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@ -11,19 +11,19 @@ FIGURE 12-1 Representing deflections as sum of modal components.
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Consider, for example, the cantilever column shown in Fig. 12-1, for which the deflected shape is expressed in terms of translational displacements at three levels. Any displacement vector $\mathbf{v}$ (static or dynamic) for this structure can be developed by superposing suitable amplitudes of the normal modes as shown. For any modal component ${\bf v}_{n}$ , the displacements are given by the product of the mode-shape vector $\phi_{n}$ and the modal amplitude $Y_{n}$ ; thus
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Consider, for example, the cantilever column shown in Fig. 12-1, for which the deflected shape is expressed in terms of translational displacements at three levels. Any displacement vector $\mathbf{v}$ (static or dynamic) for this structure can be developed by superposing suitable amplitudes of the normal modes as shown. For any modal component ${\bf v}_{n}$ , the displacements are given by the product of the mode-shape vector $\phi_{n}$ and the modal amplitude $Y_{n}$ ; thus
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以图 12-1 所示的悬臂柱为例，其变形形状可通过三个高度处的平移位移来描述。该结构的任意位移向量 **v**（静态或动态）均可通过叠加适当振幅的简正模态来表示。对于任一模态分量 **vₙ**，其位移由模态形状向量 **φₙ** 与模态振幅 **Yₙ** 的乘积给出，即：
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以图 12-1 所示的悬臂柱为例，其变形形状可通过三个高度处的平移位移来描述。该结构的任意位移向量 **v**（静态或动态）均可通过叠加适当振幅的简正模态来表示。对于任一模态分量 **vₙ**，其位移由模态形状向量 **φₙ** 与模态振幅 **Yₙ** 的乘积给出，即：
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$$
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$$
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\mathbf{v}_{n}=\phi_{n}\ Y_{n}
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\mathbf{v}_{n}=\phi_{n}\ Y_{n} \tag{12-1}
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$$
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$$
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The total displacement vector $\mathbf{v}$ is then obtained by summing the modal vectors as expressed by
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The total displacement vector $\mathbf{v}$ is then obtained by summing the modal vectors as expressed by
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总位移向量 **v** 则通过对各模态向量求和得到，具体表达式为：
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总位移向量 **v** 则通过对各模态向量求和得到，具体表达式为：
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$$
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$$
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\mathbf{v}=\phi_{1}\,Y_{1}+\phi_{2}\,Y_{2}+\cdot\cdot\cdot+\phi_{N}\,Y_{N}=\sum_{n=1}^{N}\phi_{n}\,Y_{n}
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\mathbf{v}=\phi_{1}\,Y_{1}+\phi_{2}\,Y_{2}+\cdot\cdot\cdot+\phi_{N}\,Y_{N}=\sum_{n=1}^{N}\phi_{n}\,Y_{n}\tag{12-2}
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$$
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$$
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or, in matrix notation,
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or, in matrix notation,
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$$
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$$
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\mathbf{v}=\Phi\,Y
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\mathbf{v}=\Phi\,Y\tag{12-3}
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$$
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$$
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In this equation, it is apparent that the $N\times N$ mode-shape matrix $\Phi$ serves to transform the generalized coordinate vector $Y$ to the geometric coordinate vector $\mathbf{v}$ . The generalized components in vector $Y$ are called the normal coordinates of the structure.
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In this equation, it is apparent that the $N\times N$ mode-shape matrix $\Phi$ serves to transform the generalized coordinate vector $Y$ to the geometric coordinate vector $\mathbf{v}$ . The generalized components in vector $Y$ are called the normal coordinates of the structure.
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@ -35,25 +35,25 @@ Because the mode-shape matrix consists of $N$ independent modal vectors, $\pmb{\
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由于模态形状矩阵由 $N$ 个独立的模态向量组成，即
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由于模态形状矩阵由 $N$ 个独立的模态向量组成，即
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$\pmb{\Phi} = \left[\pmb{\phi}_{1}\,\,\,\,\pmb{\phi}_{2}\,\,\,\,\cdot\,\cdot\,\,\,\pmb{\phi}_{N}\right]$，该矩阵是非奇异的，因此可以求逆。因此，对于任意给定的位移向量 $\mathbf{v}$，总是可以直接求解方程（12-3），得到与之对应的正则坐标幅值 $Y$。不过，由于模态形状具有**正交性**，无需解联立方程组。为了求取任意正则坐标（例如 $Y_n$），可将方程（12-2）左乘以 $\phi_n^T \mathbf{m}$，得到：
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$\pmb{\Phi} = \left[\pmb{\phi}_{1}\,\,\,\,\pmb{\phi}_{2}\,\,\,\,\cdot\,\cdot\,\,\,\pmb{\phi}_{N}\right]$，该矩阵是非奇异的，因此可以求逆。因此，对于任意给定的位移向量 $\mathbf{v}$，总是可以直接求解方程（12-3），得到与之对应的正则坐标幅值 $Y$。不过，由于模态形状具有**正交性**，无需解联立方程组。为了求取任意正则坐标（例如 $Y_n$），可将方程（12-2）左乘以 $\phi_n^T \mathbf{m}$，得到：
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$$
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$$
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\phi_{n}^{T}\,\mathbf{m}\,\mathbf{v}=\phi_{n}^{T}\,\mathbf{m}\,\phi_{1}\,Y_{1}+\phi_{n}^{T}\,\mathbf{m}\,\phi_{2}\,Y_{2}+\hdots+\phi_{n}^{T}\,\mathbf{m}\,\phi_{N}\,Y_{N}
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\phi_{n}^{T}\,\mathbf{m}\,\mathbf{v}=\phi_{n}^{T}\,\mathbf{m}\,\phi_{1}\,Y_{1}+\phi_{n}^{T}\,\mathbf{m}\,\phi_{2}\,Y_{2}+\dots+\phi_{n}^{T}\,\mathbf{m}\,\phi_{N}\,Y_{N}\tag{12-4}
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$$
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$$
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Because of the orthogonality property with respect to mass, i.e., $\phi_{n}^{T}\,\mathbf{m}\,\phi_{m}\,=\,0$ for $m\neq n$ , all terms on the right hand side of this equation vanish, except for the term containing $\phi_{n}^{T}\,\mathbf{m}\,\phi_{n}$ , leaving
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Because of the orthogonality property with respect to mass, i.e., $\phi_{n}^{T}\,\mathbf{m}\,\phi_{m}\,=\,0$ for $m\neq n$ , all terms on the right hand side of this equation vanish, except for the term containing $\phi_{n}^{T}\,\mathbf{m}\,\phi_{n}$ , leaving
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由于质量矩阵的正交性，即对于 $m \neq n$，有 $\phi_{n}^{T}\,\mathbf{m}\,\phi_{m} = 0$，方程右侧的所有项均消失，仅剩下包含 $\phi_{n}^{T}\,\mathbf{m}\,\phi_{n}$ 的项。
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由于质量矩阵的正交性，即对于 $m \neq n$，有 $\phi_{n}^{T}\,\mathbf{m}\,\phi_{m} = 0$，方程右侧的所有项均消失，仅剩下包含 $\phi_{n}^{T}\,\mathbf{m}\,\phi_{n}$ 的项。
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$$
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$$
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\phi_{n}^{T}\,\mathbf{m}\,\mathbf{v}=\phi_{n}^{T}\,\mathbf{m}\,\phi_{n}\,Y_{n}
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\phi_{n}^{T}\,\mathbf{m}\,\mathbf{v}=\phi_{n}^{T}\,\mathbf{m}\,\phi_{n}\,Y_{n}\tag{12-5}
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$$
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$$
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from which
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from which
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$$
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$$
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Y_{n}=\frac{\phi_{n}^{T}\,\mathbf{m}\,\mathbf{v}}{\phi_{n}^{T}\,\mathbf{m}\,\phi_{n}}\qquad\qquad n=1,2,\cdots,N
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Y_{n}=\frac{\phi_{n}^{T}\,\mathbf{m}\,\mathbf{v}}{\phi_{n}^{T}\,\mathbf{m}\,\phi_{n}}\qquad\qquad n=1,2,\cdots,N\tag{12-6}
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$$
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$$
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If vector $\mathbf{v}$ is time dependent, the $Y_{n}$ coordinates will also be time dependent; in this case, taking the time derivative of Eq. (12-6) yields
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If vector $\mathbf{v}$ is time dependent, the $Y_{n}$ coordinates will also be time dependent; in this case, taking the time derivative of Eq. (12-6) yields
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如果向量 $\mathbf{v}$ 随时间变化，则其 $Y_{n}$ 坐标也将随时间变化；此时，对式（12-6）取时间导数，可得
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如果向量 $\mathbf{v}$ 随时间变化，则其 $Y_{n}$ 坐标也将随时间变化；此时，对式（12-6）取时间导数，可得
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$$
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$$
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\dot{Y}_{n}(t)=\frac{\phi_{n}^{T}\,\mathbf{m}\,\dot{\mathbf{v}}(t)}{\phi_{n}^{T}\,\mathbf{m}\,\phi_{n}}
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\dot{Y}_{n}(t)=\frac{\phi_{n}^{T}\,\mathbf{m}\,\dot{\mathbf{v}}(t)}{\phi_{n}^{T}\,\mathbf{m}\,\phi_{n}}\tag{12-7}
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$$
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$$
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Note that the above procedure is equivalent to that used to evaluate the coefficients in the Fourier series given by Eqs. (4-3).
