有限元方法问答 14：振动与动力学

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L15a — Vibration & Dynamics（振动与动力学）🟡 样卷未考

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Q114 🟡 多自由度运动方程

EN: The equation of motion for a multi-DOF structural system is:

$$\boxed{\mathbf{M}\ddot{\mathbf{u}} + \mathbf{C}\dot{\mathbf{u}} + \mathbf{K}\mathbf{u} = \mathbf{f}(t)}$$

Physical meaning: Inertia forces + Damping forces + Elastic forces = Applied forces

• $\mathbf{M}$ — mass matrix (from kinetic energy)
• $\mathbf{C}$ — damping matrix
• $\mathbf{K}$ — stiffness matrix
• $\mathbf{u}$, $\dot{\mathbf{u}}$, $\ddot{\mathbf{u}}$ — displacement, velocity, acceleration

中文： 动力方程：$\mathbf{M}\ddot{\mathbf{u}}+\mathbf{C}\dot{\mathbf{u}}+\mathbf{K}\mathbf{u}=\mathbf{f}(t)$。惯性力+阻尼力+弹性力=外力。M是质量矩阵，C是阻尼矩阵，K是刚度矩阵。这是结构动力学的基本方程。

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Q115 🟡 无阻尼自由振动

EN: For undamped free vibration ($\mathbf{f} = \mathbf{0}$, $\mathbf{C} = \mathbf{0}$):

$$\mathbf{M}\ddot{\mathbf{u}} + \mathbf{K}\mathbf{u} = \mathbf{0}$$

Assuming harmonic motion $\mathbf{u}(t) = \boldsymbol{\phi}\sin(\omega t)$, we get the generalized eigenvalue problem:

$$\boxed{(\mathbf{K} - \omega^2\mathbf{M})\boldsymbol{\phi} = \mathbf{0}}$$

For non-trivial solutions: $\det(\mathbf{K} - \omega^2\mathbf{M}) = 0$. This yields $n$ eigenvalues $\omega_i^2$ (squared natural frequencies) and $n$ eigenvectors $\boldsymbol{\phi}_i$ (mode shapes), where $n$ is the number of DOFs.

中文： 无阻尼自由振动 → 广义特征值问题 $(\mathbf{K}-\omega^2\mathbf{M})\boldsymbol{\phi}=0$。n个DOF就有n个固有频率和n个模态。固有频率通过 $\det(\mathbf{K}-\omega^2\mathbf{M})=0$ 求解。

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Q116 🔴 集中质量矩阵 vs 一致质量矩阵

EN: Two types of mass matrices:

Lumped mass matrix: Mass is concentrated at nodes → diagonal matrix. For a bar element: $\mathbf{M} = \frac{\rho AL}{2}\begin{bmatrix}1&0\\0&1\end{bmatrix}$. Simple, computationally efficient.

Consistent mass matrix: Derived from the same shape functions as the stiffness matrix: $\mathbf{M} = \int_V \rho\mathbf{N}^T\mathbf{N}\,dV$. For a bar element: $\mathbf{M} = \frac{\rho AL}{6}\begin{bmatrix}2&1\\1&2\end{bmatrix}$. More accurate for a given mesh, but non-diagonal.

The consistent mass matrix follows the same formulation philosophy as the stiffness matrix (hence “consistent”).

中文： 集中质量矩阵 = 对角矩阵（质量集中在节点），计算效率高；一致质量矩阵 = $\int\rho\mathbf{N}^T\mathbf{N}dV$（与刚度矩阵使用相同的形函数），更准确但非对角。杆单元一致质量矩阵 = $\frac{\rho AL}{6}\begin{bmatrix}2&1\\1&2\end{bmatrix}$。

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Q117 🟡 固有频率公式

EN: For a single-DOF system, the circular natural frequency is:

$$\omega = \sqrt{\frac{k}{m}} \quad \text{[rad/s]}$$

The cyclic frequency: $f = \frac{\omega}{2\pi}$ [Hz].

For a fixed bar: $\omega_1 = \frac{\pi}{L}\sqrt{\frac{E}{\rho}}$ (first axial natural frequency).

For MDOF systems: $\omega_i = \sqrt{\lambda_i}$ where $\lambda_i$ are eigenvalues of $\mathbf{K}\boldsymbol{\phi} = \lambda\mathbf{M}\boldsymbol{\phi}$.

