# A Consistent Procedure Using Response Surface Methodology to Identify Stiffness Properties of Connections in Machine Tools

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{2}, and the t-statistic, is performed and, as a result, the number of terms in these equations is optimized. Next, using these functions, which temporarily replace the FE model, the updated values of design variables are identified by minimizing the residuals between numerical and experimental responses. In this study, a particular application of the so-called desirability function has been used to accomplish this task. Finally, the identified values are placed in the finite element model and the new dynamic responses are determined.

## 2. Dynamic Characteristics of the Machining Center

#### 2.1. Finite Element Model

#### 2.2. Experimental Modal Analysis

#### 2.3. Comparison between FE and Experimental Modal Data

_{FEA}(FEA—finite element analysis) and φ

_{exp}, as follows:

## 3. Methods: Design of Experiments, Response Surface Methodology, and Desirability Functions

#### 3.1. Two-Level Designs for Parameter Screening

^{k}design, where k is the number of variables and each of them takes an upper and a lower level. A complete trial of such a design needs 2

^{k}runs and allows for estimating the linear effects of the k variables and their interactions.

^{k−p}, where p indicates the fraction chosen (1/2

^{p}). The so-called resolution V design is especially interesting, which provides information about the contribution of variables and two-factor interactions, mixed with higher-order interactions. As these are negligible, the fractional designs are better than the complete factorial designs, because the number of runs diminishes considerably.

#### 3.2. Response Surface Methodology to Develop an Optimal Mathematical Model

_{i}, and a response, y, involved in an engineering problem. That function, preferably a low-order polynomial, is in fact a regression model, less expensive to evaluate, which can be used to predict the response developed in the system under a specific combination of variables.

_{0}is the average value of response; y, β

_{i}, β

_{ii}, and β

_{ij}are the partial regression coefficients; ε is the error term; and k is the number of variables.

^{k}) design, plus 2k axial and n

_{C}center points. The points added to factorial design allow an efficient estimation of the possible curvature of the model.

**y**is obtained on the completion the experiments of the central composite design. Then, the values of these responses and design variables are substituted in Equation (2), and rewritten in matrix form as follows:

**y**=

**X**·

**β**+

**ε**

_{i}. That leads to a least squares estimator of

**β**, as follows:

**b**=

**(X**

^{T}· X)^{−1}· X^{T}· y^{2}, the adjusted R

^{2}, and the predicted R

^{2}[17]. These coefficients are all expected to be close to 1.0, which would mean that the regression model, y

_{RSM}, explains the response, y, properly and that it also predicts adequately new responses.

#### 3.3. Identification of Updated Values of the Design Variables using the Optimum RS Model

_{RSMi}, is turned into a desirability function, d

_{i}, as follows:

_{i}= 0, y

_{RSMi}< y

_{LOWi}and y

_{UPi}< y

_{RSMi}

_{OBJi}is the target experimental response, and y

_{LOWi}and y

_{UPi}are the lower and upper limits for each response (Figure 3).

_{i}, as follows:

_{1}· d

_{2}· … · d

_{u})

^{1/u}

## 4. Case Study

#### 4.1. Initial Selection of Candidate Design Variables

- Stiffness values of the connection elements between main components of the machine tool (Figure 4);
- Stiffness values assigned to the elements that attach the machine tool to the foundation (Figure 5); and
- Stiffness value along X direction of the connection element, between the primary and secondary sections of the linear motor.

^{10−3}fractional design with 128 runs) [14]. However, it is still too laborious to manage such a number of runs. Furthermore, some parameter combinations could lead to inappropriate model responses, due to the presence of the constraints among them. Thus, in order to facilitate the analysis and gain a progressive comprehension of the significance of each design variable and interaction, instead of a single 2

^{10−3}fractional design, an alternative trial with seven 2

^{5−1}designs (16 runs each) has been performed (Table 5). Each variable has been paired up with the rest at least once, so that after the completion of the whole set of 2

^{5−1}experiments, it has been possible to look into the effects of all of the design variables and two-factor interactions by means of ANOVA.

_{T}(Figure 6), and the sum of squares of each factor and the two-factor interactions, mixed with higher interactions (SS

_{i}and SS

_{ij}) (Figure 7 and Figure 8), have been obtained as a measure of the variability, for all of the frequencies and MAC values. Firstly, for each response, the SS

_{T}has been examined, as some designs add much more variability than others, because of the variables involved. Thus, Figure 6 shows that MAC

_{1}and MAC

_{2}are not affected by the changes in the design variables, and that the variability of f

_{FEA2}is negligible. As this frequency matches its experimental pair (Table 4), it has been omitted in later analyses.

_{i}and SS

_{ij}have been gradually analyzed in each 2

^{5−1}design. As an example, Figure 7 and Figure 8 illustrate the percentage contribution of each parameter in design number 4.

