A linear equation of state model can be used to model incompressible viscous and inviscid laminar flow governed by the Navier-Stokes equation of motion. The volumetric response is governed by the equations of state, where the bulk modulus acts as a penalty parameter for the incompressible constraint.
Modeling dynamic characteristics of an automotive shock absorber is a challenging task due to its complex behavior. In the present paper, the nonparametric and hybrid approach is proposed to represent the nonlinear and hysteresis characteristics of the shock absorber. An experiment is carried out on a car damper utilizing INSTRON to obtain force-velocity characteristics of the shock absorber. The experimental data is used to devise two different models, namely, piecewise linear model and hysteresis model, to capture the damping properties of the absorber and for consequent use in simulations. The complexity involved due to certain physical phenomenon such as oil compressibility, gas entrapment etc. Gives rise to hysteresis behavior and the present paper tries to model such behavior with the help of Neural Networks. Finally, a combined (hybrid) shock absorber model (including the characteristics of both piecewise linear and hysteresis behavior) is developed in Simulink and integrated into a quarter car simulation to verify its feasibility.
The results generated by the combined (hybrid) model are compared with linear as well as piecewise linear model and the comparison shows that the proposed model substantially a better option to study the vehicle characteristics more accurately and precisely.
Linear Model Definition Math
The Mie-Gruneisen equation of state (EOS) is an analytical model for determining the pressure in a shock-compressed solid. With this model there are some important considerations that are not discussed widely. For instance there is a possibility of obtaining infinite pressure or non-physical behavior depending on the model parameters. While most of the time the given parameters are appropriate for the application of interest, sometimes in practice we must make educated guesses for missing parameters or perhaps even construct an entire model when none is available. In these scenarios it is helpful to have guiding criteria on the parameters. Below is a discussion of the key criteria to determine validity. The focus here is on the linear form of the Mie-Gruneisen model.
Mie-Gruneisen EOSIn general the Mie-Gruneisen model is only valid for shock compressed solids and is not intended for tension. If the compression form of the model was used in the expansion region, it is essentially assuming the material undergoes a tensile shock, which may lead to inaccurate results (Segletes, 1991). For this reason hydrocodes often use a close approximation to the isentrope in the tension portion of the pressure relationship that matches both the value and slope of the Hugoniot in the compression portion at ambient conditions.
Following these guidelines yields the following linear form of the Mie-Gruneisen EOS for both compression and tension,For mu = 0,P = rho0 c0^2 mu (1+(1- G0/2) mu)/(1-mu (S-1)) + G0 rho0 sieFor mu 1This criteria can also be summarizes, as shown by the shaded region in the figure below, asif S 1, then P is valid for mu 2, then P is valid for mu 0where dP/drho is taken with constant sie and dP/dsie is taken with constant rho. This expression must be greater than zero. Applying the Mie-Gruneisen pressure to this expression does not produce a simple result,c^2 = (-c0^2 (2 + 2 mu (3 + S) – A2 mu^2 + A3 mu^3) +2 G0^2 (-1 + mu (-1 + S))^3 sie)/(2 (1 + mu)^2 (-1 +mu (-1 + S))^3) 0,whereA2 = -6 + G0^2 – 4 S + 2 G0 SA3 = G0^2 (-1 + S) – 2 G0 S + 2 (1 + S)This expression needs to be checked for each cell at each time step. Some of the coefficients can be calculated before integration to speed up the calculation.
The addition of this criterion limits the range of validity on mu further as shown in the figure below with sie = 0. Plot of valid values of mu for criterion I, II, and III as a function of S and G0 with sie=0.In the case of expansion, -1 0This expression can be less than zero for a certain value of mu.
Shock Eos Linear Model Theory
Linear Model Definition
Again, it is important the check for c^2.