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Note that the above procedure is equivalent to that used to evaluate the coefficients in the Fourier series given by Eqs. (4-3).
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@ -64,38 +64,38 @@ The orthogonality properties of the normal modes will now be used to simplify th
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现在将利用简正模态的正交性质来简化多自由度（MDOF）系统的运动方程。这些方程的一般形式由式（9-13）给出（或在轴向力存在时等效的式（9-19））；对于无阻尼系统，方程则简化为：
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现在将利用简正模态的正交性质来简化多自由度（MDOF）系统的运动方程。这些方程的一般形式由式（9-13）给出（或在轴向力存在时等效的式（9-19））；对于无阻尼系统，方程则简化为：
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$$
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$$
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\mathbf{m}\;\ddot{\mathbf{v}}(t)+\mathbf{k}\;\mathbf{v}(t)=\mathbf{p}(t)
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\mathbf{m}\;\ddot{\mathbf{v}}(t)+\mathbf{k}\;\mathbf{v}(t)=\mathbf{p}(t)\tag{12-8}
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$$
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$$
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Introducing Eq. (12-3) and its second time derivative $\ddot{\mathbf{v}}=\Phi\ \ddot{\mathbf{Y}}$ (noting that the mode shapes do not change with time) leads to
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Introducing Eq. (12-3) and its second time derivative $\ddot{\mathbf{v}}=\Phi\ \ddot{\mathbf{Y}}$ (noting that the mode shapes do not change with time) leads to
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将方程（12-3）及其二阶时间导数 $\ddot{\mathbf{v}}=\Phi\ \ddot{\mathbf{Y}}$（注意模态形状不随时间变化）代入后，得到
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将方程（12-3）及其二阶时间导数 $\ddot{\mathbf{v}}=\Phi\ \ddot{\mathbf{Y}}$（注意模态形状不随时间变化）代入后，得到
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$$
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$$
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\mathbf{m}\Phi{\ddot{Y}}(t)+\mathbf{k}\;\Phi\;Y(t)=\mathbf{p}(t)
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\mathbf{m}\Phi{\ddot{Y}}(t)+\mathbf{k}\;\Phi\;Y(t)=\mathbf{p}(t)\tag{12-9}
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$$
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$$
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If Eq. (12-9) is premultiplied by the transpose of the $n$ th mode-shape vector $\phi_{n}^{T}$ , it becomes
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If Eq. (12-9) is premultiplied by the transpose of the $n$ th mode-shape vector $\phi_{n}^{T}$ , it becomes
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如果将方程（12-9）左乘第 $n$ 阶模态形状向量的转置 $\phi_{n}^{T}$，则可得到
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如果将方程（12-9）左乘第 $n$ 阶模态形状向量的转置 $\phi_{n}^{T}$，则可得到
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$$
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$$
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\pmb{\phi}_{n}^{T}\pmb{\mathrm{m}}\,\pmb{\Phi}\,\ddot{\pmb{Y}}(t)+\pmb{\phi}_{n}^{T}\,\mathbf{k}\,\pmb{\Phi}\,\pmb{Y}(t)=\pmb{\phi}_{n}^{T}\,\mathbf{p}(t)
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\pmb{\phi}_{n}^{T}\pmb{\mathrm{m}}\,\pmb{\Phi}\,\ddot{\pmb{Y}}(t)+\pmb{\phi}_{n}^{T}\,\mathbf{k}\,\pmb{\Phi}\,\pmb{Y}(t)=\pmb{\phi}_{n}^{T}\,\mathbf{p}(t)\tag{12-10}
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$$
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$$
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but if the two terms on the left hand side are expanded as shown in Eq. (12-4), all terms except the $n$ th will vanish because of the mode-shape orthogonality properties; hence the result is
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but if the two terms on the left hand side are expanded as shown in Eq. (12-4), all terms except the $n$ th will vanish because of the mode-shape orthogonality properties; hence the result is
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但如果左侧的两项按照 Eq. (12-4) 展开，由于模态形状的正交性，除了第 $n$ 项外，所有项都将消失；因此最终结果为：
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但如果左侧的两项按照 Eq. (12-4) 展开，由于模态形状的正交性，除了第 $n$ 项外，所有项都将消失；因此最终结果为：
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$$
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$$
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\pmb{\phi}_{n}^{T}\mathbf{m}\pmb{\phi}_{n}~\ddot{Y}_{n}(t)+\pmb{\phi}_{n}^{T}\mathbf{k}\pmb{\phi}_{n}~Y_{n}(t)=\pmb{\phi}_{n}^{T}\mathbf{p}(t)
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\pmb{\phi}_{n}^{T}\mathbf{m}\pmb{\phi}_{n}~\ddot{Y}_{n}(t)+\pmb{\phi}_{n}^{T}\mathbf{k}\pmb{\phi}_{n}~Y_{n}(t)=\pmb{\phi}_{n}^{T}\mathbf{p}(t)\tag{12-11}
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$$
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$$
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Now new symbols will be defined as follows:
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Now new symbols will be defined as follows:
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以下新符号定义如下：
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以下新符号定义如下：
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$$
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$$
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\begin{array}{r l}&{M_{n}\equiv\boldsymbol{\phi}_{n}^{T}\mathbf{m}\boldsymbol{\phi}_{n}}\\ &{K_{n}\equiv\boldsymbol{\phi}_{n}^{T}\mathbf{k}\boldsymbol{\phi}_{n}}\\ &{P_{n}(t)\equiv\boldsymbol{\phi}_{n}^{T}\mathbf{p}(t)}\end{array}
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\begin{array}{r l}&{M_{n}\equiv\boldsymbol{\phi}_{n}^{T}\mathbf{m}\boldsymbol{\phi}_{n}}\\ &{K_{n}\equiv\boldsymbol{\phi}_{n}^{T}\mathbf{k}\boldsymbol{\phi}_{n}}\\ &{P_{n}(t)\equiv\boldsymbol{\phi}_{n}^{T}\mathbf{p}(t)}\end{array}\tag{12-12 a b c}
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$$
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$$
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which are called the normal-coordinate generalized mass, generalized stiffness, and generalized load for mode $n$ , respectively. With them Eq. (12-11) can be written
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which are called the normal-coordinate generalized mass, generalized stiffness, and generalized load for mode $n$ , respectively. With them Eq. (12-11) can be written
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这些分别被称为第 $n$ 阶模态的**广义质量**、**广义刚度**和**广义载荷**。利用它们，方程（12-11）可以表示为：
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这些分别被称为第 $n$ 阶模态的**广义质量**、**广义刚度**和**广义载荷**。利用它们，方程（12-11）可以表示为：
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$$
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$$
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M_{n}\,\ddot{Y}_{n}(t)+K_{n}\,Y_{n}(t)=P_{n}(t)
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M_{n}\,\ddot{Y}_{n}(t)+K_{n}\,Y_{n}(t)=P_{n}(t)\tag{12-13}
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$$
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$$
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which is a SDOF equation of motion for mode $n$ . If Eq. (11-39), ${\bf k}\phi_{n}=\omega_{n}^{2}{\bf m}\phi_{n}$ , is multiplied on both sides by $\phi_{n}^{T}$ , the generalized stiffness for mode $n$ is related to the generalized mass by the frequency of vibration
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which is a SDOF equation of motion for mode $n$ . If Eq. (11-39), ${\bf k}\phi_{n}=\omega_{n}^{2}{\bf m}\phi_{n}$ , is multiplied on both sides by $\phi_{n}^{T}$ , the generalized stiffness for mode $n$ is related to the generalized mass by the frequency of vibration
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@ -104,7 +104,7 @@ which is a SDOF equation of motion for mode $n$ . If Eq. (11-39), ${\bf k}\phi_{
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or
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or
