# Matching Model of Dual Mass Flywheel and Power Transmission Based on the Structural Sensitivity Analysis Method

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## Abstract

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## 1. Introduction

## 2. Structural Sensitivity Analysis Method of Automobile Power Transmission

^{th}order natural frequency, and ${\left\{\theta \right\}}_{i}$ is the i

^{th}order modal shape. Structural damping and viscous damping still exist in the actual model; however, damping elements have little influence on the natural frequency of the system because of a small damping coefficient [4,22]. Furthermore, viscous friction and coulomb friction can cause a DMF to assume the hysteresis nonlinearity; however, the nonlinear model needs to be identified by the modified Bouc-Wen model combined with experimental data [23]. That is, the nonlinear model must be determined after a DMF is manufactured. Some studies [23] showed that the real natural frequency is approximately equal to the real natural frequency of the system without damping at low rotational speed. Therefore, the dynamic Equation (3) can be used to analyze the model in the process of matching.

#### 2.1. Sensitivity of Natural Frequencies of Torsional Vibration to Torsional Stiffness

^{th}order. Let the absolute and relative sensitivities of ${\omega}_{i}$ to the torsional stiffness of the j

^{th}unit be ${S}_{ab}\left({\omega}_{i}/{K}_{j}\right)$ and ${S}_{rt}\left({\omega}_{i}/{K}_{j}\right)$, respectively. Referring to Equations (1) and (2), the partial derivative with respect to ${K}_{j}$ in Equation (4) is operated to obtain Equation (6). Thus, ${S}_{ab}\left({\omega}_{i}/{K}_{j}\right)$ and ${S}_{rt}\left({\omega}_{i}/{K}_{j}\right)$ can be derived as:

#### 2.2. Sensitivity of Natural Frequencies of Torsional Vibration to Inertias

^{th}natural frequency, ${\omega}_{i}$ to the torsional stiffness of the j

^{th}unit be ${S}_{ab}\left({\omega}_{i}/{J}_{j}\right)$ and ${S}_{rt}\left({\omega}_{i}/{J}_{j}\right)$. By seeking the partial derivative with respect to ${J}_{j}$ in Equation (4), Equation (14) can be obtained as:

## 3. Matching Model of DMF and the Power Transmission Based on the Structural Sensitivity Analysis Method

#### 3.1. Matching Model of Inertia and Torsional Stiffness of the DMF under the Idling Condition

- (1)
- When the 1st order modal resonance speed of the power transmission is lower than the idling speed, the 0.5th and 1st harmonic resonances should be avoided. In this case, we should compare the vector sums of the relative amplitudes of the 0.5th and the 1st harmonic orders to determine the main harmonic excitations that should be avoided.
- (2)
- When the 1st order modal resonance speed of the power transmission is higher than the idling speed, the 1st, 1.5th, and 2nd harmonic resonances should be avoided. Under the idling condition, since nodes of the 1st order modal shape will not exist in the engine blocks, the main harmonic order will be the 2nd one for four-cylinder engines. In this instance, the 2nd order harmonic torsional vibration should be avoided.

^{th}and $\left(j+1\right)$

^{th}units be the primary flywheel assembly and secondary flywheel assembly, respectively, then:

#### 3.2. Matching Model of Inertia and Torsional Stiffness of the DMF under the Driving Condition

_{2}_set and λ_set, respectively. After traversing order_set2, the intersection of all values in the λ_ set and K

_{2}_set will be obtained, and the values of ${K}_{2}$ and $\lambda $ will be determined accordingly. After the above calculations under driving conditions, ${K}_{2}$ will change to be ${K}_{3}$. If ${K}_{3}>{K}_{2}$, we value the torsional stiffness of the DMF at the driving stage as ${K}_{3}$; that is ${K}_{2}={K}_{3}$. If ${K}_{3}<{K}_{2}$, ${K}_{3}$ cannot meet the requirement of torque transmission. Therefore, in this case, the intersection of the ranges of ${K}_{1}$ and ${K}_{3}$ under the idling condition should be determined firstly. In this intersection, by increasing ${K}_{1}$ and its operating angle, ${\theta}_{1}$, ${K}_{2}$ will finally be determined according to Equation (42).

## 4. Matching Example and Real Vehicle Test of DMF

#### 4.1. Matching Example of the DMF Based on the Structural Sensitivity Analysis Method

^{2}and N·m/rad, respectively.

^{2}, where the inertia of the primary flywheel assembly is not less than 0.075 kg·m

^{2}, and the secondary flywheel assembly quality should be less than 5 kg. Furthermore, $\xi =1.33$, thus $4.2<\lambda <9$.

^{2}and 0.012 kg·m

^{2}, respectively, and the torsional stiffness at the idling stage and driving stage is 160.43 N·m/rad (${\theta}_{1}=45.25\xb0$) and 733.39 N·m/rad (${\theta}_{2}=15\xb0$), respectively. In addition, the hollow travel angle is 4.75 °, and the total torsional stiffness is 65 °, that is, $\theta =65\xb0$.

