# Idle Vibration Reduction of a Diesel Sport Utility Vehicle

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## Abstract

**:**

## 1. Introduction

## 2. Experimental Analysis

## 3. Dynamic Modeling

^{2}, connecting rod length of $l=145$ mm, crank arm length of $r=48$ mm, and piston area of $A=5728$ mm

^{2}. By substituting these values into Equation (6), the magnitude of the total moment was computed as $1.72\times {10}^{6}$ Nmm from Equation (6). For the idling frequency of 27.7 Hz, the angular frequency, $2\omega $, is calculated as 348.09 rad/s. However, as reported in a study, only 1–5% of the total gas torque generally contributes to the vibration excitation torque [25]. Considering the low engine speed and its stable idle state, 1% of the gas torque presumably contributed to the excitation torque. Therefore, when the engine is in the idle state, the amplitude of the roll excitation moment finally applied to the dynamic model was $1.72\times {10}^{4}$ Nmm.

## 4. Derivation of the Equations of Motion

**M**denotes the mass matrix,

**N**represents the nonlinear internal force vector,

**q**symbolizes the displacement vector,

**f**depicts the applied load vector,

**λ**depicts the Lagrange multiplier vector,

**g**represents the constraint vector, and

**A**denotes the Jacobian matrix. $g$ can be also expressed as $g=Aq$.

**C**is the damping matrix and

**K**is the stiffness matrix. In this case, the nonlinear equations of motion expressed in Equation (17) can be reduced to the following linearized matrix–vector equation:

**M**,

**C**, and

**K**are $19\times 19$ matrices, and ${A}_{L}$ is a $6\times 19$ matrix.

^{2}). Denoting the accelerations in the dB and linear scales by ${A}_{D}$ and ${A}_{L}$, the relationship between ${A}_{D}$ and ${A}_{L}$ is given by ${A}_{D}=20\mathrm{log}\left({A}_{L}/{10}^{-6}\right)$. Although the vibration levels of the simulation and the experiment are slightly different, the trends of the overall increasing and decreasing levels are notably similar. As described earlier, the dynamic model established in this study was verified by comparing the natural frequencies and vibration levels obtained from the simulation and experiment.

## 5. Design Optimization for the Idle Vibration Reduction

^{2}on a linear scale. In fact, the idle vibration of the optimized design is reduced by 64.5% on a linear scale than that of the present design. The design variable values obtained from the design optimization are summarized in Table 4. In this table, the optimized values of the rubber bushing stiffness of the radiator and the intercooler mass represent the upper limits of the design ranges, whereas the optimized values of the lower hose stiffness of the radiator, the upper hose stiffness of the intercooler, and the lower hose stiffness of the intercooler comprise the lower limits. Conversely, the optimized values of the radiator mass, the upper hose stiffness of the radiator, and the rubber bushing stiffness of the intercooler do not exhibit any significant differences compared with the present design values.

## 6. Conclusions

- A dynamic model that can predict the idle vibration level on the driver’s seat of a diesel SUV was established, wherein not only the rigid body modes of the powertrain, radiator, and intercooler but also the deformed modes such as the bending and torsional modes of the body floor were considered.
- The design variables sensitive to the idle vibration level within the changeable variable ranges are listed in the following decreasing order of sensitivity: the rubber bushing stiffness of the radiator, the radiator mass, the lower hose stiffness of the intercooler, the intercooler mass, the lower hose stiffness of the radiator, the rubber bushing stiffness of the intercooler, the upper hose stiffness of the intercooler, and the upper hose stiffness of the radiator.
- By using design variables related to the radiator and intercooler, design optimization was performed to reduce the idle vibration on the driver’s seat, and the idle vibration was reduced by 9 dB through the optimization.
- The idle vibration level can be reduced by rendering the natural frequencies for the bounce modes of the radiator and intercooler proximate to the idle frequency. Accordingly, the radiator and intercooler act as vibration absorbers.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 5.**Operational deflection shapes of various components at the idling frequency of 27.7 Hz: (