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$$
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$$
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\begin{array}{r l}&{\phi_{n}^{T}\,\mathbf k\,\phi_{n}=\omega_{n}^{2}\phi_{n}^{T}\,\mathbf m\,\phi_{n}}\\ &{\qquad\qquad\qquad\qquad\qquad\qquad}\\ &{K_{n}=\omega_{n}^{2}M_{n}}\end{array}
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\begin{array}{r l}&{\phi_{n}^{T}\,\mathbf k\,\phi_{n}=\omega_{n}^{2}\phi_{n}^{T}\,\mathbf m\,\phi_{n}}\\ &{\qquad\qquad\qquad\qquad\qquad\qquad}\\ &{K_{n}=\omega_{n}^{2}M_{n}}\end{array}\tag{12-12d}
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$$
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$$
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(Capital letters are used to denote all normal-coordinate properties.)
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(Capital letters are used to denote all normal-coordinate properties.)
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@ -124,7 +124,7 @@ $$
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Introducing the normal-coordinate expression of Eq. (12-3) and its time derivatives and premultiplying by the transpose of the $n$ th mode-shape vector $\phi_{n}^{T}$ leads to
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Introducing the normal-coordinate expression of Eq. (12-3) and its time derivatives and premultiplying by the transpose of the $n$ th mode-shape vector $\phi_{n}^{T}$ leads to
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将方程（12-3）的正则坐标表达式及其时间导数代入，并左乘第 $n$ 阶模态形状向量的转置 $\phi_{n}^{T}$，可得到：
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将方程（12-3）的正则坐标表达式及其时间导数代入，并左乘第 $n$ 阶模态形状向量的转置 $\phi_{n}^{T}$，可得到：
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$$
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$$
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\phi_{n}^{T}\mathbf{m}\Phi\ddot{Y}(t)+\phi_{n}^{T}\mathbf{c}\Phi\dot{Y}(t)+\phi_{n}^{T}\mathbf{k}\Phi Y(t)=\phi_{n}^{T}\mathbf{p}(t)
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\phi_{n}^{T}\mathbf{m}\Phi\ddot{Y}(t)+\phi_{n}^{T}\mathbf{c}\Phi\dot{Y}(t)+\phi_{n}^{T}\mathbf{k}\Phi Y(t)=\phi_{n}^{T}\mathbf{p}(t)\tag{12-14}
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$$
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$$
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It was noted above that the orthogonality conditions
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It was noted above that the orthogonality conditions
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@ -137,31 +137,31 @@ cause all components except the nth-mode term in the mass and stiffness expressi
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在 Eq. (12-14) 的质量和刚度表达式中，**除了第 *n* 阶模态项外**，所有其他组分均消失。若假设阻尼矩阵同样满足相应的正交性条件，则阻尼表达式也将出现类似的简化，即假设：
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在 Eq. (12-14) 的质量和刚度表达式中，**除了第 *n* 阶模态项外**，所有其他组分均消失。若假设阻尼矩阵同样满足相应的正交性条件，则阻尼表达式也将出现类似的简化，即假设：
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$$
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$$
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\phi_{m}^{T}\,\mathbf{c}\,\phi_{n}=0\qquad m\neq n
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\phi_{m}^{T}\,\mathbf{c}\,\phi_{n}=0\qquad m\neq n\tag{12-15}
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$$
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$$
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In this case Eq. (12-14) may be written
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In this case Eq. (12-14) may be written
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在这种情况下，式（12-14）可表示为：
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在这种情况下，式（12-14）可表示为：
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$$
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$$
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M_{n}\ \ddot{Y}_{n}(t)+C_{n}\ \dot{Y}_{n}(t)+K_{n}\ Y_{n}(t)=P_{n}(t)
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M_{n}\ \ddot{Y}_{n}(t)+C_{n}\ \dot{Y}_{n}(t)+K_{n}\ Y_{n}(t)=P_{n}(t)\tag{12-14a}
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$$
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$$
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where the definitions of modal coordinate mass, stiffness, and load have been introduced from Eq. (12-12) and where the modal coordinate viscous damping coefficient has been defined similarly
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where the definitions of modal coordinate mass, stiffness, and load have been introduced from Eq. (12-12) and where the modal coordinate viscous damping coefficient has been defined similarly
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其中，模态坐标下的质量、刚度和载荷的定义已在式（12-12）中引入，而模态坐标下的黏性阻尼系数则类似地进行了定义。
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其中，模态坐标下的质量、刚度和载荷的定义已在式（12-12）中引入，而模态坐标下的黏性阻尼系数则类似地进行了定义。
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$$
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$$
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C_{n}=\pmb{\phi}_{n}^{T}\pmb{\mathrm{c}}\,\pmb{\phi}_{n}
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C_{n}=\pmb{\phi}_{n}^{T}\pmb{\mathrm{c}}\,\pmb{\phi}_{n}\tag{12-15a}
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$$
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$$
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If Eq. (12-14a) is divided by the generalized mass, this modal equation of motion may be expressed in alternative form:
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If Eq. (12-14a) is divided by the generalized mass, this modal equation of motion may be expressed in alternative form:
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若将式（12-14a）除以广义质量，该模态运动方程可表示为另一种形式：
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若将式（12-14a）除以广义质量，该模态运动方程可表示为另一种形式：
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$$
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$$
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\ddot{Y}_{n}(t)+2\,\xi_{n}\,\omega_{n}\,\dot{Y}_{n}(t)+\omega_{n}^{2}\,Y_{n}(t)=\frac{P_{n}(t)}{M_{n}}
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\ddot{Y}_{n}(t)+2\,\xi_{n}\,\omega_{n}\,\dot{Y}_{n}(t)+\omega_{n}^{2}\,Y_{n}(t)=\frac{P_{n}(t)}{M_{n}}\tag{12-14b}
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$$
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$$
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where Eq. (12-12d) has been used to rewrite the stiffness term and where the second term on the left hand side represents a definition of the modal viscous damping ratio
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where Eq. (12-12d) has been used to rewrite the stiffness term and where the second term on the left hand side represents a definition of the modal viscous damping ratio
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其中，式（12-12d）被用于重写刚度项，而左侧第二项则定义了模态黏性阻尼比。
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其中，式（12-12d）被用于重写刚度项，而左侧第二项则定义了模态黏性阻尼比。
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$$
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$$
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\xi_{n}={\frac{C_{n}}{2\,\omega_{n}\,M_{n}}}
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\xi_{n}={\frac{C_{n}}{2\,\omega_{n}\,M_{n}}}\tag{12-15b}
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$$
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$$
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As was noted earlier, it generally is more convenient and physically reasonable to define the damping of a MDOF system using the damping ratio for each mode in this way rather than to evaluate the coefficients of the damping matrix c because the modal damping ratios $\xi_{n}$ can be determined experimentally or estimated with adequate precision in many cases.
