Phase-Division-Based Dynamic Optimization of Linkages for Drawing Servo Presses

Abstract

Existing linkage-optimization methods are designed for mechanical presses; few can be directly used for servo presses, so development of the servo press is limited. Based on the complementarity of linkage optimization and motion planning, a phase-division-based linkage-optimization model for a drawing servo press is established. Considering the motion-planning principles of a drawing servo press, and taking account of work rating and efficiency, the constraints of the optimization model are constructed. Linkage is optimized in two modes: use of either constant eccentric speed or constant slide speed in the work segments. The performances of optimized linkages are compared with those of a mature linkage SL4-2000A, which is optimized by a traditional method. The results show that the work rating of a drawing servo press equipped with linkages optimized by this new method improved and the root-mean-square torque of the servo motors is reduced by more than 10%. This research provides a promising method for designing energy-saving drawing servo presses with high work ratings.

1 Introduction

Mechanical drawing presses are highly efficient and widely used for sheet metal drawing. With the development of automated technology, drawing and other presses with various functions have been integrated into stamping lines. The slide speed of a drawing press needs to meet the limitations of drawing speed and accommodate the high productivity of the stamping line, so the idle slide stroke time needs to be as short as possible. To fulfill this purpose, a multi-bar linkage is generally used for the drive system [1,2,3]. At present, most large drawing presses for sheet metal, both in China and abroad, adopt six- or eight-bar linkages; however, because of inherent geometric and structural limitations of the linkage mechanism, the work rating and efficiency of mechanical presses have been limited [4,5,6].

Use of a servo press can overcome the above shortcomings [7]. Along with servo-drive control, a wide range of movements can be executed with improved accuracy and the slide speed can be freely controlled, endowing the slide with variable motion features that can overcome the intrinsic limitations of the linkage mechanism in motion curves and enable the servo press to better match high-speed feeding systems [8,9,10,11]. In addition, good coordination between the servo-drive system and its linkages can decrease motor power and energy consumption of the drive system [12,13,14]. Based on these features, stamping lines equipped with drawing servo presses routinely break productivity records in various industries.

At present, the Stephenson six-bar linkage is commonly used on large drawing servo presses. Its structure is same as that of a traditional six-bar linkage mechanical press. To obtain optimum matching between the servo-drive system and linkages, linkage optimization is necessary for the drawing servo press. Zhou proposed an objective function method and used it to optimize the mechanism of a servo punch press [15, 16]. Mundo proposed a method for the synthesis of planar mechanisms: the optimized mechanism was able to move a press ram according to an optimal law of motion [17]. Du used an inheritance calculation method to optimize linkage of a press [18]. Although optimization of linkage has been studied and discussed worldwide, few results have been practically applied [19,20,21,22]. Further research on linkage optimization of drawing servo presses is still needed.

This paper aims to provide a new method for linkage optimization of a drawing servo press. The objective function and constraints were established based on a motion phase-division model of a servo press. Original data and a research methodology for the design and application of a drawing servo press are provided.

2 Methods

2.1 Phase-Division of Slide Motion

Basing on the drawing process, the slide motion of a drawing servo press was phase-divided into four segments: the work segment (SegWork), feeding segment (SegFeed), deceleration segment (SegDec), and back-acceleration segment (SegBkAcc), as shown in Figure 1. The principles corresponding to each segment were considered in the process of motion planning, as listed in Table 1.

Figure 1

Phase-division of drawing servo press slide motion

Table 1 Motion-planning principles corresponding to four defined phase-divided segments

Motion planning for the drawing servo press aimed to obtain the curves of eccentric motion, which can coordinate with other servo presses in the stamping line to achieve the set work rating. Optimization of linkage, such as implementing good technological properties, high work rating, and low vibration and noise, can enable the motion planning to more easily obtain high-quality motion curves. The linkage-optimization principles should therefore be established based on the principles of motion planning.

2.2 Constraints of Linkage Optimization

Using the principles of motion planning, linkage-optimization principles for a drawing servo press are proposed, as listed in Table 2.

Table 2 Linkage-optimization principles corresponding to motion-planning principles

Based on these principles, the objective functions and constraints were defined as follows.

