# A Novel Sensitivity Analysis Method in Structural Performance of Hydraulic Press

Authors: Peihao Zhu; Lianhong Zhang; Rui Zhou; Lihai Chen; Bing Yu; Qizhi Xie
Journal: Mathematical Problems in Engineering
 (2012)

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## Abstract

Sensitivity analysis plays a key role in structural optimization, but traditional methods of sensitivity analysis in strength and stiffness are time consuming and of high cost. In order to effectively carry out structural optimization of hydraulic press, this paper presents a novel sensitivity analysis method in structural performance of hydraulic press, which saves a great deal of time and design costs. The key dimension parameters of the optimization of design variables, which remarkably impact on the structural performance of hydraulic press, are efficiently selected. The impact order of various sensitivity parameters in strength and stiffness of machine tools is consistent with the sensitivity ranking of regression analysis. The research results provide the basis for the hydraulic machine design and references in research of machine tools and equipment.


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## Body

## 1. Introduction

Sensitivity analysis (SA) is used to explain models structure and behaviour in response to inputs variation. SA is a method proposed on the basis of the initial design proposal from professional staff. It is a fundamental tool for supporting mathematical models development [1], because of its capability of explaining the variability in the outputs of the models themselves [2, 3], which achieve optimization of the system parameters. SA is traditionally used to identify the parameters with the highest impact on model outputs; it is also increasingly used to analyze model structure and behavior [4–6]. In this context, SA is recently recommended as a tool to be used iteratively during the process of model development [7], to assure coherence in mathematical formalizations, to avoid over parameterizations by driving simplification processes [1, 8], and to support the development of balanced models [9, 10]. Advanced SA techniques are also used in the framework of finite element modeling. Sensitivity analysis in the framework of finite element modeling has been put forward by [11, 12]; it has been developed in the solution of inverse problems in shape optimization in metal forming [13–15] or in material parameter identification [16–19]. In such inverse methods, the sensitivities with respect to the unknown material parameters or shape design variables have to be calculated by the direct differentiation method or the adjoint state method [20].Hydraulic presses, both vertical and horizontal, are used in many industrial technologies. Vertical press applications include forging presses with flat dies, used for hot work to break down ingots and shape them into rolls, pressure vessels (mandrel forgings), forged bars, rods, plates, and so on. There are many structure parameters in hydraulic press. If all parameters are optimized for structural design variables of the complete machine, it leads the optimization model complex and enormous. Orthogonal design method [21, 22] is appropriate to sensitivity analysis of system parameters, which is adopted to analyze the structure parameters in hydraulic press. Since stiffness is proportional to the natural frequency in the same conditions, finite element analysis in structural performance of hydraulic press to strength and stiffness needed much computation time than the analysis of the modal analysis which extracts the natural frequency is more convenient and short time, which can achieve the same result in structural performance of hydraulic press. However, the traditional methods of the sensitivity analysis in strength and stiffness need a great amount of time and the fees. [23] made a research of the key parameters of hydraulic press with the welding composite frame. The key parameters of static and dynamic performance are obtained by using the quasi-static approach. [24] optimized the structural parameters of hydrocylinder by using improved genetic algorithm, which reduced the hydraulic cylinder weight.In order to obtain high-sensitivity parameters, a novel sensitivity analysis method in structural performance of hydraulic press uses the modal analysis instead of strength and stiffness analysis for 100MN hydraulic press and optimizes the hydraulic machine by using parametric modeling. Figure1 shows the research of the technology roadmap. The process can be described and achieved as follows.Figure 1
Research of the technology roadmap.Step 1.
The python language is used in the key parts of 100MN hydraulic press.Step 2.
The orthogonal design is adopted to analyze the higher sensitivity influencing factors in strength and stiffness by extracting the natural frequencies. According to the table of orthogonal design, the modal analysis is conducted and the natural frequency is extracted.Step 3.
According to the results of the orthogonal design, the parameters of higher sensitivity are extracted by the regression analysis.Step 4.
According to the results of the regression analysis with the key parts of 100MN hydraulic press, this paper establishes the parameters model of 100MN hydraulic press.Step 5.
The whole structural FEA model based on the key dimension parameters and other parameters is constructed and analyzed in order to obtain the influence of the sensitivity parameters for strength and stiffness of the machine.This paper is organized as follows: because the design of hydraulic presses is focused on the upper beam, lower beam, and column, Sections1, 2, and 3 introduce our parameters model of the key parts with the python language in order to obtain key structural parameters. According to parameters of the key parts, Section 5 introduces our parameters model of 100MN hydraulic press, and the results and discussion are given in Section 4. The whole structural FEA model is established in order to obtain conclusion that the impact order of various sensitivity parameters to strength and stiffness of the machine is consistent with the sensitivity ranking of regression analysis; the higher sensitivity parameters of modal analysis have the greatest impact on strength and stiffness.There is no need to directly conduct the strength and stiffness analysis of hydraulic press. The results of the whole structural FEA model showed that the impact order of various sensitivity parameters is consistent with the sensitivity ranking of regression analysis. These parameters of high sensitivity can be used as the focus of concern, such as the design variables of optimization. It is found that the higher sensitivity parameters can remarkably affect the structure performance of hydraulic press. The research results provide the basis for design of machine tool.


## 2. Structure of 100MN Hydraulic Press

Hydraulic press is the equipment of pressure working by hydraulic power, in which the pressure and speed can be regulated in a wide range. Hydraulic press plays an important role in the departments of national economy. Now the product update cycle is getting shorter, so a higher request for hydraulic product design is put forward. Therefore, the design technology for hydraulic press is one of the important problems which puzzle the development of our country’s hydraulic manufacture profession.100MN hydraulic press is one of the most common and most widely used structures, its structure is shown in Figure2. The main structure of hydraulic press is made up of the work parts, including the cylinder and the moving crossbeam; the machine parts, including the upper beam, the lower beam, and the column; auxiliary parts, including a cylinder, moving workstation. The design of this machine is focused on the upper beam, lower beam, column, moving crossbeam, and master cylinder; if these components are satisfied with the design requirements and technical standards, the design of the whole machine is fulfilled. Therefore, the characteristics and data of each part in the machine must be analyzed and calculated.Figure 2
Structure of 100MN hydraulic press. 1: tie rod, 2: upper beam, 3: auxiliary of the cylinder, 4: auxiliary of the cylinder piston, 5: master cylinder piston, 6: ram, 7: upper plate, 8: upper die, 9: lower die, 10: lower plate, 11: lower beam.


## 3. Structural Parameters Extraction of 100MN Precision Hydraulic Press

### 3.1. Extraction of Key Structural Parameters with the Upper Beam

#### 3.1.1. Parameters Model of the Upper Beam with the Python Language

Python [25] is a recent, general-purpose, high-level programming language. It is freely available and runs pretty much everywhere. Parametric modeling of the upper beam with python language is established in order to be submitted efficiently with a batch mode for the next modal analysis. Python language is a scripting language, which could obtain all the sizes of design part from the UG software, and all the geometric elements number of the part could be obtained by querying functions. Therefore, the design variables and the file of step format are exported to command stream files before starting the command stream files of python. The procedure of python command stream is shown in Figure 3. Figure 4 shows the parameters model of the upper beam.Figure 3
Flow chart of python command.Figure 4
Parameters model of the upper beam.The above-mentioned process is encapsulated as a function in order to facilitate many times analysis and calculation. Its inputs contain geometry name of design variable, current size, the relevant files path, and output variable names and other information. The outputs are the results of the upper beam performance analysis.The parameters model program of the upper beam is written as follow:X1=156 # the upper and lower plate thickness, X2=116 # the left and right plate thickness, X3=128 # the front and back plate thickness, X4=157 # the front and back rib plate thickness, X5=120 # the left and right rib plate thickness, X6=38 # the ramp plate thickness. Figure 5 illustrates the parameters model of the upper beam.Figure 5
Structure of the upper beam.


#### 3.1.2. Orthogonal Design of the Upper Beam

Orthogonal design [26–28] is a valid method which makes use of orthogonal table Ln(mk) as a tool to arrange experiments and find out the optimal design with fewer experiments. In orthogonal table Ln(mk), n represents the number of rows, representing the number of experiments; k represents the number of columns, namely, the number of factors; m represents the number of factor levels.There are six designable variables in the upper beam, every factor has five different levels, and there will be56=15625 total possible combinations of experiments. If we choose L25(56) orthogonal table, it only takes 25 combinations of experiments to obtain the desired results (Table 1).Table 1
Orthogonal experimental table of six factors and five levels.
L
25
(
5
6
)
Test no.
1
2
3
4
5
6
1
1
1
1
1
1
1
2
1
2
2
2
2
2
3
1
3
3
3
3
3
4
1
4
4
4
4
4
5
1
5
5
5
5
5
6
2
1
2
3
4
5
7
2
2
3
4
5
1
8
2
3
4
5
1
2
9
2
4
5
1
2
3
10
2
5
1
2
3
4
11
3
1
3
5
2
4
12
3
2
4
1
3
5
13
3
3
5
2
4
1
14
3
4
1
3
5
2
15
3
5
2
4
1
3
16
4
1
4
2
5
3
17
4
2
5
3
1
4
18
4
3
1
4
2
5
19
4
4
2
5
3
1
20
4
5
3
1
4
2
21
5
1
5
4
3
2
22
5
2
1
5
4
3
23
5
3
2
1
5
4
24
5
4
3
2
1
5
25
5
5
4
3
2
1The analysis of single part, such as the upper beam, lower beam, and column, are mainly discussed. Each factor of single part is determined by the initial design proposal from professional staff, because these parameters have important impact on structural performance of hydraulic press by the past practical experience. These parameters are representative. The thickness change of steel sheet in the actual production is easy to implement compare with the other factors such as length, width, and height of the whole part, which has influence on the inner structure parameters. The levels of each factor of single part are determined by steel plate thickness of the initial design, each parameter increments of the same mass increments and arithmetic sequence.


