# Effective Implementation of Elastohydrodynamic Lubrication of Rough Surfaces into Multibody Dynamics Software

^{*}

## Abstract

**:**

## Featured Application

**The research results highlight a way to effectively combine the solution of elastohydrodynamics and the commercial multibody dynamics program.**

## Abstract

## 1. Introduction

## 2. Computational Model

#### 2.1. Elastic Deformation

#### 2.2. Regime of Hydrodynamic Lubrication

_{1}and ${U}_{2}$ are the velocities for body No. 1 and No. 2 in $y$ direction. The hydrodynamic lubrication causes the shear stress:

#### 2.3. Regime of Partial Lubrication

#### 2.4. Regime of Boundary Lubrication

_{1}and ${\nu}_{2}$ represent their Poisson’s ratios. Coulomb’s law of friction is valid for the asperity contact:

#### 2.5. Numerical Solution

#### 2.6. Deformation Reconstruction

#### 2.7. Deformation and Force Mapping

## 3. Results and Discussion

#### 3.1. Design Sensitivity Analysis

**f**(

**L**(

**q**)).

#### 3.2. Perturbation Step

#### 3.3. User-Calculated Sensitivity Analysis of a Subroutine

^{−6}, step size 8.3 × 10

^{−5}s, a maximum number of iterations of 10 and the original corrector. The Intel

^{®}Core™ i7-4770S 3.1 GHz processor with four physical cores and eight virtual cores was used.

#### 3.4. Parallel Solution

^{®}Xeon

^{®}E5 2640 0 2.5 GHz with twelve virtual cores (six physical cores), and the integrator settings were the same as in the previous simulations. The elapsed time above 78 h of simulation with the Default method was caused by very low CPU usage (maximum of 5.6%). On the contrary, the Parallel method led to the maximum CPU usage of 100%, resulting in a reduction in the elapsed time, by almost ten times, to a value of approximately 8 h. In addition, the Parallel method required fewer iterations to find the converged solution: 9987 versus 11597 iterations. The Default method used perturbation steps, which were periodically updated as per a certain number of iterations. As previously mentioned, the Parallel method used perturbation steps estimated during the first call for the Jacobian unless the MBD solver struggled to converge. In this case, the perturbation steps used by the parallel solution seem to lead to a more accurate Jacobian matrix. However, this cannot be regarded as a general rule, as it is likely to be a characteristic of the current simulation and solver settings. According to several study cases it can be concluded that the Parallel method reduces the elapsed time by an order of magnitude as per the above-specified hardware and the DO loop size around 2000 runs.

#### 3.5. Hydrodynamic and Elastohydrodynamic Solution to the Piston–Liner Interaction

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**The MBD model of piston–liner interaction. Crankshaft assembly is connected to the engine block via revolute joint with motion applied. Other revolute joints are between the crankshaft asembly-connecting rod and the connecting rod–piston pin. The piston pin is connected to the piston via joint primitives. The gas force is applied at the center of the piston bowl and the modal force is applied on the entire piston flexible body. The engine block is represented as a ground.

**Figure 3.**Subroutine’s measurements not related to the deformation for the multibody dynamics model of the piston–liner interaction.

**Figure 4.**Relative perturbation steps of subroutine’s measurements not related to deformation. The values are plotted depending on the crankshaft angular position.

**Figure 5.**Scheme of sensitivity analysis of the subroutine using the User finite forward difference method for deformation-related measurements in Fortran. The modal force LOADVEC is calculated using function MFORCE_EHD based on measurements INPUT and parameters PARAM. If the multibody dynamics solver is calculating the Jacobian for the first time, the subroutine records the perturbation steps DELTA. Otherwise, the subroutine calculates the partial derivatives of Equation (29), stores them in DER, and passes them to the multibody dynamics solver.

**Figure 6.**First six mode shapes of flexible piston: (

**a**) 5,4883 Hz; (

**b**) 6320 Hz; (

**c**) 9597 Hz; (

**d**) 10,434 Hz; (

**e**) 11,370 Hz; (

**f**) 13,458 Hz.

**Figure 7.**Scheme of Parallel user finite forward difference method, illustrating the parallelization of the sensitivity analysis of the subroutine for deformation-related measurements.

**Figure 8.**The number of iterations of the piston–liner multibody dynamics simulation with elastohydrodynamic lubrication. The values are plotted depending on the crankshaft angular position.

**Figure 9.**Cumulative elapsed time of the piston–liner multibody dynamics simulation with elastohydrodynamic lubrication. The values are plotted depending on the crankshaft angular position.

**Figure 10.**Piston lateral displacement dependent on the crankshaft angular position. The firing occurs near a crank angle of 360°.

**Figure 11.**Piston thrust side minimum oil film thickness dependent on the crankshaft angular position.

**Table 1.**Computational speed of the simulation using the Default and with the User method for a reduced number of mode shapes.

Default | User | |
---|---|---|

Maximum CPU usage (%) | 18.5 | 18.0 |

Elapsed time (MM:SS) | 25:10.4 | 25:20.2 |

Simulation time acceleration (-) | 1 | 0.99 |

Iteration count (-) | 7877 | 8311 |

Iteration count increase (%) | 0 | 5.51 |

**Table 2.**Computational speed of the simulation with the Parallel method for a reduced number of mode shapes.

Parallel | |
---|---|

Maximum CPU usage (%) | 46.5 |

Elapsed time (MM:SS) | 17:38.7 |

Simulation time acceleration (-) | 1.43 |

Iteration count (-) | 8311 |

Iteration count increase (%) | 5.51 |

**Table 3.**Computational speed of the simulation with the Default and the Parallel method for the full number of mode shapes.

Default | Parallel | |
---|---|---|

Maximum CPU usage (%) | 5.6 | 100.0 |

Elapsed time (MM:SS) | 78:14:57.0 | 7:52:51.0 |

Simulation time acceleration (-) | 1.00 | 9.93 |

Iteration count (-) | 11597 | 9987 |

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**MDPI and ACS Style**

Dlugoš, J.; Novotný, P.
Effective Implementation of Elastohydrodynamic Lubrication of Rough Surfaces into Multibody Dynamics Software. *Appl. Sci.* **2021**, *11*, 1488.
https://doi.org/10.3390/app11041488

**AMA Style**

Dlugoš J, Novotný P.
Effective Implementation of Elastohydrodynamic Lubrication of Rough Surfaces into Multibody Dynamics Software. *Applied Sciences*. 2021; 11(4):1488.
https://doi.org/10.3390/app11041488

**Chicago/Turabian Style**

Dlugoš, Jozef, and Pavel Novotný.
2021. "Effective Implementation of Elastohydrodynamic Lubrication of Rough Surfaces into Multibody Dynamics Software" *Applied Sciences* 11, no. 4: 1488.
https://doi.org/10.3390/app11041488