Wave equation with damping for displacement \( \mathbf{u}(\mathbf{x}, t) \):
\[ \rho \frac{\partial^2 \mathbf{u}}{\partial t^2} + \gamma \frac{\partial \mathbf{u}}{\partial t} - \nabla \cdot \sigma = \mathbf{f} \]Cauchy Stress Tensor \( \sigma \) and Strain Tensor \( \epsilon \):
\[ \sigma = \mathbf{C} : \epsilon \quad \text{(Linear Elasticity)} \] \[ \epsilon = \frac{1}{2} (\nabla \mathbf{u} + (\nabla \mathbf{u})^T) \]Neo-Hookean strain energy density function \( W \):
\[ W = \frac{\mu}{2} (I_1 - 3) - \mu \ln(J) + \frac{\lambda}{2} (\ln(J))^2 \]Where \( I_1 = \text{tr}(\mathbf{b}) \) and \( J = \det(\mathbf{F}) \).
Alternative Mooney-Rivlin model (2 parameters):
\[ W = C_{10}(I_1 - 3) + C_{01}(I_2 - 3) + \frac{1}{D_1}(J-1)^2 \]Second Piola-Kirchhoff stress tensor \( \mathbf{S} \) from \( W \):
\[ \mathbf{S} = 2 \frac{\partial W}{\partial \mathbf{C}} \]Material Parameters Used:
Dirichlet BC: \( \mathbf{u}(\mathbf{x}, t) = \mathbf{0} \) for \( \mathbf{x} \in \Gamma_D \).
Neumann BC (Pressure): \( \sigma \cdot \mathbf{n} = -p(t) \mathbf{n} \) for \( \mathbf{x} \in \Gamma_N \).
Applied time-varying pressure load \( p(t) \):
\[ p(t) = P_0 \left( 1 + A \sin(2 \pi f t + \phi) \right) \]Parameters: \( P_0 = 5e5 \, \text{Pa} \), \( A = 0.4 \), \( f = 2 \, \text{Hz} \), \( \phi = 0 \).
Predictor step:
\[ \tilde{\mathbf{U}}_{n+1} = \mathbf{U}_n + \Delta t \dot{\mathbf{U}}_n + \frac{(\Delta t)^2}{2} (1 - 2\beta) \ddot{\mathbf{U}}_n \] \[ \dot{\tilde{\mathbf{U}}}_{n+1} = \dot{\mathbf{U}}_n + (1 - \gamma_{NM}) \Delta t \ddot{\mathbf{U}}_n \]Corrector step:
\[ \ddot{\mathbf{U}}_{n+1} = ( \mathbf{M} + \gamma_{NM} \Delta t \mathbf{C} + \beta (\Delta t)^2 \mathbf{K}_{n+1} )^{-1} ( \mathbf{F}_{ext, n+1} - \mathbf{F}_{int}(\tilde{\mathbf{U}}_{n+1}) - \mathbf{C} \dot{\tilde{\mathbf{U}}}_{n+1} ) \] /* Note: Implicit dependence on K(U) */ \[ \mathbf{U}_{n+1} = \tilde{\mathbf{U}}_{n+1} + \beta (\Delta t)^2 \ddot{\mathbf{U}}_{n+1} \] \[ \dot{\mathbf{U}}_{n+1} = \dot{\tilde{\mathbf{U}}}_{n+1} + \gamma_{NM} \Delta t \ddot{\mathbf{U}}_{n+1} \]Used: \( \beta = 0.25 \), \( \gamma_{NM} = 0.5 \), \( \Delta t = 0.001 \, \text{s} \).
Von Mises equivalent stress \( \sigma_{VM} \):
\[ \sigma_{VM} = \sqrt{\frac{1}{2} [ ((\sigma_{11}-\sigma_{22})^2 + (\sigma_{22}-\sigma_{33})^2 + (\sigma_{33}-\sigma_{11})^2 + 6(\sigma_{12}^2 + \sigma_{23}^2 + \sigma_{31}^2) ]} \]Peak stress observed: \( \sigma_{VM,max} = 485 \, \text{MPa} \) at \( t = 3.72 \, \text{s} \).
Safety Factor (SF):
\[ SF = \frac{\sigma_y}{\sigma_{VM,max}} = \frac{550 \, \text{MPa}}{485 \, \text{MPa}} \approx 1.13 \]Generalized eigenvalue problem:
\[ \mathbf{K} \mathbf{\phi}_i = \omega_i^2 \mathbf{M} \mathbf{\phi}_i \]Modal participation factor \( \Gamma_i \) for load vector \( \mathbf{P} \):
\[ \Gamma_i = \frac{\mathbf{\phi}_i^T \mathbf{M} \mathbf{P}}{\mathbf{\phi}_i^T \mathbf{M} \mathbf{\phi}_i} \]Effective modal mass \( M_{eff,i} \):
\[ M_{eff,i} = \frac{(\mathbf{\phi}_i^T \mathbf{M} \mathbf{1})^2}{\mathbf{\phi}_i^T \mathbf{M} \mathbf{\phi}_i} \]Natural Frequencies:
| Mode (i) | Freq (Hz), \( f_i \) | Effective Mass (%) |
|---|---|---|
| 1 | 5.12 | 65.2% (X-dir) |
| 2 | 5.25 | 63.8% (Y-dir) |
| 3 | 15.8 | 70.1% (Rz-dir) |
| 4 | 22.4 | 12.5% (X-dir) |
SF > 1 indicates design adequacy under specified load. Lowest natural frequency \( f_1 = 5.12 \) Hz is sufficiently above load frequency \( f = 2 \) Hz. Recommend fatigue analysis.