#### 3.1.3. Modal Analysis of the Upper Beam

The modal analysis is the most basic and important part of analysis of dynamic character [29–31]. It is the modern method to study the dynamic character of structure. And it is the powerful tool to design and evaluate structure of product. The natural frequency and deformation amplitude of system could be acquired by modal analysis.From the variation principle of elastic mechanics, the dynamic balance equation of the upper beam with multiple degree of freedom is as follows.(3.1)[M]{u¨}+[C]{u˙}+[K]{u}={P(t)}+{N}+{Q},
where [M], [C], and [K] are referred to, respectively, as the mass, damping, and stiffness matrices. The matrices are n×n square matrices, where n is the number of degrees of freedom of the system. {P(t)} is the external force function vector, {N} is the nonlinear external force vector related with {u˙} and {u}, {Q} is the boundary constrain counterforce vector, {u} is the shifting vector, {u˙} is the speed vector, and {u¨} is the acceleration vector.The natural frequency and vibration catalog represent the dynamic characters. The modal of the upper beam is analyzed. The natural frequency of theith modal is obtained. The first natural modal is shown in Figure 6 and Table 2.Table 2
The results of modal analysis and orthogonal experimental data with the upper beam.
Six factors and five levels
X
1
X
2
X
3
X
4
X
5
X
6
Y
The upper and lower plate thickness
The left and right plate thickness
The front and back plate thickness
The front and back rib plate thickness
The left and right rib plate thickness
The ramp plate thickness
The natural frequency
1
156
116
128
157
120
38
164.10
2
158
116
134
159
124
42
163.78
3
160
116
132
151
122
36
164.63
4
162
116
126
155
116
40
165.39
5
164
116
130
153
118
44
165.26
6
156
118
130
155
122
42
163.53
7
158
118
128
153
116
36
164.55
8
160
118
134
157
118
40
164.31
9
162
118
132
159
120
44
164.53
10
164
118
126
151
124
38
165.30
11
156
120
126
159
118
36
164.01
12
158
120
130
151
120
40
163.90
13
160
120
128
155
124
44
164.03
14
162
120
134
153
122
38
164.43
15
164
120
132
157
116
42
164.96
16
156
122
132
153
124
40
163.05
17
158
122
126
157
122
44
163.66
18
160
122
130
159
116
38
164.34
19
162
122
128
151
118
42
164.54
20
164
122
134
155
120
36
164.77
21
156
124
134
151
116
44
162.88
22
158
124
132
155
118
38
163.58
23
160
124
126
153
120
42
164.01
24
162
124
130
157
122
36
164.36
25
164
124
128
159
124
40
164.56Figure 6
Results of modal analysis with the upper beam.


#### 3.1.4. Regression Analysis of the Upper Beam

The regression analysis is a more commonly used quantitative analysis method [32, 33]. The so-called regression analysis is the quantity change relations between some random variables (dependent variables) and other or several variables (independent variable). The extracted relationship by the regression analysis usually is called the regression model.After we use sample data (Table2) to establish the regression equation, the general practical problems are not immediately used for the analysis. In practical problems, there are six factors (independent variable) that affect the dependent variable y; from all independent variables we hope to select the independent variables that have significant effect on y, to establish the regression equation. In the regression equation if we leave out the independent variables having significant effect on y, it will affect the fitting results.Regression flow is shown in Figure7. First of all, take coefficient of regression significance test on largest variable of partial correlation coefficient, which decide this variable whether enter regression equation. Then calculate F-numbers of each variable in regression equation, biased F test on the variable which biased F value minimum, to decide whether this variable stays in regression equation or not. Repeat the calculating, until no variable is pulled into or rejected.Figure 7
Procedure of regression equation.Let the dependent variabley be an observable random variable, and independent variables x1,x2,…,xp are generally variables. The general model of multivariate linear regression in sensitivity analysis of machining tool likes the following formula:
(3.2)y=β0+β1x1+β2x2+⋯+βpxp+ε,ε~N(0,σ2).Sample data matrix is shown in(3.3)[y1y2⋮yn]=[1x11x12⋯x1p1x21x22⋯x2p⋮⋮⋮⋮⋮1xn1xn2⋯xnp][β0β1β2⋮βp],yn=β0+β1xn1+β2xn2+⋯+βpxnp+εn,y1=β0+β1x11+β2x12+⋯+βpx1p+ε1,y2=β0+β1x21+β2x22+⋯+βpx2p+ε2,⋮⋮⋮yn=β0+β1xn1+β2xn2+⋯+βpxnp+εn,
where y represents the natural frequency (expressed by demand generally), β0,β1,…βp are unknown parameters, called regression coefficient; x1,x2,…xp are the influence factors of natural frequency (explanatory variables); ε denotes the random error, its mean is zero, and the variance is bigger than zero. That does not exist between the independent variable multicollinearity. In practice as long as the multicollinearity is weak, we can consider linear independence between the independent variables, and linear regression can be carried out.The regression equation solves for the coefficients by minimizing the sum of the squares of the deviations of the data from the model (least-square fit). The least-square fit of the model as follow:(3.4)Y=159.9456+0.1842X1+0.0940X2+0.0551X3+0.0011X4+0.0418X5+0.0468X6.According to the regression equation to know, for upper beam of hydraulic press the biggest influence for the natural frequency is the upper and lower plate thickness, influences followed are the left and right plate thickness and little influences are the front and back rib plate thickness. It can be seen through the equation that values ofX1 and X2 have greater impact on Y, which obtains the two higher sensitivity parameters of X1 and X2.


#### 3.1.5. Variance Analysis of the Upper Beam

Analysis of variance (ANOVA) [34] is a statistical procedure for summarizing a classical linear model—a decomposition of sum of squares into a component for each source of variation in the model—along with an associated test (the F test) of the hypothesis that any given source of variation in the model is zero. When applied to generalized linear models, multilevel models, and other extensions of classical regression, ANOVA can be extended in two different directions. First, the F test can be used to compare models, to test the hypothesis that the simpler of the models is sufficient to explain the data. Second, the idea of variance decomposition can be interpreted as inference for the variances of batches of parameters (sources of variation) in multilevel regressions. Variance analysis table is shown in Table 3.Table 3
Variance analysis table.
Source of variance
Sum of squares
Freedom of motion
Mean square
F ratio
Significance
Regression
U
m
U
/
m
F
=
U
/
m
Q
/
(
n
-
m
-
1
)
Residual error
Q
(
n
-
m
-
1
)
Q
/
(
n
-
m
-
1
)
Sum
s
y
y
n
-
1The average value of the natural frequency is shown in(3.5)y-=1n∑k=1nyk.
Thus, the residual sum of squares and regression sum of squares are given by
(3.6)U=∑k=1n(y^k-y-)2,Q=∑k=1n(yk-y^k)2.Total variation is the aggregate dispersion of the individual data values around the overall mean of all factor levels as shown in(3.7)syy=U+Q=∑k=1n(yk-y-)2,
where yk represents the natural frequency of upper beam, which could be acquired by modal analysis; y- is the average value of the natural frequency yk; y^k is regression value, which could be acquired by regression equation; total variation can be split into two parts: U and Q, and U is regression sum of squares; Q is residual sum of squares. Regression values mean the difference between the square and reflect the volatility caused by changes in the independent variable and its degree of freedom (since the number of variables); syy=U+Q, where syy is total sum of squares (total variation).TheF distribution is a right-skewed distribution used most commonly in Analysis of Variance (like ANOVA/MANOVA). The F distribution is a ratio of two Chi-square distributions, and a specific F distribution is denoted by the degrees of freedom for the numerator Chi-square and the degrees of freedom for the denominator Chi-square. When referencing the F distribution, the numerator degrees of freedom are always given first, as switching the order of degrees of freedom changes the distribution (e.g., F(10,12) does not equal F(12,10)). The F distribution is an asymmetric distribution that has a minimum value of 0, but no maximum value. The curve reaches a peak not far to the right of 0 and then gradually approaches the horizontal axis the larger the F value is. The F distribution approaches, but never quite touches, the horizontal axis. The formula of F is obtained by
(3.8)F=U/mQ/(n-m-1),
where m is the number of variables; n is sum of all groups.According to multiple linear regression equation, the table of variance analysis is obtained (Table4). By the F distribution table, the regression equation is remarkable (Table 4). The result of the test also proves that the equation of multiple linear regression is linear relation.Table 4
Variance analysis of the upper beam.
Source of variance
Sum of squares
Freedom of motion
Mean square
F ratio
Significance
Regression
704.4000
6
117.4000
738.3600
**
Residual error
2.8611
18
0.1590
Sum
707.2611
24
*
The regression equation is remarkable.