_{Y}

_{3}) dominates the 1st and 2nd natural frequencies, design variable 4 (k

_{X}

_{8}) has a huge influence on the 6th natural frequency and an important weight on the 3rd and 4th natural frequencies, and design variable 5 (k

_{Y}

_{9}) governs the 5th natural frequency. However, only parameter 8 (interaction k

_{X}

_{210}–k

_{X}

_{8}) slightly affects the 3rd and 4th natural frequencies. Also, design variable 3 (k

_{Z}

_{4}) does not seem to have a notable influence on any natural frequency.

_{4}is heavily affected by parameter 8 (interaction k

_{X}

_{210}–k

_{X}

_{8}), which also provides an important contribution to the variance of MAC

_{3}. Also, parameter 11 (interaction k

_{Y}

_{3}–k

_{X}

_{8}) causes approximately 35% of the total variability to MAC

_{6}. Nevertheless, in general, the individual design variables have a greater influence than the interactions.

^{5–1}design, so that it has been possible to gradually gain a better insight into the influence of the design variables and two-factor interactions on the responses. Finally, those factors providing more than 99% of the total variability for the frequencies, and 97.5% for the MAC values have been selected (Table 6 and Table 7) to continue with the improvement procedure.

- The fractional factorial experiments have allowed for finding out the design variables and interactions that affect the responses. Therefore, the screening experiment has satisfactorily achieved the initial goal.
- Two design variables do not influence the natural frequencies, namely the stiffness k
_{X}_{11}and horizontal stiffness k_{X}_{21}of the connections to the foundation. - Three design variables are only significant for one natural frequency (f
_{FEA5}), namely the transverse stiffness k_{Z}_{63}between the bed frame and foundations, and two stiffnesses between the modules of the machine tool, k_{Z}_{13}and k_{Y}_{9}. - MAC
_{5}and MAC_{6}are affected by the largest number of interactions. Furthermore, some of them include design variables that do not influence them individually, for example, interaction k_{X}_{11}-k_{X}_{21}. This situation only appears in these two responses. In addition, the total number of design variables, considered both individually and in interactions, which affect each of these responses is nine (i.e., almost all). Nevertheless, along the complete set of fractional designs, the MAC_{5}values were always larger than 80% and the MAC_{6}values ranged from 68% to 73%. Thus, it has been decided not to carry on with the study of these responses, because the number of involved variables would lead to a costly analysis in the next step, while the benefits would be quite poor. - The natural frequencies f
_{FEA3}and f_{FEA4}and the corresponding MAC values depend on the same group of design variables, k_{X}_{210}, k_{Y}_{3}, k_{Z}_{4}, and k_{X}_{8}. In addition, the natural frequency f_{FEA6}is dependent on three of these variables, k_{Y}_{3}, k_{Z}_{4}, and k_{X}_{8}. Therefore, in the next step of the improvement process, these three natural frequencies will be analyzed together, so as to reduce the number of experiments necessary to define their meta-models. In addition, it is interesting to note that the mode shapes associated to these frequencies take place in plane XZ. - The natural frequency f
_{FEA5}is affected by four variables that do not have any influence on the frequencies f_{FEA3}, f_{FEA4}, and f_{FEA6}, and, vice versa, the variables that affect these three frequencies do not provide any variability to the natural frequency f_{FEA5}. Moreover, some of the design variables representing stiffness in the X direction, k_{X}_{8}and k_{X}_{210}, do not affect the 1st and 5th mode shapes, whose principal movement is in plane YZ. Thus, it is concluded that the design variables are working collectively.

#### 4.2. Development of Explicit Relationships between Design Variables and Responses

- Central composite (CC) design 1: including f
_{FEA1}and design variables k_{Y}_{22}, k_{Y}_{3}, and k_{Z}_{4}. Although it would seem unnecessary to search for this relationship, because f_{FEA1}is already matched, as it is influenced by the design variables that also influence other frequencies, any change on them would affect this frequency too. So, it is indispensable to know this relationship. - CC design 2: with the following responses f
_{FEA3}, f_{FEA4}, f_{FEA6}, MAC_{3}, and MAC_{4}, and design variables k_{X}_{210}, k_{Y}_{3}, k_{Z}_{4}, and k_{X}_{8}. - CC design 3: including f
_{FEA5}and design variables k_{Y}_{22}, k_{Z}_{63}, k_{Z}_{13}, and k_{Y}_{9}.

^{k}points from the factorial design with k factors; 2k axial points face centered, where one variable takes the upper and lower limits and the others have mean values; and finally one central point. Thus, a total number of 65 experiments (15, 25, and 25, respectively) have been completed. Also, prior conducting the experiments, the design variables must be normalized to values (−1), (0), and (+1), which stand for the lower bound, mean value, and upper bound of each variable, respectively (Equation (9)).

_{UPi}and k

_{LOWi}are the upper and lower limits defined in Table 5.

_{FEA1}and the variables that affect it. Using the results obtained from the central composite design, an initial second-order model with all of the design variables and interactions have been developed, namely Equation (10), as follows:

_{RSM1}= 34.9265 + 1.1893 · X

_{Y}

_{3}+ 1.3810 · X

_{Y}

_{22}+ 0.1806 · X

_{Z}

_{4}+ 0.1531 · X

_{Y}

_{3}· X

_{Y}

_{22}+

+ 0.0065 · X

_{Y}

_{3}· X

_{Z}

_{4}+ 0.0177 · X

_{Y}

_{22}· X

_{Z}

_{4}− 0.4593 · (X

_{Y}

_{3})

^{2}− 0.5171 · (X

_{Y}

_{22})

^{2}− 0.0559 · (X

_{Z}

_{4})

^{2}

_{RSM1}is the estimated response corresponding to the first natural frequency, f

_{FEA1}.

