In the field of manufacturing, casting stands as a pivotal rapid prototyping method for obtaining mechanical components, extensively utilized in industrial production. Traditional casting processes often rely on a “trial-and-error” approach to determine optimal parameters, which leads to prolonged production cycles, reduced efficiency, and increased costs. Research indicates that applying controlled vibrations during molten metal filling can enhance casting quality and reduce defect rates. However, existing mechanical vibration techniques in casting often suffer from limited degrees of freedom and an inability to observe real-time fluid flow within mold cavities, thereby affecting the quality of large-scale complex castings. While experimental studies provide valuable insights, they can be costly, time-consuming, and lack observability. To address these challenges, this study employs the Discrete Element Method (DEM) to simulate the filling process of a spheroidal graphite cast iron crankshaft under multi-dimensional vibrations. By analyzing parameters such as vibration degrees of freedom, frequency, and amplitude, we aim to optimize the casting process for improved filling capability and reduced defects. The spheroidal graphite cast iron material is chosen for its excellent mechanical properties, including high strength and ductility, making it ideal for automotive components like crankshafts.
The Discrete Element Method, initially proposed by Cundall in 1971, is a numerical technique for modeling particulate systems. It discretizes materials into rigid particles and solves their motion based on Newton’s second law using time-stepping algorithms. Over decades, DEM has evolved beyond geotechnical applications to fields like mineral processing and chemical engineering, enabling detailed analysis of particle flow, shear behavior, and packing characteristics. In this context, DEM proves advantageous for simulating molten metal as a collection of fine particles, allowing for real-time visualization and multi-parameter optimization without physical trials. The fundamental equations governing particle motion in DEM are derived from Newtonian mechanics. For each particle i with mass mi, position vector xi, and velocity vi, the translational and rotational motions are described by:
$$ m_i \frac{d^2 \mathbf{x}_i}{dt^2} = \sum_j \mathbf{F}_{ij}^c + \mathbf{F}_i^{ext} $$
$$ I_i \frac{d \boldsymbol{\omega}_i}{dt} = \sum_j \mathbf{M}_{ij} $$
where Fijc represents the contact force between particle i and particle j (or walls), Fiext denotes external forces such as gravity, Ii is the moment of inertia, ωi is the angular velocity, and Mij is the torque. The contact force typically includes normal and tangential components, modeled using spring-dashpot systems or more advanced constitutive laws. For instance, the normal contact force can be expressed as:
$$ F_n = k_n \delta_n + c_n \dot{\delta}_n $$
where kn is the normal stiffness, δn is the overlap, and cn is the damping coefficient. Similarly, the tangential force incorporates frictional effects. These equations form the basis for simulating particle dynamics in casting processes, enabling the analysis of how vibrations influence fluid-like behavior in spheroidal graphite cast iron systems.
To apply DEM to casting, we focus on a spheroidal graphite cast iron crankshaft used in automotive engines. The crankshaft model has a mass of 44 kg and dimensions of Φ656 × 190 mm, with main journal diameters of Φ85 mm, connecting rod journal diameters of Φ70 mm, and balance weight thickness of 25 mm. The spheroidal graphite cast iron material is characterized by its unique microstructure containing spherical graphite nodules, which impart superior toughness and fatigue resistance. This makes it a preferred choice for high-stress components. The simulation parameters are designed to replicate real-world conditions, with particles representing molten spheroidal graphite cast iron. Key parameters include gravitational acceleration set at 9.81 m/s² in the +Y direction, particle radius of 0.2 mm, and particle injection velocity of 5 m/s in the +X direction. The total simulation time is 1 second, with a grid size of 0.8 mm to ensure computational accuracy while managing resources. Material properties for the mold cavity and particles are defined as follows:
| Parameter | Poisson’s Ratio | Shear Modulus (GPa) | Density (kg/m³) |
|---|---|---|---|
| Mold Cavity | 0.25 | 7.0 | 7800 |
| Particles (Spheroidal Graphite Cast Iron) | 0.26 | 6.1 | 7300 |
Contact parameters between particles and the mold cavity are crucial for realistic simulations. These include coefficients of restitution, static friction, and rolling friction, as summarized below:
| Contact Pair | Restitution Coefficient | Static Friction Coefficient | Rolling Friction Coefficient |
|---|---|---|---|
| Particle-Particle | 0.15 | 0.3 | 0.05 |
| Particle-Cavity | 0.1 | 0.08 | 0.3 |
These parameters ensure that the DEM simulation accurately captures the interactions between molten spheroidal graphite cast iron and the mold walls, facilitating analysis of filling patterns under vibration. The simulation process begins with particle generation at the sprue entrance, where particles flow into the crankshaft cavity under gravity and injection momentum. As particles collide with cavity walls, their velocity dissipates, leading to accumulation and gradual filling. Vibration is applied to the mold in multi-dimensional modes, including single-degree (X, Y, Z), two-degree (XY, XZ, YZ), and three-degree (XYZ) freedoms, with varying frequencies and amplitudes. This allows us to study how vibrations enhance the fluidity of spheroidal graphite cast iron during filling.