### 3.2. Extraction of Key Structural Parameters with the Lower Beam

#### 3.2.1. Parameters Model of the Lower Beam with the Python Language

The parameters model program of the lower beam as follow:X1=116 # the lower plate thickness, X2=96 # the left and right plate thickness, X3=128 # the left and right rib plate thickness, X4=142 # the front and back rib plate thickness, X5=170 # the front and back plate thickness, X6=128 # the upper plate thickness. Figure 8 illustrates the parameters model of the lower beam.Parameters model of the lower beam.
(a)
(b)


#### 3.2.2. Orthogonal Design of the Lower Beam

Orthogonal design is a statistical optimization method, which arranges multifactor test scheme based on one set of prepared standard table—orthogonal table. This paper uses the orthogonal experimental design method to analyze hydraulic press of the lower beams. Figure8 illustrates the parameters model of the lower beam. According to the influencing factors and designed level, design table is L25(56) and there are 25 kinds of calculating schemes. Statistics analysis is carried on calculating results and the significant degree of each factor is judged on the test index.


#### 3.2.3. Modal Analysis of the Lower Beam

This paper uses the python language to establish the lower beam parameters model, which is analyzed by the modal method. According to orthogonal experimental table, it only requires 25 times of the modal analysis and each modal analysis of model parameter values is based on Table5, and then submit the python command stream to the Abaqus, extract the first-order natural frequency, and use the batch mode to obtain the first-order natural frequency of 25 groups, and the results can be seen in Figure 9 and Table 5.Table 5
Data results of Modal analysis of the lower beam.
Six factors and five levels
X
1
X
2
X
3
X
4
X
5
X
6
Y
The lower plate thickness
The left and right plate thickness
The left and right rib plate thickness
The front and back rib plate thickness
The front and back plate thickness
The upper plate thickness
Natural frequency
1
116
96
128
142
170
128
175.88
2
118
96
134
144
174
132
176.27
3
120
96
132
136
172
126
176.48
4
122
96
126
140
166
130
177.33
5
124
96
130
138
168
134
178.44
6
116
98
130
140
172
132
176.11
7
118
98
128
138
166
126
175.85
8
120
98
134
142
168
130
176.98
9
122
98
132
144
170
134
177.72
10
124
98
126
136
174
128
176.98
11
116
100
126
144
168
126
174.89
12
118
100
130
136
170
130
176.15
13
120
100
128
140
174
134
176.81
14
122
100
134
138
172
128
176.73
15
124
100
132
142
166
134
177.90
16
116
102
132
138
174
130
175.41
17
118
102
126
142
172
134
176.04
18
120
102
130
144
166
128
175.89
19
122
102
128
136
168
132
176.82
20
124
102
134
140
170
126
176.51
21
116
104
134
136
166
134
175.99
22
118
104
132
140
168
128
175.33
23
120
104
126
138
170
132
175.96
24
122
104
130
142
172
126
175.57
25
124
104
128
144
174
130
176.32Figure 9
Results of modal analysis of the lower beam.


#### 3.2.4. Regression Analysis of the Lower Beam

The so-called regression analysis is based on a large number of observation data, function expression of regression relationship between the independent variable, and the dependent variable called the regression equation is established by using of mathematical statistics methods. This paper achieves the linear regression analysis.The analysis results of multiple linear regression are shown as follows:(3.9)Y=155.1471+0.1981X1+0.1343X2+0.0337X3+0.0287X4+0.0414X5+0.1353X6.According to the regression equation to know, for the lower beam of hydraulic press the biggest influence for the natural frequency is the lower plate thickness, influences followed are the upper plate thickness, and little influences are the front and back rib plate thickness.It can be seen through the equation that values ofX1 and X6 have greater impact on Y (natural frequency), then obtain the highest sensitivity of the two parameters: the lower plate thickness (X1), the upper plate thickness (X6).According to multiple linear regression equation, the table of variance analysis (Table6) can be obtained.Table 6
Regression equation of variance analysis of the lower beam.
Source of variance
Sum of squares
Freedom of motion
Mean square
F ratio
Significance
Regression
150.5780
6
25.1000
145.7610
**
Residual error
3.0990
18
0.1722
Sum
153.6770
24
*
The regression equation is remarkable.By theF distribution table, when there is remarkable degree of α=0.01, F0.01(6,18)=4.01. Because of F=145.761≫4.01, that is, the F≫F0.01, so the regression equation is very remarkable (Table 6). The results of the test prove that multiple linear regression equation is linear relation, and the regression equation is very remarkable.From the formula (3.9), we come to the conclusion that sort effects of the design parameters on the lower beam to the natural frequency: X1, X6, X2, X5, X3, X4, that is to say, X1 and X6 are the upper beam sensitive parameters.


### 3.3. Extraction of Key Structural Parameters with the Column

#### 3.3.1. Parameters Model of the Column with the Python Language

The parameters model program of the column as follow: from part import *from material import *from section import *from assembly import *from step import *from interaction import *from load import *from mesh import *from job import *from sketch import *from visualization import *from connector Behavior import*.X1=116 # The top to the first rib plates, X2=96 # The first to the second rib plates, X3=128 # The remaining distance between rib plates, X4=142 # The roof column width, X5=170 # The column plates thickness, X6=128 # The rib plates thickness. Figure 10 illustrates the parameters model of the column.Figure 10
Parameters model of the column.


#### 3.3.2. Orthogonal Design of the Column

This paper uses the orthogonal experimental design method to analyze six parameters of the columns (Figure10) and arranges five levels for each parameter as a factor. According to the six factors and five levels of orthogonal design table, obtain six factors and five levels of orthogonal experimental data of the column. Six factors contain the top to the first rib plates (X1), the first to the second rib plates (X2), the remaining distance between rib plates (X3), the roof column width (X4), the column plates thickness (X5), and the rib plates thickness (X6).


#### 3.3.3. Modal Analysis of the Column

This paper uses python language to establish the model of the column, which is analyzed by the modal method. According to orthogonal experimental table, it only requires 25 times of the modal analysis and each modal analysis of model parameter values is based on Table7, and then submit the python command stream to the Abaqus, extract the first-order natural frequency, and use the batch mode to quickly extract the first-order natural frequency of 25, and the results are shown in (Figure 10 and Table 6).Table 7
Modal analysis results of the column.
Six factors and five levels
X
1
X
2
X
3
X
4
X
5
X
6
Y
The top to the first rib plates
The first to  the second rib plates
The remaining distance between rib plates
The roof column width
The columnplates thickness
The rib plates thickness
Natural frequency
1
800
800
925
1050
135
45
17.953
2
900
800
1150
1100
150
55
18.016
3
1000
800
1075
900
143
40
18.063
4
1100
800
850
1000
120
50
17.906
5
1200
800
1000
950
127
60
17.859
6
800
900
1000
1000
143
55
17.965
7
900
900
925
950
120
40
17.903
8
1000
900
1150
1050
127
50
17.960
9
1100
900
1075
1100
135
60
17.921
10
1200
900
850
900
150
45
18.072
11
800
1000
850
1100
127
40
17.773
12
900
1000
1000
900
135
50
17.779
13
1000
1000
925
1000
150
60
17.842
14
1100
1000
1150
950
143
45
17.859
15
1200
1000
1075
1050
120
55
17.799
16
800
1100
1075
950
150
50
17.768
17
900
1100
850
1050
143
60
17.778
18
1000
1100
1000
1100
120
45
17.772
19
1100
1100
925
900
127
55
17.785
20
1200
1100
1150
1000
135
40
16.739
21
800
1200
1150
900
120
60
17.656
22
900
1200
1075
1000
127
45
17.833
23
1000
1200
850
950
135
55
17.766
24
1100
1200
1000
1050
143
40
17.941
25
1200
1200
925
1100
150
50
17.937The essence of modal analysis is to solve the matrix eigenvalues, according to the table of orthogonal design, which only needs the modal analysis of 25, times, and each model parameter is based on Table1. The python command stream is submitted to the Abaqus, which extracts the first order natural frequency (Table 7 and Figure 11). The first order natural frequency of 25 groups is quickly extracted by using the batch mode, and the results of batch mode can be seen in Table 7.Figure 11
Results of modal analysis of the column.


#### 3.3.4. Regression Analysis of the Lower Beam

Regression analysis is a parametric method that requires a specification of the analytical expression of the functional form that links the inputs and outputs. The equation of multiple linear regressionwith the lower beam is shown as follows:(3.10)Y=18.4176+0.0003X1+0.0007X2+0.0006X3+0.0002X4+0.004X5+X6.According to the regression equation to know, for the column the biggest influence for the natural frequency is the rib plates thickness, influences followed are the column plates thickness, and little influences are the roof column width.By theF distribution table, when there is remarkable degree of α=0.01, F0.01(6,18)=4.01. Because of F=169.1700≫4.01, that is, the F≫F0.01, so the regression equation is very remarkable (Table 8). The test proves that multiple linear regression equation is linear relation.Table 8
Variance analysis of the column.
Source of variance
Sum of squares
Freedom of motion
Mean square
F ratio
Significance
Regression
55.5200
6
9.250
169.1700
**
Residual error
0.9842
18
0.0547
Sum
56.5042
24
*
The regression equation is remarkable.According to multiple linear regression equation in a similar way, the results of variance analysis are shown in Table8.From the formula, we can come to the conclusion that sort effects of the design parameters on the lower beam to the natural frequency:X6, X5, X2, X3, X1, X4, that is to say, X6 and X5 are the column sensitive parameters.