^{2}and t-tests (Table 8 and Table 9).

^{2}coefficients for the initial model show that the regression function explains the observed responses in the central composite design experiment quite well. Also,

_{pred}R

^{2}suggests that the model will fit new responses remarkably.

_{j}of the initial model have been calculated (Table 9). Using a 95% confidence level (α = 0.05), these terms must be larger than the value of the t-distribution t

_{0.025,5}= 2.571, and it is shown that the corresponding t-statistics for b

_{13}, b

_{23}, and b

_{33}are smaller. Thus, these three terms are non-significant in the regression model and can be removed. As it is convenient to eliminate one term in each step, b

_{13}has been picked out first, as its t-statistic was the smallest one.

_{13}. The results in Table 8 (model 2) show that both

_{adj}R

^{2}and

_{pred}R

^{2}have increased slightly. Therefore, as expected, removing the non-significant terms in the regression model has led to a more adequate model. Nevertheless, in the regression equation still there are non-significant terms (Table 9), as some t-statistics are smaller than t

_{0.025,6}= 2.447. So, coefficient b

_{23}has been removed, and a new model (model 3) has been made. In this case,

_{adj}R

^{2}and

_{pred}R

^{2}have reduced slightly. Although the differences are totally negligible, considering that this regression model would lead to poorer results than the previous one, the iteration process has been stopped and the preceding regression model has been selected. In Table 10, the results of analysis of variance (ANOVA) of the final model are summarized.

_{RSM1}= 34.9265 + 1.1893 · X

_{Y}

_{3}+ 1.3810 · X

_{Y}

_{22}+ 0.1806 · X

_{Z}

_{4}+ 0.1531 · X

_{Y}

_{3}· X

_{Y}

_{22}+

+ 0.0177 · X

_{Y}

_{22}· X

_{Z}

_{4}− 0.4593 · (X

_{Y}

_{3})

^{2}− 0.5171 · (X

_{Y}

_{22})

^{2}− 0.0559 · (X

_{Z}

_{4})

^{2}

_{j}element has been removed and the optimum model is very similar to the initial one. As it will be shown later, in some regression equations, more b

_{j}elements will be eliminated, mainly those referred to in second-order terms (see, for example, Equation (17)).

_{FEA5}. In this case, the regression equation is as follows:

_{RSM5}= 87.4450 + 0.7551 · X

_{Y}

_{9}+ 0.5414 · X

_{Y}

_{22}+ 2.3563 · X

_{Z}

_{13}+ 0.7345 · X

_{Z}

_{63}+

+ 0.0618 · X

_{Y}

_{9}· X

_{Y}

_{22}+ 0.1096 · X

_{Y}

_{9}· X

_{Z}

_{63}+ 0.0785 · X

_{Y}

_{22}· X

_{Z}

_{63}+ 0.0725 · X

_{Z}

_{13}· X

_{Z}

_{63}−

− 0.2318 · (X

_{Y}

_{9})

^{2}− 0.1674 · (X

_{Y}

_{22})

^{2}− 0.8645 · (X

_{Z}

_{13})

^{2}− 0.2321 · (X

_{Z}

_{63})

^{2}

^{2}= 0.9989,

_{adj}R

^{2}= 0.9977, and

_{pred}R

^{2}= 0.9984 (Table 11), have led again to a reliable model.

_{FEA3}, f

_{FEA4}, and f

_{FEA6}, along with MAC

_{3}and MAC

_{4}, have been analyzed altogether, because the variables that affect them were the same. The final regression equations are shown in Equations (13)–(17), and the coefficients of determination in Table 11.

_{RSM3}= 71.1723 + 2.3373 · X

_{X}

_{8}+ 2.2295 · X

_{X}

_{210}+ 0.4979 · X

_{Y}

_{3}+ 0.4722 · X

_{Z}

_{4}+

+ 1.3206 · X

_{X}

_{8}· X

_{X}

_{210}+ 0.2019 · X

_{X}

_{8}· X

_{Y}

_{3}− 0.1702 · X

_{X}

_{210}· X

_{Y}

_{3}+ 0.1562 · X

_{X}

_{210}· X

_{Z}

_{4}−

− 2.1869 · (X

_{X}

_{8})

^{2}− 0.6804 · (X

_{X}

_{210})

^{2}

_{RSM4}= 76.7443 + 2.6736 · X

_{X}

_{8}+ 1.7965 · X

_{X}

_{210}+ 0.6450 · X

_{Y}

_{3}+ 0.5125 · X

_{Z}

_{4}−

− 1.3133 · X

_{X}

_{8}· X

_{X}

_{210}+ 0.3977 · X

_{X}

_{8}· X

_{Z}

_{4}−0.1550 · X

_{X}

_{210}· X

_{Z}

_{4}− 1.2361 · (X

_{X}

_{8})

