Stability of Linear Systems With Parametric Excitation

1970 ◽  
Vol 37 (1) ◽  
pp. 228-230 ◽  
Author(s):  
J. R. Dickerson
Keyword(s):  
Linear Systems ◽  
High Frequency ◽  
Mathieu Equation ◽  
Time Varying ◽  
Asymptotically Stable ◽  
System Matrix ◽  
Varying Coefficients ◽  
Stability Matrix ◽  
Time Varying Coefficients ◽  
Asymptotic Stability Conditions

A Lyapunov-type approach is used to develop sufficient asymptotic stability conditions for linear systems with time-varying coefficients. In particular, it is shown that parametric disturbances of high frequency cannot create instability in an already asymptotically stable system. Also it is shown that slowly varying parametric disturbances will not cause instability if the system matrix is a stability matrix for all values of time. The results are applied to the Mathieu equation to illustrate the character of the theorems. This example is chosen because of the availability of its exact stability boundaries.

On Efficient Computation of the Steady-State Response of Linear Systems With Periodic Coefficients

1996 ◽  
Vol 118 (3) ◽  
pp. 522-526 ◽  
Author(s):  
T. J. Selstad ◽  
K. Farhang
Keyword(s):  
Steady State ◽  
Linear Systems ◽  
Steady State Solution ◽  
Time Varying ◽  
Steady State Response ◽  
Efficient Computation ◽  
Periodicity Condition ◽  
Varying Coefficients ◽  
State Response ◽  
Time Varying Coefficients

An efficient method for obtaining the steady-state response of linear systems with periodically time varying coefficients is developed. The steady-state solution is obtained by dividing the fundamental period into a number of intervals and establishing, based on a fourth-order Rung-Kutta formulation, the relation between the response at the start and end of the period. Imposition of periodicity condition upon the response facilitates computation of the initial condition that yields the steady-state values in a single pass; i.e., integration over only one period. Through a practical example, the method is shown to be more accurate and computationally more efficient than other known methods for computing the steady-state response.

A Method for Analyzing Heat Conduction With High-Frequency Periodic Boundary Conditions

1990 ◽  
Vol 112 (2) ◽  
pp. 280-287 ◽  
Author(s):  
D. A. Caulk
Keyword(s):  
Surface Layer ◽  
Boundary Conditions ◽  
High Frequency ◽  
Solution Process ◽  
Interior Solution ◽  
Special Method ◽  
Time Varying ◽  
Transient Solution ◽  
Varying Coefficients ◽  
Time Varying Coefficients

A special method is developed for calculating the steady periodic temperature solution in solid bodies with high-frequency boundary conditions. The numerical difficulty associated with steep gradients and rapid temperature variation near the boundary is addressed by confining all transient temperatures to a narrow boundary layer of constant depth. The depth of the layer is specified in advance and depends only on the period of the boundary disturbance and the thermal diffusivity of the material. The transient solution in the surface layer is represented by a polynomial in its transverse coordinate, with time-varying coefficients determined by a Galerkin method. This solution is coupled with the steady interior solution by imposing continuity of temperature and time-averaged heat flux at the interface. Although the method is sufficiently general to handle nonlinear boundary conditions, it turns out to be particularly useful in the important case of a time-varying heat transfer coefficient. In the latter case, it is possible to decouple the solution process and determine the solution in the transient surface layer separately from the solution in the steady interior. This reduces the effort of determining the complete steady periodic solution to little more than a routine steady analysis. Comparison with an exact solution shows that the polynomial representation for the transient solution in the surface layer converges very rapidly with increasing order. Moreover, the solution at the surface turns out to be relatively insensitive to the choice of the layer depth as long as it is greater than a certain minimum value. An application to permanent mold casting is given, illustrating both the utility and accuracy of the method in a practical context.

