# The Influence of Servo Drive Control on the NC Vertical Milling Machine Dynamic Compliance

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Proposed Methodology

## 3. Spindle Dynamic Compliance

- ${u}_{F}$ Displacement in the force direction
- ${\mathsf{\Omega}}_{i}$ Natural frequency
- ${\varsigma}_{qi}$ Modal damping
- ${\mathit{v}}_{i}$ Eigenvectors
- ${\mathit{C}}_{\mathit{F}}$ Directional force cosine vector

## 4. Structural Model of Feed Drive Axis

#### 4.1. One-Mass Model of Linear Motor Feed Drive

#### 4.2. N-Mass Model of Linear Motor Feed Drive

#### 4.3. General Dynamic Compliance Matrix of Mechanic Structure and Feed Drive Control

**G**

_{R}(s), with inputs on the motor F

_{1ext}and on the N-mass F

_{N}and with corresponding outputs y

_{1}and y

_{N}, is expressed by the diagram in Figure 12.

- ${y}_{1}$ can be obtained from the motor current commutation position sensors.
- ${F}_{1ext}$ motor force can be simulated by the additional current source.

## 5. Feed Drive Coupled Model of the Machining Center

#### 5.1. Machine Tool Spindle

#### 5.2. Machine Tool XY Cross Table

#### 5.2.1. X Axis

- Model $1$—represents the complete system shown in Figure 16 including inner ${G}_{I}\left(s\right)$;
- Model $2$—represents the system where the current transfer function is simplified: ${G}_{I}\left(s\right)={I}_{out}/{I}_{in}=1$ and is accepted for further modelling;
- Model $3$—represents the system where the velocity transfer function is simplified: ${G}_{V}\left(s\right)={v}_{out}/{v}_{in}=1$.

#### 5.2.2. Y Axis

## 6. Impact of Feed Drive Control Parameters on Machining Stability Prediction

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

${A}_{reg}$ | Controller dynamic transfer function | |

${b}_{1},{b}_{2},\dots {b}_{N},{b}_{y}$ | $\left[\mathrm{Ns}/\mathrm{m}\right]$ | Damping coefficient |

${b}_{crit}$ | $\left[\mathrm{m}\right]$ | Critical depth of cut |

${b}_{lim}$ | $\left[\mathrm{m}\right]$ | Local limit depth of cut |

${\mathit{C}}_{\mathit{F}}$ | Directional force cosine vector | |

${F}_{F}$ | Additional force transfer function | |

${G}_{11},{G}_{12},\dots ,{G}_{1\mathrm{N}},{G}_{2\mathrm{N}},\dots ,{G}_{NN}$ | Mechanical model dynamic compliance matrix entries | |

${G}_{I}$ | Current transfer function | |

${G}_{mech}$ | Mechanical part transfer function | |

${\mathit{G}}_{mech}$ | Mechanical model dynamic compliance matrix | |

${G}_{R}$ | Cross table transfer function | |

${\mathit{G}}_{\mathit{R}}$ | General dynamic compliance matrix | |

${G}_{\mathrm{R}11},{G}_{\mathrm{R}12},{G}_{\mathrm{R}21},{G}_{R22}$ | General dynamic compliance matrix entries | |

${G}_{real}$ | Real part of G transfer function | |

${G}_{reg}$ | Feed drive transfer function | |

${G}_{S}$ | Spindle transfer function | |

${G}_{V}$ | Velocity transfer function | |

${G}_{y}$ | Machine and drive transfer function | |

${I}_{in}$ | Current input signal | |

${I}_{out}$ | Current output signal | |

${k}_{1},{k}_{2},\dots {k}_{N},{k}_{y}$ | $\left[\mathrm{N}/\mathrm{m}\right]$ | Stiffness |

${K}_{C}$ | $\left[\mathrm{MPa}\right]$ | Cutting force coefficient |

${K}_{F}$ | $\left[\mathrm{N}/\mathrm{A}\right]$ | Force constant |

${K}_{P}$ | $\left[\mathrm{Ns}/\mathrm{m}\right]$ | Proportional velocity loop gain |

${K}_{PI}$ | $\left[\mathrm{V}/\mathrm{A}\right]$ | Proportional current loop gain |

${K}_{V}$ | $\left[1/\mathrm{s}\right],\left[\mathrm{m}/\mathrm{min}/\mathrm{mm}\right]$ | Proportional position loop gain |

${T}_{d}$ | $\left[\mathrm{s}\right]$ | Time delay |

${T}_{N}$ | $\left[\mathrm{s}\right]$ | Velocity integration constant |

${T}_{NI}$ | $\left[\mathrm{s}\right]$ | Current integration constant |

${u}_{F}$ | $\left[\mathrm{m}\right]$ | Displacement in the force direction |

${\mathit{v}}_{i}$ | Eigenvectors | |

${v}_{in}$ | Velocity input signal | |

${v}_{out}$ | Velocity output signal | |

${x}_{1},{x}_{2},\dots {x}_{N},{x}_{NC}$ | $\left[\mathrm{m}\right]$ | Linear scale |