## 3.1. Extraction of Key Structural Parameters with the Upper Beam

### 3.1.1. Parameters Model of the Upper Beam with the Python Language

Python [25] is a recent, general-purpose, high-level programming language. It is freely available and runs pretty much everywhere. Parametric modeling of the upper beam with python language is established in order to be submitted efficiently with a batch mode for the next modal analysis. Python language is a scripting language, which could obtain all the sizes of design part from the UG software, and all the geometric elements number of the part could be obtained by querying functions. Therefore, the design variables and the file of step format are exported to command stream files before starting the command stream files of python. The procedure of python command stream is shown in Figure 3. Figure 4 shows the parameters model of the upper beam.Figure 3
Flow chart of python command.Figure 4
Parameters model of the upper beam.The above-mentioned process is encapsulated as a function in order to facilitate many times analysis and calculation. Its inputs contain geometry name of design variable, current size, the relevant files path, and output variable names and other information. The outputs are the results of the upper beam performance analysis.The parameters model program of the upper beam is written as follow:X1=156 # the upper and lower plate thickness, X2=116 # the left and right plate thickness, X3=128 # the front and back plate thickness, X4=157 # the front and back rib plate thickness, X5=120 # the left and right rib plate thickness, X6=38 # the ramp plate thickness. Figure 5 illustrates the parameters model of the upper beam.Figure 5
Structure of the upper beam.


### 3.1.2. Orthogonal Design of the Upper Beam

Orthogonal design [26–28] is a valid method which makes use of orthogonal table Ln(mk) as a tool to arrange experiments and find out the optimal design with fewer experiments. In orthogonal table Ln(mk), n represents the number of rows, representing the number of experiments; k represents the number of columns, namely, the number of factors; m represents the number of factor levels.There are six designable variables in the upper beam, every factor has five different levels, and there will be56=15625 total possible combinations of experiments. If we choose L25(56) orthogonal table, it only takes 25 combinations of experiments to obtain the desired results (Table 1).Table 1
Orthogonal experimental table of six factors and five levels.
L
25
(
5
6
)
Test no.
1
2
3
4
5
6
1
1
1
1
1
1
1
2
1
2
2
2
2
2
3
1
3
3
3
3
3
4
1
4
4
4
4
4
5
1
5
5
5
5
5
6
2
1
2
3
4
5
7
2
2
3
4
5
1
8
2
3
4
5
1
2
9
2
4
5
1
2
3
10
2
5
1
2
3
4
11
3
1
3
5
2
4
12
3
2
4
1
3
5
13
3
3
5
2
4
1
14
3
4
1
3
5
2
15
3
5
2
4
1
3
16
4
1
4
2
5
3
17
4
2
5
3
1
4
18
4
3
1
4
2
5
19
4
4
2
5
3
1
20
4
5
3
1
4
2
21
5
1
5
4
3
2
22
5
2
1
5
4
3
23
5
3
2
1
5
4
24
5
4
3
2
1
5
25
5
5
4
3
2
1The analysis of single part, such as the upper beam, lower beam, and column, are mainly discussed. Each factor of single part is determined by the initial design proposal from professional staff, because these parameters have important impact on structural performance of hydraulic press by the past practical experience. These parameters are representative. The thickness change of steel sheet in the actual production is easy to implement compare with the other factors such as length, width, and height of the whole part, which has influence on the inner structure parameters. The levels of each factor of single part are determined by steel plate thickness of the initial design, each parameter increments of the same mass increments and arithmetic sequence.


### 3.1.3. Modal Analysis of the Upper Beam

The modal analysis is the most basic and important part of analysis of dynamic character [29–31]. It is the modern method to study the dynamic character of structure. And it is the powerful tool to design and evaluate structure of product. The natural frequency and deformation amplitude of system could be acquired by modal analysis.From the variation principle of elastic mechanics, the dynamic balance equation of the upper beam with multiple degree of freedom is as follows.(3.1)[M]{u¨}+[C]{u˙}+[K]{u}={P(t)}+{N}+{Q},
where [M], [C], and [K] are referred to, respectively, as the mass, damping, and stiffness matrices. The matrices are n×n square matrices, where n is the number of degrees of freedom of the system. {P(t)} is the external force function vector, {N} is the nonlinear external force vector related with {u˙} and {u}, {Q} is the boundary constrain counterforce vector, {u} is the shifting vector, {u˙} is the speed vector, and {u¨} is the acceleration vector.The natural frequency and vibration catalog represent the dynamic characters. The modal of the upper beam is analyzed. The natural frequency of theith modal is obtained. The first natural modal is shown in Figure 6 and Table 2.Table 2
The results of modal analysis and orthogonal experimental data with the upper beam.
Six factors and five levels
X
1
X
2
X
3
X
4
X
5
X
6
Y
The upper and lower plate thickness
The left and right plate thickness
The front and back plate thickness
The front and back rib plate thickness
The left and right rib plate thickness
The ramp plate thickness
The natural frequency
1
156
116
128
157
120
38
164.10
2
158
116
134
159
124
42
163.78
3
160
116
132
151
122
36
164.63
4
162
116
126
155
116
40
165.39
5
164
116
130
153
118
44
165.26
6
156
118
130
155
122
42
163.53
7
158
118
128
153
116
36
164.55
8
160
118
134
157
118
40
164.31
9
162
118
132
159
120
44
164.53
10
164
118
126
151
124
38
165.30
11
156
120
126
159
118
36
164.01
12
158
120
130
151
120
40
163.90
13
160
120
128
155
124
44
164.03
14
162
120
134
153
122
38
164.43
15
164
120
132
157
116
42
164.96
16
156
122
132
153
124
40
163.05
17
158
122
126
157
122
44
163.66
18
160
122
130
159
116
38
164.34
19
162
122
128
151
118
42
164.54
20
164
122
134
155
120
36
164.77
21
156
124
134
151
116
44
162.88
22
158
124
132
155
118
38
163.58
23
160
124
126
153
120
42
164.01
24
162
124
130
157
122
36
164.36
25
164
124
128
159
124
40
164.56Figure 6
Results of modal analysis with the upper beam.


### 3.1.4. Regression Analysis of the Upper Beam

The regression analysis is a more commonly used quantitative analysis method [32, 33]. The so-called regression analysis is the quantity change relations between some random variables (dependent variables) and other or several variables (independent variable). The extracted relationship by the regression analysis usually is called the regression model.After we use sample data (Table2) to establish the regression equation, the general practical problems are not immediately used for the analysis. In practical problems, there are six factors (independent variable) that affect the dependent variable y; from all independent variables we hope to select the independent variables that have significant effect on y, to establish the regression equation. In the regression equation if we leave out the independent variables having significant effect on y, it will affect the fitting results.Regression flow is shown in Figure7. First of all, take coefficient of regression significance test on largest variable of partial correlation coefficient, which decide this variable whether enter regression equation. Then calculate F-numbers of each variable in regression equation, biased F test on the variable which biased F value minimum, to decide whether this variable stays in regression equation or not. Repeat the calculating, until no variable is pulled into or rejected.Figure 7
Procedure of regression equation.Let the dependent variabley be an observable random variable, and independent variables x1,x2,…,xp are generally variables. The general model of multivariate linear regression in sensitivity analysis of machining tool likes the following formula:
(3.2)y=β0+β1x1+β2x2+⋯+βpxp+ε,ε~N(0,σ2).Sample data matrix is shown in(3.3)[y1y2⋮yn]=[1x11x12⋯x1p1x21x22⋯x2p⋮⋮⋮⋮⋮1xn1xn2⋯xnp][β0β1β2⋮βp],yn=β0+β1xn1+β2xn2+⋯+βpxnp+εn,y1=β0+β1x11+β2x12+⋯+βpx1p+ε1,y2=β0+β1x21+β2x22+⋯+βpx2p+ε2,⋮⋮⋮yn=β0+β1xn1+β2xn2+⋯+βpxnp+εn,
where y represents the natural frequency (expressed by demand generally), β0,β1,…βp are unknown parameters, called regression coefficient; x1,x2,…xp are the influence factors of natural frequency (explanatory variables); ε denotes the random error, its mean is zero, and the variance is bigger than zero. That does not exist between the independent variable multicollinearity. In practice as long as the multicollinearity is weak, we can consider linear independence between the independent variables, and linear regression can be carried out.The regression equation solves for the coefficients by minimizing the sum of the squares of the deviations of the data from the model (least-square fit). The least-square fit of the model as follow:(3.4)Y=159.9456+0.1842X1+0.0940X2+0.0551X3+0.0011X4+0.0418X5+0.0468X6.According to the regression equation to know, for upper beam of hydraulic press the biggest influence for the natural frequency is the upper and lower plate thickness, influences followed are the left and right plate thickness and little influences are the front and back rib plate thickness. It can be seen through the equation that values ofX1 and X2 have greater impact on Y, which obtains the two higher sensitivity parameters of X1 and X2.


