# A Multibody Inertial Propulsion Drive with Symmetrically Placed Balls Rotating on Eccentric Trajectories

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Description of the Proposed IPD

- -
**Version 1**: Inner bore with a radius of R_{1}= 41 mm and the center placed eccentric at a distance e = 20 mm, relative to the center of the slotted plates;- -
**Version 2**: Inner bore consisting of a semicircle with a radius R_{2-1}= 44.6 mm, two straight portions tangent to the semicircle and an arc of radius R_{2-2}= 61 mm, concentric with the first radius semicircle;- -
**Version 3:**Inner bore consisting of two semicircles with radii R_{3}= 41 mm and centers placed at a distance of e = 20 mm, which are connected by two straight portions tangent to the semicircles.

_{i}(t), these centrifugal forces are variable in time and may be expressed in (1):

- Fc
_{i}(t) [N]—centrifugal force acting on the balls; - m
_{0}[kg]—mass of the balls; - ω [rad/s]—angular velocity of the rotating plates;
- R
_{i}(t)—trajectory radius of ball i.

## 3. Analytical Investigation of the Three Proposed Constructive Alternatives

#### 3.1. Version 1 of Retaining Disk: Cylindrical Bore Placed Eccentric Relative to the Center of the Slotted Plates

_{i}, a Cartesian system denoted with xO

_{1}y was attached to the center O

_{1}of the slotted plates (2/1 and 2/2). Furthermore, as shown in Figure 2, the circle with radius R

_{1}and center in O

_{2}, represents the inner bore of the retaining disk (3). As mentioned before, the retaining disk is placed eccentric, relative to the center of the slotted plates, at a distance e.

_{i}may be written as in (2) and (3):

#### 3.2. Version 2 of Retaining Disk: Inner Bore Consisting of a Semicircle, a Circular Arc and Two Straight Portions

_{2-1}, a circular arc with radius R

_{2-2}, concentric with the semicircle and two straight portions of length R

_{2-1}, which are tangent to the semicircle (see Figure 3).

_{x}and v

_{y}are obtained in (9) by deriving (8):

_{x}and a

_{y}are obtained in (10) by deriving (9):

#### 3.3. Version 3 of Retaining Disk: Inner Bore Consisting of Two Identical Semicircles with the Centres Located at a Distance “e” and Two Straight Portions Tangent to the Semicircles

_{3}. One of the centers of the semicircles is located in O, while the other is placed at the distance e in vertical direction. The two semicircles are connected by two vertical segments tangent to the semicircles (see Figure 4).

## 4. Motion Simulation of the IPD

- ○
- Design of the components shown in Figure 1b;
- ○
- Generation of the assembly;
- ○
- Specification of the speed for the rotary motor (1200 min
^{−1}); - ○
- Specification of gravity;
- ○
- Specification of solid body contacts;
- ○
- Specification of the mates.

^{2}, the gravitational forces acting on the mechanism were simulated. Finally, the mates between parts were applied in the motion study between the components of the assembly.

## 5. Results and Discussion

_{x}, v

_{y}, a

_{x}and a

_{y}) as functions of the rotation angle α are presented, for the three constructive versions of the retaining disk, in Figure A2, Figure A3 and Figure A4 (see Appendix A).

## 6. Experimental Proof of the Concept

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

Acronym | Meaning |

3D | Three Dimensional |

IPD | Inertial Propulsion Drive |

NASA | National Aeronautics and Space Administration |

PLA | Polylactide |

SW | SolidWorks |

## Appendix A

**Figure A1.**Constructive alternatives of the retaining disk. (

**a**) Version 1; (

**b**) Version 2; (

**c**) Version 3.

**Figure A2.**Comparison of simulation outcomes and analytical calculations for the coordinates of a ball. (

**a**) Version 1; (

**b**) Version 2; (

**c**) Version 3.

**Figure A3.**Comparison of simulation outcomes and analytical calculations for the velocities of a ball. (

**a**) Version 1; (

**b**) Version 2; (

**c**) Version 3.

**Figure A4.**Comparison of simulation outcomes and analytical calculations for the accelerations of a ball. (

**a**) Version 1; (

**b**) Version 2; (

**c**) Version 3.

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**Figure 6.**Velocity of the system. (

**a**) Entire simulation time; (

**b**) average values at last rotation of the system.

Distance [mm] | Driving Speed n [rpm] | Measured Time [s] | Average Time t _{m}[s] | Average Speed v _{m} [mm/s] | ||
---|---|---|---|---|---|---|

t_{1} | t_{2} | t_{3} | ||||

200 | 3900 | 15.0 | 15.1 | 15.0 | 15.03 | 13.31 |

7000 | 12.0 | 12.1 | 12.1 | 12.07 | 16.57 | |

9000 | 9.1 | 9.1 | 9.0 | 9.07 | 22.05 | |

10,500 | 6.0 | 6.0 | 6.0 | 6.00 | 33.33 | |

12,000 | 4.0 | 4.0 | 4.1 | 4.03 | 49.63 |

Timet [s] | 2 | 2.5 | 3 | 3.5 | 4 |

Distancey [mm] | 1.69 | 2.32 | 3.02 | 3.80 | 4.67 |

Time t [s] | Measured Distance [mm] | Average Distance y _{med} [mm] | Simulation Distance y _{sim} [mm] | Deviation Δy [%] | ||
---|---|---|---|---|---|---|

y_{1} | y_{2} | y_{3} | ||||

2 | 1.55 | 1.49 | 1.53 | 1.52 | 1.69 | 10.06 |

3 | 2.85 | 2.83 | 2.86 | 2.85 | 3.02 | 5.74 |

4 | 4.25 | 4.15 | 4.23 | 4.21 | 4.67 | 9.85 |

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**MDPI and ACS Style**

Gerocs, A.; Gillich, G.-R.; Nedelcu, D.; Korka, Z.-I.
A Multibody Inertial Propulsion Drive with Symmetrically Placed Balls Rotating on Eccentric Trajectories. *Symmetry* **2020**, *12*, 1422.
https://doi.org/10.3390/sym12091422

**AMA Style**

Gerocs A, Gillich G-R, Nedelcu D, Korka Z-I.
A Multibody Inertial Propulsion Drive with Symmetrically Placed Balls Rotating on Eccentric Trajectories. *Symmetry*. 2020; 12(9):1422.
https://doi.org/10.3390/sym12091422

**Chicago/Turabian Style**

Gerocs, Attila, Gilbert-Rainer Gillich, Dorian Nedelcu, and Zoltan-Iosif Korka.
2020. "A Multibody Inertial Propulsion Drive with Symmetrically Placed Balls Rotating on Eccentric Trajectories" *Symmetry* 12, no. 9: 1422.
https://doi.org/10.3390/sym12091422