${y}_{0}$ | $\left[\mathrm{m}\right]$ | Requested chip thickness |

${y}_{1},{y}_{2},\dots {y}_{N},{y}_{NC}$ | $\left[\mathrm{m}\right]$ | Linear scale |

${y}_{N}$ | $\left[\mathrm{m}\right]$ | Real cutting depth |

${\zeta}_{x}$ | $[\u2013]$ | Damping ratio |

${\varsigma}_{qi}$ | Modal damping | |

${\mathsf{\Omega}}_{i}$ | $\left[1/\mathrm{s}\right]$ | Natural frequencies |

$b$ | $\left[\mathrm{m}\right]$ | Chip width |

$\mathit{B}$ | Damping coefficients matrix | |

$\mathit{F}$ | Acting forces vector | |

$F,{F}_{1},\dots ,{F}_{N},{F}_{1ext},{F}_{mot\_y},{F}_{mot\_x},$ | $\left[\mathrm{N}\right]$ | Acting forces |

$FEM$ | Finite Element Method | |

$\mathbf{K}$ | Stiffness matrix | |

$L$ | $\left[\mathrm{H}\right]$ | Inductance |

$\mathbf{M}$ | Mass matrix | |

$m,{m}_{1},{m}_{2},\dots {m}_{N}$ | $\left[\mathrm{kg}\right]$ | Mass |

$R$ | $\left[\mathsf{\Omega}\right]$ | Resistance |

$RPS$ | Rounds per second | |

$s$ | Laplace operator | |

$SLD$ | Stability lobe diagram | |

$x,y,z$ | Axis direction | |

$\mathrm{y}$ | $\left[\mathrm{m}\right]$ | Actual position |

$\mathbf{y}$ | Linear scale vector | |

$\phi $ | [°] | Angle of measurement |

$\omega $ | $\left[1/\mathrm{s}\right]$ | Frequency |

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**Figure 3.**Scheme of machine tool structural parts modeled by ${G}_{S}$ and ${G}_{R}$ dynamic compliance.

**Figure 8.**Model of the direct dynamic (blue shape) and static compliance (magenta shape, as in Figure 7).

**Figure 14.**XY cross table CAD model [19].

**Figure 17.**Experimental measuring on the mid-sized vertical machine X axis (

**a**), and mathematical simulation (

**b**).

Symbol | Value | Unit | |
---|---|---|---|

Mass 1 | ${m}_{1}$ | $250$ | kg |

Mass 2 | ${m}_{2}$ | $350$ | $\mathrm{kg}$ |

Stiffness | ${k}_{x}$ | $1.6\times {10}^{6}$ | $\mathrm{N}/\mathrm{m}$ |

Damping ratio | ${\zeta}_{x}$ | $0.6$ | – |

Symbol | Value | Unit | |
---|---|---|---|

Mass 1 | ${m}_{1}$ | $250$ | $\mathrm{kg}$ |

Mass 2 | m_{2} | $350$ | $\mathrm{kg}$ |

Stiffness | ${k}_{y}$ | $9\times {10}^{6}$ | $\mathrm{N}/\mathrm{m}$ |

Damping ratio | ${\zeta}_{y}$ | $0.21$ | – |

Symbol | Value | Unit | |
---|---|---|---|

Mass | $m$ | $550$ | $\mathrm{kg}$ |

Stiffness one direction | ${k}_{1}$ | $1.875\times {10}^{8}$ | $\mathrm{N}/\mathrm{m}$ |

Stiffness in perpendicular direction | ${k}_{2}$ | $0.12\times {10}^{8}$ | $\mathrm{N}/\mathrm{m}$ |

Damping ratio one direction | ${\zeta}_{1}$ | $0.014$ | $-$ |

Damping ratio in perpendicular direction | ${\zeta}_{2}$ | $0.45$ | $-$ |

K_{v}X, K_{v}Y [(m/min)/mm] | b_{lim} [mm] | Ratio [%] |
---|---|---|

1 | 9.2 | 109.5 |

4 | 8.4 | 100 |

5 | 8.25 | 98.2 |

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**MDPI and ACS Style**

Grau, J.; Souček, P.; Sulitka, M. The Influence of Servo Drive Control on the NC Vertical Milling Machine Dynamic Compliance. *J. Manuf. Mater. Process.* **2020**, *4*, 111.
https://doi.org/10.3390/jmmp4040111

**AMA Style**

Grau J, Souček P, Sulitka M. The Influence of Servo Drive Control on the NC Vertical Milling Machine Dynamic Compliance. *Journal of Manufacturing and Materials Processing*. 2020; 4(4):111.
https://doi.org/10.3390/jmmp4040111

**Chicago/Turabian Style**

Grau, Jan, Pavel Souček, and Matěj Sulitka. 2020. "The Influence of Servo Drive Control on the NC Vertical Milling Machine Dynamic Compliance" *Journal of Manufacturing and Materials Processing* 4, no. 4: 111.
https://doi.org/10.3390/jmmp4040111