Abstract
In order to comprehensively improve the motion and mechanical performance of the clamping unit in squeeze casting equipment, a multi-objective optimization method combining the coordination curve method and layered sequential method is proposed. Firstly, the operational characteristics of the curved lever-type die-closing mechanism in squeeze casting equipment are analyzed, and a mathematical optimization model is constructed. Secondly, the harmonious correlation between stroke ratio and force ratio is established, and tolerance values are assigned for optimization objectives. The penalty function algorithm is used to optimize the multi-scheme optimization of the targets such as force increase ratio, stroke ratio, speed ratio, and member mass sum of the clamping unit, and the most suitable optimization scheme is selected. ADAMS software is leveraged for the construction of a virtual prototype model, enabling precise simulation and validation of motion. Finally, an on-site experimental platform is established for the evaluation of motion characteristics and mechanical properties, providing experimental validation for the mathematical model and simulation results. The results reveal a notable 14.4% increase in force ratio, a 6.9% enhancement in stroke ratio, and a marginal reduction in the total mass of rods compared to the pre-optimization state. The speed and acceleration performance of the moving plate are greatly improved.
1. Introduction
The clamping unit is one of the key components in squeeze casting equipment, ensuring the reliable closure of the molding die, realizing the opening action, and ejecting the casting. Its performance directly affects the quality of the castings . The structural forms mainly include hydraulic and toggle link types, among which the double-toggle structure is widely used due to its compact structure, reliable clamping, good motion characteristics, and high strength and stiffness, well meeting the production requirements of squeeze casting .
Traditional clamping unit design often relies on surveying, analogy, and experience-based design methods, which makes it difficult to obtain the optimal parameter combination of the clamping unit and fails to meet the performance requirements of force amplification, speed increase, and weight reduction, thereby reducing the practical value of the clamping unit. Therefore, the current design of clamping units adopts CAE (Computer Aided Engineering) technology. Facing the multi-objective optimization requirements of clamping units, CAE technology can greatly enhance the accuracy of optimization algorithms and significantly improve the scientific nature and design efficiency of product design . At present, researchers at home and abroad often select one or more combinations of force increase ratio, stroke ratio, speed ratio, and total rod mass as optimization objectives for clamping units. Xu Yin et al. and HUANG M S et al. applied the genetic algorithm to optimize the five-point diagonal double-toggle clamping mechanism with the force increase ratio as the optimization objective, referring to key design parameters to obtain the optimal solution of the clamping mechanism and improving the motion smoothness of the moving plate. Xu Yangyang et al. used ADAMS software to establish a parameterized model of the virtual prototype of the clamping mechanism and optimized it using the sequential quadratic programming (SQP) algorithm. Simultaneously, four optimization design schemes were proposed. Finally, it was found that the comprehensive optimization effect obtained by the unified objective function constructed by the linear combination weighting method was superior to that of the single-objective optimization function. Ji Yuliang et al. adopted the multiplication and division method to unify the objective function and optimized the performance of the force increase ratio, mechanism length, and stroke ratio of the clamping mechanism based on the genetic algorithm, enhancing the operating efficiency of the clamping mechanism. Gong Yangyang et al. used the probability cumulative function of the normal distribution to construct the optimization objective function, set the expected values and fluctuations of the force increase ratio, stroke ratio, and maximum mold opening stroke, and adopted the nonlinear constrained optimization solution method to finally obtain relatively ideal geometric structure parameters. 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 the possibility level, weighting coefficients, and penalty factors during the conversion process, thereby affecting the selection of optimal results. Therefore, considering the influence relationships between various optimization objectives for squeeze casting equipment with different extrusion forces, choosing targeted optimization design methods is crucial for improving the performance level of double-toggle clamping mechanisms.