**a**) powertrain; (

**b**) radiator; (

**c**) intercooler; and (

**d**) body floor.

**Figure 6.**Impact and measurement points for the experimental modal tests: (

**a**) powertrain; (

**b**) radiator; (

**c**) intercooler; and (

**d**) body floor.

**Figure 7.**Mode shapes of the components: (

**a**) bounce mode of the powertrain (4.8 Hz); (

**b**) pitch mode of the powertrain (13.0 Hz); (

**c**) roll mode of the powertrain (18.5 Hz); (

**d**) bounce mode of the radiator (20.0 Hz); (

**e**) roll mode of the radiator (30.3 Hz); and (

**f**) bounce mode of the intercooler (29.3 Hz).

**Figure 8.**Mode shapes of the body floor: (

**a**) bounce mode (3.2 Hz), (

**b**) pitch mode (3.7 Hz), (

**c**) bending mode (23.8 Hz), and (

**d**) torsional mode (28.4 Hz).

**Figure 10.**Equivalent stiffnesses and damping coefficients between the powertrain, radiator, intercooler, and front floor.

**Figure 11.**Measurement setup of the stiffnesses and damping coefficients for mounts, rubber bushings, and hoses: (

**a**) the photo of actual device and (

**b**) the corresponding physical model.

**Figure 12.**Dynamic stiffness for the rubber bushing of the radiator: (

**a**) the real part and (

**b**) the imaginary part.

**Figure 13.**Finite element models to obtain equivalent stiffnesses between the front and middle body floors: (

**a**) the load condition for the bending stiffness and (

**b**) the load condition for the torsional stiffness.

**Figure 14.**Rigid model of the front body floor to obtain equivalent stiffnesses between the front and middle body floors: (

**a**) the bending stiffness and (

**b**) the torsional stiffness.

**Figure 17.**Schematics of added masses to analyze the effect of radiator or intercooler mass on vibration level: (

**a**) the radiator and (

**b**) the intercooler.

**Figure 18.**Comparison of vibration levels between the simulation and experiment obtained by changing the amount of added mass: (

**a**) the radiator and (

**b**) the intercooler.

**Figure 19.**Idle vibration levels for the variations of the eight design variables: (

**a**) radiator mass; (

**b**) intercooler mass; (

**c**) rubber bushing stiffness of the radiator; (

**d**) rubber bushing stiffness of the intercooler; (

**e**) upper hose stiffness of the radiator; (

**f**) lower hose stiffness of the radiator; (

**g**) upper hose stiffness of the intercooler; and (

**h**) lower hose stiffness of the intercooler.

**Figure 21.**Convergence plots of the eight design variables in the design optimization: (

**a**) radiator mass; (

**b**) intercooler mass; (

**c**) rubber bushing stiffness of the radiator; (

**d**) rubber bushing stiffness of the intercooler; (

**e**) upper hose stiffness of the radiator; (

**f**) lower hose stiffness of the radiator; (

**g**) upper hose stiffness of the intercooler; and (

**h**) lower hose stiffness of the intercooler.

**Figure 22.**Comparison of the frequency response functions for the dynamic models with the present and optimal designs.

Component | Mass (kg) | Mass Moment of Inertia (kg·m^{2}) | |||||
---|---|---|---|---|---|---|---|

$\mathbf{m}$ | ${\mathit{I}}_{\mathit{x}}$ | ${\mathit{I}}_{\mathit{y}}$ | ${\mathit{I}}_{\mathit{z}}$ | ${\mathit{I}}_{\mathit{x}\mathit{y}}$ | ${\mathit{I}}_{\mathit{y}\mathit{z}}$ | ${\mathit{I}}_{\mathit{z}\mathit{x}}$ | |

Powertrain | 301.4 | 11.55 | 24.01 | 23.14 | 1.49 | 0.04 | −4.45 |

Radiator | 11.5 | 0.57 | 0.33 | 0.52 | 0 | 0 | 0 |

Intercooler | 5.0 | 0.13 | 0.12 | 0.02 | 0 | 0 | 0 |

Front floor | 523.5 | 119.45 | 113.37 | 171.94 | −0.59 | 1.21 | −9.43 |

Middle floor | 426.1 | 148.26 | 81.02 | 131.76 | −0.39 | 0.10 | −0.96 |

Rear floor | 474.3 | 115.50 | 120.27 | 134.17 | 0.58 | −0.45 | −1.50 |

**Table 2.**Equivalent stiffnesses and damping coefficients of the bushings, hoses, and mounts at the idle frequency of 27.7 Hz.

