tmdsimpy.postprocess.shooting.time_stability

tmdsimpy.postprocess.shooting.time_stability(vib_sys, XlamP_shoot, Fl, Nt=128)

Calculate time series and stability of a shooting solution

Parameters:

vib_systmdsimp.VibrationSystem

Object that was used for calculating the shooting solution that has N DOFs.

XlamP_shoot(M, 2*N+1) numpy.ndarray

A set of solutions to the shooting equations for vib_sys. Has N displacements, then N velocities, then frequency in rad/s. Each row is an independent solution point.

Fl(2*N) numpy.ndarray

First N entries are cosine forcing at frequency XlamP_shoot[-1]. The second N are the sine forcing terms.

Ntint, optional

Number of time steps to use in shooting calculations. The default is 128.

Returns:

y_t(N, Nt+1, M) numpy.ndarray

Time series of displacements for each solution point. Includes the first and last time point, which are numerically identical when the solution is converged.

ydot_t(N, Nt+1, M) numpy.ndarray

Time seres of velocities for each solution point.

stable(M,) numpy.ndarray of bool

Is true where the solution is stable (maximum eigenvalue of Monodromy matrix is less than 1.0).

max_eig(M,) numpy.ndarray

Maximum eigenvalue of the Monodromy matrix.

See also

tmdsimpy.VibrationSystem.shooting_res

Residual function for shooting. This postprocesses results that satisfy these equations.

Notes

For theory about shooting and stability analysis, see Section 3 of [1].

There are no formal tests for this function, but there is an example for an SDOF Duffing oscillator.

References

[1]

Peeters, M., R. Viguie, G. Sérandour, G. Kerschen, and J. -C. Golinval. 2009. “Nonlinear Normal Modes, Part II: Toward a Practical Computation Using Numerical Continuation Techniques.” Mechanical Systems and Signal Processing, Special Issue: Non-linear Structural Dynamics, 23 (1): 195–216. https://doi.org/10.1016/j.ymssp.2008.04.003.