Barna Szabó - Finite Element Analysis

	Element number
α	1	2	3	4	5
1	0	0	0	0	0	0
2	11.00	61.24	15.31	7.02	4.55	2.45
3	20.09	4.69	98.83	16.16	9.24	4.62
4	36.98	8.41	17.24	30.13	14.02	8.00

Errors in numerical integration can be particularly damaging. The reader should be mindful of this when applying the concepts and procedures discussed in this chapter to higher dimensions.

The following problem is a prototype of an important class of engineering problems which includes the undamped vibration of elastic structures:

(1.133)

where the primes represent differentiation with respect to x . For example, we may think of an elastic bar of length , cross‐section A , modulus of elasticity E , in which case given in units of Newton (N) or equivalent, the parameter is the coefficient of distributed springs ( ) and the parameter is mass per unit length (kg/m = ). The bar is vibrating in its longitudinal direction.

The boundary conditions are:

and the initial conditions are

where and are given functions in . Here we consider homogeneous Dirichlet boundary conditions. However, the boundary conditions can be homogeneous Neumann or homogeneous Robin conditions, or any combination of those.

The generalized form is obtained by multiplying eq. (1.133)by a test function and integrating by parts:

(1.134)

We now introduce where , . This is known as separation of variables. Therefore we get

(1.135)

which can be written as

(1.136)

Since the functions on the left are independent of t , the function T depends only on t , both expressions must equal some positive constant denoted by . That constant has to be positive because the expression on the left holds for all and if we select then the expression on the left is positive.

The function satisfies the ordinary differential equation

(1.137)

the solution of which is

(1.138)

where ω is the angular velocity (rad/s). Alternatively ω is written as where f is the frequency (Hz).

To find ω and U we have to solve the problem

(1.139)

which will be abbreviated as

(1.140)

There are infinitely many solutions called eigenpairs , . The set of eigenvalues is called the spectrum. If Ui is an eigenfunction and α is a real number then is also an eigenfunction. In the following we assume that the eigenfunctions have been normalized so that