|
As was noted earlier, it generally is more convenient and physically reasonable to define the damping of a MDOF system using the damping ratio for each mode in this way rather than to evaluate the coefficients of the damping matrix c because the modal damping ratios $\xi_{n}$ can be determined experimentally or estimated with adequate precision in many cases.
|
||||||
@ -174,26 +174,26 @@ As was noted earlier, it generally is more convenient and physically reasonable
|
|||||||
The normal coordinate transformation was used in Section 12-3 to convert the $N$ coupled linear damped equations of motion
|
The normal coordinate transformation was used in Section 12-3 to convert the $N$ coupled linear damped equations of motion
|
||||||
在第 12-3 节中，采用了**正常坐标变换**将由 $N$ 个耦合的线性阻尼运动方程进行转换。
|
在第 12-3 节中，采用了**正常坐标变换**将由 $N$ 个耦合的线性阻尼运动方程进行转换。
|
||||||
$$
|
$$
|
||||||
\mathbf{m}\,\ddot{\mathbf{v}}(t)+\mathbf{c}\,\dot{\mathbf{v}}(t)+\mathbf{k}\,\mathbf{v}(t)=\mathbf{p}(t)
|
\mathbf{m}\,\ddot{\mathbf{v}}(t)+\mathbf{c}\,\dot{\mathbf{v}}(t)+\mathbf{k}\,\mathbf{v}(t)=\mathbf{p}(t)\tag{12-16}
|
||||||
$$
|
$$
|
||||||
|
|
||||||
to a set of $N$ uncoupled equations given by 成由以下 $N$ 个 **解耦方程** 组成的集合：
|
to a set of $N$ uncoupled equations given by 成由以下 $N$ 个 **解耦方程** 组成的集合：
|
||||||
|
|
||||||
|
|
||||||
$$
|
$$
|
||||||
\ddot{Y}_{n}(t)+2\,\xi_{n}\,\omega_{n}\,\dot{Y}_{n}(t)+\omega_{n}^{2}\,Y_{n}(t)=\frac{P_{n}(t)}{M_{n}}\qquad n=1,2,\cdots,N
|
\ddot{Y}_{n}(t)+2\,\xi_{n}\,\omega_{n}\,\dot{Y}_{n}(t)+\omega_{n}^{2}\,Y_{n}(t)=\frac{P_{n}(t)}{M_{n}}\qquad n=1,2,\cdots,N\tag{12-17}
|
||||||
$$
|
$$
|
||||||
|
|
||||||
in which
|
in which
|
||||||
|
|
||||||
$$
|
$$
|
||||||
M_{n}=\phi_{n}^{T}\textbf{m}\phi_{n}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;P_{n}(t)=\phi_{n}^{T}\textbf{p}(t)
|
M_{n}=\phi_{n}^{T}\textbf{m}\phi_{n}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;P_{n}(t)=\phi_{n}^{T}\textbf{p}(t)\tag{12-18}
|
||||||
$$
|
$$
|
||||||
|
|
||||||
To proceed with the solution of these uncoupled equations of motion, one must first solve the eigenvalue problem
|
To proceed with the solution of these uncoupled equations of motion, one must first solve the eigenvalue problem
|
||||||
要解决这些**解耦**的运动方程，首先必须求解**特征值问题**。
|
要解决这些**解耦**的运动方程，首先必须求解**特征值问题**。
|
||||||
$$
|
$$
|
||||||
\left[\mathbf{k}-\omega^{2}\mathbf{m}\right]\,\hat{\mathbf{v}}=\mathbf{0}
|
\left[\mathbf{k}-\omega^{2}\mathbf{m}\right]\,\hat{\mathbf{v}}=\mathbf{0}\tag{12-19}
|
||||||
$$
|
$$
|
||||||
|
|
||||||
to obtain the required mode shapes $\phi_{n}$ $(n=1,2,\cdot\cdot\cdot)$ and corresponding frequencies $\omega_{n}$ . The modal damping ratios $\xi_{n}$ are usually assumed based on experimental evidence.
|
to obtain the required mode shapes $\phi_{n}$ $(n=1,2,\cdot\cdot\cdot)$ and corresponding frequencies $\omega_{n}$ . The modal damping ratios $\xi_{n}$ are usually assumed based on experimental evidence.