### 3.1.5. Variance Analysis of the Upper Beam

Analysis of variance (ANOVA) [34] is a statistical procedure for summarizing a classical linear model—a decomposition of sum of squares into a component for each source of variation in the model—along with an associated test (the F test) of the hypothesis that any given source of variation in the model is zero. When applied to generalized linear models, multilevel models, and other extensions of classical regression, ANOVA can be extended in two different directions. First, the F test can be used to compare models, to test the hypothesis that the simpler of the models is sufficient to explain the data. Second, the idea of variance decomposition can be interpreted as inference for the variances of batches of parameters (sources of variation) in multilevel regressions. Variance analysis table is shown in Table 3.Table 3
Variance analysis table.
Source of variance
Sum of squares
Freedom of motion
Mean square
F ratio
Significance
Regression
U
m
U
/
m
F
=
U
/
m
Q
/
(
n
-
m
-
1
)
Residual error
Q
(
n
-
m
-
1
)
Q
/
(
n
-
m
-
1
)
Sum
s
y
y
n
-
1The average value of the natural frequency is shown in(3.5)y-=1n∑k=1nyk.
Thus, the residual sum of squares and regression sum of squares are given by
(3.6)U=∑k=1n(y^k-y-)2,Q=∑k=1n(yk-y^k)2.Total variation is the aggregate dispersion of the individual data values around the overall mean of all factor levels as shown in(3.7)syy=U+Q=∑k=1n(yk-y-)2,
where yk represents the natural frequency of upper beam, which could be acquired by modal analysis; y- is the average value of the natural frequency yk; y^k is regression value, which could be acquired by regression equation; total variation can be split into two parts: U and Q, and U is regression sum of squares; Q is residual sum of squares. Regression values mean the difference between the square and reflect the volatility caused by changes in the independent variable and its degree of freedom (since the number of variables); syy=U+Q, where syy is total sum of squares (total variation).TheF distribution is a right-skewed distribution used most commonly in Analysis of Variance (like ANOVA/MANOVA). The F distribution is a ratio of two Chi-square distributions, and a specific F distribution is denoted by the degrees of freedom for the numerator Chi-square and the degrees of freedom for the denominator Chi-square. When referencing the F distribution, the numerator degrees of freedom are always given first, as switching the order of degrees of freedom changes the distribution (e.g., F(10,12) does not equal F(12,10)). The F distribution is an asymmetric distribution that has a minimum value of 0, but no maximum value. The curve reaches a peak not far to the right of 0 and then gradually approaches the horizontal axis the larger the F value is. The F distribution approaches, but never quite touches, the horizontal axis. The formula of F is obtained by
(3.8)F=U/mQ/(n-m-1),
where m is the number of variables; n is sum of all groups.According to multiple linear regression equation, the table of variance analysis is obtained (Table4). By the F distribution table, the regression equation is remarkable (Table 4). The result of the test also proves that the equation of multiple linear regression is linear relation.Table 4
Variance analysis of the upper beam.
Source of variance
Sum of squares
Freedom of motion
Mean square
F ratio
Significance
Regression
704.4000
6
117.4000
738.3600
**
Residual error
2.8611
18
0.1590
Sum
707.2611
24
*
The regression equation is remarkable.


## 3.1.1. Parameters Model of the Upper Beam with the Python Language

Python [25] is a recent, general-purpose, high-level programming language. It is freely available and runs pretty much everywhere. Parametric modeling of the upper beam with python language is established in order to be submitted efficiently with a batch mode for the next modal analysis. Python language is a scripting language, which could obtain all the sizes of design part from the UG software, and all the geometric elements number of the part could be obtained by querying functions. Therefore, the design variables and the file of step format are exported to command stream files before starting the command stream files of python. The procedure of python command stream is shown in Figure 3. Figure 4 shows the parameters model of the upper beam.Figure 3
Flow chart of python command.Figure 4
Parameters model of the upper beam.The above-mentioned process is encapsulated as a function in order to facilitate many times analysis and calculation. Its inputs contain geometry name of design variable, current size, the relevant files path, and output variable names and other information. The outputs are the results of the upper beam performance analysis.The parameters model program of the upper beam is written as follow:X1=156 # the upper and lower plate thickness, X2=116 # the left and right plate thickness, X3=128 # the front and back plate thickness, X4=157 # the front and back rib plate thickness, X5=120 # the left and right rib plate thickness, X6=38 # the ramp plate thickness. Figure 5 illustrates the parameters model of the upper beam.Figure 5
Structure of the upper beam.


## 3.1.2. Orthogonal Design of the Upper Beam

Orthogonal design [26–28] is a valid method which makes use of orthogonal table Ln(mk) as a tool to arrange experiments and find out the optimal design with fewer experiments. In orthogonal table Ln(mk), n represents the number of rows, representing the number of experiments; k represents the number of columns, namely, the number of factors; m represents the number of factor levels.There are six designable variables in the upper beam, every factor has five different levels, and there will be56=15625 total possible combinations of experiments. If we choose L25(56) orthogonal table, it only takes 25 combinations of experiments to obtain the desired results (Table 1).Table 1
Orthogonal experimental table of six factors and five levels.
L
25
(
5
6
)
Test no.
1
2
3
4
5
6
1
1
1
1
1
1
1
2
1
2
2
2
2
2
3
1
3
3
3
3
3
4
1
4
4
4
4
4
5
1
5
5
5
5
5
6
2
1
2
3
4
5
7
2
2
3
4
5
1
8
2
3
4
5
1
2
9
2
4
5
1
2
3
10
2
5
1
2
3
4
11
3
1
3
5
2
4
12
3
2
4
1
3
5
13
3
3
5
2
4
1
14
3
4
1
3
5
2
15
3
5
2
4
1
3
16
4
1
4
2
5
3
17
4
2
5
3
1
4
18
4
3
1
4
2
5
19
4
4
2
5
3
1
20
4
5
3
1
4
2
21
5
1
5
4
3
2
22
5
2
1
5
4
3
23
5
3
2
1
5
4
24
5
4
3
2
1
5
25
5
5
4
3
2
1The analysis of single part, such as the upper beam, lower beam, and column, are mainly discussed. Each factor of single part is determined by the initial design proposal from professional staff, because these parameters have important impact on structural performance of hydraulic press by the past practical experience. These parameters are representative. The thickness change of steel sheet in the actual production is easy to implement compare with the other factors such as length, width, and height of the whole part, which has influence on the inner structure parameters. The levels of each factor of single part are determined by steel plate thickness of the initial design, each parameter increments of the same mass increments and arithmetic sequence.


## 3.1.3. Modal Analysis of the Upper Beam

The modal analysis is the most basic and important part of analysis of dynamic character [29–31]. It is the modern method to study the dynamic character of structure. And it is the powerful tool to design and evaluate structure of product. The natural frequency and deformation amplitude of system could be acquired by modal analysis.From the variation principle of elastic mechanics, the dynamic balance equation of the upper beam with multiple degree of freedom is as follows.(3.1)[M]{u¨}+[C]{u˙}+[K]{u}={P(t)}+{N}+{Q},
where [M], [C], and [K] are referred to, respectively, as the mass, damping, and stiffness matrices. The matrices are n×n square matrices, where n is the number of degrees of freedom of the system. {P(t)} is the external force function vector, {N} is the nonlinear external force vector related with {u˙} and {u}, {Q} is the boundary constrain counterforce vector, {u} is the shifting vector, {u˙} is the speed vector, and {u¨} is the acceleration vector.The natural frequency and vibration catalog represent the dynamic characters. The modal of the upper beam is analyzed. The natural frequency of theith modal is obtained. The first natural modal is shown in Figure 6 and Table 2.Table 2
The results of modal analysis and orthogonal experimental data with the upper beam.
Six factors and five levels
X
1
X
2
X
3
X
4
X
5
X
6
Y
The upper and lower plate thickness
The left and right plate thickness
The front and back plate thickness
The front and back rib plate thickness
The left and right rib plate thickness
The ramp plate thickness
The natural frequency
1
156
116
128
157
120
38
164.10
2
158
116
134
159
124
42
163.78
3
160
116
132
151
122
36
164.63
4
162
116
126
155
116
40
165.39
5
164
116
130
153
118
44
165.26
6
156
118
130
155
122
42
163.53
7
158
118
128
153
116
36
164.55
8
160
118
134
157
118
40
164.31
9
162
118
132
159
120
44
164.53
10
164
118
126
151
124
38
165.30
11
156
120
126
159
118
36
164.01
12
158
120
130
151
120
40
163.90
13
160
120
128
155
124
44
164.03
14
162
120
134
153
122
38
164.43
15
164
120
132
157
116
42
164.96
16
156
122
132
153
124
40
163.05
17
158
122
126
157
122
44
163.66
18
160
122
130
159
116
38
164.34
19
162
122
128
151
118
42
164.54
20
164
122
134
155
120
36
164.77
21
156
124
134
151
116
44
162.88
22
158
124
132
155
118
38
163.58
23
160
124
126
153
120
42
164.01
24
162
124
130
157
122
36
164.36
25
164
124
128
159
124
40
164.56Figure 6
Results of modal analysis with the upper beam.


