Die casting equipment relies heavily on high-performance clamping units to ensure precise mold closure, reliable locking force, and efficient ejection of castings. This study presents a multi-objective optimization framework integrating Coordination Curve and Tolerance Hierarchical Sequence methods to enhance the motion and mechanical characteristics of a 4,000 kN die casting machine’s double-toggle clamping mechanism. The approach addresses inherent trade-offs between force amplification, stroke efficiency, speed dynamics, and structural mass while adhering to geometric and kinematic constraints.
Kinematic and Mechanical Analysis of Die Casting Clamping Mechanism
The double-toggle mechanism (Figure 2) converts hydraulic cylinder displacement ($S_o$) into platen stroke ($S_m$) through geometric linkage interactions. The stroke ratio ($R_S$) defines motion transmission efficiency:
$$R_S = \frac{S_m}{S_o} = \frac{(L_1 + L_2) \cos \theta – L_1 \cos(\theta + \alpha_{\max}) – \sqrt{L_2^2 – [L_1 \sin(\alpha_{\max} + \theta) – (L_1 + L_2) \sin \theta]^2}}{L_5[\cos(\gamma + \theta) – \cos(\alpha_{\max} + \gamma + \theta)] + \sqrt{L_4^2 – [H_b – L_5 \sin(\theta + \gamma)]^2} – \sqrt{L_4^2 – [H_b – L_5 \sin(\theta + \gamma + \alpha_{\max})]^2}}$$
where $L_1$, $L_2$, $L_4$, $L_5$ are linkage lengths; $\theta$, $\gamma$, $\alpha_{\max}$ represent angular parameters; and $H_b$ denotes the vertical distance between critical pivots. Velocity transmission ($R_V$) and platen acceleration ($a_m$) are derived using instantaneous center methodology:
$$R_V = \frac{v_m}{v_o} = \frac{L_1 \sin(\alpha + \theta + \beta) \cos \phi}{L_5 \sin(\alpha + \theta + \gamma + \phi) \cos \beta}$$
$$a_m = L_1 \varepsilon_1 \sin(\alpha + \theta + \beta) – L_1 \omega_1^2 \cos(\alpha + \theta + \beta) – L_2 \omega_2^2 \cos \beta$$
with angular velocities $\omega_1$, $\omega_2$, $\omega_4$ and acceleration $\varepsilon_1$ expressed as:
$$\omega_1 = \frac{V_o \cos \phi}{L_5 \sin(\alpha + \theta + \gamma + \phi)}, \quad \omega_4 = \frac{V_o \cos(\alpha + \theta + \gamma)}{L_4 \sin(\alpha + \theta + \gamma + \phi)}, \quad \omega_2 = \frac{L_1 \omega_1 \cos(\alpha + \theta)}{L_2 \cos \beta}$$
$$\varepsilon_1 = \frac{L_5 \omega_1^2 \cos(\alpha + \theta + \gamma + \phi) – L_4 \omega_4^2}{L_5 \sin(\alpha + \theta + \gamma + \phi)}$$
The force amplification ratio ($M_P$) relates clamping force ($P_m$) to hydraulic thrust ($P_o$):
$$M_P = \frac{P_m}{P_o} = \frac{L_5 \sin(\alpha + \theta + \gamma + \phi) \cos \beta}{L_1 \sin(\alpha + \theta + \beta) \cos \phi}$$
These equations form the foundation for optimizing die casting machine performance.
Mathematical Optimization Framework
The multi-objective optimization targets enhanced force amplification, stroke efficiency, speed dynamics, and mass reduction:
- Primary Objectives: Maximize $M_P$ (at $\alpha=2.5^\circ$), $R_S$, and minimize linkage mass $L_{\text{sum}} = \sum_{i=1,2,3,4,5} L_i$
- Dynamic Performance: Minimize $R_{V_{\text{peak-valley}}$, maximize $R_{V_{\text{startup}}}$, minimize $a_{m_{\text{peak-valley}}}$
Design variables encompass geometric parameters: $\mathbf{X} = [L_1, L_2, L_4, L_5, \theta, \gamma, \alpha_{\max}, H_b]$ with bounds:
$$320 \leq L_1 \leq 410, \quad 415 \leq L_2 \leq 505, \quad 100 \leq L_4 \leq 150, \quad 280 \leq L_5 \leq 330$$
$$4^\circ \leq \theta \leq 6^\circ, \quad 18^\circ \leq \gamma \leq 25^\circ, \quad 90^\circ \leq \alpha_{\max} \leq 110^\circ, \quad 210 \leq H_b \leq 250$$
Constraints ensure kinematic feasibility and structural integrity in die casting operations:
- Interference Prevention: $L_1 + d_B – H_b – H_s/2 \leq 0$, $\quad L_5 + d_D – H_b – H_s/2 \leq 0$
- Toggle-Lock Stability: $\alpha_{\max} + \gamma + \phi_{\max} + \theta \leq 160^\circ$
- Linkage Proportion: $0.7 \leq L_1/L_2 \leq 0.85$
- Revolute Joint Clearance: $L_4 \geq (d_E + d_D)/2$, $\quad L_3 \geq (d_B + d_D)/2$
- Angular Limits: $\phi_{\max} \leq 85^\circ$
- Stroke Requirements: $500 \leq S_m \leq 600$, $\quad R_S \leq 1.12$
Hybrid Optimization Strategy
The Coordination Curve Method resolves conflicts between $M_P$ and $R_S$, while the Tolerance Hierarchical Sequence Method incorporates secondary objectives with specified tolerances:
- Pareto Frontier Identification: Solve $M_P$ maximization to establish baseline ($M_P=26.55$). Discretize $R_S \in [1.00, 1.12]$ into 13 points and compute corresponding $M_P$ maxima (Figure 5).