In view of this, the double-toggle clamping mechanism of a 4000 kN squeeze casting machine is optimized and analyzed. The combined optimization design method of the coordination curve method and the tolerance hierarchical sequence method is adopted, which avoids converting the interval optimization problem into a deterministic optimization problem. By comparing the parameters such as the force increase ratio, stroke ratio, speed ratio, and total rod mass in various optimization schemes, the optimal design scheme suitable for this machine model is selected. The research results aim to provide a reference for the mechanism design of the squeezer.
2. Analysis of Clamping Unit Working Characteristics
The schematic diagram of the double-toggle clamping mechanism of a 4000 kN squeeze casting machine is shown in Figure 1.
2.1 Motion Characteristic Analysis
The simplified motion principle diagram of the lower half of the clamping unit is shown in Figure 2. The dashed line represents the limit position of the clamping unit in the mold opening state, and the solid line represents the limit position in the mold closing state. Sₒ is the stroke of the oil cylinder, and Sₘ is the stroke of the moving plate. Its working principle is as follows: At the beginning of mold closing, the clamping cylinder pushes the crosshead E to move and links the small connecting rod L₄ to drive the toggle lever to rotate around the hinge point A at the same angular velocity, gradually straightening the mechanism components and pushing the moving plate C to move until it contacts the mold. Then, the clamping cylinder continues to drive the crosshead, forcing the components of the clamping mechanism to produce elastic deformation, thereby generating a clamping force (locking force) on the mold, causing the clamping mechanism to be in a self-locking state. At this time, even if the cylinder thrust is removed, the locking force still exists . Among them, β and φ are the angles between the large connecting rod L₂, small connecting rod L₄, and the horizontal direction, respectively; θ is the angle between the toggle lever L₁ and the horizontal direction when the mold is fully closed, known as the oblique arrangement angle; γ is the angle between the elbow rods L₁ and L₅; αₘₐₓ is the maximum toggle angle, which is the angle between the toggle lever L₁ at the mold closing limit and mold opening limit.
The stroke Sₒ of the oil cylinder is the movement stroke of the crosshead E, and the stroke Sₘ of the moving plate is the movement stroke of the hinge connection point. According to the geometric relationship of the mechanism in Figure 2, the ratio of the movement stroke Sₘ of the moving plate to the movement stroke Sₒ of the crosshead, namely the stroke ratio Rₛ, can be derived.
The speed analysis of the clamping unit is shown in Figure 3. The clamping unit is regarded as a crank-connecting rod mechanism, and the instantaneous center method of velocity is adopted to obtain the speed ratio of the moving plate C to the crosshead E. Firstly, find the two instantaneous centers of the mechanism, the velocity instantaneous center G of the toggle lever L₁ and the large connecting rod L₂, and the velocity instantaneous center F of the toggle lever L₅ and the small connecting rod L₄. Then, obtain the speed relationship between the moving plate C and the hinge point B, the hinge point D, and the crosshead E, and finally convert it to obtain the ratio of the moving speed of the moving plate to the moving speed of the crosshead. By differentiating the instantaneous speed vₘ of the moving plate with respect to time t, the acceleration aₘ of the moving plate with the toggle angle α as a variable can be obtained.
RV=vovm=vEvC=vBvC×vDvB×vEvD=L5sin(α+θ+γ+φ)cosβL1sin(α+θ+β)cosφ
am=L5sin(α+θ+γ+φ)cosβL1ε1sin(α+θ+β)−L1ω12cos(α+θ+β)−L2ω22cosβ
varepsilon1=L5sin(α+θ+γ+φ)L5ω21cos(α+θ+γ+φ)−L4ω42
omega1=L5sin(α+θ+γ+φ)cosφvo
omega4=L4sin(α+θ+γ)cos(α+θ+γ+φ)vo
omega2=L2cosβL1ω1cos(α+θ)
2.2 Mechanical Characteristic Analysis
When the clamping unit enters the mold locking stage, the clamping cylinder continues to drive the moving plate to move forward, forcing the components of the clamping mechanism to produce elastic deformation, thereby generating a clamping force Pₘ (locking force) on the mold to achieve the pre-tightening effect on the mold. Figure 4 shows the force situation of the mechanism without considering friction during the mold closing process.