## 3.1.4. Regression Analysis of the Upper Beam

The regression analysis is a more commonly used quantitative analysis method [32, 33]. The so-called regression analysis is the quantity change relations between some random variables (dependent variables) and other or several variables (independent variable). The extracted relationship by the regression analysis usually is called the regression model.After we use sample data (Table2) to establish the regression equation, the general practical problems are not immediately used for the analysis. In practical problems, there are six factors (independent variable) that affect the dependent variable y; from all independent variables we hope to select the independent variables that have significant effect on y, to establish the regression equation. In the regression equation if we leave out the independent variables having significant effect on y, it will affect the fitting results.Regression flow is shown in Figure7. First of all, take coefficient of regression significance test on largest variable of partial correlation coefficient, which decide this variable whether enter regression equation. Then calculate F-numbers of each variable in regression equation, biased F test on the variable which biased F value minimum, to decide whether this variable stays in regression equation or not. Repeat the calculating, until no variable is pulled into or rejected.Figure 7
Procedure of regression equation.Let the dependent variabley be an observable random variable, and independent variables x1,x2,…,xp are generally variables. The general model of multivariate linear regression in sensitivity analysis of machining tool likes the following formula:
(3.2)y=β0+β1x1+β2x2+⋯+βpxp+ε,ε~N(0,σ2).Sample data matrix is shown in(3.3)[y1y2⋮yn]=[1x11x12⋯x1p1x21x22⋯x2p⋮⋮⋮⋮⋮1xn1xn2⋯xnp][β0β1β2⋮βp],yn=β0+β1xn1+β2xn2+⋯+βpxnp+εn,y1=β0+β1x11+β2x12+⋯+βpx1p+ε1,y2=β0+β1x21+β2x22+⋯+βpx2p+ε2,⋮⋮⋮yn=β0+β1xn1+β2xn2+⋯+βpxnp+εn,
where y represents the natural frequency (expressed by demand generally), β0,β1,…βp are unknown parameters, called regression coefficient; x1,x2,…xp are the influence factors of natural frequency (explanatory variables); ε denotes the random error, its mean is zero, and the variance is bigger than zero. That does not exist between the independent variable multicollinearity. In practice as long as the multicollinearity is weak, we can consider linear independence between the independent variables, and linear regression can be carried out.The regression equation solves for the coefficients by minimizing the sum of the squares of the deviations of the data from the model (least-square fit). The least-square fit of the model as follow:(3.4)Y=159.9456+0.1842X1+0.0940X2+0.0551X3+0.0011X4+0.0418X5+0.0468X6.According to the regression equation to know, for upper beam of hydraulic press the biggest influence for the natural frequency is the upper and lower plate thickness, influences followed are the left and right plate thickness and little influences are the front and back rib plate thickness. It can be seen through the equation that values ofX1 and X2 have greater impact on Y, which obtains the two higher sensitivity parameters of X1 and X2.


## 3.1.5. Variance Analysis of the Upper Beam

Analysis of variance (ANOVA) [34] is a statistical procedure for summarizing a classical linear model—a decomposition of sum of squares into a component for each source of variation in the model—along with an associated test (the F test) of the hypothesis that any given source of variation in the model is zero. When applied to generalized linear models, multilevel models, and other extensions of classical regression, ANOVA can be extended in two different directions. First, the F test can be used to compare models, to test the hypothesis that the simpler of the models is sufficient to explain the data. Second, the idea of variance decomposition can be interpreted as inference for the variances of batches of parameters (sources of variation) in multilevel regressions. Variance analysis table is shown in Table 3.Table 3
Variance analysis table.
Source of variance
Sum of squares
Freedom of motion
Mean square
F ratio
Significance
Regression
U
m
U
/
m
F
=
U
/
m
Q
/
(
n
-
m
-
1
)
Residual error
Q
(
n
-
m
-
1
)
Q
/
(
n
-
m
-
1
)
Sum
s
y
y
n
-
1The average value of the natural frequency is shown in(3.5)y-=1n∑k=1nyk.
Thus, the residual sum of squares and regression sum of squares are given by
(3.6)U=∑k=1n(y^k-y-)2,Q=∑k=1n(yk-y^k)2.Total variation is the aggregate dispersion of the individual data values around the overall mean of all factor levels as shown in(3.7)syy=U+Q=∑k=1n(yk-y-)2,
where yk represents the natural frequency of upper beam, which could be acquired by modal analysis; y- is the average value of the natural frequency yk; y^k is regression value, which could be acquired by regression equation; total variation can be split into two parts: U and Q, and U is regression sum of squares; Q is residual sum of squares. Regression values mean the difference between the square and reflect the volatility caused by changes in the independent variable and its degree of freedom (since the number of variables); syy=U+Q, where syy is total sum of squares (total variation).TheF distribution is a right-skewed distribution used most commonly in Analysis of Variance (like ANOVA/MANOVA). The F distribution is a ratio of two Chi-square distributions, and a specific F distribution is denoted by the degrees of freedom for the numerator Chi-square and the degrees of freedom for the denominator Chi-square. When referencing the F distribution, the numerator degrees of freedom are always given first, as switching the order of degrees of freedom changes the distribution (e.g., F(10,12) does not equal F(12,10)). The F distribution is an asymmetric distribution that has a minimum value of 0, but no maximum value. The curve reaches a peak not far to the right of 0 and then gradually approaches the horizontal axis the larger the F value is. The F distribution approaches, but never quite touches, the horizontal axis. The formula of F is obtained by
(3.8)F=U/mQ/(n-m-1),
where m is the number of variables; n is sum of all groups.According to multiple linear regression equation, the table of variance analysis is obtained (Table4). By the F distribution table, the regression equation is remarkable (Table 4). The result of the test also proves that the equation of multiple linear regression is linear relation.Table 4
Variance analysis of the upper beam.
Source of variance
Sum of squares
Freedom of motion
Mean square
F ratio
Significance
Regression
704.4000
6
117.4000
738.3600
**
Residual error
2.8611
18
0.1590
Sum
707.2611
24
*
The regression equation is remarkable.


## 3.2. Extraction of Key Structural Parameters with the Lower Beam

### 3.2.1. Parameters Model of the Lower Beam with the Python Language

The parameters model program of the lower beam as follow:X1=116 # the lower plate thickness, X2=96 # the left and right plate thickness, X3=128 # the left and right rib plate thickness, X4=142 # the front and back rib plate thickness, X5=170 # the front and back plate thickness, X6=128 # the upper plate thickness. Figure 8 illustrates the parameters model of the lower beam.Parameters model of the lower beam.
(a)
(b)


### 3.2.2. Orthogonal Design of the Lower Beam

Orthogonal design is a statistical optimization method, which arranges multifactor test scheme based on one set of prepared standard table—orthogonal table. This paper uses the orthogonal experimental design method to analyze hydraulic press of the lower beams. Figure8 illustrates the parameters model of the lower beam. According to the influencing factors and designed level, design table is L25(56) and there are 25 kinds of calculating schemes. Statistics analysis is carried on calculating results and the significant degree of each factor is judged on the test index.


### 3.2.3. Modal Analysis of the Lower Beam

This paper uses the python language to establish the lower beam parameters model, which is analyzed by the modal method. According to orthogonal experimental table, it only requires 25 times of the modal analysis and each modal analysis of model parameter values is based on Table5, and then submit the python command stream to the Abaqus, extract the first-order natural frequency, and use the batch mode to obtain the first-order natural frequency of 25 groups, and the results can be seen in Figure 9 and Table 5.Table 5
Data results of Modal analysis of the lower beam.
Six factors and five levels
X
1
X
2
X
3
X
4
X
5
X
6
Y
The lower plate thickness
The left and right plate thickness
The left and right rib plate thickness
The front and back rib plate thickness
The front and back plate thickness
The upper plate thickness
Natural frequency
1
116
96
128
142
170
128
175.88
2
118
96
134
144
174
132
176.27
3
120
96
132
136
172
126
176.48
4
122
96
126
140
166
130
177.33
5
124
96
130
138
168
134
178.44
6
116
98
130
140
172
132
176.11
7
118
98
128
138
166
126
175.85
8
120
98
134
142
168
130
176.98
9
122
98
132
144
170
134
177.72
10
124
98
126
136
174
128
176.98
11
116
100
126
144
168
126
174.89
12
118
100
130
136
170
130
176.15
13
120
100
128
140
174
134
176.81
14
122
100
134
138
172
128
176.73
15
124
100
132
142
166
134
177.90
16
116
102
132
138
174
130
175.41
17
118
102
126
142
172
134
176.04
18
120
102
130
144
166
128
175.89
19
122
102
128
136
168
132
176.82
20
124
102
134
140
170
126
176.51
21
116
104
134
136
166
134
175.99
22
118
104
132
140
168
128
175.33
23
120
104
126
138
170
132
175.96
24
122
104
130
142
172
126
175.57
25
124
104
128
144
174
130
176.32Figure 9
Results of modal analysis of the lower beam.


