1 Introduction
2 Simulation model
2.1 Cutting process dynamics
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Flexibility of the tool and flexibility of the workpiece are considered. The latter especially concerns a large-size flexible workpiece [30].
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Coupling Elements (CEs) are applied for modeling the cutting process dynamic interaction [23, 28]. It is a phenomenological model taking into account the properties of the processed material, as well as the relationship between the instantaneous values of the components of cutting forces and the geometry of the cutting layer. Modeling using CE well reflects the physical nature of the material being processed.
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An effect of first pass of the edge along cutting layer causes proportional feedback, and the effect of multiple passes causes delayed feedback additionally. The latter makes it possible to include the multiple regenerative effect in the calculation model, which is one of the potential causes of loss of tool-workpiece vibrations stability.
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modal subsystem, i.e., a stationary model of the Finite Element Method (FEM) of a flexible workpiece supported by Elastic-Damping Elements (EDE), which moves at the desired feed speed vf. At first, the subsystem is idealized as a set of tetragonal 10-node Finite Elements (FE) [7, 23] and has a large number of degrees of freedom. However, after modal transformation [7, 23], the behavior of this subsystem is described by a vector of its modal coordinates a, whose number is in practice much smaller than the corresponding number of degrees of freedom. Therefore, when we consider a finite number of subsystem normal modes mod, we define its dynamic properties using:
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Ω = diag(ω0i)—matrix of natural angular frequencies of the modal subsystem; i = 1, …, mod. This is also called the stiffness modal matrix;
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\({\varvec{\Psi}} = \left[ {\begin{array}{*{20}c} {{\varvec{\Psi}}_{1} } & \ldots & {{\varvec{\Psi}}_{mod} } \\ \end{array} } \right]\)—matrix of considered mass-normalised normal modes of the modal subsystem; i = 1, …, mod;
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\({\mathbf{Z}} = diag\left( {\zeta_{i} } \right)\)—matrix of dimensionless damping coefficients (also called modal damping) of the modal subsystem; i = 1, …, mod;
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structural subsystem, i.e., a non-stationary discrete model of a rotating milling cutter with a given spindle speed n (i.e., a flexible finite element as Euler–Bernoulli Bar (E-BB) no. e [7, 23] having a local coordinate system xe1, xe2, xe3) and the cutting process (i.e., Coupling Element (CE) no l [23, 28] placed in the momentary position of the “active” cutting edge [23]). The cutting edges are “active” when they come into contact with the workpiece and the others are called “inactive”. The subsystem’s behavior is described by a vector of its generalized coordinates q. The dynamic properties of the decoupled structural subsystem (i.e., the E-BB modeling the tool itself) are determined by the matrices of inertia M, damping L and stiffness K;
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abstractive connecting subsystem as a conventional contact point S between tool and workpiece. Its generalized coordinates are related to other coordinates using time-dependent constraints equations [23, 24]. The latter allows us to eliminate these generalized coordinates from the description of the behavior of the hybrid system.
2.2 Dynamics of flexible details in hybrid system coordinates
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computer software for calculating eigenfrequencies and corresponding normal modes of discretely idealized systems. In practice, high-degree-of-freedom calculation models, created using the Finite Element Method (FEM), are applied;
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Experimental Modal Analysis (EMA) methods. For example, impact modal tests are performed on the workpiece actually installed on the machine tool’s table.
3 Selection of the best spindle speed based on simulation of cutting process and full FEM model of the workpiece
4 Simulation and experimental results
4.1 The workpiece
4.2 Standard parameters
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for surface 1 − n = 1300 rev/min, vf = 600 mm/min, ap = 1 mm;
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for surface 2 − n = 560 rev/min, vf = 1233 mm/min, ap = 1 mm.