中文： 单DOF固有频率 $\omega=\sqrt{k/m}$（单位rad/s）；频率 $f=\omega/(2\pi)$（Hz）。多DOF系统：$\omega_i$ 通过特征值求解。

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Q118 🔴 低阶模态比高阶模态更准确

EN: In FEM dynamic analysis, lower modes (smaller $\omega_i$) are more accurate than higher modes. This is because:

• Lower modes have longer wavelengths, requiring fewer elements per wavelength for accurate representation
• Higher modes have shorter wavelengths, and the FE discretization cannot capture them well unless the mesh is very fine
• A rule of thumb: at least 6-10 elements per wavelength are needed for acceptable accuracy

For a given mesh, only the lowest $n/3$ to $n/2$ modes are typically considered reliable ($n$ = number of DOFs).

中文： 低阶模态（小频率）比高阶模态更准确。因为低阶模态波长大，有限元离散容易捕捉；高阶模态波长短，需要极密网格才能准确。经验法则：每个波长至少6-10个单元。给定网格下，通常只有最低的1/3到1/2阶模态可靠。

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Q119 🔴 刚体模态(Rigid Body Modes)

EN: Rigid body modes correspond to $\omega = 0$. These represent motions that produce no strain energy — pure translation or rotation of the unconstrained (or insufficiently constrained) structure. In a free-free vibration analysis:

• A 2D structure has 3 rigid body modes (2 translations + 1 rotation)
• A 3D structure has 6 rigid body modes (3 translations + 3 rotations)

Practical use: Rigid body modes can be used to check the FE model — unexpected zero-frequency modes indicate mechanisms, missing constraints, or disconnected parts.

中文： 刚体模态对应 $\omega=0$（无应变能的运动）。2D有3个刚体模态（2平动+1转动），3D有6个。可用于检查模型——多余的零频率说明有机构、缺少约束或连接错误。

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Q120 🔴 模态位移的大小无物理意义

EN: In normal mode analysis (eigenvalue analysis), the magnitude of the eigenvector has NO physical meaning — only the shape (relative proportions between DOFs) matters. Eigenvectors can be arbitrarily scaled. Common normalization: $\boldsymbol{\phi}_i^T\mathbf{M}\boldsymbol{\phi}_i = 1$ (mass normalization) or set the maximum component to 1.

中文： 模态分析中，位移大小无物理意义，只有形状（各DOF的相对比例）有意义。可以任意缩放。常见归一化：质量归一化 $\boldsymbol{\phi}_i^T\mathbf{M}\boldsymbol{\phi}_i=1$，或最大分量为1。

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Q121 🟡 模态正交性

EN: Mode shapes satisfy orthogonality with respect to $\mathbf{M}$ and $\mathbf{K}$:

$$\boldsymbol{\phi}_i^T \mathbf{M} \boldsymbol{\phi}_j = 0, \quad \boldsymbol{\phi}_i^T \mathbf{K} \boldsymbol{\phi}_j = 0 \quad \text{for } i \neq j$$

Mass-normalized: $\boldsymbol{\phi}_i^T \mathbf{M} \boldsymbol{\phi}_i = 1$ and $\boldsymbol{\phi}_i^T \mathbf{K} \boldsymbol{\phi}_i = \omega_i^2$.

Orthogonality enables modal superposition — the response can be expressed as a linear combination of mode shapes.

中文： 不同模态关于M和K正交。正交性使得模态叠加法成为可能（系统响应 = 各模态贡献的线性组合）。这是模态分析的理论基础。

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Q122 🟡 阻尼比和Rayleigh阻尼

EN: Damping ratio: $\zeta = c/c_c$, where $c_c = 2\sqrt{km} = 2m\omega$ is the critical damping. Structural damping: $0 < \zeta < 0.15$, typically 1–5%. For small damping: $\omega_d \approx \omega$.

Rayleigh (proportional) damping: $\mathbf{C} = \alpha\mathbf{M} + \beta\mathbf{K}$, where $\alpha$ and $\beta$ are determined from: $\zeta_i = \frac{\alpha}{2\omega_i} + \frac{\beta\omega_i}{2}$. This form preserves modal orthogonality and is convenient for FEM implementation.

中文： 阻尼比 $\zeta=c/c_c$，结构阻尼约1-5%，小阻尼下 $\omega_d\approx\omega$。Rayleigh阻尼 $\mathbf{C}=\alpha\mathbf{M}+\beta\mathbf{K}$ 保持了模态正交性，是FEM中最常用的阻尼模型。

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