Combining the static equilibrium theory and moment equilibrium theory, the force analysis of the clamping mechanism is carried out, and the mathematical relationship between the clamping force Pₘ and the cylinder thrust Pₒ and the expression of the force amplification coefficient Mᵢ are obtained [14]:
MP=PoPm=L1sin(α+θ+β)cosφL5sin(α+θ+γ+φ)cosβ
3. Establishment of Mathematical Model Optimization
Before optimizing the clamping unit with multi-objectives, a complete mathematical model must be established, including determining the optimization performance criteria, design variables, and analyzing various constraints in the model.
3.1 Objective Function
- Force Increase Ratio Mᵢ: To satisfy a large force amplification multiple, which can generate a large clamping force. According to previous design experience, the Mᵢ value when α = 2.5° is selected as the optimization objective .
- Stroke Ratio Rₛ: Within the specifications of the clamping cylinder and moving plate stroke of the 4000 kN squeeze casting equipment, a larger stroke ratio is satisfied, with Rₛ as the optimization objective.
- Speed Ratio Rᵥ and Acceleration aᵢ: To have good speed characteristics, considering that the optimized machine model is a 4000 kN squeeze casting machine, a more rapid mold opening and closing process and better acceleration and deceleration performance of the moving plate are required. The difference between the peak and valley values of the speed ratio, the starting value of the speed ratio, and the difference between the peak and valley values of the acceleration are taken as the optimization objectives.
- Total Rod Mass Lᵢ: To satisfy a compact structure and a small total rod length, the sum of the lengths of the toggle lever, small connecting rod, and large connecting rod is taken as the optimization objective, with the function expression as Lᵢ = L₁ + L₂ + L₃ + L₄ + L₅.
3.2 Design Variables
According to the structural analysis of the toggle-type clamping unit, with the longitudinal dimensions of the three major plates remaining unchanged, L₁, L₂, L₄, L₅, θ, γ, αₘₐₓ, and Hₐ 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,Ha]
3.3 Constraint Conditions
- Interference and Self-Locking Constraint Conditions: The toggle-type clamping unit is an upper and lower symmetrical structure. To avoid interference between the upper and lower toggle levers during mold opening and closing:
G1(x)=L1+dB−Ha−Hs≤0
G2(x)=L5+dD−Ha−Hs≤0
To avoid self-locking during the mold opening stage, i.e., ∠EDB = 180°, and considering the friction circle theory of the elbow and rod, it is necessary to satisfy:
G3(x)=αmax+γ+φmax+θ−160°≤0
- Rod Length Constraint Conditions: To achieve a compact structure and improve system rigidity, according to design experience, the rod length ratio λ = L₁/L₂ is generally within the range of [0.7, 0.85], so:
G4(x)=0.7−λ≤0
G5(x)=λ−0.85≤0
To satisfy the rod rotation requirements, it is ensured that the small connecting rod L₄ and the toggle lever L₃ have sufficient lengths to satisfy the rotation conditions, i.e., L₄ ≥ (dₑ + dₐ)/2, L₃ ≥ (dᵦ + dₐ)/2. Then:
G6(x)=2dE+dD−L4≤0
G7(x)=2dB+dD−L3≤0
- Angle Constraint Conditions: Relevant research shows that to ensure the smoothness of the moving plate speed and the mechanical properties of the clamping mechanism, the maximum vertex angle φₘₐₓ should be ≤ 85°, so:
G8(x)=φmax−85°≤0
- Definition Domain Constraint Conditions: From the geometric relationship of the mechanism, for Mᵢ and Rₛ to be meaningful, it is required that:
L24−[Ha−L5sin(θ+γ+\2≥0
L22−[L1sin(α+θ)−(L1+L2)sin2≥0
Then the constraints are:
G9(x)=[Ha−L5sin(θ+γ+2−L24≤0
G10(x)=[L1sin(α+θ)−(L1+L2)\2−L22≤0
For the clamping unit of the 4000 kN model of squeeze casting equipment, the stroke of the moving plate needs to satisfy Sₘ ∈ [500, 600], and the stroke ratio Rₛ should not exceed 1.12, so:
G11(x)=Sm−600≤0
G12(x)=500−Sm≤0
G13(x)=RS−1.12≤0
