Chaos Control Of Geared System And Its Simulation

Chaos control research based on the chaotic dynamics is the frontier problem inmodem nonlinear dynamics. The chaotic dynamics and chaos control issue ofnonlinear geared system with time-varying mesh stiffness and backlash has beenextensively studied in this dissertation, the main research works are as follows:1. For single-stage geared rotor-bearing system, a single-degree-of-freedomnonlinear dynamics model with piecewise linearity characteristic, and a3-degree-of-freedom nonlinear dynamics model involving gear backlash andtime-varying mesh stiffness are respectively developed. The chaotic responses of thetwo systems are calculated through numerical integration. The Lyapunov exponentspectrum of 3-d.o.f system is also calculated and plotted.2. The application and the special treatment process for piecewise linear items ofincremental harmonic balance method(IHB) in solving periodic motions of gearedsystem are studied. The method to determine the stability of the periodic motion andits bifurcation types by Floquet theory is also investigated. The effects of theparameters such as exciting frequency and bearing damping on the system dynamicsare studied, the response diagrams for exciting frequency are plotted. Period-doublingbifurcation route to chaos, quasi-periodic bifurcation route to chaos and multiplesteady state periodic solutions caused by saddle-node bifurcation are observed in someparameter regions. It is shown that IHB method can be used to seek both the stable andunstable periodic solutions which compare very well with the results obtained usingnumerical integration. The example shows the effectiveness and accuracy of IHBmethod for computing the periodic solutions of 3-d.o.f. geared system with piecewiselinearity characteristic.3. The principle of traditional OGY chaos control method and its relationship withmodem control theory are studied. It has been proved that OGY method is a specialpole placement technique in essence. The procedures to extend and adapt the OGYmethod into continuous system such as geared system are investigated, including theprocess of locating unstable period orbits(UPO) by shooting method, computing the Jacobian matrices and sensitivity vectors by variational form of differential equations.The stabilization of high period UPOs is achieved through numerical simulation. Theextensive effect analysis of chaos control of s.d.o.f geared system is presented.4. A multi-step chaos control method is presented to realize the stabilization ofUPOs embedded in the chaotic attractor of 3.d.o.f. geared system. By means of theanalysis of the structure of system's UPOs. the existence of complex-conjugateeigenvalues on the unit circle along UPOs and unstable dimension or stable dimensionvariability of the UPOs indicate the nonhyperbolicity of the geared system. The controlmethod is set up for the situations of unstable dimension and stable dimensionvariability respectively to drive the system state to lie on the local stable manifold ofUPOs through continuous parametric perturbations. The method is essentially thegeneralization of classical OGY method in nonhyperbolic and high-dimensionalsystem. Numerical simulation verifies the effectiveness of the control method for theUPOs even with long period.5. Assuming the prior knowledge of the geared system's dynamics is unknown, acomplete chaos control scheme is derived using delay coordinates technique. Theoptimal delay time and minimum embedding dimension for phase space reconstructionare acquired respectively by averaged mutual information method and averaged falsenearest neighbors method, the whole scheme for constructing delay coordinate vectorsof geared system is set up. Considering the dynamical dependence on the pastparameters, the chaos control method is constructed based on the combined dynamicsof state-plus-parameter system, the procedures to obtain the parameters forimplementing chaos control from experimental time series are investigated. Numericalsimulation shows the effectiveness of the method in practical applications.6. The dynamic response of geared system is tested in a testing machine ofnonlinear spur gear pair. The vibration acceleration of the gear pair in differentparameter conditions is obtained in the testing process. The testing results are analyzedand compared with the theoretical results, which verifies the effectiveness of theanalyzing model and the calculating methods.This work is supported by the National Natural Science Foundation of China (No.50075070).