2.2.1 Constraints of Slide Displacement

The relationship between slide displacement and velocity was defined as follows:

$$f_{1} (X) = \int_{{\varphi_{{S_{c} }} }}^{{\varphi_{{S_{p} }} }} {[Vs(\varphi )} - Vs_{\text{WkC}} ]^{2} {\text{d}}\varphi ,$$

(1)

where $\varphi_{{S_{c} }}$ is the angle corresponding to the maximum work stroke, $\varphi_{{S_{p} }}$ is the angle corresponding to the nominal stroke. Vs(φ) is the slide velocity function, and X is an optimized parameter vector.

S _c and S _p meet the following constraints respectively:

$$h_{1} (X) = s(\varphi_{{S_{c} }} ) - S_{c} = 0,$$

(2)

$$h_{2} (X) = s(\varphi_{{S_{p} }} ) - S_{p} = 0,$$

(3)

where S _p is the nominal stroke, S _c is the maximum work stroke, and s(φ) is the slide displacement function.

2.2.2 Constraints of Slide Velocity

The friction loss of the linkage joint in SegWork can be expressed using the following function:

$$f_{2} (X) = \sum\limits_{j = 1}^{{N_{j} }} {\int_{{\varphi_{Sc} }}^{{\varphi_{\text{BDC}} }} {\frac{{\mu_{j} D_{j} }}{2}} } F_{R,j} (\varphi )\varOmega_{j} (\varphi ){\text{d}}\varphi ,$$

(4)

where F _R,j (φ) is the radial force of joint j, $\varOmega_{j} (\varphi ) = \omega_{j} (t)/\omega_{\text{ECC}}$ is the similar velocity of joint j, φ _BDC is the angle at bottom dead center (BDC), μ _j is the friction coefficient of joint j, and D _j is the diameter of joint j.

The constraints at BDC and top dead center (TDC) are defined as:

$$h_{3} (X) = Vs(\varphi_{\text{BDC}} ) = 0,$$

(5)

$$h_{4} (X) = Vs(\varphi {}_{\text{TDC}}) = 0,$$

(6)

where φ _TDC is the angle at TDC. The friction between the slide and the gib can be ignored for two-point and four-point presses.

2.2.3 Constraints of Ideal Equivalent Force Arm

The ideal equivalent force arm (m _IE) is given by:

$$m_{\text{IE}} = \frac{{M_{I} }}{{P_{n} }} = Vs(\varphi_{{S_{p} }} ),$$

(7)

where M _I is the ideal press nominal torque and P _n is the press capacity.

Ideal equivalent force arm can significantly influence the gear inertia and servo motor torque. The constraint is defined as:

$$g_{1} (X) = m_{\text{IE}} - C_{1} \le 0,$$

(8)

where C ₁ is a constant.

2.2.4 Constraints in SegFeed

The open height (h _A ) and closed height (h _p ) of the die meet the following constraints:

$$h_{5} (X) = s(\varphi_{p} ) - h_{p} = 0,$$

(9)

$$h_{6} (X) = s(\varphi {}_{A}) - h_{A} = 0,$$

(10)

where φ _p and φ _A are the eccentric angles corresponding to die closure and opening, respectively.

The constraint of φ _A and φ _p can be defined as follows (C ₂ is a constant):

$$g_{2} (X) = C_{2} - (\varphi_{A} - \varphi_{P} ) \le 0.$$

(11)

Maximum acceleration of the slide near TDC meets the following constraint (C ₃ is a constant):

$$g_{3} (X) = C_{3} - \inf \left\{ {g(\varphi )\left| {\varphi \in \left[ {0,2\pi } \right]} \right.} \right\} \le 0.$$

(12)

The constraints of the linkage pressure angles are given by:

$$g_{4} (X) = C_{4} - \inf \left\{ {\theta_{J7} (\varphi )\left| {\varphi \in \left[ {\varphi_{{S_{c} }} ,\varphi_{\text{BDC}} } \right]} \right.} \right\} \le 0,$$

(13)

$$g_{5} (X) = \sup \left\{ {\theta_{J7} (\varphi )\left| {\varphi \in \left[ {\varphi_{{S_{c} }} ,\varphi_{\text{BDC}} } \right]} \right.} \right\} - C_{5} \le 0,$$

(14)

where $\theta_{J7} (\varphi )$ is the inclination between the linkage axis and vertical direction and C ₄ and C ₅ are constant values.

