# 第 2 部分：非线性系统的谐波激励建模

2016年 8月 11日

### 带有时间谐波分量的载荷和非线性系统

1. 系统上的所有随时间变化的载荷和约束必须以相同的固定频率正弦变化。
2. 所有的载荷、约束和材料属性必须与解无关。

(1)

\begin{align}
M u_{tt} + C u_t + \nabla \cdot (-K \nabla u) = F_0+\tilde F sin(\omega t) & \text {在域}  \Omega  \text {上，满足} \\
\mathbf{n} \cdot (K \nabla u) + Au = f _0 + \tilde f sin(\omega t ) & \text {在边界}  \Gamma_1  \text{上，满足}\\
u = g_0 + \tilde g sin ( \omega t) & \text{ 在边界 }  \Gamma_2  \text{上，满足}
\end{align}

(2)

\begin{align}
\nabla \cdot (-K \nabla u_0) = F_0 &\text { on } \Omega \\
\mathbf{n} \cdot (K \nabla u_0) + Au_0 = f _0 &\text{ on } \Gamma_1 \\
u_0 = g_0 &\text{ on } \Gamma_2
\end{align}

(3)

\begin{align}
-\omega^2 M(u_0) \tilde u + j \omega C(u_0) \tilde u + \nabla \cdot (-K(u_0) \nabla \tilde u) = \tilde F &\text { on } \Omega \\
\mathbf{n} \cdot (K(u_0) \nabla \tilde u) + A(u_0) \tilde u = \tilde f &\text{ on } \Gamma_1 \\
\tilde u = \tilde g &\text{ on } \Gamma_2
\end{align}

### 频域激励引起的非线性问题怎么解决？

(4)

\begin{align}
-\omega^2 M(|\tilde u|) \tilde u + j \omega C(|\tilde u|) \tilde u + \nabla \cdot (-K(|\tilde u|) \nabla \tilde u) = \tilde F(|\tilde u|) &\text { on } \Omega \\
\mathbf{n} \cdot (K(|\tilde u|) \nabla \tilde u) + A(|\tilde u|) \tilde u = \tilde f(|\tilde u|) &\text{ on } \Gamma_1 \\
\tilde u = \tilde g(|\tilde u|) &\text{ on } \Gamma_2
\end{align}

(5)

\begin{align}
-\omega^2 M \tilde u_1 -j \omega C \tilde u_1 + \nabla \cdot (-K \nabla \tilde u_1) = \tilde F_1 -Q &\text { on } \Omega \\
\mathbf{n} \cdot (K \nabla \tilde u_1) + A \tilde u_1 = \tilde f_1 &\text{ on } \Gamma_1 \\
\tilde u = \tilde g_1 &\text{ on } \Gamma_2 \\
-4 \omega^2 M \tilde u_2 -j 2 \omega C \tilde u_2 + \nabla \cdot (-K \nabla \tilde u_2) = + Q &\text { on } \Omega \\
\mathbf{n} \cdot (K \nabla \tilde u_2) + A \tilde u_2 = 0 &\text{ on } \Gamma_1 \\
\tilde u_2 = 0 &\text{ on } \Gamma_2
\end{align}