Component | Part | Stiffness (N/mm) | Damping Coefficient (kg/s) |
---|---|---|---|

Radiator | Rubber bushing | 53.3 | 57.5 |

Upper hose | 5.0 | 157.7 | |

Lower hose | 12.8 | 88.9 | |

Intercooler | Rubber bushing | 55.2 | 54.4 |

Upper hose | 10.2 | 150.3 | |

Lower hose | 13.6 | 189.8 | |

Powertrain | Engine mount | 500.2 | 1223.8 |

Transmission mount | 497.5 | 375.1 | |

Roll mount | 2911.6 | 2553.4 |

Component | Mode | Natural Frequency (Hz) | ||
---|---|---|---|---|

Simulation | Experiment | Difference | ||

Powertrain | Bounce mode | 6.7 | 4.8 | −1.9 |

Pitch mode | 15.3 | 13.0 | −2.3 | |

Roll mode | 21.1 | 18.5 | −2.6 | |

Radiator | Bounce mode | 19.4 | 20.0 | 0.6 |

Roll mode | 30.3 | 30.3 | 0.0 | |

Intercooler | Bounce mode | 31.0 | 29.3 | −1.7 |

Body floor | Bounce mode | 1.2 | 3.2 | 2 |

Pitch mode | 1.7 | 3.7 | 2 | |

Bending mode | 25.2 | 23.8 | −1.4 | |

Torsional mode | 28.0 | 28.4 | 0.4 |

Component | Design Variables | Present Value | Lower Limit | Upper Limit | Optimized Value |
---|---|---|---|---|---|

Radiator | Mass (kg) | 11.5 | 9.2 | 13.8 | 11.2 |

Stiffness of the rubber bushing (N/mm) | 53.3 | 26.7 | 106.6 | 106.6 | |

Stiffness of the upper hose (N/mm) | 57.1 | 28.6 | 114.2 | 54.8 | |

Stiffness of the lower hose (N/mm) | 79.9 | 40.0 | 159.8 | 40.0 | |

Intercooler | Mass (kg) | 5.0 | 4.0 | 6.0 | 6.0 |

Stiffness of the rubber bushing (N/mm) | 98.1 | 49.1 | 196.2 | 102.2 | |

Stiffness of the upper hose (N/mm) | 65.5 | 32.8 | 131.0 | 32.8 | |

Stiffness of the lower hose (N/mm) | 63.0 | 31.5 | 126.0 | 31.5 |

Component | Mode | Natural Frequency (Hz) | ||
---|---|---|---|---|

Present Design | Optimal Design | Difference | ||

Powertrain | Bounce mode | 6.7 | 6.6 | −0.1 |

Pitch mode | 15.3 | 15.3 | 0 | |

Roll mode | 21.1 | 20.9 | −0.2 | |

Radiator | Bounce mode | 19.4 | 26.7 | 7.3 |

Roll mode | 30.3 | 34.1 | 3.8 | |

Intercooler | Bounce mode | 31.0 | 28.9 | −2.1 |

Body floor | Bounce mode | 1.2 | 1.2 | 0 |

Pitch mode | 1.7 | 1.7 | 0 | |

Bending mode | 25.2 | 24.1 | −1.1 | |

Torsional mode | 28.0 | 28.1 | 0.1 |

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**MDPI and ACS Style**

Ryu, S.; Kim, D.; Lee, B.; Han, D.; Jung, I.; Chung, J.
Idle Vibration Reduction of a Diesel Sport Utility Vehicle. *Appl. Sci.* **2022**, *12*, 5448.
https://doi.org/10.3390/app12115448

**AMA Style**

Ryu S, Kim D, Lee B, Han D, Jung I, Chung J.
Idle Vibration Reduction of a Diesel Sport Utility Vehicle. *Applied Sciences*. 2022; 12(11):5448.
https://doi.org/10.3390/app12115448

**Chicago/Turabian Style**

Ryu, Seokwon, Dongju Kim, Booyoung Lee, Doohee Han, Insoo Jung, and Jintai Chung.
2022. "Idle Vibration Reduction of a Diesel Sport Utility Vehicle" *Applied Sciences* 12, no. 11: 5448.
https://doi.org/10.3390/app12115448