Abstract
In order to comprehensively enhance the kinematic and mechanical performance of the clamping unit of squeeze casting equipment, a multi-objective optimization method combining the Coordination Curve Method and the Hierarchical Sequence Method is proposed. Firstly, the operational characteristics of the curved lever-type clamping unit of squeeze casting equipment are analyzed, and a mathematical optimization model is constructed. Secondly, the coordinated curve relationship between stroke ratio and force multiplication ratio is determined. The results aim to provide references for the mechanical design of squeeze casting machines.
1. Introduction
The clamping unit is one of the key components of squeeze casting equipment, ensuring the reliable closing and opening of the molding die and the ejection of castings. Its performance directly affects the quality of castings. Structural forms mainly include hydraulic type and toggle link type, among which the double-toggle structure is widely used due to its compact structure, reliable die locking, good kinematic characteristics, and high strength and stiffness of the mechanism.
To meet the performance requirements of force multiplication, speed increase, and weight reduction, the practical value of the clamping unit is reduced. Therefore, CAE (Computer Aided Engineering) technology is now adopted in the design of clamping units. Faced with the multi-objective optimization requirements of clamping units, CAE technology can greatly improve the accuracy of optimization algorithms and significantly enhance the scientific nature and efficiency of product design.
Previous optimization models mainly converted multi-objective interval optimization functions into single-objective deterministic optimization functions. However, it is difficult to give specific standards for parameters such as possibility levels, weighting coefficients, and penalty factors during the conversion process, thereby affecting the selection of optimal results.
Therefore, for squeeze casting equipment with different extrusion forces, considering the influence relationship between various optimization objectives, selecting targeted optimization design methods is crucial for improving the performance level of double-toggle clamping units.
In view of this, the double-toggle clamping unit of a 4000 kN squeeze casting machine is optimized using a combination of the Coordination Curve Method and the Tolerance Hierarchical Sequence Method, avoiding converting interval optimization problems into deterministic optimization problems. The parameters such as force multiplication ratio, stroke ratio, speed ratio, and total rod mass of various optimization schemes are compared to select the optimal design scheme suitable for this machine type.
2. Analysis of Clamping Unit Operating Characteristics
2.1 Kinematic Characteristics Analysis
The schematic diagram of the double-toggle clamping unit of a 4000 kN squeeze casting machine. The simplified kinematic principle diagram of the lower part of the clamping unit. The dashed lines represent the extreme open position of the clamping unit, and the solid lines represent the extreme closed position. So is the oil cylinder stroke, and Sm is the moving platen stroke.
The working principle is as follows: at the beginning of clamping, the clamping cylinder pushes the crosshead E to move, and the small connecting rod L4 drives the toggle lever to rotate at the same angular velocity around the hinge point A, gradually straightening the mechanism components and pushing the moving platen C to move until it contacts the mold. Then, the clamping cylinder continues to drive the crosshead, forcing the clamping unit components to produce elastic deformation, thereby generating a clamping force (locking force) on the mold. At this point, even if the cylinder thrust is removed, the locking force still exists.
3. Mathematical Model Optimization
Before conducting multi-objective optimization of the clamping unit, a complete mathematical model must be established, including determining optimization performance criteria, design variables, and analyzing various constraints in the model.
3.1 Objective Functions
(1) Force Multiplication Ratio (MP): To achieve a large force multiplication factor and generate a larger clamping force, the MP value at α=2.5° is selected as the optimization target based on previous design experience.
(2) Stroke Ratio (RS): Within the specifications of the clamping cylinder and moving platen stroke of the 4000 kN squeeze casting machine, a larger stroke ratio is desired, with RS as the optimization target.
(3) Speed Ratio (RV) and Acceleration Value: To achieve better kinematic characteristics, the difference between peak and trough values of the speed ratio, the starting value of the speed ratio, and the difference between peak and trough values of the acceleration are considered as optimization targets.
3.2 Design Variables
Based on the structural analysis of the toggle-type clamping unit, with the longitudinal dimensions of the three main plates remaining unchanged, L1, L2, L4, L5, θ, γ, αmax, and Hb are taken as design variables. The design variables for multi-objective function optimization are:
X = [X1, X2, X3, X4, X5, X6, X7, X8] = [L1, L2, L4, L5, θ, γ, αmax, Hb]
3.3 Constraint Conditions
(Mathematical expressions and inequalities for constraint conditions omitted for brevity)
4. Optimization Calculation Method
According to the established mathematical model, the structural optimization of the toggle-type clamping unit is a nonlinear optimization problem with multiple constraints, variables, and objectives. For such problems, the interior point penalty function method in the MATLAB Optimization Toolbox for nonlinear programming is used for solving.
4.1 Optimization Design Method
After analyzing the kinematic and mechanical characteristics of the clamping unit, it is found that the force multiplication ratio and stroke ratio are a pair of mutually contradictory objective function values. During optimization, optimizing the force multiplication ratio objective function will lead to the deterioration of other objective functions. Therefore, the current single-objective function optimization design method for the force multiplication ratio, stroke ratio, total rod mass, and speed ratio of the clamping unit is too direct.
The proposed method combines the Coordination Curve Method and the Tolerance Hierarchical Sequence Method to optimize the clamping unit.
4.1.1 Coordination Curve Method
During the optimization process, the Coordination Curve Method coordinates the various objective function values of the clamping unit to compromise with each other, ultimately obtaining the most reasonable scheme acceptable to all sub-objectives from the perspective of engineering practicability.
4.1.2 Tolerance Hierarchical Sequence Method
The Tolerance Hierarchical Sequence Method ranks the objective functions such as force multiplication ratio, stroke ratio, and total rod mass of the clamping unit during the optimization process according to their importance. Firstly, the optimal solution for the important objective function is sought, and a given tolerance is taken for the optimal value. Then, the next objective is optimized, avoiding poor calculation results caused by different magnitudes of objective function values.
5. Optimization Result Analysis
Thirteen sets of data are obtained through optimization, and their performance parameters are compared. Combining the values of the objective functions, it can be found that Scheme 4 is a better design. When the force multiplication ratio is greater than 23.1 and the stroke ratio is less than 1.09, parameters such as the difference between peak and trough values of the speed ratio and the stroke ratio decline rapidly, and the acceleration and deceleration performance of the moving platen decreases linearly, reducing the operating efficiency of the clamping unit. Meanwhile, the rod length increases, leading to higher mass and a larger footprint for the clamping unit. Conversely, when the force multiplication ratio is less than 23.1 and the stroke ratio is greater than 1.09, the force multiplication performance and speed ratio starting value are poor.
6. Optimization Model Verification
To verify the accuracy and rationality of the optimized toggle-type clamping unit, kinematic simulations are performed using ADAMS software. To simplify the calculation and analysis process, some structures and components that have no influence on the kinematic analysis are simplified in the virtual prototype model. The simplified simulation model.
The curves of the speed ratio, acceleration target, and force multiplication ratio coefficient of the clamping unit before and after optimization.
The simulated speed performance is close to the mathematical model calculations. After optimization, the peak acceleration of the moving platen is higher, and the acceleration performance is effectively improved.