^{th}order and j represents the j

^{th}unit. According to this calculation flow, the program of m file is coded by using MATLAB, and the corresponding program code is shown in Appendix A. The absolute sensitivities of the 1st order natural frequency to the inertias and torsional stiffness can be obtained based on the torsional vibration model under the driving condition, as shown in Table 4 and Table 5 and Figure 11 and Figure 12.

^{2}, ${J}_{9}=0.018$ kg·m

^{2}, ${K}_{8}=257.8$ N·m/rad (${\theta}_{1}=54.25\xb0$) (the torsion stiffness under idling condition), ${K}_{8}=710.5$ N·m/rad (${\theta}_{2}=6\xb0$) (the torsion stiffness under driving condition), ${\theta}_{0}=4.75\xb0$ (the hollow travel angle), and $\theta =65\xb0$ (the total torsion angle). The rematching DMF is shown as Figure 14.

#### 4.2. Real Vehicle Test

## 5. Conclusions

- (1)
- The absolute structural sensitivity can effectively isolate key structural parameters of the vibration modal at each stage and the resultant parameters can be quantitatively revised by the relative structural sensitivity. Under the driving condition, the inertia of the secondary flywheel assembly and torsional stiffness of DMF have a significant influence on the 1st order natural frequency of automotive power transmission. The inertia of the secondary flywheel assembly is inversely proportional to the 1st order natural frequency. In contrast, the torsional stiffness is positively proportional to the 1st order and the inertia ratio of the primary and secondary assembly is positively associated to the 1st order.
- (2)
- Given that the resonance speed is higher than the idle speed under the idling condition, the 1st order natural frequency of the system should be increased through enhancement of the torsional stiffness of the DMF at the idling stage to decrease the angular acceleration amplitude of the input shaft of the transmission. In contrast, the 1st order natural frequency of the system should be decreased through reduction of the torsional stiffness of the DMF at the idling stage to attenuate the angular acceleration amplitude of the input shaft of the transmission.
- (3)
- Under the driving condition, the 1st order natural frequency of the system should be decreased by reduction of the inertia ratio and the torsional stiffness of the DMF at the driving stage, which appears to protect resonances in low speed zones and attenuates the angular acceleration amplitude of the input shaft of the transmission.
- (4)
- Given that the torsional stiffness at the driving stage cannot meet the requirements of the matching model, the operation range of the torsional stiffness at the idling stage should be enlarged to make it work under driving conditions.

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

## References

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**Figure 1.**The power transmission with the DMF (Dual Mass Flywheel). 1. Engine; 2. Primary flywheel assembly; 3. Secondary flywheel assembly; 4. Clutch and gear box; 5. Transmission shaft; 6. Vehicle load.

**Figure 4.**Structure parameters matching method of the dual mass flywheel under the idle condition based on structural sensitivity.

**Figure 5.**Structure parameters matching method of the dual mass flywheel under the driving condition based on structural sensitivity.

**Figure 11.**The absolute sensitivities of the 1st order natural frequency to inertias under the driving condition.

**Figure 12.**The absolute sensitivity of the 1st order natural frequency to torsional stiffness under the driving condition.

**Figure 13.**The relative sensitivities of the 1st order natural frequency to torsional stiffness under the driving condition.

**Figure 19.**Measurement tracking setting: (

**a**) tracking setting of the rotating speed signal channel of the primary flywheel and (

**b**) tracking setting of the rotating speed signal channel of the input shaft of the transmission.

**Figure 22.**The angular acceleration of the primary flywheel and the transmission input under the idling condition for the engine with the original DMF.

**Figure 23.**Time-domain speed signal of the engine with the original DMF under the driving condition.

**Figure 24.**The angular acceleration of the primary flywheel and the transmission input under the driving condition for the engine with the original DMF.

**Figure 25.**Time-domain speed signal of the engine with the rematching DMF under the idling condition.

**Figure 26.**The angular acceleration of the primary flywheel and the transmission input under the idling condition for the engine with the rematching DMF.

**Figure 27.**Time-domain speed signal of the engine with the rematching DMF under the driving condition.

**Figure 28.**The angular acceleration of the primary flywheel and the transmission input under the driving condition for the engine with the rematching DMF.

Names of Elements | Inertia | Value of Inertia (kg·m^{2}) | Torsional Stiffness | Value of Torsional Stiffness (N·m/rad) |
---|---|---|---|---|

