# Prediction of Tool Point Frequency Response Functions within Machine Tool Work Volume Considering the Position and Feed Direction Dependence

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Background

#### 2.1. Modal Theory and Matrix Transformation

_{r}, ξ

_{r}, M

_{r}and K

_{er}are the rth modal natural frequency, damping ratio, mass and stiffness respectively.

^{rth}

_{max}can be equal to (1 − ξ

_{r}) if ξ

_{r}

^{2}is ignored; the minimum real part value occurs when ${\lambda}^{{r}_{th}}{}_{\mathrm{min}}=\sqrt{1+2{\xi}_{r}}$, and λ

^{rth}

_{min}can be equal to (1 + ξ

_{r}) if ξ

_{r}

^{2}is ignored. Therefore, the vibration frequencies corresponding to the maximum and minimum real part values are ω

_{1}

^{r}= ω

_{r}(1 − ξ

_{r}) and ω

_{2}

^{r}= ω

_{r}(1 + ξ

_{r}) respectively, and then the ξ

_{r}can be obtained by ω

_{1}

^{r}and ω

_{2}

^{r}using the following Equation (4).

^{rth}

_{i}

_{min}= 1, and the corresponding vibration frequency ω

_{r}and amplitude A

^{r}are represented in Equation (5).

_{1}

^{r}, ω

_{2}

^{r}and A

^{r}for each mode are labeled in Figure 2a and used to identify related modal parameters. The identified modal parameters are used to reorganize the measured FRF and well consistence is observed in Figure 2a.

_{MT}and Y

_{MT}) shown in Figure 2b, and the x and y directional tool point FRFs in the machine tool coordinate system are measured to analyze vibrations in related directions. The direct and cross FRFs at the tool point in x and y directions compose the frequency response function matrix

**G**

_{xy}described in Equation (6). However, in real machining process, the feed direction is not always consistent with the x and y axes defined in machine tool coordinate system (X

_{MT}and Y

_{MT}). Therefore, based on the machine tool coordinate system, a rotation angle θ and the feeding directions (u and v) are defined to establish a feed direction coordinate system (U

_{feed}and V

_{feed}) as shown in Figure 2b.

**G**

_{xy}in machine tool coordinate system is modified by a transformation matrix

**R**described in Equation (6) to obtain the frequency response function matrix

**G**

_{uv}in feed direction coordinate system [13,21]. Therefore, if the position-dependent matrix

**G**

_{xy}is measured or simulated, it can be further used to calculate the position and feed direction-dependent matrix

**G**

_{uv}with Equation (6).

#### 2.2. BP Neural Network and PSO Algorithm

_{R}and Λ

_{I}are the real and imaginary parts of the complex eigenvalue Λ, N

_{t}is the tool tooth number, K

_{t}is the tangential cutting force coefficient, a

_{p}is the axial cutting depth, ω

_{c}is the chatter frequency and T is the tooth passing period determined by the spindle speed n. The

**G**

_{0}(iω

_{c}) is the directional matrix can be written as Equation (8).

_{p}

_{lim-feed}and the related spindle speed n can be analytically calculated using Equation (9).

**G**

_{uv}. Since the matrix

**G**

_{uv}is composed by the tool point FRFs, its variations can be represented by the variations of the corresponding FRFs. Considering that the FRFs are organized by the modal parameters as described in the aforementioned Section 2.1, variations of the tool point FRFs can be represented by the variations of modal parameters, and further ascribed to the position and feed direction changes. Therefore, a mathematic model needs to be established to predict the modal parameters at any combination of machining position and feed direction.

