# Verification Manual

Verifies the validity of the bearing elements provided in Bearings.

## Introduction

The verification of two machinery bearings, deep groove ball bearing (DGBB) and cylindrical roller bearing (CRB), is achieved by comparing the results of their respective full multibody dynamics (MBD) representation against their reduced order model (ROM).

The MBD representation defines both bearing rings (inner and outer) and each rolling element as a rigid body with an accurate graphical representation. Rigid body contact is defined between the graphics of every rolling element and the two bearing rings. The cage, what keeps the rolling elements in place, is replaced by kinematic constraints to keep the distance of the rolling elements constant during rotation.

The ROM uses an analytical formulation to calculate bearing forces. This is implemented using a general force element (Gforce) with a MotionSolve subroutine (GFOSUB). The ROM details are described in Bearings.

For the verification, each model approach is used with the same bearing parameters. Differences occur only if the model methodology requires it. Contact stiffness between the rings and rolling elements in the MBD model is calculated similar to the analytical formulation inside the ROM model. For ball bearings (point contact), the contact stiffness is calculated according to the Hertzian approach, and for roller bearings (line contact), it is calculated according to the Palmgren formula. Small damping proportional to stiffness is introduced to simulate the material and oil/grease damping effects in the contact areas. Damping effects due to oil slushing is ignored. The bearings are connected to a shaft on the inner ring and to the ground on the outer ring.

## Test Subjects

DGBB 6005 | ||
---|---|---|

Parameter | Value | Units |

number_of_rollers | 10 | - |

pitch_diameter | 36.0 | mm |

width | 12.0 | mm |

inner_diameter | 25.0 | mm |

inner_shoulder_diameter | 32.04 | mm |

outer_diameter | 47.0 | mm |

outer_shoulder_diameter | 39.96 | mm |

roller_diameter | 6.746 | mm |

bearing_clearance | 10 | μm |

inner_race_conformity | 0.52 | - |

outer_race_conformity | 0.53 | - |

bearing_density | 7.85e-6 | Kg/mm^3 |

NLC Parameters (DGBB) | ||
---|---|---|

Parameter | Value | Units |

young_modulus | 210000 | N/mm’^2 |

poisson_ratio | 0.3 | - |

damping_force | True | - |

friction_torque | True | - |

damping_ratio | 0.1 | - |

static_load_rating | 6.55e3 | N |

lubricant_viscosity | 80 | cSt |

lubrication_method | ‘grease’ | - |

`na`,

`nc`,

`no`) express the fidelity of the ring's triamesh. Due to the small clearances in bearings, the graphical tessellation needs to be very fine to obtain realistic bearing results.

Contact Parameters (DGBB) | ||
---|---|---|

Parameter | Value | Units |

use_contact_stiffness | True | - |

damping_factor | 2.5e-5 | - |

method.stiffness | 730424.83 | N/mm |

method.exponent | 1.5 | - |

method.damping | 18.26 | Ns/mm |

method.dmax | 0.013 | mm |

method.coulomb_friction | “On” | - |

method.vd | 110 | mm/s |

method.md | 0.03 | - |

method.ms | 0.05 | - |

method.vs | 100 | mm/s |

na | 2000 | - |

nc | 80 | - |

no | 2 | - |

CRB NU 1005 | ||
---|---|---|

Parameter | Value | Units |

number_of_rollers | 12 | - |

pitch_diameter | 36.0 | mm |

width | 13.0 | mm |

inner_diameter | 25.0 | mm |

inner_shoulder_diameter | 30.5 | mm |

outer_diameter | 47.0 | mm |

outer_shoulder_diameter | 38.8 | mm |

roller_diameter | 5.5 | mm |

roller_length | 5.67 | mm |

bearing_clearance | 20 | μm |

effective_roller_length | 5 | mm |

bearing_density | 7.85e-6 | Kg/mm^3 |

ROM Parameters (CRB) | ||
---|---|---|

Parameter | Value | Units |

axial_constraint | True | - |

damping_force | True | - |

friction_torque | True | - |

damping_ratio | 0.15 | - |

lubricant_viscosity | 80 | cSt |

lubrication_method | ‘grease’ | - |

Contact Parameters (CRB) | ||
---|---|---|

Parameter | Value | Units |

axial_constraint | True | - |

use_contact_stiffness | True | - |

damping_factor | 10e-3 | - |

method.stiffness | 336980.92 | N/mm |

method.exponent | 10/9 | - |

method.damping | 3369.8 | Ns/mm |

method.dmax | 0.013 | mm |

method.coulomb_friction | “On” | - |

method.vd | 110 | mm/s |

method.md | 0.03 | - |

method.ms | 0.05 | - |

method.vs | 100 | mm/s |

na | 2000 | - |

nc | 80 | - |

no | 10 | - |

The damping factor chosen is higher compared to the DGBB. The extra damping helps reduce the numerical noise coming from the line contact.