|
||||||
@ -202,19 +202,19 @@ to obtain the required mode shapes $\phi_{n}$ $(n=1,2,\cdot\cdot\cdot)$ and corr
|
|||||||
The total response of the MDOF system now can be obtained by solving the $N$ uncoupled modal equations and superposing their effects, as indicated by Eq. (12-3). Each of Eqs. (12-17) is a standard SDOF equation of motion and can be solved in either the time domain or the frequency domain by the procedures described in Chapter 6. The time-domain solution is expressed by the Duhamel integral [see Eq. (6-7)]
|
The total response of the MDOF system now can be obtained by solving the $N$ uncoupled modal equations and superposing their effects, as indicated by Eq. (12-3). Each of Eqs. (12-17) is a standard SDOF equation of motion and can be solved in either the time domain or the frequency domain by the procedures described in Chapter 6. The time-domain solution is expressed by the Duhamel integral [see Eq. (6-7)]
|
||||||
多自由度（MDOF）系统的总响应可通过求解 $N$ 个**解耦**的**模态**方程并叠加其效应来获得，如式（12-3）所示。式（12-17）中的每一个方程均为标准的单自由度（SDOF）运动方程，可通过第 6 章所述的方法在**时域**或**频域**中求解。时域解可用**杜哈美Duhamel积分**（见式（6-7））表示。
|
多自由度（MDOF）系统的总响应可通过求解 $N$ 个**解耦**的**模态**方程并叠加其效应来获得，如式（12-3）所示。式（12-17）中的每一个方程均为标准的单自由度（SDOF）运动方程，可通过第 6 章所述的方法在**时域**或**频域**中求解。时域解可用**杜哈美Duhamel积分**（见式（6-7））表示。
|
||||||
$$
|
$$
|
||||||
Y_{n}(t)=\frac{1}{M_{n}\omega_{n}}\,\int_{0}^{t}P_{n}(\tau)\;\exp\big[-\xi_{n}\omega_{n}\left(t-\tau\right)\big]\;\sin\omega_{D n}(t-\tau)\;d\tau
|
Y_{n}(t)=\frac{1}{M_{n}\omega_{n}}\,\int_{0}^{t}P_{n}(\tau)\;\exp\big[-\xi_{n}\omega_{n}\left(t-\tau\right)\big]\;\sin\omega_{D n}(t-\tau)\;d\tau\tag{12-20}
|
||||||
$$
|
$$
|
||||||
|
|
||||||
which also may be written in standard convolution integral form
|
which also may be written in standard convolution integral form
|
||||||
也可表示为标准卷积积分形式。
|
也可表示为标准卷积积分形式。
|
||||||
$$
|
$$
|
||||||
Y_{n}(t)=\int_{0}^{t}P_{n}(\tau)\;h_{n}(t-\tau)\;d\tau
|
Y_{n}(t)=\int_{0}^{t}P_{n}(\tau)\;h_{n}(t-\tau)\;d\tau\tag{12-21}
|
||||||
$$
|
$$
|
||||||
|
|
||||||
in which
|
in which
|
||||||
|
|
||||||
$$
|
$$
|
||||||
h_{n}(t-\tau)=\frac{1}{M_{n}\omega_{D n}}\ \sin\omega_{D n}(t-\tau)\ \exp\big[-\xi_{n}\omega_{n}\left(t-\tau\right)\big]\ \ \ 0<\xi_{n}<1
|
h_{n}(t-\tau)=\frac{1}{M_{n}\omega_{D n}}\ \sin\omega_{D n}(t-\tau)\ \exp\big[-\xi_{n}\omega_{n}\left(t-\tau\right)\big]\ \ \ 0<\xi_{n}<1\tag{12-22}
|
||||||
$$
|
$$
|
||||||
|
|
||||||
is the unit-impulse response function, similar to Eq. (6-8).
|
is the unit-impulse response function, similar to Eq. (6-8).
|
||||||
@ -222,38 +222,41 @@ is the unit-impulse response function, similar to Eq. (6-8).
|
|||||||
In the frequency domain, the response is obtained similarly to Eq. (6-24) from
|
In the frequency domain, the response is obtained similarly to Eq. (6-24) from
|
||||||
在频域中，响应的获得方式与方程（6-24）类似。
|
在频域中，响应的获得方式与方程（6-24）类似。
|
||||||
$$
|
$$
|
||||||
Y_{n}(t)=\frac{1}{2\pi}\;\int_{-\infty}^{\infty}\mathrm{H}_{n}(i\overline{{{\omega}}})\;\mathrm{P}_{n}(i\overline{{{\omega}}})\;\exp i\overline{{{\omega}}}t\;d\overline{{{\omega}}}
|
Y_{n}(t)=\frac{1}{2\pi}\;\int_{-\infty}^{\infty}\mathrm{H}_{n}(i\overline{{{\omega}}})\;\mathrm{P}_{n}(i\overline{{{\omega}}})\;\exp i\overline{{{\omega}}}t\;d\overline{{{\omega}}}\tag{12-23}
|
||||||
$$
|
$$
|
||||||
|
|
||||||
In this equation, the complex load function $\mathrm{P}_{n}(i\overline{{\omega}})$ is the Fourier transform of the modal loading $P_{n}(t)$ , and similar to Eq. (6-23) it is given by
|
In this equation, the complex load function $\mathrm{P}_{n}(i\overline{{\omega}})$ is the Fourier transform of the modal loading $P_{n}(t)$ , and similar to Eq. (6-23) it is given by
|
||||||
在此方程中，复杂载荷函数 $\mathrm{P}_{n}(i\overline{{\omega}})$ 是模态载荷 $P_{n}(t)$ 的傅里叶变换，与式（6-23）类似，其表达式为
|
在此方程中，复杂载荷函数 $\mathrm{P}_{n}(i\overline{{\omega}})$ 是模态载荷 $P_{n}(t)$ 的傅里叶变换，与式（6-23）类似，其表达式为
|
||||||
$$
|
$$
|
||||||
\mathbf{P}_{n}(i\overline{{\omega}})=\int_{-\infty}^{\infty}P_{n}(t)\,\exp(-i\overline{{\omega}}t)\;d t
|
\mathbf{P}_{n}(i\overline{{\omega}})=\int_{-\infty}^{\infty}P_{n}(t)\,\exp(-i\overline{{\omega}}t)\;d t\tag{12-24}
|
||||||
$$
|
$$
|
||||||
|
|
||||||
Also in Eq. (12-23), the complex frequency response function, $\mathrm{H}_{n}(i\overline{{\omega}})$ , may be expressed similarly to Eq. (6-25) as follows:
|
Also in Eq. (12-23), the complex frequency response function, $\mathrm{H}_{n}(i\overline{{\omega}})$ , may be expressed similarly to Eq. (6-25) as follows:
|
||||||
在方程（12-23）中，复频响应函数 $\mathrm{H}_{n}(i\overline{{\omega}})$ 也可类似于方程（6-25）表示如下：
|
在方程（12-23）中，复频响应函数 $\mathrm{H}_{n}(i\overline{{\omega}})$ 也可类似于方程（6-25）表示如下：
|
||||||
$$
|
$$
|
||||||
\begin{array}{r l r}{\mathrm{H}_{n}(i\overline{{\omega}})=\frac{1}{\omega_{n}^{2}M_{n}}\left[\frac{1}{\left(1-\beta_{n}^{2}\right)+i(2\xi_{n}\beta_{n})}\right]}&{}&\\ {=\frac{1}{\omega_{n}^{2}M_{n}}\left[\frac{\left(1-\beta_{n}^{2}\right)-i(2\xi_{n}\beta_{n})}{(1-\beta_{n}^{2})^{2}+(2\xi_{n}\beta_{n})^{2}}\right]}&{}&{\xi_{n}\ge0}\end{array}
|
\begin{array}{r l r}{\mathrm{H}_{n}(i\overline{{\omega}})=\frac{1}{\omega_{n}^{2}M_{n}}\left[\frac{1}{\left(1-\beta_{n}^{2}\right)+i(2\xi_{n}\beta_{n})}\right]}&{}&\\ {=\frac{1}{\omega_{n}^{2}M_{n}}\left[\frac{\left(1-\beta_{n}^{2}\right)-i(2\xi_{n}\beta_{n})}{(1-\beta_{n}^{2})^{2}+(2\xi_{n}\beta_{n})^{2}}\right]}&{}&{\xi_{n}\ge0}\end{array}\tag{12-25}