#### 2.2.1. BP Neural Network

**X**= [x

_{1}, x

_{2},…,x

_{m}], and its neurons number equals the number of input variables. The output layer contains the output variable vector

**Y**= [y

_{1}, y

_{2},…,y

_{t}], and its neuron number equals the number of output variables. Since this paper aims to obtain the modal parameter at any position and feed direction, the displacements of the moving parts in x, y and z directions and the feed angle θ described in Figure 2b are taken as the inputs of the BP neural network, and the modal parameters are taken as the outputs of the BP neural network. The neurons number in the hidden layer usually determined according to the empirical equation:

_{ik}is the output of the kth neuron in ith layer, f(·) is the excitation function, w

_{kj}is the linking weight between the jth neuron in i − 1th layer and the kth neuron in ith layer, Ψ

_{ik}is the threshold of the kth neuron in ith layer. The linking weight is modified based on the error function descried by the actual and predicted output values in Equation (12).

_{aq}and y

_{pq}are the qth actual and predicted output value respectively.

#### 2.2.2. Particle Swarm Optimization Algorithm

**X**

_{i}= (x

_{i}

_{1}, x

_{i}

_{2},…,x

_{iD}). During the searching process, each current particle position is a candidate solution to the corresponding optimization problem, and the fitness function is determined to judge the historical optimal position of each particle and the historical optimal position of the population. Each current individual optimal position and the global optimal position are further used to determine the corresponding velocity vector of each particle. For the ith particle, the determined velocity vector

**V**

_{i}= (v

_{i}

_{1}, v

_{i}

_{2},…,v

_{i}

_{D}) is used to modify the current particle position to close to the historical optimal one. Then, the new position for each particle is obtained to continue the next iteration. During the iteration, the position and velocity of each particle are repeatedly updated using the following Equation (13). The iteration terminates until the best global fitness value or the total iteration number meets the termination condition.

_{1}and c

_{2}are the acceleration factors, r

_{1}and r

_{2}are the random numbers between 0 and 1.

**Step 1:**Initialize the BP neural network. Define the neuron numbers of the input layer, hidden layer and output layer according to Section 2.2.1, and normalize the input and output sample data between −1 and 1. Other basic parameters should also be determined, such as the learning rate, excitation function, training goal and so on.

**Step 2:**Initialize required parameters of the PSO algorithm. First, the particle dimension D is determined by the total number of the weights and thresholds using Equation (14).

_{input}and N

_{output}are the neuron numbers in the input and output layers, N

_{ht}is the total number of hidden layers, and N

_{hi}is the neuron number of the ith hidden layer. Then, the population size, iteration number, inertia weight, initial positions and velocities, acceleration factors, and rand numbers are initialized. Moreover, the lower and upper boundaries of these positions and velocities are also defined. The error function described in Equation (12) is taken as the fitness function.

**Step 3:**Obtain the optimal initial weights and thresholds of the BP neural network. During each iteration of the PSO algorithm, the individual and global optimal positions are updated and recoded by comparing the so far calculated particle fitness values, and the corresponding individual and global optimal fitness values are also updated and recoded. When the iteration terminates, the global optimal position is taken as the optimal initial weights and thresholds to train the BP neural network according to Section 2.2.1. In addition, the accuracy of the trained BP neural network is verified by the testing sample including the information of the input and output variables.

## 3. Case Study

#### 3.1. Impact Testing Based on Orthogonal Experiment Design

^{4}) is used to determine 64 experiment schemes shown in Table 2.

^{−8}m/N and for feed direction angles θ = 150° and θ = 330 ° are 3.85 × 10

^{−8}m/N; in Figure 6b, when the natural frequency is 636 Hz, the corresponding amplitudes for feed direction angles θ = 60° and θ = 240° are 1.00 × 10

^{−7}m/N and for the feed direction angles θ = 150° and θ = 330° are 2.52 × 10

^{−8}m/N. Thus, effects of the feed direction variations on tool point FRFs can only consider the feed direction angle within the range [0°, 180°].

#### 3.2. BP-PSO Method for Modal Parameters Prediction

## 4. Chatter Stability Analysis Considering Uncertain Position and Feed Direction

_{t}= 4, tangential cutting force coefficient K

_{t}= 1799 MPa and radial cutting force coefficient K

_{r}= 760 MPa.

_{p}

_{lim}_

_{feed}values are plotted in Figure 8.