Driven part of rubber damper | J_{1} | 4.795 × 10^{−3} | K_{1} | 14,320.0 |

Driving part of rubber damper | J_{2} | 2.038 × 10^{−3} | K_{2} | 74,636.1 |

Accessories | J_{3} | 9.74 × 10^{−5} | K_{3} | 356,080.5 |

Cylinder 1 | J_{4} | 4.669 × 10^{−3} | K_{4} | 358,856.9 |

Cylinder 2 | J_{5} | 4.712 × 10^{−3} | K_{5} | 360,638.8 |

Cylinder 3 | J_{6} | 4.712 × 10^{−3} | K_{6} | 359,640 |

Cylinder 4 | J_{7} | 4.686 × 10^{−3} | K_{7} | 1,871,080 |

Primary flywheel assembly | J_{8} | 0.08 | K_{8} | 160.43; 733.39 |

Secondary flywheel assembly | J_{9} | 0.012 | K_{9} | 98,731.62 |

Input shaft of CVT | J_{10} | 7.312 × 10^{−3} | K_{10} | 48,483.78 |

Driving cone of CVT | J_{11} | 0.0268368 |

f_{1} | f_{2} | f_{3} | f_{4} | f_{5} | f_{6} | f_{7} | f_{8} | f_{9} |
---|---|---|---|---|---|---|---|---|

15.8 | 239.8 | 603.8 | 742 | 1053 | 1731 | 2481 | 3602 | 10,667 |

f_{1} | f_{2} | f_{3} | f_{4} | f_{5} | f_{6} | f_{7} | f_{8} | f_{9} | f_{10} |
---|---|---|---|---|---|---|---|---|---|

22.5 | 239.8 | 300 | 603.8 | 821.79 | 1053 | 1731 | 2481 | 3602 | 10,667 |

**Table 4.**The absolute sensitivities of the 1st order natural frequency to inertias under the driving condition.

J_{1} | J_{2} | J_{3} | J_{4} | J_{5} | J_{6} | J_{7} | J_{8} | J_{9} | J_{10} | J_{11} |
---|---|---|---|---|---|---|---|---|---|---|

−221.583 | −218.24 | −217.333 | −217.14 | −216.821 | −216.376 | −215.803 | −215.668 | −1104.44 | −1122.1 | −1150.73 |

**Table 5.**The absolute sensitivities of the 1st order natural frequency to torsional stiffness under the driving condition.

K_{1} | K_{2} | K_{3} | K_{4} | K_{5} | K_{6} | K_{7} | K_{8} | K_{9} | K_{10} |
---|---|---|---|---|---|---|---|---|---|

5.62 × 10^{−7} | 4.18 × 10^{−8} | 1.89 × 10^{−9} | 5.18 × 10^{−9} | 1.01 × 10^{−8} | 1.68 × 10^{−8} | 9.29 × 10^{−10} | 0.101543 | 3.1 × 10^{−6} | 7.97 × 10^{−6} |

**Table 6.**The relative sensitivities of the 1st order natural frequency to torsional stiffness under the driving condition.

K_{1} | K_{2} | K_{3} | K_{4} | K_{5} | K_{6} | K_{7} | K_{8} | K_{9} | K_{10} |
---|---|---|---|---|---|---|---|---|---|

5.35 × 10^{−5} | 2.08 × 10^{−5} | 4.47 × 10^{−6} | 1.24 × 10^{−5} | 2.42 × 10^{−5} | 4.03 × 10^{−5} | 1.16 × 10^{−5} | 0.495229 | 0.002033 | 0.00257 |

f_{1} | f_{2} | f_{3} | f_{4} | f_{5} | f_{6} | f_{7} | f_{8} | f_{9} |
---|---|---|---|---|---|---|---|---|

18.8 | 240.1 | 605.1 | 713.4 | 1053.6 | 1731.9 | 2481.7 | 3604.7 | 10,667.8 |

f_{1} | f_{2} | f_{3} | f_{4} | f_{5} | f_{6} | f_{7} | f_{8} | f_{9} | f_{10} |
---|---|---|---|---|---|---|---|---|---|

15 | 240.1 | 282.9 | 605.1 | 801.7 | 1053.6 | 1731.9 | 2481.7 | 3604.7 | 10,667.8 |

Items | Under Idling Condition | Under Driving Conditon | ||
---|---|---|---|---|

Original DMF | Rematching DMF | Original DMF | Rematching DMF | |

The maximum angular acceleration (rad/s^{2}) | 83 | 25 | 203 | 184 |

Measured resonance speed (r/min) | 1210 | 836 | ||

Theoretical resonance speed (r/min) | 940 | 1128 | 1350 | 900 |

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## Share and Cite

**MDPI and ACS Style**

Chen, L.; Zhang, X.; Yan, Z.; Zeng, R.
Matching Model of Dual Mass Flywheel and Power Transmission Based on the Structural Sensitivity Analysis Method. *Symmetry* **2019**, *11*, 187.
https://doi.org/10.3390/sym11020187

**AMA Style**

Chen L, Zhang X, Yan Z, Zeng R.
Matching Model of Dual Mass Flywheel and Power Transmission Based on the Structural Sensitivity Analysis Method. *Symmetry*. 2019; 11(2):187.
https://doi.org/10.3390/sym11020187

**Chicago/Turabian Style**

Chen, Lei, Xiao Zhang, Zhengfeng Yan, and Rong Zeng.
2019. "Matching Model of Dual Mass Flywheel and Power Transmission Based on the Structural Sensitivity Analysis Method" *Symmetry* 11, no. 2: 187.
https://doi.org/10.3390/sym11020187