4.3 Modal identification and spindle speed selection
Mode no. | EMA identification (ERA) | FEM (PERMAS + MEDINA software) | MAC | ||
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Frequency (Hz) | Modal damping (%) | Frequency (Hz) | Modal damping (%) | ||
1 | 91.2 | 10.19 | 90.3123 | 10.19 | 0.96 |
2 | 111.2 | 5.92 | 110.978 | 5.92 | – |
3 | 151.18 | 2.0 | – | ||
4 | 151.3 | 4.21 | 153.184 | 4.21 | 0.95 |
170.9 | 6.30 | – | |||
5 | 196.85 | 2.02 | – | ||
216.1 | 4.36 | – | |||
6 | 234.5 | 2.05 | 235.861 | 2.05 | 0.98 |
253.3 | 0.84 | – | |||
7 | 266.7 | 1.63 | 263.49 | 1.63 | 0.95 |
8 | 279.4 | 1.62 | 286.333 | 1.62 | 0.82 |
9 | 313.6 | 1.63 | 312.817 | 1.63 | 0.92 |
10 | 307.1 | 1.61 | 331.486 | 1.61 | 0.82 |
342.0 | 1.21 | – | |||
11 | 377.8 | 1.35 | 373.387 | 1.35 | 0.89 |
396.0 | 1.48 | – | |||
12 | 409.794 | 1.48 | – | ||
13 | 426.212 | 1.48 | – | ||
14 | 452.0 | 2.02 | 468.307 | 2.02 | 0.96 |
15 | 497.005 | 0.5 | – |
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RMS95%—RMS of tool-workpiece relative displacements calculated for 95% of the whole cutting time. The remaining 5% consist of transient effects of the tool entrance and exit out of the workpiece;
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Amax—maximum amplitude of tool-workpiece relative displacements calculated for the same period as for RMS95%;
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RMS95%MR—RMS of tool-workpiece relative displacements calculated for 95% of the whole cutting time (similarly to RMS95%), but relatively to the mean value of the considered vibrations (MR—mean related). Thus, this indicator is equivalent to standard deviation of vibrations. This can be interpreted as an indicator of the vibration level after correcting the surface displacement caused by tool deflection in the workpiece. This corresponds better to the way how vibrations are measured during the real milling process, because during acceleration measurements “static” components of the machined surface deflection are omitted.
4.4 Real milling results
Speed selection | ap (mm) | Spindle speed n (rev/min) | Feed speed vf (mm/min) | ||||||||
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A22 | A23 | A32 | A24 | A25 | A22 | A23 | A32 | A24 | A25 | ||
Workpiece, surface 1 | |||||||||||
Standard | 1 | 1300 | 600 | ||||||||
Modal | 1 | 1380 | 637 |
A18 | A19 | A20 | A21 | A18 | A19 | A20 | A21 | ||
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Workpiece, surface 2 | |||||||||
Standard | 1 | 560 | 1233 | ||||||
Modal | 1 | 700 | 800 | 1541 | 1761 |
Milling type | Speed selection | Displacements RMS (mm) | |||||
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A22 | A23 | A32 | A24 | A25 | Average | ||
Workpiece, surface 1 | |||||||
Full | Standard | 0.000128 | 0.000340 | 0.000432 | 0.000390 | 0.000144 | 0.000287 |
Down | Standard | 0.000275 | 0.001376 | 0.001581 | 0.001281 | 0.000261 | 0.000955 |
Full | Modal | 0.000174 | 0.000232 | 0.000320 | 0.000305 | 0.000203 | 0.000247 |
Down | Modal | 0.000298 | 0.000500 | 0.000671 | 0.000584 | 0.000327 | 0.000476 |
A18 | A19 | A20 | A21 | Average | ||
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Workpiece, surface 2 | ||||||
Down | Standard | 0.003524 | 0.002460 | 0.002310 | 0.002556 | 0.002992 0.002433 |
Down | Modal | 0.001913 | 0.001546 | 0.002486 | 0.001708 | 0.001730 0.002097 |
Milling type | Speed selection | Change in RMS values (%) | |||||
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A22 | A23 | A32 | A24 | A25 | Average | ||
Workpiece, surface 1 | |||||||
Full | Standard | – | – | – | – | – | – |
Down | Standard | – | – | – | – | – | – |
Full | Modal | 35.9 | − 31.8 | − 25.9 | − 21.8 | 41.0 | − 13.9 |
Down | Modal | 8.4 | − 63.7 | − 57.6 | − 54.4 | 25.3 | − 50.1 |
A18 | A19 | A20 | A21 | Average | ||
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Workpiece, surface 2 | ||||||
Down | Standard | – | – | – | – | – – |
Down | Modal | − 45.7 | − 37.2 | 7.6 | − 33.2. | − 42.2 − 13.8 |