_{p}

_{lim}_

_{feed}value changing with the feed directions when the worktable, saddle and spindle system are at three positions along x, y and z directions respectively, and origin-symmetric distributions of a

_{p}

_{lim}_

_{feed}values for each position are observed. In Figure 8a, when the feed direction angle varies from 0° to 360° at the three positions, the maximum and minimum a

_{p}

_{lim}_

_{feed}values are 7.20 mm and 6.42 mm respectively, and the variation rate is 12.15%. In Figure 8b, the maximum and minimum a

_{p}

_{lim}_

_{feed}values are 5.66 mm and 5.24 mm respectively, and the variation rate is 8.02%. In Figure 8c, the maximum and minimum a

_{p}

_{lim}_

_{feed}values are 9.08 mm and 6.02 mm respectively, and the variation rate is 50.83%. Comparing three figures, Figure 8a,b show that a

_{p}

_{lim}_

_{feed}values are mainly affected by the feeding directions, and the machining position variations in x and z directions show slight effects on the a

_{p}

_{lim}_

_{feed}values; however, in Figure 8c, not only the feed direction variations but also the machining position variations have significant effects on the a

_{p}

_{lim}_

_{feed}values. Thus, the machining position and feed direction should be considered when designing the process plan.

_{p}

_{lim}_

_{feed}values are shown in Table 2. Thus, the range analysis and variance analysis described as follows were adopted to comprehensively study the effects of the machining position (x, y, z), feed directions θ and spindle speed n on the milling stability.

_{p}

_{lim}_

_{feed}and determine the optimal level of each factor. A factor with higher range value indicates that it shows a greater effect on the a

_{p}

_{lim}_

_{feed}[29].

_{j}is the range value of the jth factor, k

_{ij}is the sum of the a

_{p}

_{lim}_

_{feed}values calculated using the ith level of the jth factor, i = 1, 2, …, m, j = 1, 2, …, n, and m and n are the level and factor numbers of the orthogonal table respectively.

_{p}

_{lim}_

_{feed}, and the parameters needed to perform the F-test were calculated according to Equation (16) [30].

_{p}is the a

_{p}

_{lim}_

_{feed}value calculated using the pth scheme designed in the orthogonal table, N

_{ump}is the total number of designed schemes, SS

_{j}is the sum of squared deviation of the jth factor, df

_{T}is the total degree of freedom, df

_{j}is the degree of freedom for the jth factor, S

_{j}is the variance of the jth factor, and F

_{j}is the F value of the jth factor. The significance level of each factor is determined by comparing the obtained F value with the standard F value (F

_{α}(df

_{j}, df

_{e})). When the significant level α is given, the standard value F

_{α}(df

_{j}, df

_{e}) can be attained from the F-table in accordance with the degree of freedom [30]. If the F value is higher than the standard F

_{α}value, the factor is regarded as significant. And in addition, the factor with a higher F value means that it has more evident impact on the a

_{p}

_{lim}_

_{feed}.

_{ij}value of each factor listed in Table 5, the level with a higher k

_{ij}value is regarded as the optimal level of the related factor. Thus, the 4th, 4th, 1st, 6th, 8th levels of the factor x, y, z, θ and n are determined as the optimal combination to have a bigger a

_{p}

_{lim}_

_{feed}, and specific values of the subscripts are listed in Table 1. For the variance analysis, the calculated SS

_{T}is 1029.10, SS

_{e}is 83.31, df

_{T}is 63, df

_{e}is 28, and other corresponding parameters are listed in Table 5. In this current research, the significant level α is defined as 0.05, and the obtained standard F values F

_{α}are listed in Table 5. Comparing the F and F

_{α}values of each factor listed in Table 5, the F values of the spindle speed n, z directional displacement, feed direction angle θ and x directional displacement are bigger than the F

_{0.05}(7, 28) = 2.359, which indicates that these four factors have significant effects on a

