# On the Dynamics of an Enhanced Coaxial Inertial Exciter for Vibratory Machines

^{1}

^{2}

^{*}

## Abstract

**:**

_{2}/ω

_{1}= 1, ω

_{2}/ω

_{1}= –1, and ω

_{2}/ω

_{1}= 2, are considered. Based on these relations, the circular, elliptical, and complex motion trajectories of the working members are implemented. In the first two cases, single-frequency harmonic oscillations take place. In the latter case, the double-frequency periodic oscillations are excited. The dynamic behavior of the motor’s shaft during its running-up and running-out is considered. The influence of the inertial parameters of the unbalanced rotors and the relative phase shift angle between them on the elliptical trajectories of the vibratory system’s mass center motion is investigated. The use of forced kinematic synchronization provides the motion stability of the vibratory system for all considered working regimes.

## 1. Introduction

## 2. Design of Vibrator and Methods of Research

#### 2.1. Dynamical Model of the Inertial Vibration Exciter with a Single Asynchronous Electric Motor

_{2}= ω

_{1}, ω

_{2}= −ω

_{1}, and ω

_{2}= 2ω

_{1}, under the specific values of the relative phase shift angle $\phi $ between the unbalanced rotors. In order to provide these relations, two belt transmissions 1 and 2 are used for actuating two coaxially installed unbalanced rotors from a single electric motor (Figure 3).

_{m}is the mutual inductance; ${i}_{s\alpha}$, ${i}_{s\beta}$, ${i}_{r\alpha}$, and ${i}_{r\beta}$ are the projections of the stator’s and rotor’s currents on the coordinate axes $\alpha -\beta $, respectively.

#### 2.2. Dynamical Model of the Vibratory System with Kinematically Synchronized Unbalanced Rotors

## 3. Results

_{x}, k

_{y}) are assigned as equal (3.944 × 10

^{5}N/m), as they are similar in real machines. The observed effect is a consequence of the assigned phase-shift angle ($\phi ={90}^{\xb0}$) between rotating unbalanced masses and the Sommerfeld effect, which plays a more significant role in the second case.

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Kinematic diagram of the inertial vibration exciter with a single asynchronous electric motor, as follows: 1, 2—internal and external unbalanced rotors, respectively; 3—exciter’s body; 4—bearings; 5, 6—belt transmissions; 7—electric motor.

**Figure 3.**Schemes of kinematic synchronization of the unbalanced rotors of the inertial vibration exciter. (

**a**) ${\omega}_{2}={\omega}_{1}$; (

**b**) ${\omega}_{2}=-{\omega}_{1}$; (

**c**) ${\omega}_{2}=2{\omega}_{1}$.

**Figure 4.**Time series of the electric motor’s shaft angular speeds under different synchronization conditions of the unbalanced rotors.

**Figure 5.**Time dependence of the electric motor’s shaft torque at different synchronization conditions of the unbalanced rotors. (

**a**) ${\omega}_{2}={\omega}_{1}$; (

**b**) ${\omega}_{2}=-{\omega}_{1}$; (

**c**) ${\omega}_{2}=2{\omega}_{1}$

**Figure 6.**Trajectories of the mass center motion at different ratios between the angular speeds of the unbalanced rotors.

**Figure 7.**Time dependencies of the disturbing force of the inertial vibration exciter at different ratios between the angular speeds of the unbalanced rotors.

**Figure 8.**Time dependencies of the working member displacements at different synchronization conditions of the unbalanced rotors. (

**a**) ${\omega}_{2}={\omega}_{1}$; (

**b**) ${\omega}_{2}=-{\omega}_{1}$; (

**c**) ${\omega}_{2}=2{\omega}_{1}$

**Figure 9.**Trajectories of the working member’s mass center motion at different values of the phase shift angle $\phi $ and static moments of the unbalanced rotors ${m}_{1}{r}_{1}/{m}_{2}{r}_{2}$ under their counter-rotation conditions, as follows: ${\omega}_{2}=-{\omega}_{1}$.