### 3.2.4. Regression Analysis of the Lower Beam

The so-called regression analysis is based on a large number of observation data, function expression of regression relationship between the independent variable, and the dependent variable called the regression equation is established by using of mathematical statistics methods. This paper achieves the linear regression analysis.The analysis results of multiple linear regression are shown as follows:(3.9)Y=155.1471+0.1981X1+0.1343X2+0.0337X3+0.0287X4+0.0414X5+0.1353X6.According to the regression equation to know, for the lower beam of hydraulic press the biggest influence for the natural frequency is the lower plate thickness, influences followed are the upper plate thickness, and little influences are the front and back rib plate thickness.It can be seen through the equation that values ofX1 and X6 have greater impact on Y (natural frequency), then obtain the highest sensitivity of the two parameters: the lower plate thickness (X1), the upper plate thickness (X6).According to multiple linear regression equation, the table of variance analysis (Table6) can be obtained.Table 6
Regression equation of variance analysis of the lower beam.
Source of variance
Sum of squares
Freedom of motion
Mean square
F ratio
Significance
Regression
150.5780
6
25.1000
145.7610
**
Residual error
3.0990
18
0.1722
Sum
153.6770
24
*
The regression equation is remarkable.By theF distribution table, when there is remarkable degree of α=0.01, F0.01(6,18)=4.01. Because of F=145.761≫4.01, that is, the F≫F0.01, so the regression equation is very remarkable (Table 6). The results of the test prove that multiple linear regression equation is linear relation, and the regression equation is very remarkable.From the formula (3.9), we come to the conclusion that sort effects of the design parameters on the lower beam to the natural frequency: X1, X6, X2, X5, X3, X4, that is to say, X1 and X6 are the upper beam sensitive parameters.


## 3.2.1. Parameters Model of the Lower Beam with the Python Language

The parameters model program of the lower beam as follow:X1=116 # the lower plate thickness, X2=96 # the left and right plate thickness, X3=128 # the left and right rib plate thickness, X4=142 # the front and back rib plate thickness, X5=170 # the front and back plate thickness, X6=128 # the upper plate thickness. Figure 8 illustrates the parameters model of the lower beam.Parameters model of the lower beam.
(a)
(b)


## 3.2.2. Orthogonal Design of the Lower Beam

Orthogonal design is a statistical optimization method, which arranges multifactor test scheme based on one set of prepared standard table—orthogonal table. This paper uses the orthogonal experimental design method to analyze hydraulic press of the lower beams. Figure8 illustrates the parameters model of the lower beam. According to the influencing factors and designed level, design table is L25(56) and there are 25 kinds of calculating schemes. Statistics analysis is carried on calculating results and the significant degree of each factor is judged on the test index.


## 3.2.3. Modal Analysis of the Lower Beam

This paper uses the python language to establish the lower beam parameters model, which is analyzed by the modal method. According to orthogonal experimental table, it only requires 25 times of the modal analysis and each modal analysis of model parameter values is based on Table5, and then submit the python command stream to the Abaqus, extract the first-order natural frequency, and use the batch mode to obtain the first-order natural frequency of 25 groups, and the results can be seen in Figure 9 and Table 5.Table 5
Data results of Modal analysis of the lower beam.
Six factors and five levels
X
1
X
2
X
3
X
4
X
5
X
6
Y
The lower plate thickness
The left and right plate thickness
The left and right rib plate thickness
The front and back rib plate thickness
The front and back plate thickness
The upper plate thickness
Natural frequency
1
116
96
128
142
170
128
175.88
2
118
96
134
144
174
132
176.27
3
120
96
132
136
172
126
176.48
4
122
96
126
140
166
130
177.33
5
124
96
130
138
168
134
178.44
6
116
98
130
140
172
132
176.11
7
118
98
128
138
166
126
175.85
8
120
98
134
142
168
130
176.98
9
122
98
132
144
170
134
177.72
10
124
98
126
136
174
128
176.98
11
116
100
126
144
168
126
174.89
12
118
100
130
136
170
130
176.15
13
120
100
128
140
174
134
176.81
14
122
100
134
138
172
128
176.73
15
124
100
132
142
166
134
177.90
16
116
102
132
138
174
130
175.41
17
118
102
126
142
172
134
176.04
18
120
102
130
144
166
128
175.89
19
122
102
128
136
168
132
176.82
20
124
102
134
140
170
126
176.51
21
116
104
134
136
166
134
175.99
22
118
104
132
140
168
128
175.33
23
120
104
126
138
170
132
175.96
24
122
104
130
142
172
126
175.57
25
124
104
128
144
174
130
176.32Figure 9
Results of modal analysis of the lower beam.


## 3.2.4. Regression Analysis of the Lower Beam

The so-called regression analysis is based on a large number of observation data, function expression of regression relationship between the independent variable, and the dependent variable called the regression equation is established by using of mathematical statistics methods. This paper achieves the linear regression analysis.The analysis results of multiple linear regression are shown as follows:(3.9)Y=155.1471+0.1981X1+0.1343X2+0.0337X3+0.0287X4+0.0414X5+0.1353X6.According to the regression equation to know, for the lower beam of hydraulic press the biggest influence for the natural frequency is the lower plate thickness, influences followed are the upper plate thickness, and little influences are the front and back rib plate thickness.It can be seen through the equation that values ofX1 and X6 have greater impact on Y (natural frequency), then obtain the highest sensitivity of the two parameters: the lower plate thickness (X1), the upper plate thickness (X6).According to multiple linear regression equation, the table of variance analysis (Table6) can be obtained.Table 6
Regression equation of variance analysis of the lower beam.
Source of variance
Sum of squares
Freedom of motion
Mean square
F ratio
Significance
Regression
150.5780
6
25.1000
145.7610
**
Residual error
3.0990
18
0.1722
Sum
153.6770
24
*
The regression equation is remarkable.By theF distribution table, when there is remarkable degree of α=0.01, F0.01(6,18)=4.01. Because of F=145.761≫4.01, that is, the F≫F0.01, so the regression equation is very remarkable (Table 6). The results of the test prove that multiple linear regression equation is linear relation, and the regression equation is very remarkable.From the formula (3.9), we come to the conclusion that sort effects of the design parameters on the lower beam to the natural frequency: X1, X6, X2, X5, X3, X4, that is to say, X1 and X6 are the upper beam sensitive parameters.


## 3.3. Extraction of Key Structural Parameters with the Column

### 3.3.1. Parameters Model of the Column with the Python Language

The parameters model program of the column as follow: from part import *from material import *from section import *from assembly import *from step import *from interaction import *from load import *from mesh import *from job import *from sketch import *from visualization import *from connector Behavior import*.X1=116 # The top to the first rib plates, X2=96 # The first to the second rib plates, X3=128 # The remaining distance between rib plates, X4=142 # The roof column width, X5=170 # The column plates thickness, X6=128 # The rib plates thickness. Figure 10 illustrates the parameters model of the column.Figure 10
Parameters model of the column.


### 3.3.2. Orthogonal Design of the Column

This paper uses the orthogonal experimental design method to analyze six parameters of the columns (Figure10) and arranges five levels for each parameter as a factor. According to the six factors and five levels of orthogonal design table, obtain six factors and five levels of orthogonal experimental data of the column. Six factors contain the top to the first rib plates (X1), the first to the second rib plates (X2), the remaining distance between rib plates (X3), the roof column width (X4), the column plates thickness (X5), and the rib plates thickness (X6).


### 3.3.3. Modal Analysis of the Column

This paper uses python language to establish the model of the column, which is analyzed by the modal method. According to orthogonal experimental table, it only requires 25 times of the modal analysis and each modal analysis of model parameter values is based on Table7, and then submit the python command stream to the Abaqus, extract the first-order natural frequency, and use the batch mode to quickly extract the first-order natural frequency of 25, and the results are shown in (Figure 10 and Table 6).Table 7
Modal analysis results of the column.
Six factors and five levels
X
1
X
2
X
3
X
4
X
5
X
6
Y
The top to the first rib plates
The first to  the second rib plates
The remaining distance between rib plates
The roof column width
The columnplates thickness
The rib plates thickness
Natural frequency
1
800
800
925
1050
135
45
17.953
2
900
800
1150
1100
150
55
18.016
3
1000
800
1075
900
143
40
18.063
4
1100
800
850
1000
120
50
17.906
5
1200
800
1000
950
127
60
17.859
6
800
900
1000
1000
143
55
17.965
7
900
900
925
950
120
40
17.903
8
1000
900
1150
1050
127
50
17.960
9
1100
900
1075
1100
135
60
17.921
10
1200
900
850
900
150
45
18.072
11
800
1000
850
1100
127
40
17.773
12
900
1000
1000
900
135
50
17.779
13
1000
1000
925
1000
150
60
17.842
14
1100
1000
1150
950
143
45
17.859
15
1200
1000
1075
1050
120
55
17.799
16
800
1100
1075
950
150
50
17.768
17
900
1100
850
1050
143
60
17.778
18
1000
1100
1000
1100
120
45
17.772
19
1100
1100
925
900
127
55
17.785
20
1200
1100
1150
1000
135
40
16.739
21
800
1200
1150
900
120
60
17.656
22
900
1200
1075
1000
127
45
17.833
23
1000
1200
850
950
135
55
17.766
24
1100
1200
1000
1050
143
40
17.941
25
1200
1200
925
1100
150
50
17.937The essence of modal analysis is to solve the matrix eigenvalues, according to the table of orthogonal design, which only needs the modal analysis of 25, times, and each model parameter is based on Table1. The python command stream is submitted to the Abaqus, which extracts the first order natural frequency (Table 7 and Figure 11). The first order natural frequency of 25 groups is quickly extracted by using the batch mode, and the results of batch mode can be seen in Table 7.Figure 11
Results of modal analysis of the column.


