Response Sensitivity Of Nonlinear Vibration And Dynamic Modification Of Vehicle Transmission System

The eigensensitivity analysis does not meet the increasing industrial requirements of thedynamic performance of a vehicle transmission system. To reduce vibration, it is necessaryto include response sensitivity in the guideline in the design stage.This paper develops a universal nonlinear lateral–torsional coupling vehicletransmission system model including shafts, bearings, single/planetary gear pairs, clutches,and inertia plate. The first-and second-order response sensitivity are obtained by thenonlinear dynamical equations including the effect of time-varying mesh stiffness, clearance,mass eccentricity, and transmission error. The sensitivity parameters which are selected outbased on the response sensitivity analysis are used for dynamic modification for the purposeof optimum dynamic response.The dynamic characteristics analysis of vehicle transmission system is transferring fromthe analysis of modal characteristics to the analysis of the dynamic response. Firstly, thedynamic equations and the sensitivity equations of linear lateral–torsional coupling vehicletransmission system model are developed. The comparison of similarities and differences ofidentical sensitivity between torsional model and lateral–torsional coupling model and thecomparison of the sensitivity of different parts in system with respect to identical parameterare studied. Basing on the analysis of problems mentioned above, the relationship betweenresponse sensitivity and the energy of vibration is uncovered. Furthermore, the effect of time-varying mesh stiffness in response sensitivity is studied by the comparison of the responsesensitivity result of the model which taking constant mesh stiffness and the model whichtaking time-varying mesh stiffness.As the derivative of discontinuity point of backlash function is not existed, the methodof polynomial fit is adopted in deducing the response sensitivity of nonlinear dynamic model.The first-order response sensitivity equations which being fit for nonlinear model are deduced.In allusion to the characteristic of the aperiodicity of response sensitivity curves of thenonlinear system in the time domain, a novel assessment method——differential sensitivitybased on the root mean square of response is proposed. This method provides statisticalresults in a certain range, thus avoiding the inaccuracy of the partial amplitude. The vibrational energy of modified system can also be estimated. The abovementionedcharacteristics make it possible to provide the theoretical support for dynamic modification.The target object whose dynamic characteristics needing modification are determinedby the result of dynamic equations while the modified parameters determined by the resultof first-order sensitivity equations. The second-order sensitivity equations are deduced aswell. Two techniques for predicting the forced response of mechanical components for localsingle parameter changes in properties based on first-order multi-step iterative prediction andsecond-order iterative sensitivity functions are developed. The techniques which do not needupdating the sensitivity and recalculating the dynamic equation of modification model canobtain good accuracy in engineering make it possible to obtain local optimum response inone parameter’s perturbation. The techniques mentioned above provide theoretical supportto acquire local optimum vibration response characteristics for NVH engineers who do notpossess enough time and authority of structural modification.Due to the situation of the multi-parameter being modified which is common inengineering for the purpose of the optimal dynamic response, the technique of predictingchanges in vibration response expand to the multi-parameter perturbation. The second-ordermixed partial derivatives which are deduced from the first-order derivatives combining withunitary first-order partial derivatives, unitary second-order partial derivatives and multistepmethod being taken into multi taylor series expansion to predict the changes in vibrationresponse in the situation of multi-parameter perturbation. The technique mentioned abovecan meet the requirements of accuracy in engineering and provide local optimal response inthe situation of multi-parameter perturbation. The technique of predicting the changes invibration response in the situation of multi-parameter perturbation is more suitable inengineering when comparing to unitary parameter’s perturbation.The nonlinear lateral–torsional coupling dynamic equations and sensitivity equations ofsingle stage planetary gear which are used to predict changes in vibration response in thesituation of perturbation of the amplitude of transmission error are developed. The test bedsof original and low level accuracy of gear are set up. From which the displacement of ringgear, the acceleration of bearing block and torsional shear stress in outout shaft can beacquired. And the correctness of the techniques of predicting changes in vibration response is verified by variation of physical quantity mentioned above in the test data.