# Modeling of Boring Mandrel Working Process with Vibration Damper

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Design of Mathematical Model

- machine support with a tool holder;
- boring mandrel;
- inertial body (damper).

_{y}, which, for simplicity, will be denoted by P. The deformation in the direction of the force P

_{y}is the most critical from the point of the view of item size accuracy, waviness and surface roughness. The designed scheme of the system is presented in Figure 2. Figure 2 presents the parallel effect of masses m

_{2}and m

_{3}on mass m

_{1}.

_{1}—the movement of the mandrel flexible part (the movement of the cutting part).

- the natural vibrations frequency (approximately 150–250 Hz) that arise due to the occurrence of self-oscillations characteristic of the sharpening process;
- the forced vibrations frequency (approximately 400–600 Hz) that arises as a result of the chip formation process.

- the elastic resistance force, proportional to the movement,
- the viscous resistance force, proportional to the speed of movement.

_{i}< 1). These parameters, when jointly considering Equations (2) and (3) make the following:

_{1}, obtains the general transfer Equation (11) for the force P and disturbance V, that is, the solution of Equation (1) in an operator form.

## 3. Results

_{1}= 2 kg, m

_{2}= 2 kg, m

_{3}= 6 kg, m

_{4}= 20 kg, c

_{1}= 6 × 106 N/m, c

_{2}= 2 × 106 N/m, c

_{3}= 12 × 106 N/m, c

_{4}= 25 × 106 N/m, a

_{1}= 50 kg/s, a

_{2}= 1000 kg/s, a

_{3}= 50 kg/s, a

_{4}= 100 kg/s.

_{1}) for a single impulse action of a force P = 50 kN with a duration of 0.00002 s (the rectangular impulse integral is 50,000 × 0.00002 = 1). Figure 6 presents a comparison of the working process under the frequency action of the force with an amplitude of P = 500 N at a frequency of ω = 900 rad/s (≈ 140 Hz). The comparison data clearly confirm the vibration damper effectiveness in the tool holder.

_{i}, one can study the dynamics and frequency characteristics of any of the four masses participating in the calculation model (Figure 2).

## 4. Discussion

## 5. Conclusions

- This dynamic model was made including the mandrel separation into segments with different parameters of:
- -
- mass,
- -
- stiffness and damping.

- The segmentation allowed to describe the operation of the hole boring process with such a mandrel with a damper more accurately.
- The proposed representation of a mathematical model of differential equations system in the form of a structural diagram of interacting dynamic elements with transfer functions allowed:
- -
- to obtain a common frequency complex transfer function of the entire system,
- -
- for its subsequent transformation into an amplitude-frequency characteristic.

- This solution allows to find a safe frequency range with a minimum amplitude of the instrumental system vibration.
- Important for the technological process for this dynamic mathematical model of the technological system is the finding that the vibration amplitude:
- -
- can be reduced 2–3 times during impulse action on the tool,
- -
- at steady state during a frequency exposure also decreases significantly up to 4 times.

## Author Contributions

## Funding

## Conflicts of Interest

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**MDPI and ACS Style**

Sentyakov, K.; Peterka, J.; Smirnov, V.; Bozek, P.; Sviatskii, V.
Modeling of Boring Mandrel Working Process with Vibration Damper. *Materials* **2020**, *13*, 1931.
https://doi.org/10.3390/ma13081931

**AMA Style**

Sentyakov K, Peterka J, Smirnov V, Bozek P, Sviatskii V.
Modeling of Boring Mandrel Working Process with Vibration Damper. *Materials*. 2020; 13(8):1931.
https://doi.org/10.3390/ma13081931

**Chicago/Turabian Style**

Sentyakov, Kirill, Jozef Peterka, Vitalii Smirnov, Pavol Bozek, and Vladislav Sviatskii.
2020. "Modeling of Boring Mandrel Working Process with Vibration Damper" *Materials* 13, no. 8: 1931.
https://doi.org/10.3390/ma13081931