Formula Expression
Parameters
| Symbol | Name | Unit |
|---|---|---|
| De | De | mm |
| Di | Di | mm |
| h0 | h0 | mm |
| t | t | mm |
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Contact Engineering TeamDetailed Calculation Guide
DIN 2093 Explicit Dynamics Analysis: Impact and Transient Strong Nonlinear Problems
1. Fundamental Concept of Explicit Dynamics
Explicit dynamics uses the central difference method to directly integrate over time, without assembling or inverting the tangent stiffness matrix, and without iteration convergence issues. Therefore, it is particularly suitable for disc springs under transient impact, high-speed collision, forming processes, and highly discontinuous nonlinear problems (such as complex contact, material fracture).
2. Explicit Integration of the Equations of Motion
At time step $n$, the discrete dynamic equation is:
where $\mathbf{M}$ is the diagonal mass matrix, and $\mathbf{a}_n$ is the nodal acceleration vector. The acceleration can be solved directly in a decoupled manner. Velocity and displacement are updated via the central difference scheme:
The entire process requires no formation of a global stiffness matrix and no iteration, making the computational cost per step extremely low.
3. Critical Time Step (Courant‑Friedrichs‑Lewy Condition)
The explicit method is conditionally stable; the time step $\Delta t$ must be smaller than the critical value $\Delta t_{crit}$ to ensure numerical stability. Physically, this means information cannot travel across a single element within one time step. The condition is expressed as:
where: - $L_{min}$ — Characteristic length of the smallest element in the model (mm). For hexahedral elements, typically the shortest edge; for tetrahedral elements, the shortest height. - $c_d$ — One-dimensional longitudinal wave speed of the material (mm/s), calculated as:
- $E$ — Elastic modulus (MPa)
- $\nu$ — Poisson's ratio
- $\rho$ — Material density (t/mm³ or kg/mm³), ensure unit consistency.
For steel disc springs: $E \approx 206\,000\ \text{MPa}$, $\nu=0.3$, $\rho \approx 7.85\times10^{-9}\ \text{t/mm}^3$, yielding $c_d \approx 5.2\times10^6\ \text{mm/s}$.
If the smallest element size in the disc spring finite element mesh is $L_{min} = 0.2\ \text{mm}$, then:
This is why the time step in explicit dynamics typically falls in the range of $10^{-8}\sim10^{-7}$ seconds. Smaller elements and higher material sound speeds lead to shorter critical time steps and greater computational effort.
4. Typical Settings for Disc Spring Impact Analysis
| Item | Recommended Setting |
|---|---|
| Analysis Type | Explicit Dynamics |
| Time Step Control | Automatic (based on smallest element), with possible slight mass scaling to increase step size |
| Element Type | First-order reduced-integration hexahedral (handles large deformation, efficient) |
| Mesh Density | ≥ 4 layers through thickness, refine at inner and outer edges to prevent excessively small elements |
| Material Model | Elastic-plastic, can include strain rate effects (e.g., Cowper‑Symonds) |
| Contact | Automatic surface-to-surface contact, friction coefficient $\mu_f$ (0.03~0.10) |
| Loading | Define impact initial velocity $v_0$ (or drop hammer mass), applied to the disc spring support structure |
| Output | Total reaction force vs. time, stress vs. time, disc spring compression vs. time, kinetic energy vs. internal energy conversion |
5. Mass Scaling and Computational Efficiency
To increase computational speed, mass scaling can be used to artificially increase the time step while maintaining accuracy. The basic relationship is: since $\Delta t_{crit} \propto L / \sqrt{E/\rho}$, increasing the density of the smallest element by a factor $\alpha$ increases the critical time step by $\sqrt{\alpha}$. The mass increase should be controlled to < 2% of the total mass to avoid significantly altering the dynamic response.
6. Comparison of Explicit and Implicit Methods
| Item | Explicit Dynamics | Implicit Static/Dynamics (Newton‑Raphson) |
|---|---|---|
| Applicable Problems | Short-duration transients, impact, strong nonlinearity, wave propagation | Static, quasi-static, low-frequency vibration, moderate nonlinearity |
| Time Step Size | Very small ($10^{-8}\sim10^{-7}$ s), determined by element size | Larger, load steps can be arbitrary (determined by iteration convergence) |
| Cost per Step | Low (no matrix inversion) | High (assemble and factorize tangent stiffness matrix) |
| Total Time | Suitable for millisecond events; extremely time-consuming for second-scale processes | Suitable for second-scale and above static or slow events |
| Stability | Conditionally stable | Unconditionally stable (standard methods) |
| Typical Disc Spring Application | Drop hammer impact, explosion, crash cushioning | Quasi-static compression, preload application, modal analysis |
7. Calculation Example (Conceptual)
A disc spring stack is impacted by a mass $m = 50\ \text{kg}$ with an initial velocity $v_0 = 5\ \text{m/s}$.
Smallest element size $L_{min} = 0.15\ \text{mm}$, steel $c_d \approx 5.2\times10^6\ \text{mm/s}$ → $\Delta t_{crit} \approx 2.88\times10^{-8}\ \text{s}$.
Total impact duration is approximately 2 ms, requiring ≈ $2\times10^{-3} / (2.88\times10^{-8}) \approx 69\,400$ increments. This scale can be completed within a few hours. Outputs include maximum compression, peak force, and energy absorption ratio, which can be verified against energy methods.
8. Important Considerations
- Avoid excessively fine meshes: Very small elements severely reduce the time step. Reasonably control the minimum element size to balance accuracy and efficiency.
- Avoid hourglass modes: When using reduced-integration elements, check the hourglass energy (< 5% of internal energy) and apply hourglass control if necessary.
- Damping and material rate effects: For impact problems, it is recommended to include material strain rate sensitivity; otherwise, peak stresses may be underestimated.
- Post-processing focus: Pay attention to the force-displacement hysteresis loop, maximum compression, stress wave propagation, and possible localized plastic deformation.
Summary: Explicit dynamics analysis, centered on the central difference method and critical time step control, can accurately simulate the transient response of disc springs under millisecond-scale impacts. Properly setting element size, material parameters, and contact yields reliable force, stress, and energy data, providing a basis for cushioning and protection design.