2.2.5 Constraints of Equivalent Inertia

The linkage inertia equation is given by:

$$f_{3} (X) = \sup \left\{ {Je_{\text{Lk}} (\varphi )\left| {\varphi \in \left[ {0,2\pi } \right]} \right.} \right\} - C_{\text{JeL}} ,$$

(15)

where $Je_{\text{LK}} \left( \varphi \right)$ is the equivalent inertial of linkage as a function of eccentric angle C _JeL is a constant.

Maximum equivalent downward inertia with slide and die is given by:

$$f_{4} (X) = \sup \left\{ {Je_{s} (\varphi )\left| {\varphi \in \left[ {\varphi_{\text{TDC}} - 2\pi ,\varphi_{\text{BDC}} } \right]} \right.} \right\} - C_{\text{Jd}} ,$$

(16)

where $Je_{s} \left( \varphi \right)$ is the equivalent inertial of the linkage, slide, and tool as a function of eccentric angle and C _Jd is a constant.

Maximum equivalent inertia in the back stroke with slide and die (C _Ju is a constant) is given by:

$$f_{5} (X) = \sup \left\{ {Je_{s} (\varphi )\left| {\varphi \in \left[ {\varphi_{\text{BDC}} ,\varphi_{\text{TDC}} } \right]} \right.} \right\} - C_{\text{Ju}} .$$

(17)

2.2.6 Slide Velocity in SegWork

Slide constant velocity, Vs _WkC , in SegWork was taken as a design parameter.

2.2.7 Other Constraints

Other constraints included: a size constraint for the linkage component; a space-size constraint for linkage components at limited position; a total stroke constraint for the slide; a constraint for the slide in the linkage group [23].

2.3 Objectives of Linkage Optimization

The unknown variable X included nine size variables of the six-bar linkage (R, X ₂, Y ₂, L ₂, X ₃, L ₄₂, L ₄₅, Ang ₄, L ₅), where R is crank length, X ₂ is horizontal distance from the pull-bar shaft to the eccentric shaft, Y ₂ is vertical distance from pull-bar shaft to eccentric shaft, L ₂ is pull-bar length, X ₃ is the horizontal deviation of the small-end axis of the link to the eccentric shaft, L ₄₂ is the upper-arm length of the angle plate, L ₄₅ is the lower-arm length of the angle plate, Ang ₄ is the inclination between the upper and lower arms of the angle plate, and L ₅ is the length of connecting rod, as shown in Figure 2. The phase-division angles included φ _TDC, φ _P, $\varphi_{{S_{c} }}$, $\varphi_{{S_{p} }}$, φ _BDC, and φ _A .

Figure 2

Structure and size parameters of six-bar linkage used for drawing servo press

The objectives of six-bar linkage optimization were defined as follows:

$$\mathop {\hbox{min} }\limits_{{X \in R^{15} }} F(X) = \sum\limits_{i} {w_{i} } f_{i} (X),$$

(18)

$$s.t. g_{u} (X) \le 0 (u = 1 , 2 , \ldots 5,\ldots ),$$

(19)

$$h_{v} (X) = 0 (v = 1 , 2 , \ldots 5,\ldots ).$$

(20)

Owing to the large number and complexity of the constraints, it is difficult to find suitable results for certain conditions. The interior-point algorithm of the non-linear constraint optimization function, fmincon, in MATLAB ^TM was therefore used to solve the above problems.

2.4 Method for Calculation of Equivalent Inertia

The linkage components used on drawing servo presses are often irregularly shaped, as shown in Figure 2. The mass and inertia of each component needs to be calculated at each iteration in the process of dynamic optimization. It is therefore essential to efficiently and precisely obtain the mass, mass center position, and rotary inertia of each component [24]. In this paper, a convenient method is proposed to obtain these parameters. The components are divided into trapezoid and truncated cones, where the areas of the new parts can be either positive or negative (negative means that material is removed), as shown in Figure 3. It is easier to obtain analytical expressions for the first and second moments to origin of these two shapes.

