# Modeling dissipative heating of hydraulic dampers under consideration of stochastic uncertainties in their geometric parameters

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## Abstract

To obtain a hydraulic damper thermodynamic model that is much more realistic than the thermodynamic model of literature, research is carried out by introducing the stochastic uncertainty theory. The geometric variables of the hydraulic damper are defined as uncertain variables. The stochastic variables are defined with the stochastic factor method, and the stochastic thermodynamic model is established with the algebraic synthesis method. The results of the experiment are considered as the standards for verifying the correctness and superiority of the stochastic thermodynamic model. The results of the stochastic thermodynamic model and the thermodynamic model of literature are compared with the experimental results, respectively. The oil temperature at the stable state of the stochastic thermodynamic model is close to that of the experiment, indicating that the stochastic thermodynamic model is correct. The oil temperature curve of the stochastic thermodynamic model is much closer to that of the experiment than that of the thermodynamic model of the literature, indicating that the stochastic thermodynamic model is superior to the thermodynamic model of literature.

## Keywords

Hydraulic damper Stochastic uncertainty theory Thermodynamic model Stochastic factor The experimental results## List of symbols

*m*_{o}Mass of the hydraulic oil

*c*_{o}Specific heat capacity of the hydraulic oil

*T*Oil temperature

*F*_{d}Damping force

*t*Time

*v*(*t*)Velocity of the oil

*T*_{∞}Temperature of the air

*R*_{1}Heat transfer resistance of the forced convection in the working cylinder

*R*_{c}Heat transfer resistance from the working cylinder inner wall to the storage cylinder outer wall

*R*_{4}Heat transfer resistance of the forced convection out of the storage cylinder

*ρ*_{o}Density of the hydraulic oil

*V*_{r}Piston velocity of the recovery process

*A*_{p}Cross-sectional area of the piston

*A*_{po}Cross-sectional area of the piston rod

*A*_{r}Area of thin holes in the recovery process

*V*_{c}Piston velocity of the compression process

*A*_{c}Area of thin holes in the compression process

*C*_{q}Flow coefficient of the thin holes

*l*_{r}Symbol for replacing an expression

*l*_{c}Symbol for replacing an expression

*k*_{1}Symbol for replacing an expression

*k*_{2}Symbol for replacing an expression

*b*_{1}Symbol for replacing an expression

*b*_{2}Symbol for replacing an expression

*W*Thermal energy generated in a cycle

*A*Vibration amplitude of the outer excitation

*f*Vibration frequency

*T*_{n}End oil temperature of the

*n*th cycle*T*_{n−1}Initial oil temperature of the

*n*th cycle*T*_{p}Iterative period

*h*_{t}Heat transfer coefficient in the working cylinder

*A*_{1}Heat transfer area in the working cylinder

*λ*_{o}Heat conductivity coefficient of the hydraulic oil

*Re*_{o}Reynolds number of the hydraulic oil

*Pr*_{o}Prandtl number of the hydraulic oil

*u*_{o}Average velocity of the hydraulic oil

*v*_{o}Kinematic viscosity of the hydraulic oil

*h*_{r}Radiation heat transfer coefficient of the storage cylinder outer wall

*δ*Stefan–Boltzmann constant

*ε*Radiation emissivity of the storage cylinder outer wall

- \(T_{{d_{4} }}\)
Temperature of the storage cylinder outer wall

*h*_{a}Heat transfer coefficient of the forced convection out of the storage cylinder

*A*_{4}Heat transfer area out of the storage cylinder

*λ*_{a}Heat conductivity coefficient of the air

*Re*_{a}Reynolds number of the air

*Pr*_{a}Prandtl number of the air

*u*_{a}Average velocity of the air

*v*_{a}Kinematic viscosity of the air

*λ*_{c}Heat conductivity coefficient of the cylinder material

*λ*_{q}Heat conductivity coefficient of the nitrogen

*d*_{1}Inner diameter of the working cylinder

*d*_{2}Outer diameter of the working cylinder

*d*_{3}Inner diameter of the storage cylinder

*d*_{4}Outer diameter of the storage cylinder

*L*Length of the heat transfer

*d*_{p}Diameter of the piston

*d*_{po}Diameter of the piston rod

- \(\bar{d}_{1}\)
Mean value of the working cylinder inner diameter

- \(\bar{d}_{2}\)
Mean value of the working cylinder outer diameter

- \(\bar{d}_{3}\)
Mean value of the storage cylinder inner diameter

- \(\bar{d}_{4}\)
Mean value of the storage cylinder outer diameter

- \(\bar{L}\)
Mean value of the heat transfer length

- \(\bar{d}_{\text{p}}\)
Mean value of the piston diameter

- \(\bar{d}_{\text{po}}\)
Mean value of the piston rod diameter

*r*_{1}Inner radius of the working cylinder

*r*_{2}Outer radius of the working cylinder

*r*_{3}Inner radius of the storage cylinder

*r*_{4}Outer radius of the storage cylinder

*T*_{e}Oil temperature at the stable state

*E*Expected value

*σ*Mean square error

*i*Variation coefficient

## Notes

### Acknowledgements

We thank the Digital Design and Manufacture Key Laboratory of Anhui Province at the Hefei University of Technology for supporting this research.

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

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