# Vibration Damping Analysis of Lightweight Structures in Machine Tools

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

^{2}and speeds from 3 m/s [7]. The utilization of large quantities of steel may generate undesired vibrations in high transfer speed conditions, seriously compromising the quality of the workpiece [8,9]. In this way, hybrid materials that are able to dampen mechanical vibrations may be preferred in manufacturing MT structural parts.

## 2. Materials and Methods

#### 2.1. Analysis of Vibration Damping in Machine Tools

_{1}, has three rotational degrees of freedom (DOFs) constrained to the basement. These DOFs allow evaluating its natural frequencies. In this way, a simple finite segment method (FSM) may model the stiffness using one rigid mass and three springs for each degree of freedom. This approach is based on modelling a flexible body as a finite number of rigid elements that are linked by dampers and springs. This requires the use of rigid multibody formulations. The spring s

_{1}defines the bending in the Z-axis, s

_{2}describes the torsion around the Y-axis, while s

_{3}shows the inflection in the X-axis. The dampers d

_{i}(i = 1, 2, 3) regulate the resonance, without influencing the natural frequencies of the body. It is assumed that the translational movement of body A is not permitted.

_{2}, is the ram of the machine tool. It has a translation DOF in the Z-axis that represents the rigid movement of the slide. It also has a DOF, defined by s

_{3}and d

_{3}in the Y-axis.

_{x}, F

_{y}, F

_{z}) is applied to the machine ram (body B), the dynamic behaviour of the system may be evaluated by applying the Newton-Euler equations for a general coordinate z = [α, β, γ, L

_{Y}, L

_{Z}], where L

_{Y}and L

_{Z}are the distances between the body’s barycenters (in the Y-Z plane).

- ${J}_{{R}_{i}},{J}_{{T}_{i}}$: Jacobian matrices of rotation and translation of the i-th body,
- ${I}_{i}$: Inertia matrix of the i-th body,
- ${\overline{a}}_{i}$: Acceleration of the i-th body,
- ${\omega}_{i}$: Angular rate of the i-th body,
- ${\tilde{\omega}}_{i}$: Rotation matrix of the i-th body,
- ${f}_{i},{h}_{i}$: External forces and moments of the i-th body,
- m: Mass matrix

_{i}, acting on the body. In particular, the model parameters (e.g., Jacobian matrixes, speeds, accelerations) depend on the position and they need to be calculated from theoretical considerations. However, it is also interesting to highlight that the active forces are not pose-dependent [41,48]. This parameter model may be useful both for the numerical studies (e.g., finite element analysis—FEA) and for the experimental campaigns (e.g., static and dynamic tests). It underlines those variables that describe the machine dynamic behaviour and, as a consequence, how to compensate for the undesired effects of vibrations. For example, assuming a prescribed mass to guarantee the lightweight characteristics of a structure, it is noted that the stiffness may be improved through the increase of the structure inertia, working on ram thicknesses and dimensions. This study may be developed using FE simulations that produce an accurate representation of distributed stiffness and inertia properties.

#### 2.2. The Lighweight Structure Prototypes

## 3. The Stiffness Analysis of the Lightweight Structures

#### 3.1. Numerical Analysis of the Structure Stiffness

#### 3.2. Experimental Test Campaign: The Stiffness Analysis of the Lightweight Structures

## 4. The Modal Analysis of the Lightweight Structures

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Lightweight MT structure materials: Al foam sandwiches (AFS) (

**a**); Al corrugated-core sandwiches (

**b**); and composite materials reinforced by carbon fibres (CFRP) (

**c**).

**Figure 3.**Lightweight MT structure prototypes: Al metal foams (AFS) (

**a**); Al corrugated-core sandwiches (ACS) (

**b**); and carbon fibre reinforced polymer (CFRP) (

**c**).

**Figure 4.**FE model of AFS structure: the deformed shape (μm) in the X orientation (

**a**) and in Y orientation (

**b**), applying a static load of 60 N at tool tip point.

**Figure 5.**Test bench setup overview (

**a**) and the experimental results of the statistic tests performed on a ram (

**b**).

**Figure 7.**The frequency response function (FRF) of the excited point (centre of ram) in the Y direction.

**Figure 8.**The result comparison: X-axis stiffness (

**a**); Y-axis stiffness (

**b**); mass (

**c**); and damping (

**d**).

Vibration Damping | Characteristics | References |
---|---|---|

Active methods | - Installation of additional devices, such as actuators
- Use of advanced and complex control algorithms
- Knowledge of vibrations’ eigen-frequencies
- Model-based strategy
| [1,18,19,20,21,22,23,24] |