_{p}

_{lim}_

_{feed}with a confidence level of 95%. Further comparing the F values of the five factors, the influence degree of the machining position (x, y, z), feed directions θ and spindle speed n on axial limiting cutting depth a

_{p}

_{lim}_

_{feed}is n > z > θ > x > y. Analyzing the R and F values in Table 5, the spindle speed shows the dominant effects on the a

_{p}

_{lim}_

_{feed}, the displacement in z direction and the feed direction angle θ have similarly significant effects on the a

_{p}

_{lim}_

_{feed}, the displacement in x direction shows less evident influence on the a

_{p}

_{lim}_

_{feed}, and the displacement in y direction has no evident effect on the a

_{p}

_{lim}_

_{feed}. Accordingly, when designing the tool path, an optimal combination of z directional position and feed direction needs to be determined, and the movements in x and y directions should be first taken into consideration.

## 5. Conclusions

- (1)
- The displacements of the worktable, saddle and spindle system along the x, y and z directions represents the machining position, and the feed direction is represented by the rotation angle θ around the machine tool coordinate system. 8 levels of each factor were determined within its variation range, and then an orthogonal table with 64 schemes was designed. For each scheme, the machine tool was driven to the related spatial structure to measure the x and y directional tool point FRFs, and the matrix transformation method was used to calculate the related FRFs in u and v directions.
- (2)
- Three obvious modes for each tool point FRF were observed, and the related modal parameters were identified according to modal theory. Thus, considering the u and v directions, 18 modal parameters should be predicted for one machining position and feed direction combination. To avoid the complex training process and improve the convergence speed, 6 BP neural networks were established for predicting the modal parameters of six modes respectively. Then these BP neural networks were trained with the information of the randomly selected 58 schemes from the Table 2 and the optimized initial linking weights and thresholds using the PSO algorithm. The accuracies of established BP neural networks were verified by the measured modal parameters related to the combinations of machining position and feed direction included in the testing sample.
- (3)
- Moreover, these BP neural networks were used to research the milling stability within the machine tool work volume. Distributions of the axial limiting cutting depth within the feed direction angle range [0°, 360°] at 9 selected machining positions show that the z directional displacement and feed direction θ have significant effects on the limiting axial cutting depth. The same phenomenon was also observed from results of the range analysis and variance analysis performed according to the information in orthogonal Table 2. The optimal level for each factor was determined to obtain an optimal combination of machining position and feed direction with higher limiting axial cutting depth.

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

ω_{r} | rth modal natural frequency (Hz) |

ξ_{r} | rth modal damping ratio (%) |

M_{r} | rth modal mass (kg) |

K_{er} | rth modal stiffness respectively (×10^{9} N/m) |

A^{r} | rth amplitude of the imaginary part of tool point FRF (m/N) |

X_{MT} and Y_{MT} | machine tool coordinate system |

θ | feed direction angle (°) |

U_{feed} and V_{feed} | feed direction coordinate system |

G_{xy} | position and feed direction-dependent matrix |

Λ_{R} | real part of the complex eigenvalue Λ |

Λ_{I} | imaginary part of the complex eigenvalue Λ |

N_{t} | tool tooth number |

K_{t} | tangential cutting force coefficient (MPa) |

K_{r} | radial cutting force coefficient (MPa) |

a_{p} | axial cutting depth (mm) |

ω_{c} | chatter frequency (Hz) |

T | tooth passing period (s) |

n | spindle speed (rpm) |

G_{0}(iω_{c}) | directional matrix |

a_{plim-feed} | limiting axial cutting depth considering the feed direction (mm) |

X | input variable vector of a BP neural network |

Y | output variable vector of a BP neural network |

O_{ik} | output of the kth neuron in ith layer of a BP neural network |

f(·) | excitation function of a BP neural network |

w_{kj} | linking weight between the jth neuron in i − 1th layer and the kth neuron in ith layer |

