$u=u(x,t)$$, so it is

And we know Hook's law:

where


 * $$E$$ || is Young's modulus.
 * }
 * }

Substituting Eq(8) into Eq(9)

Substituting in (5)

Substituting Eq(11) into Eq(4)

Eq(12) is same with the inertia force according to balance of forces. Therefore, it can be written as

where


 * $$\mathbf \rho $$ || is the mass density.
 * }
 * }

Rewriting the above equation

where


 * $$ m(x) $$ || is the linear mass density. (in the units: mass/unit length)
 * }
 * }

Finally, we got the dynamic response of an elastic bar of variable cross section. From Eq(12) and Eq(14), we obtain