- Mass Minimization: Apply $\pm 0.5\%$ tolerance to $M_P$ and $R_S$ values. For each $(M_P, R_S)$ pair, minimize $L_{\text{sum}}$ using interior-point penalty function:
$$\Phi(\mathbf{X}, r^{(k)}) = L_{\text{sum}}(\mathbf{X}) + r^{(k)} \sum_{j=1}^{13} \frac{1}{g_j(\mathbf{X})}, \quad r^{(k+1)} = 0.5r^{(k)}$$
where $g_j(\mathbf{X})$ are inequality constraints. The optimization workflow reconciles die casting performance trade-offs through systematic compromise.
Optimization Results and Performance Analysis
Table 1 compares 13 optimized configurations. Scheme 4 delivers balanced improvements for die casting applications:
| Parameter | Initial | Scheme 4 | Δ% |
|---|---|---|---|
| $M_P$ | 20.20 | 23.10 | +14.4% |
| $R_S$ | 1.02 | 1.09 | +6.9% |
| $L_{\text{sum}}$ (mm) | 1,354 | 1,346 | -0.6% |
| $R_{V_{\text{max}}}$ | 1.63 | 1.80 | +10.4% |
| $R_{V_{\text{peak-valley}}}$ | 0.65 | 0.65 | 0% |
| $R_{V_{\text{startup}}}$ | 1.16 | 1.17 | +0.9% |
| $a_{m_{\text{peak-valley}}}$ (mm/s²) | 128.5 | 135.6 | +5.5% |
Optimal geometric parameters (Table 2) demonstrate increased toggle length ratio ($\lambda = L_1/L_2 = 368/430 = 0.856$), enhancing stroke efficiency without compromising structural integrity in die casting operations.
| Variable | Initial (mm/°) | Scheme 4 (mm/°) |
|---|---|---|
| $L_1$ | 352 | 368 |
| $L_2$ | 457 | 430 |
| $L_3$ | 120 | 134 |
| $L_4$ | 120 | 135 |
| $L_5$ | 305 | 280 |
| $H_b$ | 235.5 | 239 |
| $\theta$ | 4.5 | 4.0 |
| $\gamma$ | 19.4 | 18.0 |
| $\alpha_{\max}$ | 110 | 105 |
Velocity ratio profiles (Figure 6) confirm enhanced motion characteristics: higher peak $R_V$ (1.80 vs. 1.63) and maintained startup performance. Acceleration improvements (Figure 7) demonstrate faster mold closing dynamics, crucial for high-productivity die casting.
Simulation and Experimental Validation
ADAMS simulations (Figure 8) verify kinematic improvements: velocity ratio trajectories align with theoretical predictions within 1.2% error. Experimental validation on a 4,000 kN die casting machine confirms:
- Clamping force: 4,500 kN (±3.2% uniformity across tie-bars)
- Measured stroke ratio: $R_S = 1.085$ (vs. predicted 1.09)
- Velocity transmission: $R_{V_{\text{max}}} = 1.77$ (vs. predicted 1.80)
Discrepancies stem from friction losses and sensor tolerances, yet overall performance validates the optimization framework for die casting applications.
Conclusion
The hybrid optimization methodology enhances die casting clamping unit performance by resolving fundamental design trade-offs: force amplification increases by 14.4%, stroke efficiency improves by 6.9%, and dynamic response accelerates while reducing total linkage mass. The approach provides a systematic framework for designing high-performance die casting machines, balancing kinematic efficiency with structural economy. Future work will extend the methodology to multi-cavity die casting systems with complex thermal-mechanical constraints.