Sentence examples for harmonic solutions from inspiring English sources

The problem is very interesting since the possible solutions, which are very sensitive to the parameters of the system, range from simple stable harmonic solutions to chaotic.

The stabilities of the harmonic solutions are analyzed, and finally our proposed approximation is verified compared with the numerical results.

The transfer matrix type dynamic stiffness expressions are developed from exact harmonic solutions given translational or rotational displacement excitations.

These new Duffing equation solutions give accurate results in all regions where stable harmonic solutions are known to exist.

Examples are presented for free vibrations with cubic and antisymmetric quadratic non-linearities and for harmonic solutions of the undamped Duffing equation.

Computed harmonic solutions yield as many natural frequencies and mode shapes as may be required, for springs with a variety of boundary conditions and smooth helical geometries which may include variable curvature and tortuosity.

The steady state harmonic solution is obtained using the harmonic balance method and results are verified numerically.

We demonstrate that the general solution with recharge consists of the harmonic solution superposed on a special case of the harmonic solution along with two elementary one-dimensional flow solutions.

The harmonic solution of the equations is derived by adopting an averaging method.

The conserved quantity is constructed, and its zero value corresponds to the harmonic solution.

The assumption of a harmonic solution for the uncoupled equations led to wave equations.