Differential Equations and Linear Algebra
Math 2150, Fall 2001
Lab 3 : Transverse Vibrations of a Beam
(A Boundary Value Problem)

This is essentially Project B on page 367, with more elaborate instructions. To study the transverse vibrations of a beam of length L we need to consider the differential equation

y'''' - k^4 y = 0,
that is,
y^(4) - k^4 y = 0,

where k is a constant determined by the mechanical properties of the beam and a parameter which we seek to determine. The independent variable x describes the position along the beam, so lies in the interval [0,L], and the dependent variable y is related to the displacement of the beam.

If the ends of the beam are clamped, then the solutions must satisfy

y(0) = y'(0) = 0     and       y(L) = y'(L) = 0.

This is called a boundary value problem because we require the solutions to satisfy conditions at the boundary points x = 0 and x = L , unlike an initial value problem which restricts the value of the solution (and its derivatives) at one initial point.

We want to determine those values of k for which there are nontrivial solutions ( y not identically 0).
Proceed as follows.

(a) Show that if k = 0 then there are no nonzero solutions to the boundary value problem.

(b) Assume that k > 0 and show that the general solution can be written as a sum of sines. cosines, hyperbolic sines, and hyperbolic cosines.

(c) Use the boundary conditions to obtain four linear equations that the constants in the general solution must satisfy if the boundary conditions hold.

(d) Show that these equations have a nonzero solution only when k satisfies the equation

cosh(kL) = sec(kL)

(e) Plotting the graphs of cosh and sec on the same axes, show that there are an infinite number of values of k for which there are nontrivial solutions to the boundary value problem.

Due Wednesday, December 12, 2001, in class.