^{2}

_{RSM6}= 118.5952 + 8.1498 · X

_{X}

_{8}+ 0.3207 · X

_{Y}

_{3}+ 1.0787 · X

_{Z}

_{4}− 0.1247 · X

_{X}

_{8}· X

_{Y}

_{3}−

− 0.2232 · X

_{X}

_{8}· X

_{Z}

_{4}+ 0.1697 · X

_{Y}

_{3}· X

_{Z}

_{4}− 3.2813 · (X

_{X}

_{8})

^{2}− 0.2193 · (X

_{Y}

_{3})

^{2}− 0.3628 · (X

_{Z}

_{4})

^{2}

_{RSM3}= 63.4984 − 9.4610 · X

_{X}

_{8}+ 8.9519 · X

_{X}

_{210}+ 2.1027 · X

_{Y}

_{3}− 3.8195 · X

_{Z}

_{4}+

+ 10.4715 · X

_{X}

_{8}· X

_{X}

_{210}+ 1.1365 · X

_{X}

_{8}· X

_{Y}

_{3}− 2.6332 · X

_{X}

_{8}· X

_{Z}

_{4}+ 3.0198 · (X

_{Z}

_{4})

^{2}

_{RSM4}= 84.2937 + 5.9049 · X

_{X}

_{8}− 5.2622 · X

_{X}

_{210}− 1.0676 · X

_{Z}

_{4}+ 12.2980 · X

_{X}

_{8}· X

_{X}

_{210}−

− 9.2492 · (X

_{X}

_{8})

^{2}

_{RSM1}, f

_{RSM5}, and f

_{RSM6}are very close to 1.0, while the coefficients for f

_{RSM3}and f

_{RSM4}are slightly lower, although greater than 0.974, and all of them are similar or better than those attained by the authors of [22,23,24]. On the other hand, the coefficients of determination for MAC

_{RSM3}and MAC

_{RSM4}are lower, in some cases under 0.9. In this case, it is not possible to compare them to others, because, to the best of our knowledge, in the literature, there are no results using RSM to simulate MAC responses. Nevertheless, those values are also superior to the coefficients obtained by the authors of [22,23,24] for other responses. So, in conclusion, the approximate functions in Equations (11)–(17) were judged as good enough to accurately relate the design variables and responses, and are adequate to use in the subsequent phase of the improvement procedure.

#### 4.3. Determination of Updated Values of Design Variables

_{expi}− 1 Hz) < f

_{FEAi}< (f

_{expi}+ 1 Hz), and where MAC values higher than the initial ones have been bolded.

- The natural frequency f
_{FEA3}approximately matches its experimental pair and, at the same time, the corresponding MAC value is higher than the initial one, only when the design variable k_{X}_{8}is at its lower boundary. If k_{X}_{8}takes the central or upper values, it is not possible to adequately accomplish the pairing. - Also, the natural frequency, f
_{FEA6}, needs lower k_{X}_{8}values to match its experimental pair. - However, on the other side, at lower k
_{X}_{8}values, it is not viable to adjust the natural frequency f_{FEA4}while maintaining accurate values of MAC. Intermediate or upper values of k_{X}_{8}are necessary to improve f_{FEA4}, although they give rise to MAC values slightly poorer than initially.

_{X}

_{8}. In fact, Wu et al. [36] have also addressed a similar behavior in other machine tool with roller type linear guideways. Therefore, it will be necessary to identify one k

_{X}

_{8}value to match, in combination with k

_{X}

_{210}, k

_{Y}

_{3}, and k

_{Z}

_{4}; natural frequencies f

_{FEA3}, f

_{FEA6}; and necessarily MAC

_{3}, as well as other k

_{X}

_{8}value to match f

_{FEA4}and MAC

_{4}, taking into account that design variable k

_{X}

_{8}does not affect the rest of the responses (Table 6).

_{i}= 1 − 20 · ABS(f

_{RSMi}− f

_{expi}), (f

_{expi}− 0.05 Hz) < f

_{RSMi}< (f

_{expi}+ 0.05 Hz)

_{i}= 0, f

_{RSMi}< (f

_{expi}− 0.05 Hz), f

_{RSMi}> (f

_{expi}+ 0.05 Hz)

_{Mi}= 0, MAC

_{RSMi}< MAC

_{0}

_{0}has been selected taking into consideration Table 4 and Table 12.

_{f}, and another function for the MAC values, D

_{M}, and applying weighting coefficients w

_{f}and w

_{M}to each of them.

_{f}· D

_{f}+ w

_{M}· D

_{M}= w

_{f}· (d

_{1}· d

_{3}· d

_{4}· d

_{5}· d

_{6})

^{1/5}+ w

_{M}· (d

_{M}

_{3}· d

_{M}

_{4})

^{1/2}

_{i}, and the corresponding natural values k

_{i}. As mentioned before, two stiffness values k

_{X}

_{8}have been estimated, namely: (1) is valid for the frequency ranges from 0 Hz to 72 Hz, and from 100 Hz to the upper limit of the range of interest, and (2) is adequate for the remaining range, which includes the 4th natural frequency.