Ψ_{ik} | threshold of the kth neuron in ith layer |

Δw | linking weight adjustment |

η | learning rate of a BP neural network |

y_{aq} | qth actual output value of a BP neural network |

y_{pq} | qth predicted output value of a BP neural network |

X_{i} | ith position vector of a PSO algorithm |

V_{i} | ith velocity vector of a PSO algorithm |

ω | inertia weight of a PSO algorithm |

c | acceleration factor of a PSO algorithm |

r | random number of a PSO algorithm |

N_{input} | neuron number of the input layer |

N_{output} | neuron number of the output layer |

N_{hi} | neuron number of the ith hidden layer |

R_{j} | range value of the jth factor |

k_{ij} | sum of the a_{p}_{lim}__{feed} values calculated using the ith level of the jth factor |

x_{p} | a_{p}_{lim}__{feed} value calculated using the pth scheme designed in the orthogonal table |

N_{ump} | total number of designed schemes in the orthogonal table |

SS_{j} | sum of squared deviation of the jth factor |

df_{T} | total degree of freedom |

df_{j} | degree of freedom for the jth factor |

S_{j} | variance of the jth factor |

F_{j} | F value of the jth factor |

F_{α}(df_{j}, df_{e}) | standard F value |

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**Figure 2.**(

**a**) Comparisons between the original and fitted FRFs; (

**b**) The machine tool and feed direction coordinate system.

**Figure 3.**(

**a**) A two DOF dynamic model of milling process; (

**b**) Structure of a BP neural network with three layers.

**Figure 6.**(

**a**) Tool point FRFs in different u directions; (

**b**) Tool point FRFs in different v directions.

**Figure 7.**(

**a**) An example tool point FRF in v direction and the identified 3 modes; (

**b**) Comparisons of the predicted and measured FRFs of u and v directions at one position.

**Figure 8.**The feed direction-dependent limiting axial cutting depths for different machining positions. (

**a**) a

_{p}

_{lim}_

_{feed}values for three positions along x direction; (

**b**) a

_{p}

_{lim}_

_{feed}values for three positions along y direction; (

**c**) a

_{p}

_{lim}_

_{feed}values for three positions along z direction.

Factors | Levels | |||||||
---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |

x/mm | 70 | 140 | 210 | 280 | 350 | 420 | 490 | 550 |

y/mm | 50 | 100 | 150 | 200 | 250 | 300 | 350 | 400 |

z/mm | 50 | 100 | 150 | 200 | 250 | 300 | 350 | 400 |

θ/° | 22.5 | 45 | 67.5 | 90 | 112.5 | 135 | 157.5 | 180 |

n/rpm | 2000 | 4000 | 6000 | 8000 | 10,000 | 12,000 | 14,000 | 15,000 |

**Table 2.**64 orthogonal experiment schemes for obtaining the tool point FRFs and analyzing the machining stability.

No. | x | y | z | θ | n | a_{p}_{lim-feed}/mm | No. | x | y | z | θ | n | a_{p}_{lim-feed}/mm |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 7 | 4 | 6 | 3 | 7 | 7.53 | 33 | 1 | 2 | 2 | 2 | 2 | 4.74 |

