dyna2f¶

Purpose:

Compute the dynamic solution to a set of uncoupled second-order differential equations.

Syntax:

Y = dyna2f(w2, xi, f, p, dt)

Description:

dyna2f computes the solution to the set of differential equations:

\[\ddot{x}_i + 2 \xi_i\omega_i \dot{x}_i + \omega^{2}_i x_i = f_i g(t), \qquad i=1,\ldots,m\]

The vectors w2, xi and f are the squared circular frequencies \(\omega_i^2\), the damping ratios \(\xi_i\), and the applied forces \(f_i\), respectively. The force vector p contains the Fourier coefficients \(p(k)\) formed by the command fft.

The solution in the frequency domain is computed at equal time increments defined by dt. The result is stored in the \(m \times n\) matrix Y, where m is the number of equations and n is the number of frequencies resulting from the fft command. The dynamic solution in the time domain is achieved by the use of the command ifft.

Example:

The dynamic solution to a set of uncoupled second-order differential equations can be computed by the following sequence of commands:

>> g = gfunc(G, dt);
>> p = fft(g);
>> Y = dyna2f(w2, xi, f, p, dt);
>> X = (real(ifft(Y.')))';

where it is assumed that the input variables G, dt, w2, xi and f are properly defined. Note that the ifft command operates on column vectors if Y is a matrix; therefore use the transpose of Y. The output from the ifft command is complex. Therefore use Y.' to transpose rows and columns in Y in order to avoid the complex conjugate transpose of Y.

The time response is represented by the real part of the output from the ifft command. If the transpose is used and the result is stored in a matrix X, each row will represent the time response for each equation as the output from the command dyna2.

See also:

gfunc, fft, ifft