Figure 3

Area moment of trapezoid and truncated cone to origin: (a) origin moment of trapezoid; (b) origin moment of truncated cone; (c) linkage shape division by different axis zone; (d) division of axis zones

This division method is suitable for planar linkages and even for eccentric complex shapes. The inertia calculation based on this method achieves a high level of accuracy: it has better accuracy than three-dimensional inertia calculation models. Eqs. (15)–(17), used for calculation of inertia, are based on this method.

2.5 Treatment of Linkage Joint Friction Loss

The force analysis of a six-bar linkage is a nonlinear problem: iteration is required to obtain a solution. Taking friction into account in the process of force analysis will decrease the efficiency of solution of linkage optimization. To improve the efficiency of the solution process, a dynamic static force analysis method was employed to calculate the radial forces of joints, ignoring the joint friction. After optimization, dynamic simulation could then be used to verify the reduction of friction loss due to optimization.

2.6 Assignment of Initial Value

To test the stability of this proposed calculation method and evaluate its capability to obtain a solution from any initial value, the size of a similar six-bar linkage obtained by ruler measurement on a technical document was used as the initial value, as shown in Figure 4. Because the initial value was roughly measured by ruler, the data error was large, as listed in Table 3.

Figure 4

Image of six-bar linkage in technical document

Table 3 Sizes of optimized linkage based on constant-eccentric- speed mode

3 Case Study

The first 20 MN drawing servo press (SL4-2000A) in China, made by JIER (JIER Machine-Tool Group Co., Ltd), was used to demonstrate this optimization method. The capacity of the press is 20 MN, its nominal working stroke is 8 mm, the slide stroke is 1300 mm, and there are 15 slide strokes per minute. Its transmission system uses a six-bar linkage (Figure 2) that was precisely optimized by traditional methods during the design stage [25, 26]. In this paper, the six-bar linkage was again optimized using the as-proposed method.

4 Results and Discussion

4.1 Optimization Based on Constant-eccentric-speed Mode

Use of constant eccentric speed at SegWork is the basic work mode (constant-eccentric-speed mode) of this drawing servo press, which is beneficial to the full conversion of motor torque to forming force in the drawing process. If a six-bar linkage is designed in constant-eccentric-speed mode, the ideal motion curve is that in which the slide velocity is kept constant in SegWork, as shown in Table 2.

The validity of the optimization method in constant-eccentric-speed mode was verified in two ways: by optimizing under the same design parameters as those of the SL4-2000A and by optimizing under different design parameters.

4.1.1 Optimization under Same Design Parameters as SL4-2000A

When the SL4-2000A eccentric rotates at a constant speed of 1 rad/s, the corresponding slide velocity at SegWork is about 303 mm/s. Linkage optimization in this section was therefore carried out using S _c = 300 mm (maximum work stroke) and Vs_WkC=300 mm/s (slide velocity in Segwork). The simulated load in the work strokes used for motion planning is shown in Figure 5.

Figure 5

Simulated load for work stroke of 300 mm

The sizes of linkage Lk₁ optimized by the above method are shown in Table 3. The actual SL4-2000A linkage size is proprietary knowledge of JIER, so cannot be revealed here; however, the performances of the two linkages were compared as follows.

The slide velocity in SegWork and slide displacement of optimized linkage Lk₁ coincided well with those of SL4-2000A, as shown in Figs. 6 and 7, respectively. Basic parameters of linkages Lk₁ and SL4-2000A are listed in Table 4, from which it is easily shown that, even if starting from a rough initial value, the proposed optimization method can still obtain a solution that is extremely close to that of the mature linkage of SL4-2000A.