**Figure 10.**Amplitude–frequency characteristics of the vibratory system at the phase-shift angle $\phi =90\xb0$ and under different synchronization conditions of (

**a**) ${\omega}_{2}={\omega}_{1}$; (

**b**) ${\omega}_{2}=-{\omega}_{1}$; (

**c**) ${\omega}_{2}=2{\omega}_{1}$.

**Figure 11.**Amplitude–frequency characteristics of the vibratory system for under synchronization condition ${\omega}_{2}=2{\omega}_{1}$ at the phase-shift angles of (

**a**) $\phi =0\xb0$ and (

**b**) $\phi =180\xb0$.

**Table 1.**Technical characteristics of the asynchronous electric motor [41].

Parameters | Symbol | Values |
---|---|---|

Electric power | P | 1.1 kW |

Nominal voltage | U_{0} | 230 V |

Nominal speed | n | 1420 rpm |

Number of poles | p | 2 |

Stator resistance | R_{s} | 7.6 Ω |

Rotor resistance | R_{r} | 3.6 Ω |

Stator inductance | L_{s} | 0.6015 H |

Rotor inductance | L_{r} | 0.6015 H |

Mutual inductance | L_{m} | 0.58 H |

Moment of inertia | J | 0.005 kg⋅m^{2} |

Parameters | Symbol | Values |
---|---|---|

Total mass of exciter | m | 100 kg |

Unbalanced mass 1 | m_{1} | 5 kg |

Unbalanced mass 2 | m_{2} | 5 kg |

Springs’ stiffness in horizontal and vertical directions | k_{x}, k_{y} | 3.944 × 10^{5} N/m |

Coefficient of viscous damping in horizontal and vertical directions | c_{x}, c_{y} | 2512 N⋅s/m |

Coefficient of viscous friction in bearings | γ | 0.01 N m s/rad |

Static moment of the internal unbalanced rotor | m_{1}r_{1} | 0.15 kg⋅m |

Static moment of the external unbalanced rotor | m_{2}r_{2} | 0.06 kg⋅m |

Total inertial moment of rotating masses | I | 0.015 kg⋅m^{2} |

Bearing inner diameters | d_{01}d _{02} | 0.050 m 0.120 m |

Friction coefficient in bearings | f | 0.004 |

Parameters | Angular Velocities Ratio ω_{2}/ω_{1} | ||
---|---|---|---|

1 | −1 | 2 | |

Static moments of the unbalanced rotors 1/2, kg⋅m | 0.15/0.06 | ||

Angular speed of the unbalanced rotors 1/2, rad/s | 148.7/148.7 | 148.7/−148.7 | 148.7/297.4 |

Disturbing force, kN | 3.6 | 2–4.5 | 2–8.2 |

Total mass, kg | 100 | ||

Displacement amplitude, mm | 2 | ||

Acceleration, m/s^{2} | 40.7 | 53.2 | 89.4 |

Power, kW | 1.1 |

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**MDPI and ACS Style**

Gurskyi, V.; Korendiy, V.; Krot, P.; Zimroz, R.; Kachur, O.; Maherus, N.
On the Dynamics of an Enhanced Coaxial Inertial Exciter for Vibratory Machines. *Machines* **2023**, *11*, 97.
https://doi.org/10.3390/machines11010097

**AMA Style**

Gurskyi V, Korendiy V, Krot P, Zimroz R, Kachur O, Maherus N.
On the Dynamics of an Enhanced Coaxial Inertial Exciter for Vibratory Machines. *Machines*. 2023; 11(1):97.
https://doi.org/10.3390/machines11010097

**Chicago/Turabian Style**

Gurskyi, Volodymyr, Vitaliy Korendiy, Pavlo Krot, Radosław Zimroz, Oleksandr Kachur, and Nadiia Maherus.
2023. "On the Dynamics of an Enhanced Coaxial Inertial Exciter for Vibratory Machines" *Machines* 11, no. 1: 97.
https://doi.org/10.3390/machines11010097