Piecewise Linear Systems With Time Varying Coefficients

1997 ◽  
Author(s):  
S. Natsiavas ◽  
S. Theodossiades
Keyword(s):  
Linear Systems ◽  
Piecewise Linear ◽  
Duffing Oscillator ◽  
Constant Load ◽  
Time Varying ◽  
Gear Pair ◽  
Linear Dynamical Systems ◽  
Varying Coefficients ◽  
Piecewise Linear Systems ◽  
Time Varying Coefficients

Abstract A new method is presented for determining periodic steady state response of piecewise linear dynamical systems with time varying coefficients. As an example mechanical model, a gear-pair system with backlash is examined, under the action of a constant torque. Originally, some useful insight is gained on the type of motions expected by investigating the response of a weakly nonlinear Mathieu-Duffing oscillator, subjected to a constant external load. The information obtained is then used in seeking the appropriate form of approximate periodic solutions of the piecewise linear system. Finally, these solutions are determined by developing a new analytical method. This method combines elements from approaches applied for piecewise linear systems with constant coefficients as well as classical perturbation techniques applied for systems with time varying coefficients. The validity and accuracy of the approach is verified by numerical results. In addition, response diagrams are presented, illustrating the effect of the constant load and the damping on the gear-pair response.

Time Varying Hydrodynamics Identification of a Flexible Riser Under Multi-Frequency Vortex-Induced Vibrations

2017 ◽  
Author(s):  
Chang Liu ◽  
Shixiao Fu ◽  
Mengmeng Zhang ◽  
Haojie Ren
Keyword(s):  
High Frequency ◽  
Coupling Effect ◽  
Time Varying ◽  
Flexible Riser ◽  
Force Coefficients ◽  
Basic Frequency ◽  
Varying Coefficients ◽  
Time Varying Parameters ◽  
Frequency Coupling ◽  
Time Varying Coefficients

In this paper, Forgetting Factor Recursive Least Squares (FF-LS) method is proposed to identify time-varying vortex-induced force coefficients of the flexible riser under multi-frequency VIV. FF-LS method introduces the forgetting factor, which gives more weight to the data closer to the present. This modification improves the method’s sensitivity to the time-varying parameters, and enables it to identify the time-varying parameters under multi-frequency coupling. In this paper, the mass-spring-dashpot model is used to verify FF-LS method’s ability to accurately identify time-varying parameters under multi-frequency coupling. Then, this methodology is used to identify the vortex-induced force coefficients of flexible riser at the basic frequency when vortex-induced vibration occurs. Identified coefficients are consistent with the result obtained from Least Squares method, which indicates that the proposed FF-LS method retrogrades to the Least Squares method when VIV response and the vortex-induced force contains only one single frequency. Finally, this methodology is used to identify the time-varying vortex-induced force coefficients of a flexible riser under VIV considering the coupling effect between the basic frequency and high frequency. The vortex-induced force reconstructed from identified time-varying coefficients is consistent with real vortex-induced force, which verifies the validity and applicability of this methodology in identifying time-varying vortex-induced force coefficients considering multi-frequency coupling effect. The results show that when the flexible riser is subjected to multi-frequency VIV, its vortex-induced force coefficients change periodically, the time-averaged values of these time-varying coefficients are different from the vortex-induced force coefficients at the basic frequency, which results from the coupling effect between the basic frequency and high frequency. Linear superposition of vortex-induced force coefficient at basic frequency and high frequency are different from time-varying coefficients considering coupling effect of multi-frequency and cannot reconstruct vortex-induced force correctly.

scholarly journals Model of time-varying linear systems and Kolmogorov equations

2015 ◽  
Vol 25 (2) ◽  
pp. 201-214
Author(s):  
Assen V. Krumov
Keyword(s):  
Analytical Solution ◽  
Linear Systems ◽  
Sufficient Conditions ◽  
Original System ◽  
Approximate Model ◽  
Time Varying ◽  
Kolmogorov Equations ◽  
Varying Coefficients ◽  
Method Of Solution ◽  
Time Varying Coefficients

Abstract In the paper an approximate model of time-varying linear systems using a sequence of time-invariant systems is suggested. The conditions for validity of the approximation are proven with a theorem. Examples comparing the numerical solution of the original system and the analytical solution of the model are given. For the system under the consideration a new criterion giving sufficient conditions for robust Lagrange stability is suggested. The criterion is proven with a theorem. Examples are given showing stable and non stable solutions of a time-varying system and the results are compared with the numerical Runge-Kutta solution of the system. In the paper an important application of the described method of solution of linear systems with time-varying coefficients, namely analytical solution of the Kolmogorov equations is shown.

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