Trigonometric Collocation

2002 ◽  
pp. 293-324
Author(s):  
Jukka Saranen ◽  
Gennadi Vainikko
Keyword(s):  
Trigonometric Collocation

Periodic Solutions in Rotor Dynamic Systems With Nonlinear Supports: A General Approach

1989 ◽  
Vol 111 (2) ◽  
pp. 187-193 ◽  
Author(s):  
C. Nataraj ◽  
H. D. Nelson
Keyword(s):  
Dynamic Systems ◽  
Original Problem ◽  
Squeeze Film ◽  
Algebraic Equations ◽  
Trigonometric Collocation ◽  
Nonlinear Algebraic Equations ◽  
The Stability ◽  
Future Work ◽  
Rotor Dynamic ◽  
Large Rotor

A new quantitative method of estimating steady state periodic behavior in nonlinear systems, based on the trigonometric collocation method, is outlined. A procedure is developed to analyze large rotor dynamic systems with nonlinear supports by the use of the above method in conjunction with Component Mode Synthesis. The algorithm discussed is seen to reduce the original problem to solving nonlinear algebraic equations in terms of only the coordinates associated with the nonlinear supports and is a big improvement over commonly used integration methods. The feasibility and advantages of the procedure so developed are illustrated with the help of an example of a typical rotor dynamic system with an uncentered squeeze film damper. Future work on the investigation of the stability of the periodic response so obtained is outlined.

Trigonometric Collocation for Computation of Steady State Responses of Cracked Structures

2012 ◽  
Author(s):  
Bakeer Bakeer ◽  
Oleg Shiryayev ◽  
Ammaar Tahir
Keyword(s):  
Steady State ◽  
Fatigue Cracks ◽  
Equations Of Motion ◽  
Dynamic Responses ◽  
Vibration Period ◽  
State Response ◽  
Trigonometric Collocation ◽  
Steady State Responses ◽  
Cracked Structures ◽  
State Responses

Development of vibration-based structural health monitoring techniques requires the use of various computational methods to predict dynamic responses of damaged structures. The method described in this work can be used for prediction of steady state harmonic responses for structures with fatigue cracks and may have several advantages over alternative techniques. The method appears to be relatively easy to implement and computationally inexpensive. The steady state response of the system at a given number of time points distributed over one vibration period is represented in terms of Fourier series containing higher frequency harmonics. Equations of motion are formulated in the form that allows for easy computation of Fourier coefficients for all terms in the series. Iterative procedure is used for determining the time of stiffness change in order to capture bilinear dynamic behavior. We present results of initial investigation by applying the method to a model of a cantilever beam with a crack.

Stability and Bifurcation of Unbalanced Response of a Squeeze Film Damped Flexible Rotor

1994 ◽  
Vol 116 (2) ◽  
pp. 361-368 ◽  
Author(s):  
J. Y. Zhao ◽  
I. W. Linnett ◽  
L. J. McLean
Keyword(s):  
Power Spectra ◽  
Squeeze Film ◽  
Integration Scheme ◽  
Flexible Rotor ◽  
Jump Phenomenon ◽  
Stability And Bifurcation ◽  
Trigonometric Collocation ◽  
Numerical Integration Scheme ◽  
The Stability ◽  
Nonsynchronous Vibrations

The stability and bifurcation of the unbalance response of a squeeze film damper-mounted flexible rotor are investigated based on the assumption of an incompressible lubricant together with the short bearing approximation and the “π” film cavitation model. The unbalanced rotor response is determined by the trigonometric collocation method and the stability of these solutions is then investigated using the Floquet transition matrix method. Numerical examples are given for both concentric and eccentric damper operations. Jump phenomenon, subharmonic, and quasi-periodic vibrations are predicted for a range of bearing and unbalance parameters. The predicted jump phenomenon, subharmonic and quasi-periodic vibrations are further examined by using a numerical integration scheme to predict damper trajectories, calculate Poincare´ maps and power spectra. It is concluded that the introduction of unpressurized squeeze film dampers may promote undesirable nonsynchronous vibrations.

Optimal Control of a Nonlinear Magnetic Bearing System Using the Trigonometric Collocation Method

2008 ◽  
Author(s):  
C. Nataraj ◽  
Steven Marx
Keyword(s):  
Optimal Control ◽  
Collocation Method ◽  
Degrees Of Freedom ◽  
Magnetic Bearing ◽  
Degree Of Freedom ◽  
Lookup Table ◽  
Base Excitation ◽  
Strongly Nonlinear ◽  
Trigonometric Collocation ◽  
Bearing System

Magnetic bearings are non-contacting, with the rotor being suspended between electromagnets, and therefore they can eliminate the need for lube oil and reduce machinery wear. The magnetic bearing is naturally unstable, and very nonlinear. This paper proposes a method designed to suppress the motion of a nonlinear magnetic bearing system rotor due to base excitation. The method combines PD feedback with feedforward optimal control, where a measured base motion is used to select a control signal designed to suppress the rotor response. The signal is generated from a combination of subharmonic frequencies and optimized coefficients stored in a lookup table. The trigonometric collocation method (TCM) is used to generate solutions for the four degree-of-freedom system made up of a shaft suspended at each end by a magnetic bearing. The TCM method uses a trigonometric series to simulate the multiharmonic behavior of each degree-of-freedom of strongly nonlinear systems. The method is easy to use and its advantage over numerical methods is that it demands less computation, particularly with higher numbers of degrees-of-freedom.

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