2 | 3 | 8 | 6 | 4 | 5 | 7.27 | 34 | 5 | 5 | 1 | 2 | 3 | 4.14 |

3 | 7 | 1 | 7 | 2 | 6 | 7.48 | 35 | 3 | 4 | 2 | 8 | 1 | 2.57 |

4 | 6 | 5 | 2 | 4 | 6 | 8.02 | 36 | 8 | 2 | 7 | 3 | 4 | 4.80 |

5 | 3 | 3 | 1 | 7 | 2 | 1.65 | 37 | 6 | 1 | 6 | 8 | 2 | 2.33 |

6 | 4 | 4 | 1 | 6 | 8 | 23.3 | 38 | 2 | 8 | 7 | 6 | 1 | 1.53 |

7 | 5 | 7 | 3 | 4 | 1 | 1.77 | 39 | 4 | 8 | 5 | 2 | 4 | 6.95 |

8 | 5 | 3 | 7 | 8 | 5 | 7.29 | 40 | 1 | 3 | 3 | 3 | 3 | 1.87 |

9 | 4 | 3 | 2 | 5 | 7 | 10.7 | 41 | 1 | 6 | 6 | 6 | 6 | 15.1 |

10 | 2 | 7 | 8 | 5 | 2 | 1.53 | 42 | 2 | 3 | 4 | 1 | 6 | 7.64 |

11 | 4 | 6 | 7 | 4 | 2 | 1.68 | 43 | 8 | 3 | 6 | 2 | 1 | 5.83 |

12 | 2 | 2 | 1 | 4 | 7 | 8.80 | 44 | 8 | 1 | 8 | 4 | 3 | 1.37 |

13 | 4 | 1 | 4 | 7 | 5 | 7.12 | 45 | 3 | 6 | 8 | 2 | 7 | 7.17 |

14 | 6 | 6 | 1 | 3 | 5 | 6.94 | 46 | 1 | 4 | 4 | 4 | 4 | 4.02 |

15 | 8 | 7 | 2 | 6 | 5 | 8.92 | 47 | 7 | 6 | 4 | 5 | 1 | 1.55 |

16 | 3 | 1 | 3 | 5 | 4 | 3.87 | 48 | 8 | 4 | 5 | 1 | 2 | 5.14 |

17 | 8 | 5 | 4 | 8 | 7 | 6.98 | 49 | 5 | 1 | 5 | 6 | 7 | 12.75 |

18 | 4 | 2 | 3 | 8 | 6 | 6.80 | 50 | 4 | 7 | 6 | 1 | 3 | 4.66 |

19 | 7 | 5 | 3 | 6 | 2 | 1.49 | 51 | 2 | 5 | 6 | 7 | 4 | 3.97 |

20 | 3 | 5 | 7 | 1 | 8 | 9.23 | 52 | 7 | 3 | 5 | 4 | 8 | 7.74 |

21 | 6 | 4 | 7 | 5 | 3 | 1.21 | 53 | 3 | 2 | 4 | 6 | 3 | 1.19 |

22 | 2 | 4 | 3 | 2 | 5 | 9.63 | 54 | 6 | 3 | 8 | 6 | 4 | 4.10 |

23 | 5 | 4 | 8 | 7 | 6 | 8.05 | 55 | 6 | 7 | 4 | 2 | 8 | 8.87 |

24 | 8 | 6 | 3 | 7 | 8 | 7.13 | 56 | 7 | 2 | 8 | 1 | 5 | 9.90 |

25 | 5 | 2 | 6 | 5 | 8 | 13.91 | 57 | 1 | 5 | 5 | 5 | 5 | 10.08 |

26 | 7 | 7 | 1 | 8 | 4 | 4.88 | 58 | 5 | 8 | 4 | 3 | 2 | 2.26 |

27 | 4 | 5 | 8 | 3 | 1 | 2.49 | 59 | 1 | 8 | 8 | 8 | 8 | 6.11 |

28 | 2 | 6 | 5 | 8 | 3 | 1.93 | 60 | 2 | 1 | 2 | 3 | 8 | 9.33 |

29 | 8 | 8 | 1 | 5 | 6 | 12.6 | 61 | 6 | 8 | 3 | 1 | 7 | 6.64 |

30 | 6 | 2 | 5 | 7 | 1 | 1.78 | 62 | 3 | 7 | 5 | 3 | 6 | 7.24 |

31 | 5 | 6 | 2 | 1 | 4 | 6.69 | 63 | 1 | 7 | 7 | 7 | 7 | 6.76 |

32 | 1 | 1 | 1 | 1 | 1 | 6.60 | 64 | 7 | 8 | 2 | 7 | 3 | 1.35 |

Mode No. | 1 | 2 | 3 |
---|---|---|---|

Natural frequency ω (Hz) | 384 | 636 | 1428 |

Modal damping ratio ζ (%) | 4.17 | 5.35 | 4.20 |

Modal stiffness K (×10^{9} N/m) | 2.02 | 0.11 | 1.23 |

**Table 4.**Comparisons between the measured and predicted modal parameters for the second mode in u direction for the six testing samples.