Figure 6

Comparison of slide velocity in SegWork

Figure 7

Comparison of slide displacement

Table 4 Comparison of basic parameters for optimized linkage Lk₁ and SL4-2000A

When the slide work stroke was 300 mm (S _wk = 300 mm), the maximum work rating of the drawing servo press equipped with linkage Lk₁ was about 15 s/min, which was equivalent to that of the 20 MN drawing servo press (SL4-2000A); however, the root-mean-square value of torque of linkage Lk₁ was 3.4% lower, as shown in Table 5. Under the conditions of S _wk = 300 mm and Nc = 15 s/min, the motion-planning curves of linkage Lk₁ are shown in Figure 8. The above results indicate the validity of the optimization method in constant-eccentric-speed mode.

Table 5 Motion-planning results for optimized linkage Lk₁

Figure 8

Motion-planning curves based on S _wk = 300 mm and Nc = 15 r/min

4.1.2 Optimization under Design Parameters Differing from SL4-2000A

In Section 4.1.1, the major difference identified between optimized linkage Lk₁ and that of SL4-2000A was that the former had slightly lower velocity Vs_WkC in SegWork, which is beneficial to improving stamping quality. This study therefore aimed to lower velocity Vs_WkC while maintaining the equivalent inertia as low as possible, thereby decreasing energy consumption during the acceleration and deceleration segments.

Under the conditions of S _c = 300 mm, Vs_WkC = 280 mm/s, and a different die open angle, optimized linkages Lk₂ and Lk₃ were obtained, the sizes and basic parameters of which are shown in Tables 3 and 6, respectively. Compared with SL4-2000A, the inertias of linkages Lk₂ and Lk₃ slightly increased due to the decrease of Vs_WkC (Table 6); however, the root-mean-square value of torque was significantly decreased, as shown in Table 7. Under the condition of S _wk=300 mm and Nc=15 s/min, the root-mean-square torque of linkage Lk₂ was more than 10% smaller than that of SL4-2000A; for linkage Lk₃, the value was 5.2% smaller. Moreover, compared with SL4-2000A, the ideal equivalent force arm of linkages Lk₂ and Lk₃ also decreased, resulting in a decrease of the gear system design load, which is beneficial to lifetime extension and weight reduction of the gear system. Taking linkage Lk₂ as an example, the motion-planning curves are shown in Figure 9.

Table 6 Comparison of basic parameters for optimized linkages Lk₂ and Lk₃ and SL4-2000A

Table 7 Motion-planning results for optimized linkages Lk₂ and Lk₃

Figure 9

Motion-planning curves based on S _wk = 300 mm and Nc = 15 s/min

According to the above results and analyses, the following conclusions can be drawn. If the basic work mode of a drawing servo press employs constant-eccentric- speed in SegWork, the slide speed of a drawing servo press equipped with a linkage optimized by the above method can approach ideal during the work phase. This optimization method can precisely control Vs_WkC, the ideal force arm, and the friction consumption of the linkage in SegWork, without obviously increasing the moments of inertia. During the process of optimization, the speed and ideal force arm can be selected as needed. This method has advantages over the traditional method in its ability to improve energy saving and running efficiency of a linkage. In addition, servo motion planning techniques can be merged into the linkage-optimization process as a reference for selection of linkage-optimization parameters, thereby comprehensively improving the properties of the different phases.

4.2 Optimization Based on Constant-slide-speed Mode

The typical characteristic of slide velocity in SegWork is shown in Figure 6: slide velocity Vs_WkC fluctuates in the segment and the slide stroke corresponding to constant velocity Vs_WkC is too short (only about half of the maximum work stroke). Owing to the limitations of a six-bar linkage, traditional mechanical presses cannot overcome these two shortcomings; however, the drawing servo press can overcome these by motion planning in constant-slide-speed mode. A schematic diagram of slide speed control is given in Figure 10.

Figure 10

Extension of constant-velocity region by use of servo-control system

Using a servo-controlled system, the constant-slide-speed region can be extended and speed fluctuations in this region will decrease. Nevertheless, control of slide speed in SegWork should not affect the nominal torque output capability of the press: only surplus torque of the motor is allowed to be used for controlling the slide speed in SegWork, as shown in Figure 11.