_{RSMi}and MAC

_{RSMi}, obtained in Equations (11)–(17), when the updated values of the design variables are substituted, are also shown.

^{2}and t-statistic, have performed adequately.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 6.**Total corrected sum of squares (SS

_{T}) for natural frequencies (

**a**) and modal assurance criterion (MAC) values (

**b**).

**Figure 7.**Percentage contribution for natural frequencies in design number 4: 1–5, design variables k

_{X}

_{210}, k

_{Y}

_{3}, k

_{Z}

_{4}, k

_{X}

_{8}, and k

_{Y}

_{9}, respectively; 6–15, two factor interactions k

_{X}

_{21}–k

_{Y}

_{3}, k

_{X}

_{210}–k

_{Z}

_{4}, k

_{X}

_{210}–k

_{X}

_{8}, k

_{X}

_{210}–k

_{Y}

_{9}, k

_{Y}

_{3}–k

_{Z}

_{4}, k

_{Y}

_{3}–k

_{X}

_{8}, k

_{Y}

_{3}–k

_{Y}

_{9}, k

_{Z}

_{4}–k

_{X}

_{8}, k

_{Z}

_{4}–k

_{Y}

_{9}, and k

_{X}

_{8}–k

_{Y}

_{9}, respectively.

**Figure 8.**Percentage contribution for MAC values in design number 4: 1–5, design variables k

_{X}

_{210}, k

_{Y}

_{3}, k

_{Z}

_{4}, k

_{X}

_{8}, and k

_{Y}

_{9}, respectively; 6–15, two factor interactions k

_{X}

_{210}–k

_{Y}

_{3}, k

_{X}

_{210}–k

_{Z}

_{4}, k

_{X}

_{210}–k

_{X}

_{8}, k

_{X}

_{210}–k

_{Y}

_{9}, k

_{Y}

_{3}–k

_{Z}

_{4}, k

_{Y}

_{3}–k

_{X}

_{8}, k

_{Y}

_{3}–k

_{Y}

_{9}, k

_{Z}

_{4}–k

_{X}

_{8}, k

_{Z}

_{4}–k

_{Y}

_{9}, and k

_{X}

_{8}–k

_{Y}

_{9}, respectively.

**Figure 10.**Test frequency response functions (FRFs) and synthetized FE FRF in point 5. (

**a**) Response X, impact X; (

**b**) response Y, impact Y.

Parameter | Value(s) | Description |
---|---|---|

Stiffness X,Y,Z | 750,750,750 N/μm | Connections foundation-bed frame. |

Stiffness X,Y,Z | 1,720,750 N/μm | Connections bed frame-column (guideway). |

Stiffness X,Y,Z | 720,1,750 N/μm | Connections column-framework (guideway). |

Stiffness X,Y,Z | 560,750,1 N/μm | Connections framework-ram (guideway). |

Stiffness X | 110 N/μm | Connection between primary and secondary sections of the linear motor. |

Lumped mass | 120 kg | Spindle. |

Lumped mass | 1.5 kg | Face milling cutter. |

Stiffness Y | 176.7 N/μm | Y ball-screw. |

Lumped mass | 100 kg | Servo motor Y. |

Stiffness Z | 172.7 N/μm | Z ball-screw. |

Lumped mass | 100 kg | Servo motor Z. |

E, ρ | 125 GPa, 7100 kg/m^{3} | Young’s modulus (E) and mass density (ρ) of the bed frame and column (cast iron). |

E, ρ | 175 GPa, 7100 kg/m^{3} | Young’s modulus (E) and mass density (ρ) of the framework and ram (cast iron GGG70). |

E, ρ | 210 GPa, 7850 kg/m^{3} | Young’s modulus (E) and mass density (ρ) of specific parts of the machine tool. |

f_{FEA1} | f_{FEA2} | f_{FEA3} | f_{FEA4} | f_{FEA5} | f_{FEA6} |
---|---|---|---|---|---|

33.7 | 60.4 | 69.7 | 73.9 | 87.5 | 112.3 |

Mode Order | f_{exp} (Hz) | Damping Ratio (%) | Description of the Mode Shape |
---|---|---|---|

1 | 33.7 | 4.8 | Rotation of the whole structure around the X-axis. |

2 | 60.5 | 3.3 | Translation along Y of the framework and ram. |

3 | 65.9 | 3.5 | Rotation of the upper part of the machine around the Y-axis. |

4 | 77.2 | 5.4 | Rotation of the upper part of the machine around the Y-axis, but now ram is in counter-phase. |

5 | 84.0 | 5.1 | Rotation of framework and ram around the X-axis. |

6 | 106.5 | 3.3 | Rotation of the whole structure around the Y-axis. Ram is in counter-phase. |

FEA Order | f_{FEA} (Hz) | f_{exp1} 33.7 | f_{exp2} 60.5 | f_{exp3} 65.9 | f_{exp4} 77.2 | f_{exp5} 84.0 | f_{exp6} 106.5 | Diff. (Hz) | Diff. (%) | Pair Number |
---|---|---|---|---|---|---|---|---|---|---|