The effect of vibration parameters on filling capability is evaluated using a controlled variable approach. Filling distance, defined as the maximum distance particles travel within the cavity under fixed injection conditions, serves as a proxy for molten metal fluidity. For spheroidal graphite cast iron, this metric indicates how well the material fills complex geometries, reducing defects like cold shuts or misruns. First, we examine vibration degrees of freedom (DOF). With frequency fixed at 50 Hz and amplitude at 0.5 mm, results show that increasing DOF from 1 to 3 enhances filling distance, as particles experience multi-directional agitation that reduces stagnation. The data is summarized below:
| Vibration DOF | 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, vibration frequency is analyzed with DOF set to 3 and amplitude at 0.5 mm. Frequency influences the energy input into the system; optimal values promote particle movement without causing turbulence that impedes flow. As shown in the table, filling distance peaks at 50 Hz for spheroidal graphite cast iron, beyond which excessive vibration may degrade filling performance.
| 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, vibration amplitude is studied with DOF=3 and frequency=50 Hz. Amplitude determines the displacement magnitude of vibrations, affecting particle momentum transfer. Results indicate that an amplitude of 0.5 mm yields the maximum filling distance for spheroidal graphite cast iron, as it provides sufficient agitation without causing particle scattering.
| 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 assess the impact of vibration parameters, an orthogonal experimental design L9(3^4) is employed. Factors include vibration DOF (B), frequency (C), and amplitude (D), with filling distance as the response. The experimental layout and results are presented below, where K values represent sums of responses for each factor level, k values are averages, and R indicates the range of k values reflecting factor influence.
| Experiment No. | B (DOF) | C (Frequency) | D (Amplitude) | Empty Column | Filling Distance (mm) |
|---|---|---|---|---|---|
| 1 | 1 | 1 | 1 | 1 | 300 |
| 2 | 1 | 2 | 2 | 2 | 336 |
| 3 | 1 | 3 | 3 | 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 |
| 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 |
Based on the R values, the order of influence on filling distance is: vibration frequency (C) > vibration DOF (B) > vibration amplitude (D). This highlights the critical role of frequency in modulating particle dynamics for spheroidal graphite cast iron. The optimal combination derived from orthogonal analysis is B3C3D1, corresponding to DOF=3, frequency=50 Hz, and amplitude=0.5 mm, which maximizes filling distance and potentially improves casting quality. To further elucidate the underlying mechanics, we can model the effect of vibrations on particle motion. The equation of motion for a particle under sinusoidal vibration in the X-direction can be expressed as:
$$ m \frac{d^2 x}{dt^2} = F_{contact} – mg + F_{vibration} $$
$$ F_{vibration} = A \omega^2 \sin(\omega t) $$
where A is amplitude, ω is angular frequency (ω = 2πf), and t is time. This external force term enhances particle kinetic energy, promoting flow in spheroidal graphite cast iron simulations. Similarly, for multi-dimensional vibrations, the force vector incorporates components in X, Y, and Z directions, leading to more complex interactions that can be solved iteratively in DEM. The benefits of vibrations in casting spheroidal graphite cast iron include reduced viscosity effects, improved heat transfer, and minimized gas entrapment, all contributing to denser and more uniform castings.
In conclusion, this study demonstrates the efficacy of DEM in simulating the multi-dimensional vibration casting filling process for spheroidal graphite cast iron crankshafts. Through controlled simulations and orthogonal experimentation, we identify vibration frequency as the most influential parameter, followed by degrees of freedom and amplitude. The optimal conditions—frequency of 50 Hz, three degrees of freedom, and amplitude of 0.5 mm—yield superior filling performance, enhancing the fluidity of spheroidal graphite cast iron in complex molds. Compared to traditional casting methods, DEM offers a cost-effective and observable approach to optimizing vibration parameters, reducing reliance on physical trials. Future work could explore finer particle resolutions, advanced contact models, and thermal coupling to further refine simulations for spheroidal graphite cast iron applications. This research provides a theoretical foundation for leveraging DEM and multi-dimensional vibrations in casting processes, ultimately contributing to higher-quality automotive components made from spheroidal graphite cast iron.