Figure 11

Torque compatibility principle for constant-slide speed control

The implication of using constant-slide-speed mode lies in the fact that the work rating can be greatly increased and energy consumption of the motor and drive system decreased, while the load-bearing capability of the servo press and the motor capability remain almost the same. Motion-planning results of optimized linkages Lk₁ to Lk₃ in the different modes are shown in Table 8. Under the same work stroke, slide work velocity, and feeding time, one additional work rating was achieved per minute and energy consumption of the motor and drive decreased by 40% in constant-slide-speed mode.

Table 8 Motion-planning results of for different modes

The above results indicate that motion planning in constant-slide-speed mode can overcome the inherent shortcomings of linkages, offering more choices with respect to increasing the speed and energy saving.

To obtain consistency with the constant-slide-speed mode in SegWork, the following constraint can be added to the linkage inertia optimization, improving the acceleration performance of the linkage in SegWork:

$$f_{3b} (X) = \frac{1}{{\varphi_{\text{BDC}} - \varphi_{\text{WkS}} }}\int_{{\varphi_{\text{WkS}} }}^{{\varphi_{\text{BDC}} }} {Je_{\text{Lk}} } (\varphi ){\text{d}}\varphi .$$

(21)

Linkage sizes for Lk₄ and Lk₅, optimized in constant-slide-speed mode, are shown in Table 9 for conditions of different work strokes, slide speeds in SegWork, and die open angles.

Table 9 Sizes of optimized linkages in constant-slide-speed mode

Features of linkages Lk₄ and Lk₅ are shown in Table 10. Optimization in constant-slide-speed mode can lower the slide velocity in SegWork and ideal equivalent force arm of linkage, while keeping the equivalent inertia almost constant.

Table 10 Features of linkages optimized in constant-slide-speed mode

Motion-planning results for linkages Lk₄ and Lk₅ optimized in constant-slide-speed mode are shown in Table 11, which shows that the work rating of a drawing servo press equipped with linkages Lk₄ or Lk₅ is equal to that of SL4-2000A, but the motor and drive system energy consumption is more than 11% lower. Compared with SL4-2000A in constant-eccentric-speed mode (S _wk = 300 mm and Nc = 15 s/min), the energy consumption of linkage Lk₅ was reduced by 34.3% and the work rating was increased by 1.12 times per minute. A comparison of motion-planning results between linkage Lk₅ (in constant-slide-speed mode) and SL4-2000A (in constant-eccentric-speed mode) is shown in Fig. 12.

Table 11 Motion-planning results for linkages optimized in constant-slide-speed mode

Figure 12

Comparison of motion planning between linkage Lk₅ and SL4-2000A (constant-eccentric-speed mode)

Figure 13 shows that the constant slide speed in SegWork of optimized linkage Lk₅ was 250 mm/s and the upper limit of the stroke to maintain this speed was about 260 mm. However, Table 11 shows that linkage Lk₅ can be used for a drawing stroke of 300 mm and the energy consumption is lower than that of a SL4-2000A (this product was designed for a drawing stroke of 300 mm) linkage system, implying that use of this optimization method in constant-slide-speed mode can extend the work capability of a small-stroke linkage.

Figure 13

Motion curves for optimized linkage Lk₅ and L4-2000A

The results of motion planning for both constant-eccentric-speed and constant-slide-speed modes proved that this method of linkage based on phase division is efficient in improving energy saving and work rating. Using this method, the root-mean-square torque can be lowered, which can enable large decreases in drawing servo press manufacturing costs. Although use of constant-slide-speed mode in the SegWork phase is an advanced motion-planning technique, it is not yet widely used in practice: this paper could be helpful to its applications in drawing servo presses.

5 Conclusions

• 1. A new linkage-optimization method for drawing servo presses is proposed, based on phase-division. The objective functions and constraints of the method are established. Starting from any initial value, this method can obtain solutions with high accuracy.

• 2. Optimization based on constant-eccentric-speed mode was investigated. Linkages were optimized under both the same and different design parameters to those of the SL4-2000A. Comparative results showed that optimization in constant-eccentric-speed mode can lower the root-mean-square torque of the motor, which has implications for significantly reducing the manufacturing costs of drawing servo presses.

• 3. Optimization in constant-slide-speed mode was systematically investigated. The results showed that optimization in this mode has obvious advantages over constant-eccentric-speed mode in respect of work rating and energy saving.