### 3.3.4. Regression Analysis of the Lower Beam

Regression analysis is a parametric method that requires a specification of the analytical expression of the functional form that links the inputs and outputs. The equation of multiple linear regressionwith the lower beam is shown as follows:(3.10)Y=18.4176+0.0003X1+0.0007X2+0.0006X3+0.0002X4+0.004X5+X6.According to the regression equation to know, for the column the biggest influence for the natural frequency is the rib plates thickness, influences followed are the column plates thickness, and little influences are the roof column width.By theF distribution table, when there is remarkable degree of α=0.01, F0.01(6,18)=4.01. Because of F=169.1700≫4.01, that is, the F≫F0.01, so the regression equation is very remarkable (Table 8). The test proves that multiple linear regression equation is linear relation.Table 8
Variance analysis of the column.
Source of variance
Sum of squares
Freedom of motion
Mean square
F ratio
Significance
Regression
55.5200
6
9.250
169.1700
**
Residual error
0.9842
18
0.0547
Sum
56.5042
24
*
The regression equation is remarkable.According to multiple linear regression equation in a similar way, the results of variance analysis are shown in Table8.From the formula, we can come to the conclusion that sort effects of the design parameters on the lower beam to the natural frequency:X6, X5, X2, X3, X1, X4, that is to say, X6 and X5 are the column sensitive parameters.


## 3.3.1. Parameters Model of the Column with the Python Language

The parameters model program of the column as follow: from part import *from material import *from section import *from assembly import *from step import *from interaction import *from load import *from mesh import *from job import *from sketch import *from visualization import *from connector Behavior import*.X1=116 # The top to the first rib plates, X2=96 # The first to the second rib plates, X3=128 # The remaining distance between rib plates, X4=142 # The roof column width, X5=170 # The column plates thickness, X6=128 # The rib plates thickness. Figure 10 illustrates the parameters model of the column.Figure 10
Parameters model of the column.


## 3.3.2. Orthogonal Design of the Column

This paper uses the orthogonal experimental design method to analyze six parameters of the columns (Figure10) and arranges five levels for each parameter as a factor. According to the six factors and five levels of orthogonal design table, obtain six factors and five levels of orthogonal experimental data of the column. Six factors contain the top to the first rib plates (X1), the first to the second rib plates (X2), the remaining distance between rib plates (X3), the roof column width (X4), the column plates thickness (X5), and the rib plates thickness (X6).


## 3.3.3. Modal Analysis of the Column

This paper uses python language to establish the model of the column, which is analyzed by the modal method. According to orthogonal experimental table, it only requires 25 times of the modal analysis and each modal analysis of model parameter values is based on Table7, and then submit the python command stream to the Abaqus, extract the first-order natural frequency, and use the batch mode to quickly extract the first-order natural frequency of 25, and the results are shown in (Figure 10 and Table 6).Table 7
Modal analysis results of the column.
Six factors and five levels
X
1
X
2
X
3
X
4
X
5
X
6
Y
The top to the first rib plates
The first to  the second rib plates
The remaining distance between rib plates
The roof column width
The columnplates thickness
The rib plates thickness
Natural frequency
1
800
800
925
1050
135
45
17.953
2
900
800
1150
1100
150
55
18.016
3
1000
800
1075
900
143
40
18.063
4
1100
800
850
1000
120
50
17.906
5
1200
800
1000
950
127
60
17.859
6
800
900
1000
1000
143
55
17.965
7
900
900
925
950
120
40
17.903
8
1000
900
1150
1050
127
50
17.960
9
1100
900
1075
1100
135
60
17.921
10
1200
900
850
900
150
45
18.072
11
800
1000
850
1100
127
40
17.773
12
900
1000
1000
900
135
50
17.779
13
1000
1000
925
1000
150
60
17.842
14
1100
1000
1150
950
143
45
17.859
15
1200
1000
1075
1050
120
55
17.799
16
800
1100
1075
950
150
50
17.768
17
900
1100
850
1050
143
60
17.778
18
1000
1100
1000
1100
120
45
17.772
19
1100
1100
925
900
127
55
17.785
20
1200
1100
1150
1000
135
40
16.739
21
800
1200
1150
900
120
60
17.656
22
900
1200
1075
1000
127
45
17.833
23
1000
1200
850
950
135
55
17.766
24
1100
1200
1000
1050
143
40
17.941
25
1200
1200
925
1100
150
50
17.937The essence of modal analysis is to solve the matrix eigenvalues, according to the table of orthogonal design, which only needs the modal analysis of 25, times, and each model parameter is based on Table1. The python command stream is submitted to the Abaqus, which extracts the first order natural frequency (Table 7 and Figure 11). The first order natural frequency of 25 groups is quickly extracted by using the batch mode, and the results of batch mode can be seen in Table 7.Figure 11
Results of modal analysis of the column.


## 3.3.4. Regression Analysis of the Lower Beam

Regression analysis is a parametric method that requires a specification of the analytical expression of the functional form that links the inputs and outputs. The equation of multiple linear regressionwith the lower beam is shown as follows:(3.10)Y=18.4176+0.0003X1+0.0007X2+0.0006X3+0.0002X4+0.004X5+X6.According to the regression equation to know, for the column the biggest influence for the natural frequency is the rib plates thickness, influences followed are the column plates thickness, and little influences are the roof column width.By theF distribution table, when there is remarkable degree of α=0.01, F0.01(6,18)=4.01. Because of F=169.1700≫4.01, that is, the F≫F0.01, so the regression equation is very remarkable (Table 8). The test proves that multiple linear regression equation is linear relation.Table 8
Variance analysis of the column.
Source of variance
Sum of squares
Freedom of motion
Mean square
F ratio
Significance
Regression
55.5200
6
9.250
169.1700
**
Residual error
0.9842
18
0.0547
Sum
56.5042
24
*
The regression equation is remarkable.According to multiple linear regression equation in a similar way, the results of variance analysis are shown in Table8.From the formula, we can come to the conclusion that sort effects of the design parameters on the lower beam to the natural frequency:X6, X5, X2, X3, X1, X4, that is to say, X6 and X5 are the column sensitive parameters.


## 4. The Finite Element Analysis of Hydraulic Press

The natural frequency, also known as the fundamental frequency, refers to the number of times a given event will happen in a second. According to fundamentals of vibration analysis, the stiffness of machine tools is proportional to the natural frequency in the same conditions. Enhancing the natural frequency improves the dynamic performance of machine tools to satisfy the performance requirement of high stiffness and light mass of the 100MN hydraulic press. So the natural frequency can be used to evaluate the performances of strength and stiffness in a sense. Enhancing the natural frequency, reducing the mass, and improving the stiffness are very important to improve the machine performance.According to the novel method of sensitivity analysis in structural performance of hydraulic press, the key dimension parameters are efficiently obtained: the column ofX5 (the column plates thickness), X6 (the rib plates thickness), the lower beam of X1 (the lower plate thickness), X6 (the upper plate thickness), the upper beam of X1 (the upper and lower plate thickness), and X4 (the front and back rib plate thickness). The 100MN hydraulic press of the whole structural model based on the key dimension parameters and other parameters is analyzed in order to verify the influence of the sensitivity parameters on strength and stiffness of the machine and identify the impact order of various sensitivity parameters to strength and stiffness of the machine which is consistent with the sensitivity ranking of regression analysis. The whole structural FEA model based on key dimension parameters and other dimension parameters is determined by modal analysis, regression analysis, and random selection.The results are shown in Figure12. The higher sensitivity parameters of modal analysis compared with other parameters have greater impact on strength and stiffness. The results of the whole structural FEA model have showed that the impact order of various sensitivity parameters to strength and stiffness of the machine has been consistent with the sensitivity ranking of regression analysis.Figure 12
Stress analysis of hydraulic press.


## 5. Conclusion

This paper has conducted the study on the higher sensitivity parameters of hydraulic press with a novel sensitivity analysis method in structural performance. The results have shown that(1) the higher sensitivity parameters had the greatest impact on strength and stiffness and increasing the higher sensitivity parameters remarkably has enhanced the strength and stiffness of machine tools (2). The results of the whole structural FEA model have showed that the impact order of various sensitivity parameters has been consistent with the sensitivity ranking of regression analysis. These parameters of high sensitivity have been used as the focus of concern, such as the design variables of optimization. It has been found that the higher sensitivity parameters remarkably affected the structure performance of hydraulic press (3). The finite element analysis in structural performance of hydraulic press to strength and stiffness has needed much computation time than the modal analysis, and the modal analysis which has extracted the natural frequency has been more convenient and of short time in structural performance of hydraulic press, which has achieved the same result in structural performance of hydraulic press (4). There has been no need to conduct directly the strength and stiffness analysis in structural performance of hydraulic press by novel sensitivity analysis method. The research results have provided the basis for the forging machine design, and the methods can also provide reference to machine tools and equipment research.



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*Source: 647127-2012-10-18.xml*