|
||||||
$$
|
$$
|
||||||
|
|
||||||
In these functions, $\beta_{n}\equiv\,\overline{{\omega}}/\omega_{n}$ and $\omega_{D n}\,=\,\omega_{n}\sqrt{1-\xi_{n}^{2}}$ . As indicated previously by Eqs. (6-53) and (6-54), $h_{n}(t)$ and $\mathrm{H}_{n}(i\overline{{\omega}})$ are Fourier transform pairs. Solving Eq. (12-20) or (12-23) for any general modal loading yields the modal response $Y_{n}(t)$ for $t\,\geq\,0$ , assuming zero initial conditions, i.e., $Y_{n}(0)\,=\,\dot{Y}_{n}(0)\,=\,0$ . Should the initial conditions not equal zero, the damped free-vibration response [Eq. (2-49)]
|
In these functions, $\beta_{n}\equiv\,\overline{{\omega}}/\omega_{n}$ and $\omega_{D n}\,=\,\omega_{n}\sqrt{1-\xi_{n}^{2}}$ . As indicated previously by Eqs. (6-53) and (6-54), $h_{n}(t)$ and $\mathrm{H}_{n}(i\overline{{\omega}})$ are Fourier transform pairs. Solving Eq. (12-20) or (12-23) for any general modal loading yields the modal response $Y_{n}(t)$ for $t\,\geq\,0$ , assuming zero initial conditions, i.e., $Y_{n}(0)\,=\,\dot{Y}_{n}(0)\,=\,0$ . Should the initial conditions not equal zero, the damped free-vibration response [Eq. (2-49)]
|
||||||
|
|
||||||
在这些函数中，$\beta_{n}\equiv\,\overline{{\omega}}/\omega_{n}$ 且 $\omega_{D n}\,=\,\omega_{n}\sqrt{1-\xi_{n}^{2}}$。如前所述（式（6-53）和（6-54）所示），$h_{n}(t)$ 和 $\mathrm{H}_{n}(i\overline{{\omega}})$ 是傅里叶变换对。对于任意一般模态载荷，通过求解方程（12-20）或（12-23）可得到模态响应 $Y_{n}(t)$（$t\,\geq\,0$），假设初始条件为零，即 $Y_{n}(0)\,=\,\dot{Y}_{n}(0)\,=\,0$。若初始条件不为零，则需考虑阻尼自由振动响应（式（2-49））。
|
在这些函数中，$\beta_{n}\equiv\,\overline{{\omega}}/\omega_{n}$ 且 $\omega_{D n}\,=\,\omega_{n}\sqrt{1-\xi_{n}^{2}}$。如前所述（式（6-53）和（6-54）所示），$h_{n}(t)$ 和 $\mathrm{H}_{n}(i\overline{{\omega}})$ 是傅里叶变换对。对于任意一般模态载荷，通过求解方程（12-20）或（12-23）可得到模态响应 $Y_{n}(t)$（$t\,\geq\,0$），假设初始条件为零，即 $Y_{n}(0)\,=\,\dot{Y}_{n}(0)\,=\,0$。若初始条件不为零，则需考虑阻尼自由振动响应（式（2-49））。
|
||||||
$$
|
$$
|
||||||
Y_{n}(t)=\left[Y_{n}(0)\,\cos{\omega_{D n}t}+\left(\frac{\dot{Y}_{n}(0)+Y_{n}(0)\xi_{n}\omega_{n}}{\omega_{D n}}\right)\,\sin{\omega_{D n}t}\right]\,\exp(-\xi_{n}\omega_{n}t)
|
Y_{n}(t)=\left[Y_{n}(0)\,\cos{\omega_{D n}t}+\left(\frac{\dot{Y}_{n}(0)+Y_{n}(0)\xi_{n}\omega_{n}}{\omega_{D n}}\right)\,\sin{\omega_{D n}t}\right]\,\exp(-\xi_{n}\omega_{n}t)\tag{12-26}
|
||||||
$$
|
$$
|
||||||
|
|
||||||
must be added to the forced-vibration response given by Eqs. (12-20) or (12-23). The initial conditions $Y_{n}(0)$ and $\dot{Y}_{n}(0)$ in this equation are determined from $\mathbf{v}(0)$ and $\dot{\mathbf{v}}(0)$ using Eqs. (12-6) and (12-7) in the forms
|
must be added to the forced-vibration response given by Eqs. (12-20) or (12-23). The initial conditions $Y_{n}(0)$ and $\dot{Y}_{n}(0)$ in this equation are determined from $\mathbf{v}(0)$ and $\dot{\mathbf{v}}(0)$ using Eqs. (12-6) and (12-7) in the forms
|
||||||
必须将其添加到方程（12-20）或（12-23）给出的强迫振动响应中。该方程中的初始条件 $Y_{n}(0)$ 和 $\dot{Y}_{n}(0)$ 可通过方程（12-6）和（12-7）的形式，利用 $\mathbf{v}(0)$ 和 $\dot{\mathbf{v}}(0)$ 确定。
|
必须将其添加到方程（12-20）或（12-23）给出的强迫振动响应中。该方程中的初始条件 $Y_{n}(0)$ 和 $\dot{Y}_{n}(0)$ 可通过方程（12-6）和（12-7）的形式，利用 $\mathbf{v}(0)$ 和 $\dot{\mathbf{v}}(0)$ 确定。
|
||||||
$$
|
$$
|
||||||
\begin{array}{l}{\displaystyle Y_{n}(0)=\frac{\phi_{n}^{T}\textbf{m v}(0)}{\phi_{n}^{T}\textbf{m}\phi_{n}}}\\ {\displaystyle\dot{Y}_{n}(0)=\frac{\phi_{n}^{T}\textbf{m}\dot{\mathbf{v}}(0)}{\phi_{n}^{T}\textbf{m}\phi_{n}}}\end{array}
|
{\displaystyle Y_{n}(0)=\frac{\phi_{n}^{T}\textbf{m v}(0)}{\phi_{n}^{T}\textbf{m}\phi_{n}}}\tag{12-27}
|
||||||
|
$$
|
||||||
|
$$
|
||||||
|
{\displaystyle\dot{Y}_{n}(0)=\frac{\phi_{n}^{T}\textbf{m}\dot{\mathbf{v}}(0)}{\phi_{n}^{T}\textbf{m}\phi_{n}}}\tag{12-28}
|
||||||
$$
|
$$
|
||||||
|
|
||||||
Having generated the total response for each mode $Y_{n}(t)$ using either Eq. (12- 20) or Eq. (12-23) and Eq. (12-26), the displacements expressed in the geometric coordinates can be obtained using Eq. (12-2), i.e.,
|
Having generated the total response for each mode $Y_{n}(t)$ using either Eq. (12- 20) or Eq. (12-23) and Eq. (12-26), the displacements expressed in the geometric coordinates can be obtained using Eq. (12-2), i.e.,
|
||||||
在使用公式（12-20）或公式（12-23）和（12-26）生成各模态 $Y_{n}(t)$ 的总响应后，可通过几何坐标下的位移表达式（即公式（12-2））得到各模态的位移，即：
|
在使用公式（12-20）或公式（12-23）和（12-26）生成各模态 $Y_{n}(t)$ 的总响应后，可通过几何坐标下的位移表达式（即公式（12-2））得到各模态的位移，即：
|
||||||
$$
|
$$
|
||||||
\mathbf{v}(t)=\phi_{1}\,\,Y_{1}(t)+\phi_{2}\,\,Y_{2}(t)+\cdot\,\cdot\,\cdot+\phi_{N}\,\,Y_{N}(t)
|
\mathbf{v}(t)=\phi_{1}\,\,Y_{1}(t)+\phi_{2}\,\,Y_{2}(t)+\cdot\,\cdot\,\cdot+\phi_{N}\,\,Y_{N}(t)\tag{12-29}
|
||||||
$$
|
$$
|
||||||
|
|
||||||
which superposes the separate modal displacement contributions; hence, the commonly referred to name mode superposition method. It should be noted that for most types of loadings the displacement contributions generally are greatest for the lower modes and tend to decrease for the higher modes. Consequently, it usually is not necessary to include all the higher modes of vibration in the superposition process; the series can be truncated when the response has been obtained to any desired degree of accuracy. Moreover, it should be kept in mind that the mathematical idealization of any complex structural system also tends to be less reliable in predicting the higher modes of vibration; for this reason, too, it is well to limit the number of modes considered in a dynamic-response analysis.
