In modern manufacturing, casting remains a pivotal rapid prototyping technique for producing mechanical components, widely employed across industrial sectors. Traditional casting processes often rely on a “trial-and-error” approach to determine optimal parameters, which leads to extended production cycles, low efficiency, and high costs. Research has demonstrated that applying vibrations during metal filling can enhance casting quality and reduce defect rates. However, conventional vibration methods and existing simulation tools frequently suffer from limitations such as single-degree-of-freedom vibrations and an inability to observe real-time molten metal flow within molds, particularly affecting the成型 quality of large, complex castings like those made from nodular cast iron. While experimental studies provide valuable insights, they often entail high costs, prolonged validation periods, and limited observability. To address these challenges, this study leverages the Discrete Element Method (DEM) to simulate the filling process under multi-dimensional vibrations, focusing on nodular cast iron crankshafts for automotive engines. By configuring vibration parameters—including degrees of freedom, frequency, and amplitude—and using fine particles to represent molten metal flow, this approach overcomes the shortcomings of traditional casting and existing software. In this paper, I employ DEM-based numerical simulation to analyze the filling process of a nodular cast iron crankshaft under multi-dimensional vibrations, utilizing controlled variable methods and orthogonal experiments to assess the influence of various vibration parameters.
The Discrete Element Method (DEM), first proposed by Cundall P.A. in 1971, is a numerical technique designed for analyzing discrete particulate materials. Initially applied in geotechnical and geological engineering, DEM has evolved over five decades to model particle systems in fields such as mineral processing, civil engineering, and chemical engineering, enabling the analysis of particle flow, shear behavior, and packing characteristics. The core principle of DEM involves discretizing a system into rigid elements, then solving their equations of motion using Newton’s second law and iterative time-stepping algorithms like dynamic or static relaxation methods. For a particle i, the governing equation is:
$$m_i \frac{d^2 \mathbf{x}_i}{dt^2} = \sum_j \mathbf{F}_{ij} + \mathbf{F}_{\text{ext}}$$
where \(m_i\) is the particle mass, \(\mathbf{x}_i\) is the position vector, \(\mathbf{F}_{ij}\) represents the contact forces between particle \(i\) and neighboring particles \(j\), and \(\mathbf{F}_{\text{ext}}\) denotes external forces such as gravity. The contact forces often incorporate spring-dashpot models, expressed as:
$$\mathbf{F}_{ij} = k \delta_{ij} \mathbf{n}_{ij} + c \mathbf{v}_{ij}$$
Here, \(k\) is the stiffness coefficient, \(\delta_{ij}\) is the overlap distance, \(\mathbf{n}_{ij}\) is the unit normal vector, \(c\) is the damping coefficient, and \(\mathbf{v}_{ij}\) is the relative velocity. This framework allows DEM to accurately simulate granular flows, making it suitable for modeling molten metal behavior in casting processes, especially for materials like nodular cast iron known for their ductility and strength.
In this study, I focus on a nodular cast iron crankshaft used in automotive engines. The crankshaft model, designed with a mass of 44 kg and dimensions of Φ656 mm × 190 mm, features main journal diameters of Φ85 mm, connecting rod journal diameters of Φ70 mm, and balance weight thicknesses of 25 mm. The three-dimensional model was created in SolidWorks and exported in IGS format for DEM simulation. To reflect real-world conditions, the simulation parameters were carefully set, as summarized in Table 1. These include gravitational acceleration (9.81 m/s² in the +Y direction), particle properties (radius of 0.2 mm, quantity of 20,000), injection velocity (5 m/s in the +X direction), simulation time (1 s), and mesh size (0.8 mm). The materials for both the mold cavity and particles, representing nodular cast iron, were assigned specific mechanical properties, such as Poisson’s ratio, shear modulus, and density. Contact parameters, including restitution, static friction, and rolling friction coefficients, were also defined to model interactions accurately.
