Research On Modeling And Stochastic Optimal Control For Viscoelastic Elastomer Damper’s Nonlinear Dynamical Systems Involving Time Delays

Viscoelastic elastomer damper is a component of tracked vehicle suspensionsystem, which plays a great importance on the vehicle’s ride and dampening per-formance and suspension system’s working well. Viscoelastic elastomer damperconsists of viscoelastic elastomer, piston and cylinder etc. Viscoelastic elastomer is acompressed polyorganosiloxane material, which performance is between hydraulicoil and rubber.When elastomer damper is subjected to external force, the pistoncompresses the viscoelastic elastomer and the energy is absorbed and the potentialenergy is stored.When external forces disappear; the viscoelastic elastomer expandsto push the piston back.When the viscoelastic elastomer damper is designed, the key point is to buildthe mathematical model between the resistance force and the pistion’s displacementand velocity, of which stiffness and damping varies with the pistion’s displacementand velocity. In mathematics, the relationship between the resistance force and thepiston’s displacement, velocity, stiffness and damping is the constitutive relation ofnonlinear dynamical systems involving time delays; their constitutive relation is anonlinear function and has the obvious characteristic of non-smooth and stronglynonlinear. The mathematical model describing the dynamical systems involving timedelays are nonlinear time delay differential equation (set).The theoretical analysis iscombined with the experimental research in this paper, the solution andmathematical model and stochastic optimal control of nonlinear dynamical systemsinvolving time delays were mainly studied in this paper. The damper’s stiffnessand dampening ratio’s adjusting ranges were determined, the relationship betweenthe resistance force and the piston’s displacement, velocity, stiffness and damping were built, the optimum dampening ratios between the nonlinear suspension systemand the linear one were inferred when the vehicle velocity and road condition weregiven, all of which will provide the theoretical guide and approach evidence for thedynamical matching research between the elastomer damper and the vehicle’ssuspension system.On the solution research of nonlinear dynamical systems involving time delays,time delay differential equations were made a classification, and their solution’sexistence and uniqueness theorem were set forth. The oscillation of first order andsecond order time delay differential equations were anylized and the consistencecondition of their oscillation solution was explained. The stability conception of timedelay differential equations was set forth, and the essential difference on the stabilitybetween the time delay differential equations and the ordinary differential equationswas anylized. A new stability anylysis approach that Nyquist stability criterion incombination with the stability transmission anylysis was used to explore the stabilityzone of nonlinear dynamical systems involving time delays was proposed. All ofwhich laid foundations for the farther studies and discussions.Aiming at the problem that the solution of nonlinear time delay differentialequation have not uniform and universal approach, Initial value transformationmethod and incremental harmonic balance nonlinear parameter identification methodwere used to discuss the periodic solution of the nonlinear time delay differentialequation, and two methods’ advantages and disadvantages were anylized. Based onthe above research, Initial value transformation method’s advantage of nonlinearequivalent processing and incremental harmonic balance nonlinear parameteridentification method’s advantage of the flexible control algorithm’s convergencewere adopted, a new method of linear acceleration successive integration methodwith no condition stability was proposed to make the numerical solution for thenonlinear equation set.The nonlinear suspension systems installing the elastomerdampet was referred as an exmple to study the numerical solution process ofdynamical differential equation of the nonlinear suspension systems in detail, and. Finally the relationship between the elastomer damper’s stiffness and compresseddisplacement had been gained, which provided a theoretical basis for the design anddynamicals modeling of elastomer damper.On the mathematical modeling, the dynamical stiffness and dampingcharacteristics of nonlinear dynamical systems involving time delays were studied, anew mathematical model of time domain and frequency domain was proposed. theuniform mathematical model between the resistance force and displacement, velocity,frequency, amplitude and damping forces was built, and mathematical model’sparameters’ identification was carried on by using the nonlinear parameters optimalseparated identification method. Based on which, a variable rigidity Maxwell fluidmodel was proposed to model viscoelastic elastomer damper. The resistanceformulae of elastomer damper in the input process and return process wereconducted respectively.Finally, the shock experiments validate the reliability of themodel.Based on the stochastic optimal control theory, The nonlinear system’sstochasitic averaging method under the white noise or non-white noise’s excitementwere studied, The stochastic averaging method in combination with dynamicalalprogramming principle was proposed to study the stochastic optimal control problemof nonlinear dynamical systems involving time delays. Based on the above theory,the elastomer damper’s application in the vhicle’s suspension system was studied.The time delay dynamical system’s optimal control law was combined withsuspension system’s ride and damping performance index, the relationship betweenroot mean square value of absolute acceleration and relative displacement andelastomer damper’s damping ratio for suspension system was deduced, whichprovided the theoretical evidence for the dynamical matching between the elastomerdamper and the suspension system.The time delay dynamical system’s optimalcontrol law was combined with the tracked vehicle’s suspension system’s dynamicalequation, relationship between the vertical acceleration and vehicle’s velocity, roadcondition and elastomer damper’s dampening ratio was deduced, when vehicle’s velocity and road conditon were given, the relationship between the verticalacceleration and elastomer damper’s dampening ratio was quadratic function ofopening up, which had shown the existence of the optimum dampening ratio intheory. Finally, Dynamicals model and formula for the total suspension system of thetracked vehicle were built. Modeling and simulation analysis of the total trackedvehicle in the E and G class road surface have been carried on by use of Recurdynsoftware, which had shown that the elastomer damper designed in the paper canimprove the suspension system’s performance.The static pressure and dynamical pressure laboratory were carried on afterdesigning the elastomer damper. And many times’ experiments were set forth and thelaboratory curves were analyzed to validate the elastomer damper met with theperformance index required in the working conditions and impact resistance abilityand stability of nonlinear dynamical systems involving time delays.