Finite Element Analysis. Barna Szabó

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

If the eigenvalues are distinct then the corresponding eigenfunctions are orthogonal: Let and be eigenpairs, . Then from eq. (1.140) we have

      Subtracting the second equation from the first we see that if then Ui and Uj are orthogonal functions:

      (1.141)

      and hence .

      Importantly, it can be shown that any function

Скачать книгу