| Parameter | Value |
|---|---|
| Gravity Acceleration | 9.81 m/s² (+Y direction) |
| Particle Radius | 0.2 mm |
| Particle Quantity | 20,000 |
| Injection Velocity | 5 m/s (+X direction) |
| Simulation Time | 1 s |
| Mesh Size | 0.8 mm |
| Material Properties | |
| Poisson’s Ratio (Mold) | 0.25 |
| Poisson’s Ratio (Particles) | 0.26 |
| Shear Modulus (Mold) | 7 GPa |
| Shear Modulus (Particles) | 6.1 GPa |
| Density (Mold) | 7800 kg/m³ |
| Density (Particles) | 7300 kg/m³ |
| Contact Parameters | |
| Restitution Coefficient (Particle-Particle) | 0.15 |
| Restitution Coefficient (Particle-Mold) | 0.1 |
| Static Friction Coefficient (Particle-Particle) | 0.3 |
| Static Friction Coefficient (Particle-Mold) | 0.08 |
| Rolling Friction Coefficient (Particle-Particle) | 0.05 |
| Rolling Friction Coefficient (Particle-Mold) | 0.3 |
The simulation process began with particles, representing molten nodular cast iron, entering the crankshaft mold cavity from the flange end connected to the riser. Under gravity and vibration effects, particles flowed downward, colliding with cavity walls and gradually accumulating. As particles advanced, their velocity decreased due to interactions, but with appropriate vibration parameters, they eventually filled the entire cavity. This process was monitored in real-time to observe flow patterns and filling distances, which serve as indirect indicators of molten metal filling capability. The filling distance, defined as the maximum travel of particles along the cavity within the simulation time, was used to evaluate the influence of vibration parameters. For instance, the filling distance \(L\) can be related to particle velocity \(v\) and time \(t\) by:
$$L = \int_0^t v(t) \, dt$$
where \(v(t)\) is affected by vibration-induced forces. To visualize this, the following figure illustrates the multi-dimensional vibration filling process, showing particle distribution at different stages.
To analyze the impact of vibration parameters on the filling ability of nodular cast iron, I employed a controlled variable approach, varying one parameter at a time while keeping others constant. The vibration parameters included degrees of freedom (DOF), frequency (\(f\)), and amplitude (\(A\)). The vibration motion was modeled as harmonic oscillation, with the displacement \(x(t)\) given by:
$$x(t) = A \sin(2\pi f t)$$
for a single direction. For multi-dimensional vibrations, this extends to vector form, e.g., \(\mathbf{x}(t) = [A_x \sin(2\pi f_x t), A_y \sin(2\pi f_y t), A_z \sin(2\pi f_z t)]\), where subscripts denote directions. In the simulation, vibrations were applied to the mold, affecting particle dynamics through contact forces.
First, I investigated the effect of vibration degrees of freedom. With frequency fixed at 50 Hz and amplitude at 0.5 mm, different DOF configurations—single (X, Y, Z), dual (XY, XZ, YZ), and triple (XYZ)—were tested. The filling distances and particle counts at the farthest cavity positions are listed in Table 2. The results show that filling distance increases with higher DOF, peaking at DOF=3 (XYZ), indicating enhanced flowability for nodular cast iron under multi-dimensional vibrations.
| Degrees of Freedom | Filling Distance (mm) | Particle Count at Farthest Position |
|---|---|---|
| X | 300 | 76 |
| Y | 361 | 2 |
| Z | 300 | 12 |
| XY | 361 | 7 |
| XZ | 401 | 4 |
| YZ | 401 | 7 |
| XYZ | 446 | 10 |
Next, the influence of vibration frequency was examined. Based on Table 2, DOF=3 yielded the best performance, so I set DOF=3 and A=0.5 mm while varying frequency from 10 Hz to 60 Hz. The filling distances and particle counts are presented in Table 3. The maximum filling distance occurred at 50 Hz, suggesting an optimal frequency for promoting nodular cast iron flow. Beyond this, higher frequencies may induce turbulence, reducing filling ability. This can be explained by the resonance concept, where the system’s natural frequency aligns with external vibrations, maximizing energy transfer. The natural frequency \(f_n\) of a particle system can be approximated as:
$$f_n = \frac{1}{2\pi} \sqrt{\frac{k_{\text{eff}}}{m_{\text{eff}}}}$$
where \(k_{\text{eff}}\) is the effective stiffness from contacts and \(m_{\text{eff}}\) is the effective mass. At 50 Hz, vibrations likely match this frequency, enhancing particle mobility.