1 | 33.7 | 96.6 | 0.6 | 0.3 | 0.0 | 1.9 | 0.1 | 0.0 | 0.0 | 1 |

2 | 60.4 | 1.7 | 98.8 | 1.2 | 0.0 | 1.3 | 0.1 | −0.1 | −0.2 | 2 |

3 | 69.7 | 0.0 | 0.0 | 76.3 | 4.3 | 0.0 | 1.5 | 3.8 | 5.8 | 3 |

4 | 73.9 | 0.1 | 0.0 | 30.3 | 89.0 | 1.3 | 3.7 | −3.3 | −4.3 | 4 |

5 | 87.5 | 1.9 | 0.1 | 1.2 | 0.9 | 91.0 | 0.1 | 3.5 | 4.2 | 5 |

6 | 112.3 | 0.0 | 1.0 | 0.1 | 0.0 | 0.1 | 70.3 | 5.8 | 5.4 | 6 |

Connection | Design Variable | Code | Lower Bound | Nominal Value | Upper Bound | 2^{5–1} Design |
---|---|---|---|---|---|---|

Foundation—bed frame | Stiffness X | k_{X}_{21} | 600 | 750 | 1050 | 1,3,5,6 |

Foundation—bed frame | Stiffness Y | k_{Y}_{22} | 600 | 750 | 1500 | 1,3,5,6 |

Foundation—bed frame | Stiffness Z | k_{Z}_{63} | 600 | 750 | 1050 | 1,3,5,6 |

Linear motor (inner) | Stiffness X | k_{X}_{210} | 80 | 110 | 160 | 2,4,6 |

Bed frame—column | Stiffness Y | k_{Y}_{3} | 450 | 720 | 1125 | 2,4,5,7 |

Bed frame—column | Stiffness Z | k_{Z}_{4} | 400 | 750 | 900 | 2,4,5,6 |

Column—framework | Stiffness X | k_{X}_{11} | 450 | 720 | 1125 | 2,3,7 |

Column—framework | Stiffness Z | k_{Z}_{13} | 400 | 750 | 900 | 2,3,7 |

Framework—ram | Stiffness X | k_{X}_{8} | 210 | 560 | 900 | 1,4,7 |

Framework—ram | Stiffness Y | k_{Y}_{9} | 450 | 750 | 1000 | 1,4,7 |

Connection | Design Variable | Code | f_{FEA1} | f_{FEA3} | f_{FEA4} | f_{FEA5} | f_{FEA6} | MAC_{3} | MAC_{4} | MAC_{5} | MAC_{6} |
---|---|---|---|---|---|---|---|---|---|---|---|

Foundation—bed frame | Stiffness X | k_{X}_{21} | |||||||||

Foundation—bed frame | Stiffness Y | k_{Y}_{22} | X | X | X | X | |||||

Foundation—bed frame | Stiffness Z | k_{Z}_{63} | X | X | X | ||||||

Linear motor (inner) | Stiffness X | k_{X}_{210} | X | X | X | X | X | ||||

Bed frame—column | Stiffness Y | k_{Y}_{3} | X | X | X | X | X | X | X | X | |

Bed frame—column | Stiffness Z | k_{Z}_{4} | X | X | X | X | X | X | X | ||

Column—framework | Stiffness X | k_{X}_{11} | |||||||||

Column—framework | Stiffness Z | k_{Z}_{13} | X | X | X | ||||||

Framework—ram | Stiffness X | k_{X}_{8} | X | X | X | X | X | X | |||

Framework—ram | Stiffness Y | k_{Y}_{9} | X | X | X |

Responses | Interactions |
---|---|

f_{FEA1} | k_{Y}_{22}–k_{Y}_{3} |

f_{FEA3} | k_{X}_{210}–k_{Y}_{3}, k_{X}_{210}–k_{Z}_{4}, k_{X}_{210}–k_{X}_{8}, k_{Y}_{3}–k_{X}_{8} |

f_{FEA4} | k_{X}_{210}–k_{Z}_{4}, k_{X}_{210}–k_{X}_{8}, k_{Z}_{4}–k_{X}_{8} |

f_{FEA5} | k_{Y}_{22}–k_{Z}_{63}, k_{Y}_{22}–k_{Y}_{9}, k_{Z}_{63}–k_{Z}_{13}, k_{Z}_{63}–k_{Y}_{9} |

f_{FEA6} | k_{Y}_{3}–k_{Z}_{4}, k_{Y}_{3}–k_{X}_{8}, k_{Z}_{4}–k_{X}_{8} |

MAC_{3} | k_{X}_{210}–k_{Y}_{3}, k_{X}_{210}–k_{Z}_{4}, k_{X}_{210}–k_{X}_{8}, k_{Y}_{3}–k_{Z}_{4}, k_{Y}_{3}–k_{X}_{8}, k_{Z}_{4}–k_{X}_{8} |

