Project Omega: Supplemental FEA Calculations

Dr. E. Noether | Rev: 3.2

1. Governing Equations

Wave equation with damping for displacement $\mathbf{u}(\mathbf{x},t)$:

$$\rho \frac{{\partial }^2\mathbf{u}}{\partial t^2}+\gamma \frac{\partial \mathbf{u}}{\partial t}-\mathrm{\nabla }\cdot \sigma =\mathbf{f}$$

Cauchy Stress Tensor $\sigma$ and Strain Tensor $ϵ$:

$$\sigma =\mathbf{C}:ϵ\text{(Linear Elasticity)}$$ $$ϵ=\frac{1}{2}(\mathrm{\nabla }\mathbf{u}+(\mathrm{\nabla }\mathbf{u}{)}^T)$$

2. Constitutive Models

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:

• $E=2.1\times {10}^{11}\text{Pa}$, $\nu =0.3$, $\rho =7850{\text{kg/m}}^3$, $\gamma =5.5\times {10}^4{\text{Ns/m}}^3$
• \( \mu = \frac{E}{2(1+\nu)} \approx 8.08 \times 10^{10} \, \text{Pa} \)
• \( \lambda = \frac{E\nu}{(1+\nu)(1-2\nu)} \approx 1.21 \times 10^{11} \, \text{Pa} \)

3. Loads & Boundary Conditions

Dirichlet BC: $\mathbf{u}(\mathbf{x},t)=\mathbf{0}$ for $\mathbf{x}\in {\mathrm{\Gamma }}_D$.

Neumann BC (Pressure): $\sigma \cdot \mathbf{n}=-p(t)\mathbf{n}$ for $\mathbf{x}\in {\mathrm{\Gamma }}_N$.

Applied time-varying pressure load $p(t)$:

$$p(t)=P_0(1+A\sin (2\pi ft+\varphi ))$$

Parameters: $P_0=5e5\text{Pa}$, $A=0.4$, $f=2\text{Hz}$, $\varphi =0$.

4. Numerical Integration (Newmark-beta)

Predictor step:

$${\tilde{\mathbf{U}}}_{n+1}={\mathbf{U}}_n+\mathrm{\Delta }t{\dot{\mathbf{U}}}_n+\frac{(\mathrm{\Delta }t{)}^2}{2}(1-2\beta ){\ddot{\mathbf{U}}}_n$$ $${\dot{\tilde{\mathbf{U}}}}_{n+1}={\dot{\mathbf{U}}}_n+(1-{\gamma }_{NM})\mathrm{\Delta }t{\ddot{\mathbf{U}}}_n$$

Corrector step:

$${\ddot{\mathbf{U}}}_{n+1}=(\mathbf{M}+{\gamma }_{NM}\mathrm{\Delta }t\mathbf{C}+\beta (\mathrm{\Delta }t{)}^2{\mathbf{K}}_{n+1}{)}^{-1}({\mathbf{F}}_{ext,n+1}-{\mathbf{F}}_{int}({\tilde{\mathbf{U}}}_{n+1})-\mathbf{C}{\dot{\tilde{\mathbf{U}}}}_{n+1})$$ /* Note: Implicit dependence on K(U) */ $${\mathbf{U}}_{n+1}={\tilde{\mathbf{U}}}_{n+1}+\beta (\mathrm{\Delta }t{)}^2{\ddot{\mathbf{U}}}_{n+1}$$ $${\dot{\mathbf{U}}}_{n+1}={\dot{\tilde{\mathbf{U}}}}_{n+1}+{\gamma }_{NM}\mathrm{\Delta }t{\ddot{\mathbf{U}}}_{n+1}$$

Used: $\beta =0.25$, ${\gamma }_{NM}=0.5$, $\mathrm{\Delta }t=0.001\text{s}$.

5. Stress Calculation & Results

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):

\[ SF = \frac{\sigma_y}{\sigma_{VM,max}} = \frac{550 \, \text{MPa}}{485 \, \text{MPa}} \approx 1.13 \]

6. Modal Analysis

Generalized eigenvalue problem:

$$\mathbf{K}{\varphi }_i={\omega }_i^2\mathbf{M}{\varphi }_i$$

Modal participation factor ${\mathrm{\Gamma }}_i$ for load vector $\mathbf{P}$:

$${\mathrm{\Gamma }}_i=\frac{{\varphi }_i^T\mathbf{M}\mathbf{P}}{{\varphi }_i^T\mathbf{M}{\varphi }_i}$$

Effective modal mass $M_{eff,i}$:

$$M_{eff,i}=\frac{({\varphi }_i^T\mathbf{M}\mathbf{1}{)}^2}{{\varphi }_i^T\mathbf{M}{\varphi }_i}$$

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)

7. Conclusion Summary

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.