| Vibration Frequency (Hz) | Filling Distance (mm) | Particle Count at Farthest Position |
|---|---|---|
| 10 | 235 | 32 |
| 20 | 300 | 59 |
| 30 | 361 | 6 |
| 40 | 426 | 38 |
| 50 | 446 | 3 |
| 60 | 401 | 2 |
Finally, I studied the effect of vibration amplitude. With DOF=3 and f=50 Hz, amplitude was varied from 0.1 mm to 0.7 mm. The results in Table 4 reveal that filling distance peaks at A=0.5 mm, after which further increases provide diminishing returns or even reductions. This trend aligns with the idea that moderate amplitudes improve particle fluidization, while excessive amplitudes may cause scattering or energy dissipation. The relationship between filling distance \(L\) and amplitude \(A\) can be empirically modeled as:
$$L = L_0 + \alpha A – \beta A^2$$
where \(L_0\) is the baseline distance without vibration, and \(\alpha\) and \(\beta\) are coefficients dependent on material properties like those of nodular cast iron.
| Vibration Amplitude (mm) | Filling Distance (mm) | Particle Count at Farthest Position |
|---|---|---|
| 0.1 | 235 | – |
| 0.2 | 361 | 16 |
| 0.3 | 426 | 8 |
| 0.5 | 491 | 38 |
| 0.6 | 466 | 2 |
| 0.7 | 466 | 1 |
To comprehensively evaluate the significance of each vibration parameter and minimize experimental runs, I conducted an orthogonal experiment using an L9(3^4) design. The factors were: B (vibration degrees of freedom), C (vibration frequency), and D (vibration amplitude), each at three levels. An empty column was included for error estimation. The experimental design and results, including filling distances, are shown in Table 5. The levels for B were 1 (X), 2 (XY), and 3 (XYZ); for C, 10 Hz, 30 Hz, and 50 Hz; and for D, 0.1 mm, 0.3 mm, and 0.5 mm. The response variable was filling distance in mm. I calculated the sum of responses for each level (K1, K2, K3), averages (k1, k2, k3), and ranges (R) to determine factor influence. The range analysis indicates that factor C (vibration frequency) has the largest range (104.3), followed by B (82.4) and D (65), meaning the order of influence is: vibration frequency > vibration degrees of freedom > vibration amplitude. The optimal combination, based on highest average filling distances, is B3C3D1, corresponding to DOF=3, f=50 Hz, and A=0.5 mm. This aligns with the earlier controlled variable results, confirming that these parameters maximize filling ability for nodular cast iron crankshafts.
| Experiment No. | B (DOF) | C (Frequency, Hz) | D (Amplitude, mm) | Empty Column | Filling Distance (mm) |
|---|---|---|---|---|---|
| 1 | 1 | 1 (10) | 1 (0.1) | 1 | 300 |
| 2 | 1 | 2 (30) | 2 (0.3) | 2 | 336 |
| 3 | 1 | 3 (50) | 3 (0.5) | 3 | 361 |
| 4 | 2 | 1 | 2 | 3 | 300 |
| 5 | 2 | 2 | 3 | 1 | 361 |
| 6 | 2 | 3 | 1 | 2 | 426 |
| 7 | 3 | 1 | 3 | 2 | 300 |
| 8 | 3 | 2 | 1 | 3 | 491 |
| 9 | 3 | 3 | 2 | 1 | 426 |
| Analysis | |||||
| K1 | 997 | 900 | 1217 | 1087 | – |
| K2 | 1087 | 1108 | 1062 | 1062 | – |
| K3 | 1217 | 1213 | 1022 | 1152 | – |
| k1 | 332.3 | 300.0 | 405.7 | 362.3 | – |
| k2 | 362.3 | 369.3 | 354.0 | 354.0 | – |
| k3 | 405.7 | 404.3 | 340.7 | 384.0 | – |
| R | 73.4 | 104.3 | 65.0 | 30.0 | – |
| Factor Order: C > B > D (Frequency > DOF > Amplitude) | |||||
The findings from this DEM-based simulation highlight the efficacy of multi-dimensional vibrations in enhancing the filling process for nodular cast iron components. The orthogonal experiment revealed that vibration frequency exerts the strongest influence on filling distance, followed by degrees of freedom and amplitude. Optimal parameters—specifically, a vibration frequency of 50 Hz, three degrees of freedom, and an amplitude of 0.5 mm—yield the greatest filling distance, indicative of superior molten metal flow and potential for high-quality nodular cast iron crankshafts. Compared to traditional casting approaches, DEM not only enables realistic visualization of metal flow within mold cavities but also facilitates the analysis of complex vibration effects without costly physical trials. This study underscores the value of DEM as a tool for optimizing vibration-assisted casting processes, particularly for advanced materials like nodular cast iron. Future work could explore additional factors, such as particle size distribution or temperature effects, to further refine the simulation accuracy and expand applications in industrial casting of nodular cast iron parts.