MAC_{4} | k_{X}_{210}–k_{Y}_{3}, k_{X}_{210}–k_{Z}_{4}, k_{X}_{210}–k_{X}_{8}, k_{Y}_{3}–k_{X}_{8} |

MAC_{5} | k_{X}_{21}–k_{X}_{11}, k_{X}_{21}–k_{X}_{8}, k_{Y}_{22}–k_{Z}_{63}, k_{Y}_{22}–k_{Z}_{13}, k_{Y}_{22}–k_{Y}_{9}, k_{Z}_{63}–k_{Z}_{13}, k_{Z}_{63}–k_{Y}_{9}, k_{Y}_{3}–k_{Y}_{9}, k_{Z}_{13}–k_{Y}_{9} |

MAC_{6} | k_{Y}_{22}–k_{Z}_{63}, k_{Y}_{22}–k_{Z}_{4}, k_{X}_{210}–k_{Y}_{3}, k_{X}_{210}–k_{X}_{8}, k_{Y}_{3}–k_{Z}_{13}, k_{Y}_{3}–k_{X}_{8}, k_{Z}_{4}–k_{Y}_{9}, k_{X}_{11}–k_{Y}_{9}, k_{Z}_{13}–k_{X}_{8} |

Coef. | Initial Model | Model 2 | Model 3 |
---|---|---|---|

R^{2} | 0.9997 | 0.9997 | 0.9996 |

_{adj}R^{2} | 0.9984 | 0.9986 | 0.9985 |

_{pred}R^{2} | 0.9974 | 0.9982 | 0.9981 |

Term | Coef. | Initial Model | Model 2 | Model 3 |
---|---|---|---|---|

Constant | b_{0} | 1410.7606 | 1521.1830 | 1482.9372 |

k_{Y}_{3} | b_{1} | 81.6470 | 88.0377 | 85.8242 |

k_{Y}_{22} | b_{2} | 94.8073 | 102.2280 | 99.6577 |

k_{Z}_{4} | b_{3} | 12.3961 | 13.3664 | 13.0303 |

k_{Y}_{3}–k_{Y}_{22} | b_{12} | 9.4043 | 10.1404 | 9.8854 |

k_{Y}_{3}–k_{Z}_{4} | b_{13} | 0.4007 | - | - |

k_{Y}_{22}–k_{Z}_{4} | b_{23} | 1.0838 | 1.1686 | - |

(k_{Y}_{3})^{2} | b_{11} | −5.9900 | −17.2415 | −16.8080 |

(k_{Y}_{22})^{2} | b_{22} | −18.0039 | −19.4131 | −18.9250 |

(k_{Z}_{4})^{2} | b_{33} | −1.9461 | −2.0984 | −2.0457 |

Source | Sum of Squares | Degree of Freedom | Mean Squares | F Value | p Value |
---|---|---|---|---|---|

Regression | 36.114 | 8 | 4.514 | 2473.8 | 0.000 |

Residual | 0.011 | 6 | 0.002 | - | - |

Total | 36.125 | 14 | 2.580 | - | - |

Coefficients of Determination | f_{RSM1} | f_{RSM3} | f_{RSM4} | f_{RSM5} | f_{RSM6} | MAC_{RSM3} | MAC_{RSM4} |
---|---|---|---|---|---|---|---|

R^{2} | 0.9997 | 0.9910 | 0.9870 | 0.9989 | 0.9995 | 0.9304 | 0.9271 |

_{adj}R^{2} | 0.9986 | 0.9845 | 0.9805 | 0.9977 | 0.9993 | 0.8956 | 0.9080 |

_{pred}R^{2} | 0.9982 | 0.9746 | 0.9750 | 0.9948 | 0.9986 | 0.8692 | 0.8816 |

Run | f_{FEA3} | MAC_{3} | f_{FEA4} | MAC_{4} | f_{FEA6} | k_{X}_{8} | k_{X}_{210} | k_{Y}_{3} | k_{Z}_{4} |
---|---|---|---|---|---|---|---|---|---|