For both the MBD model and the ROM model, the `axial_constraint`
flag is used to prevent axial displacement due to the lack of axial stiffness in the
models (lack of flanges on the inner/outer ring).

## Experiments

Six different experiments are considered, three for each bearing.

### Static Load for DGBB

Both models behave very similar statically, particularly up to the static load rating for the bearing (6550N). Above the static load rating, the results diverge, since the ROM’s equations are derived from empirical data that is only valid up to the static load rating. This is acceptable since bearings should never experience loads above their static load rating in real applications. Up to the static load rating, the relative error is below 4% and measured at low forces. This is expected due to the discontinuous behavior created by the bearing clearances.

### Static Load for CRB

Like the DGBB example, the maximum error is identified again in the lower range of the external load, which is less than 4%. Furthermore, the two models correlate very well, even above the static load rating.

Opposed to the DGBB, the slope of the bearing yield is almost linear for the CRB. This is due to the different contact phenomena between those bearings. While the DGBB is experiencing a point contact between rolling elements and rings that spread into an elliptical contact area, the CRB experiences more of a line contact that spreads into a rectangular shape. In Hertzian Contact methodology, the difference between point and line contact is primarily expressed by the contact exponent (3/2 for point and 10/9 for line contact).

### Dynamic Response for DGBB

Both models, ROM and the MBD model, showcase similar frequency responses. Two regions of critical frequencies behavior can be noticed. The first one is between 824 Hz (ROM) and 949 Hz (MBD), while the second one is between 3559 Hz (ROM) and 3925 Hz (MBD). Frequency-wise, the relative error between the ROM and the MBD model is around 9-13%, which is acceptable. Larger differences occur in the magnitude of the response, particularly for larger frequencies. The magnitude of the response is primarily driven by the damping, which is implemented differently for both models.

It should be noted that a bearing’s frequency response changes based on how loads are applied to it. This is primarily due to the highly nonlinear characteristic of the bearing. Both models, ROM and the full MBD model, can successfully capture the shift in frequency response due to changes in the direction of the load.

### Dynamic Response for CRB

### Fundamental Frequency for DGBB

Both models, ROM and the full MBD model, show an oscillation with the same frequency that is caused by the rolling elements passing through the loading zone. Although the ROM model does not include rolling element parts, their theoretical move is calculated inside the force subroutine of the machinery bearings. Here, this calculation is clearly validated.

Looking at the computational performance, the ROM performs significantly better in a transient analysis compared to the full MBD model.

Simulation Time Comparison | ||
---|---|---|

Method | Simulation time [sec] | Simulation time [min] |

ROM | 1.513E+01 | 0.25 |

MBD | 1.814E+03 | 30.23 |

### Fundamental Frequency for CRB

In term of computational performance, the ROM is significantly superior compared to the respective full MBD model for transient analysis.

Simulation Time Comparison | ||
---|---|---|

Method | Simulation time [sec] | Simulation time [min] |

ROM | 1.485E+01 | 0.25 |

MBD | 5.852E+03 | 97.53 |

## Conclusion

Comparisons between full multibody dynamic (MBD) bearing models and reduced order bearing models (ROMs) from the MSolve API was conducted. The comparison was first performed on a Deep Groove Ball Bearing (DGBB), followed by a Cylindrical Roller Bearing (CRB). The verification shows that the results of the ROM models are similar to their respective MBD model. Furthermore, it highlights the greater computational performance of ROMs compared to full MBD models.

While the MSolve API includes ROMs for several bearing types, the verification was performed only on two bearing families, ball bearings and roller bearings. The remaining bearing types uses the same theory principles and a similar outcome as shown here can be expected.