|
which superposes the separate modal displacement contributions; hence, the commonly referred to name mode superposition method. It should be noted that for most types of loadings the displacement contributions generally are greatest for the lower modes and tend to decrease for the higher modes. Consequently, it usually is not necessary to include all the higher modes of vibration in the superposition process; the series can be truncated when the response has been obtained to any desired degree of accuracy. Moreover, it should be kept in mind that the mathematical idealization of any complex structural system also tends to be less reliable in predicting the higher modes of vibration; for this reason, too, it is well to limit the number of modes considered in a dynamic-response analysis.
|
||||||
@ -262,26 +265,26 @@ which superposes the separate modal displacement contributions; hence, the commo
|
|||||||
The displacement time-histories in vector $\mathbf{v}(t)$ may be considered to be the basic measure of a structure’s overall response to dynamic loading. In general, other response parameters such as stresses or forces developed in various structural components can be evaluated directly from the displacements. For example, the elastic forces $\mathbf{f}_{S}$ which resist the deformation of the structure are given directly by
|
The displacement time-histories in vector $\mathbf{v}(t)$ may be considered to be the basic measure of a structure’s overall response to dynamic loading. In general, other response parameters such as stresses or forces developed in various structural components can be evaluated directly from the displacements. For example, the elastic forces $\mathbf{f}_{S}$ which resist the deformation of the structure are given directly by
|
||||||
结构在动载作用下的整体响应可通过向量形式的位移时程 **$\mathbf{v}(t)$** 来衡量。通常，其他响应参数（如各结构构件中的应力或作用力）均可直接由位移推导得出。例如，抵抗结构变形的弹性力 **$\mathbf{f}_{S}$** 可直接表示为：
|
结构在动载作用下的整体响应可通过向量形式的位移时程 **$\mathbf{v}(t)$** 来衡量。通常，其他响应参数（如各结构构件中的应力或作用力）均可直接由位移推导得出。例如，抵抗结构变形的弹性力 **$\mathbf{f}_{S}$** 可直接表示为：
|
||||||
$$
|
$$
|
||||||
\mathbf{f}_{S}(t)=\mathbf{k}\;\mathbf{v}(t)=\mathbf{k}\;\Phi\;Y(t)
|
\mathbf{f}_{S}(t)=\mathbf{k}\;\mathbf{v}(t)=\mathbf{k}\;\Phi\;Y(t)\tag{12-30}
|
||||||
$$
|
$$
|
||||||
|
|
||||||
An alternative expression for the elastic forces may be useful in cases where the frequencies and mode shapes have been determined from the flexibility form of the eigenvalue equation [Eq. (11-17)]. Writing Eq. (12-30) in terms of the modal contributions
|
An alternative expression for the elastic forces may be useful in cases where the frequencies and mode shapes have been determined from the flexibility form of the eigenvalue equation [Eq. (11-17)]. Writing Eq. (12-30) in terms of the modal contributions
|
||||||
在某些情况下，当频率和模态形状是通过柔度形式的特征值方程（式（11-17））确定的，弹性力的另一种表达方式可能更为有用。将式（12-30）用模态贡献表示。
|
在某些情况下，当频率和模态形状是通过柔度形式的特征值方程（式（11-17））确定的，弹性力的另一种表达方式可能更为有用。将式（12-30）用模态贡献表示。
|
||||||
$$
|
$$
|
||||||
{\mathbf{f}}_{S}(t)={\mathbf{k}}\;\phi_{1}\;Y_{1}(t)+{\mathbf{k}}\;\phi_{2}\;Y_{2}(t)+{\mathbf{k}}\;\phi_{3}\;Y_{3}(t)+\cdot\cdot\cdot
|
{\mathbf{f}}_{S}(t)={\mathbf{k}}\;\phi_{1}\;Y_{1}(t)+{\mathbf{k}}\;\phi_{2}\;Y_{2}(t)+{\mathbf{k}}\;\phi_{3}\;Y_{3}(t)+\cdot\cdot\cdot\tag{12-31}
|
||||||
$$
|
$$
|
||||||
|
|
||||||
and substituting Eq. (11-39) leads to
|
and substituting Eq. (11-39) leads to
|
||||||
|
|
||||||
$$
|
$$
|
||||||
\mathbf{f}_{S}(t)=\omega_{1}^{2}\,\mathbf{m}\,\phi_{1}\,Y_{1}(t)+\omega_{2}^{2}\,\mathbf{m}\,\phi_{2}\,Y_{2}(t)+\omega_{3}^{2}\,\mathbf{m}\,\phi_{3}\,Y_{3}(t)+\cdot\cdot\cdot
|
\mathbf{f}_{S}(t)=\omega_{1}^{2}\,\mathbf{m}\,\phi_{1}\,Y_{1}(t)+\omega_{2}^{2}\,\mathbf{m}\,\phi_{2}\,Y_{2}(t)+\omega_{3}^{2}\,\mathbf{m}\,\phi_{3}\,Y_{3}(t)+\cdot\cdot\cdot\tag{12-32}
|
||||||
$$
|
$$
|
||||||
|
|
||||||
Writing this series in matrix form gives
|
Writing this series in matrix form gives
|
||||||
将该系列以矩阵形式表示，得到：
|
将该系列以矩阵形式表示，得到：
|
||||||
|
|
||||||
$$
|
$$
|
||||||
\mathbf{f}_{S}(t)=\mathbf{m}\boldsymbol{\Phi}\left\{\omega_{n}^{2}\,Y_{n}(t)\right\}
|
\mathbf{f}_{S}(t)=\mathbf{m}\boldsymbol{\Phi}\left\{\omega_{n}^{2}\,Y_{n}(t)\right\}\tag{12-33}
|
||||||
$$
|
$$
|
||||||
|
|
||||||
where $\{\omega_{n}^{2}\,Y_{n}(t)\}$ represents a vector of modal amplitudes each multiplied by the square of its modal frequency.
|
where $\{\omega_{n}^{2}\,Y_{n}(t)\}$ represents a vector of modal amplitudes each multiplied by the square of its modal frequency.