1 | 75.2 | 71.8 | 80.0 | 86.8 | 124.3 | 900 | 160 | 1125 | 900 |

2 | 73.8 | 84.0 | 78.8 | 88.1 | 121.9 | 900 | 160 | 1125 | 400 |

3 | 74.2 | 64.4 | 78.8 | 83.6 | 123.3 | 900 | 160 | 450 | 900 |

4 | 73.0 | 81.3 | 77.4 | 87.6 | 122.1 | 900 | 160 | 450 | 400 |

5 | 68.3 | 37.5 | 79.1 | 75.0 | 124.2 | 900 | 80 | 1125 | 900 |

6 | 67.9 | 47.6 | 76.9 | 78.7 | 121.8 | 900 | 80 | 1125 | 400 |

7 | 66.4 | 32.0 | 78.2 | 71.9 | 123.2 | 900 | 80 | 450 | 900 |

8 | 66.1 | 40.0 | 75.9 | 74.9 | 122.0 | 900 | 80 | 450 | 400 |

9 | 67.5 | 73.0 | 76.8 | 50.2 | 108.5 | 210 | 160 | 1125 | 900 |

10 | 66.3 | 72.8 | 76.8 | 51.3 | 105.7 | 210 | 160 | 1125 | 400 |

11 | 67.2 | 72.8 | 75.3 | 49.2 | 107.5 | 210 | 160 | 450 | 900 |

12 | 66.0 | 72.5 | 75.3 | 50.2 | 104.9 | 210 | 160 | 450 | 400 |

13 | 66.1 | 79.1 | 70.5 | 88.3 | 108.3 | 210 | 80 | 1125 | 900 |

14 | 65.0 | 80.3 | 70.3 | 86.4 | 105.5 | 210 | 80 | 1125 | 400 |

15 | 65.2 | 75.2 | 69.0 | 88.1 | 107.3 | 210 | 80 | 450 | 900 |

16 | 64.3 | 79.8 | 68.6 | 89.3 | 104.7 | 210 | 80 | 450 | 400 |

17 | 66.7 | 76.4 | 73.0 | 67.7 | 107.2 | 210 | 120 | 787.5 | 650 |

18 | 67.5 | 43.9 | 76.3 | 77.7 | 118.5 | 555 | 80 | 787.5 | 650 |

19 | 70.4 | 56.8 | 76.0 | 81.8 | 118.0 | 555 | 120 | 450 | 650 |

20 | 70.7 | 74.8 | 75.8 | 88.3 | 117.5 | 555 | 120 | 787.5 | 400 |

21 | 71.6 | 52.8 | 78.4 | 80.4 | 123.5 | 900 | 120 | 787.5 | 650 |

22 | 73.8 | 84.1 | 78.0 | 88.7 | 118.6 | 555 | 160 | 787.5 | 650 |

23 | 71.7 | 66.8 | 77.1 | 86.0 | 118.8 | 555 | 120 | 1125 | 650 |

24 | 71.5 | 58.3 | 77.3 | 82.8 | 119.0 | 555 | 120 | 787.5 | 900 |

25 | 71.3 | 63.8 | 76.8 | 84.7 | 118.5 | 555 | 120 | 787.5 | 650 |

Initial | 69.7 | 76.3 | 73.9 | 89.0 | 112.3 | ||||

Obj | 65.9 | 100 | 77.2 | 100 | 106.5 |

X_{Y}_{22} | X_{Z}_{63} | X_{X}_{210} | X_{Y}_{3} | X_{Z}_{4} | X_{Z}_{13} | X_{X}_{8} (1) | X_{Y}_{9} | X_{X}_{8} (2) |
---|---|---|---|---|---|---|---|---|

−0.920 | −0.250 | −0.695 | 1.000 | −0.540 | −0.800 | −1.000 | −0.300 | 0.485 |

k_{Y}_{22} | k_{Z}_{63} | k_{X}_{210} | k_{Y}_{3} | k_{Z}_{4} | k_{Z}_{13} | k_{X}_{8} (1) | k_{Y}_{9} | k_{X}_{8} (2) |

636 | 769 | 92 | 1125 | 515 | 450 | 210 | 642 | 722 |

FEA Order | f_{RSM} | f_{FEA} | f_{exp} | Diff. (Hz) | Diff. (%) | MAC | MAC_{RSM} | k_{X}_{8} |
---|---|---|---|---|---|---|---|---|

1 | 33.7 | 33.7 | 33.7 | 0.0 | 0.0 | 96.7 | - | (1) |

2 | - | 60.5 | 60.5 | 0.0 | 0.0 | 98.7 | - | (1) |

3 | 65.9 | 65.8 | 65.9 | −0.1 | −0.2 | 78.9 | 76.5 | (1) |

4 | 77.2 | 77.0 | 77.2 | −0.2 | −0.3 | 80.6 | 83.9 | (2) |

5 | 84.0 | 84.1 | 84.0 | 0.1 | 0.1 | 80.9 | - | (2) |

6 | 106.5 | 106.1 | 106.5 | −0.4 | −0.4 | 70.0 | - | (1) |

© 2018 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**

Hernandez-Vazquez, J.-M.; Garitaonandia, I.; Fernandes, M.H.; Muñoa, J.; Lacalle, L.N.L.d.
A Consistent Procedure Using Response Surface Methodology to Identify Stiffness Properties of Connections in Machine Tools. *Materials* **2018**, *11*, 1220.
https://doi.org/10.3390/ma11071220

**AMA Style**

Hernandez-Vazquez J-M, Garitaonandia I, Fernandes MH, Muñoa J, Lacalle LNLd.
A Consistent Procedure Using Response Surface Methodology to Identify Stiffness Properties of Connections in Machine Tools. *Materials*. 2018; 11(7):1220.
https://doi.org/10.3390/ma11071220

**Chicago/Turabian Style**

Hernandez-Vazquez, Jesus-Maria, Iker Garitaonandia, María Helena Fernandes, Jokin Muñoa, and Luis Norberto López de Lacalle.
2018. "A Consistent Procedure Using Response Surface Methodology to Identify Stiffness Properties of Connections in Machine Tools" *Materials* 11, no. 7: 1220.
https://doi.org/10.3390/ma11071220