- Design Variable Boundary Constraint Conditions: Setting reasonable boundary conditions can improve optimization efficiency and reduce computational complexity. Among them, the inclination angle θ is taken as 4°∼6°, which can slightly increase the size of the head plate while significantly increasing the stroke of the moving plate without significantly affecting the force amplification and speed increase performances. From an economic perspective, the size of the head plate should not be too large, and the upper boundary of the distance Hₐ between the connection point E of the crosshead and the small connecting rod and the lower support point A is set to 250 mm. Without affecting the convergence of the optimal solution, the boundary conditions of each design variable are obtained through repeated experiments as follows:
L1∈[320,410],L2∈[415,505],L4∈[100,150],L5∈[280,330]
Based on the provided information and the need for comprehensive boundary constraints in the design optimization process, the following detailed boundary conditions for each design variable are established to ensure both efficiency and effectiveness:
- Inclination Angle (θ):
- Lower Boundary: 4°
- Upper Boundary: 6°
- Reasoning: This range allows for a balance between increasing the head plate size slightly and significantly enhancing the moving plate’s stroke, while maintaining acceptable force amplification and speed increase performances.
- Distance between Connection Point E and Lower Support Point A (Hₐ):
- Upper Boundary: 250 mm
- Reasoning: To control economic costs and prevent excessive size of the head plate, the maximum allowable distance is set at 250 mm. The lower boundary can be determined based on practical constraints of the mechanism’s functional requirements, ensuring it does not interfere with other components or compromise structural integrity.
- Length of Small Connecting Rod (Lₛ):
- Lower Boundary: Determined through experimental iteration to ensure proper linkage and mechanism operation without interference.
- Upper Boundary:Constrained by the overall design envelope and the need to maintain a compact, efficient design. Typically, this would be set to a value that ensures the connecting rod does not exceed practical limits of material strength and manufacturability.
- Length of Crosshead (Lₖ):
- Lower Boundary: Sufficient length to provide stable connection and proper leverage.
- Upper Boundary: Limited by the overall dimensions of the system and the need to avoid excessive weight and complexity. This should be experimentally determined to strike a balance between structural integrity and performance.
- Angle of Crosshead relative to Horizontal (α):
- Lower Boundary: Determined by the mechanical requirements for efficient force transmission and stability.
- Upper Boundary:Constrained by the need to avoid excessive stress concentrations and maintain a practical range of motion.
- Pivot Point Positions (e.g., positions of A, E, and other critical points):
- Boundaries: Defined by the geometry of the surrounding structure and the need for smooth, efficient operation of the mechanism. These should be experimentally optimized to ensure proper alignment and minimize wear and tear.
- Material Properties (if considered as design variables):
- Boundaries: Based on available materials, cost considerations, and required mechanical properties such as strength, ductility, and corrosion resistance. The selection should aim for a balance between performance and economy.
- Force and Load Capacity:
- Lower Boundary: Must meet the minimum requirements for the intended application.
- Upper Boundary:Constrained by the need to avoid over-engineering and excessive costs. The upper limit should be determined by safety factors and practical operating conditions.
By setting these boundary conditions through a combination of theoretical analysis, experimental iteration, and practical considerations, the optimization process can proceed efficiently, reducing computational complexity while ensuring that the resulting design meets all performance, economic, and manufacturability requirements. Regular review and adjustment of these boundaries during the design process may be necessary to adapt to new findings or changing design constraints.