Sample No. | Natural Frequency ω (Hz) | Modal Damping Ratio ζ (%) | Modal Stiffness K (×10^{9} N/m) | ||||||
---|---|---|---|---|---|---|---|---|---|

Measured | Predicted | Error (%) | Measured | Predicted | Error (%) | Measured | Predicted | Error (%) | |

1 | 1228 | 1231 | 0.24 | 3.74 | 3.76 | 0.53 | 3.96 | 3.99 | 0.76 |

2 | 1236 | 1232 | 0.32 | 3.56 | 3.65 | 2.52 | 2.62 | 2.54 | 3.05 |

3 | 1232 | 1233 | 0.08 | 3.57 | 3.53 | 1.12 | 1.05 | 1.03 | 1.90 |

4 | 1240 | 1234 | 0.48 | 3.71 | 3.69 | 0.54 | 2.37 | 2.28 | 3.79 |

5 | 1236 | 1233 | 0.24 | 3.72 | 3.64 | 2.15 | 7.07 | 7.08 | 0.14 |

6 | 1220 | 1221 | 0.08 | 3.44 | 3.32 | 3.48 | 0.844 | 0.850 | 0.71 |

x/mm | y/mm | z/mm | θ/° | n/rpm | x/mm | |
---|---|---|---|---|---|---|

Range analysis | K_{1} | 55.23 | 50.84 | 68.84 | 56.49 | 24.12 |

K_{2} | 44.35 | 51.91 | 52.34 | 54.80 | 20.82 | |

K_{3} | 40.20 | 46.84 | 39.20 | 42.46 | 17.71 | |

K_{4} | 63.66 | 61.39 | 39.63 | 40.66 | 39.29 | |

K_{5} | 56.85 | 46.41 | 53.60 | 55.46 | 67.14 | |

K_{6} | 39.88 | 48.14 | 60.54 | 68.29 | 72.88 | |

K_{7} | 41.92 | 44.62 | 39.98 | 37.81 | 67.32 | |

K_{8} | 52.75 | 44.70 | 40.71 | 38.88 | 85.56 | |

Range R | 23.47 | 16.77 | 29.65 | 30.48 | 67.85 | |

Optimal levels | x_{4} | y_{4} | z_{1} | θ_{6} | n_{8} | |

Variance analysis | SS_{j} | 70.10 | 26.77 | 111.52 | 105.29 | 632.10 |

Freedom df_{j} | 7 | 7 | 7 | 7 | 7 | |

Variance S_{j} | 10.01 | 3.82 | 15.93 | 15.04 | 90.30 | |

F value | 3.37 | 1.29 | 5.35 | 5.06 | 30.35 | |

Standard F_{α} value | 2.359 | 2.359 | 2.359 | 2.359 | 2.359 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Deng, C.; Feng, Y.; Shu, J.; Huang, Z.; Tang, Q.
Prediction of Tool Point Frequency Response Functions within Machine Tool Work Volume Considering the Position and Feed Direction Dependence. *Symmetry* **2020**, *12*, 1073.
https://doi.org/10.3390/sym12071073

**AMA Style**

Deng C, Feng Y, Shu J, Huang Z, Tang Q.
Prediction of Tool Point Frequency Response Functions within Machine Tool Work Volume Considering the Position and Feed Direction Dependence. *Symmetry*. 2020; 12(7):1073.
https://doi.org/10.3390/sym12071073

**Chicago/Turabian Style**

Deng, Congying, Yi Feng, Jie Shu, Zhiyu Huang, and Qian Tang.
2020. "Prediction of Tool Point Frequency Response Functions within Machine Tool Work Volume Considering the Position and Feed Direction Dependence" *Symmetry* 12, no. 7: 1073.
https://doi.org/10.3390/sym12071073