Passive methods | - Based on viscoelastic materials, viscous fluids, magnetic or passive piezoelectric, lightweight materials
- Vibration energy dissipation or redirection
- Cost effective
- Dampers are usually small size and easy to install
| [16,25,26,27,28,29,30,31,32] |

Material | E^{1/3}/ρ (GPa^{1/3}/(Mg/m^{3})) | η |
---|---|---|

Cast iron | 0.63 | 1.2 × 10^{−3}–1.7 × 10^{−3} |

Steel | 0.77 | 6.0 × 10^{−4}–1.0 × 10^{−3} |

Al alloys | 1.50 | 2.0 × 10^{−4}–4.0 × 10^{−4} |

Mg alloys | 1.90 | 1.0 × 10^{−3}–1.0 × 10^{−2} |

Al corrugated sandwich | 2.52 | 1.0 × 10^{−3}–1.0 × 10^{−2} |

Al foams | 2.67 | 4.0 × 10^{−3}–1.0 × 10^{−2} |

CFRP (unidirectional) | 4.00 | 1.5 × 10^{−3}–3.0 × 10^{−3} |

Configuration | Mass (kg) | Conventional Mass (kg) | Mass Variation (%) |
---|---|---|---|

AFS ram | 116.0 | 97.0 | +19.6% |

Al corrug. ram | 78.0 | 97.0 | −19.5% |

CFRP ram | 50.0 | 97.0 | −48.5% |

Configuration | Kx (kg/µm) | Ky (kg/µm) | Kz (kg/µm) |
---|---|---|---|

Conventional (steel) RAM | 2.20 | 3.43 | 22.05 |

AFS RAM | 2.48 | 3.95 | 55.62 |

Al corrug. RAM | 1.90 | 4.41 | 42.81 |

CFRP RAM | 2.65 | 5.10 | 39.85 |

Configuration | Direction | K (N/µm) | Simulation Error | Stiffness Comparison with a Conventional RAM |
---|---|---|---|---|

AFS ram | X | 2.50 | 0.80% | +13.64% |

Y | 4.64 | 14.87% | +35.29% | |

Al corrug. ram | X | 1.75 | 8.57% | −20.45% |

Y | 4.44 | 0.63% | +29.43% | |

CFRP ram | X | 2.30 | 14.97% | +4.78% |

Y | 5.55 | 8.09% | +61.38% |

Mode | FE Model Frequency (Hz) | Description |
---|---|---|

1 | 650 | Breathing |

2 | 677 | Breathing and Bending Z-X |

3 | 709 | Breathing |

4 | 782 | Bending mode Z-Y |

5 | 816 | Torsion Z axis |

Configuration | 1st Frequency (Hz) | Loss Factor (%) |
---|---|---|

Conventional (steel) RAM | 670 | 0.08 |

AFS RAM | 667 | 1.70 |

Al corrug. RAM | 745 | 0.17 |

CFRP RAM | 1286 | 0.23 |

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**MDPI and ACS Style**

Aggogeri, F.; Borboni, A.; Merlo, A.; Pellegrini, N.; Ricatto, R.
Vibration Damping Analysis of Lightweight Structures in Machine Tools. *Materials* **2017**, *10*, 297.
https://doi.org/10.3390/ma10030297

**AMA Style**

Aggogeri F, Borboni A, Merlo A, Pellegrini N, Ricatto R.
Vibration Damping Analysis of Lightweight Structures in Machine Tools. *Materials*. 2017; 10(3):297.
https://doi.org/10.3390/ma10030297

**Chicago/Turabian Style**

Aggogeri, Francesco, Alberto Borboni, Angelo Merlo, Nicola Pellegrini, and Raffaele Ricatto.
2017. "Vibration Damping Analysis of Lightweight Structures in Machine Tools" *Materials* 10, no. 3: 297.
https://doi.org/10.3390/ma10030297