|
||||||
@ -426,19 +429,19 @@ As pointed out in Section 3-7, damping of the linear viscous form represented in
|
|||||||
Making this change in type of damping by using a complex-generalized-stiffness of the form given by Eq. (3-79), that is, using
|
Making this change in type of damping by using a complex-generalized-stiffness of the form given by Eq. (3-79), that is, using
|
||||||
将阻尼类型的变化通过采用 Eq. (3-79) 中给出的复数广义刚度形式实现，即采用
|
将阻尼类型的变化通过采用 Eq. (3-79) 中给出的复数广义刚度形式实现，即采用
|
||||||
$$
|
$$
|
||||||
\hat{K}_{n}=K_{n}\,\left[1+i\,2\,\xi_{n}\right]
|
\hat{K}_{n}=K_{n}\,\left[1+i\,2\,\xi_{n}\right]\tag{12-34}
|
||||||
$$
|
$$
|
||||||
|
|
||||||
in which
|
in which
|
||||||
|
|
||||||
$$
|
$$
|
||||||
K_{n}=\omega_{n}^{2}\:M_{n}
|
K_{n}=\omega_{n}^{2}\:M_{n}\tag{12-35}
|
||||||
$$
|
$$
|
||||||
|
|
||||||
the response will be given by Eq. (12-23) using the complex-frequency-response transfer function
|
the response will be given by Eq. (12-23) using the complex-frequency-response transfer function
|
||||||
响应将由式（12-23）给出，采用复频域响应传递函数。
|
响应将由式（12-23）给出，采用复频域响应传递函数。
|
||||||
$$
|
$$
|
||||||
\mathrm{H}_{n}(i\overline{{\omega}})=\frac{1}{\omega_{n}^{2}\,M_{n}}\left[\frac{1}{\left(1-\beta_{n}^{2}\right)+i\left(2\xi_{n}\right)}\right]=\frac{1}{\omega_{n}^{2}\,M_{n}}\left[\frac{(1-\beta_{n}^{2})-i\left(2\xi_{n}\right)}{(1-\beta_{n}^{2})^{2}+(2\xi_{n})^{2}}\right]
|
\mathrm{H}_{n}(i\overline{{\omega}})=\frac{1}{\omega_{n}^{2}\,M_{n}}\left[\frac{1}{\left(1-\beta_{n}^{2}\right)+i\left(2\xi_{n}\right)}\right]=\frac{1}{\omega_{n}^{2}\,M_{n}}\left[\frac{(1-\beta_{n}^{2})-i\left(2\xi_{n}\right)}{(1-\beta_{n}^{2})^{2}+(2\xi_{n})^{2}}\right]\tag{12-36}
|
||||||
$$
|
$$
|
||||||
|
|
||||||
rather than the corresponding transfer function given by Eq. (12-25) for viscous damping; see Eq. (6-46). All quantities in this transfer function are defined the same as those in the transfer function of Eq. (12-25).
|
rather than the corresponding transfer function given by Eq. (12-25) for viscous damping; see Eq. (6-46). All quantities in this transfer function are defined the same as those in the transfer function of Eq. (12-25).
|
||||||
@ -526,25 +529,25 @@ Clearly the simplest way to formulate a proportional damping matrix is to make i
|
|||||||
显然，构造一个比例阻尼矩阵最简单的方法是将其与质量矩阵或刚度矩阵成比例关系，因为无阻尼模态形状在这两种矩阵下都是正交的。因此，阻尼矩阵可以表示为：
|
显然，构造一个比例阻尼矩阵最简单的方法是将其与质量矩阵或刚度矩阵成比例关系，因为无阻尼模态形状在这两种矩阵下都是正交的。因此，阻尼矩阵可以表示为：
|
||||||
|
|
||||||
$$
|
$$
|
||||||
\mathbf{c}=a_{0}\;\mathbf{m}\qquad{\mathrm{or}}\qquad\mathbf{c}=a_{1}\;\mathbf{k}
|
\mathbf{c}=a_{0}\;\mathbf{m}\qquad{\mathrm{or}}\qquad\mathbf{c}=a_{1}\;\mathbf{k}\tag{12-37a}
|
||||||
$$
|
$$
|
||||||
|
|
||||||
in which the proportionality constants $a_{0}$ and $a_{1}$ have units of $s e c^{-1}$ and $s e c$ , respectively. These are called mass proportional and stiffness proportional damping, and the damping behavior associated with them may be recognized by evaluating the generalized modal damping value for each [see Eq. (12-15a)],
|
in which the proportionality constants $a_{0}$ and $a_{1}$ have units of $s e c^{-1}$ and $s e c$ , respectively. These are called mass proportional and stiffness proportional damping, and the damping behavior associated with them may be recognized by evaluating the generalized modal damping value for each [see Eq. (12-15a)],
|
||||||
其中，比例常数$a_{0}$ 和$a_{1}$ 的单位分别为$\mathrm{sec}^{-1}$ 和$\mathrm{sec}$。它们被称为质量比例阻尼和刚度比例阻尼，其对应的阻尼行为可通过计算各自的广义模态阻尼值来识别（见式（12-15a））。
|
其中，比例常数$a_{0}$ 和$a_{1}$ 的单位分别为$\mathrm{sec}^{-1}$ 和$\mathrm{sec}$。它们被称为质量比例阻尼和刚度比例阻尼，其对应的阻尼行为可通过计算各自的广义模态阻尼值来识别（见式（12-15a））。
|
||||||
$$
|
$$
|
||||||
\begin{array}{r}{C_{n}=\phi_{n}^{T}\,c\,\phi_{n}=a_{0}\,\phi_{n}^{T}\,{\bf m}\,\phi_{n}\qquad\mathrm{or}\qquad a_{1}\,\phi_{n}^{T}\,{\bf k}\,\phi_{n}}\end{array}
|
\begin{array}{r}{C_{n}=\phi_{n}^{T}\,c\,\phi_{n}=a_{0}\,\phi_{n}^{T}\,{\bf m}\,\phi_{n}\qquad\mathrm{or}\qquad a_{1}\,\phi_{n}^{T}\,{\bf k}\,\phi_{n}}\end{array}\tag{12-37b}
|
||||||
$$
|
$$
|
||||||
|
|
||||||
or combining with Eq. (12-15b)
|
or combining with Eq. (12-15b)
|
||||||
|
|
||||||
$$
|
$$
|
||||||
2\omega_{n}\,M_{n}\,\xi_{n}=a_{0}\;M_{n}\quad\mathrm{or}\quad a_{1}\,K_{n}\qquad\mathrm{(where}\quad K_{n}=\omega_{n}^{2}\,M_{n})
|
2\omega_{n}\,M_{n}\,\xi_{n}=a_{0}\;M_{n}\quad\mathrm{or}\quad a_{1}\,K_{n}\qquad\mathrm{(where}\quad K_{n}=\omega_{n}^{2}\,M_{n})\tag{12-37c}
|
||||||
$$
|
$$
|
||||||
|
|
||||||
from which
|
from which
|
||||||
|
|
||||||
$$
|
$$
|
||||||
\xi_{n}=\frac{a_{0}}{2\omega_{n}}\qquad\mathrm{or}\qquad\xi_{n}=\frac{a_{1}\omega_{n}}{2}
|
\xi_{n}=\frac{a_{0}}{2\omega_{n}}\qquad\mathrm{or}\qquad\xi_{n}=\frac{a_{1}\omega_{n}}{2}\tag{12-37d}
|
||||||
$$
|
$$
|
||||||
|
|
||||||
These expressions show that for mass proportional damping, the damping ratio is inversely proportional to the frequency while for stiffness proportional damping it is directly in proportion with the frequency. In this regard it is important to note that the dynamic response generally will include contributions from all $N$ modes even though only a limited number of modes are included in the uncoupled equations of motion. Thus, neither of these types of damping matrix is suitable for use with an MDOF system in which the frequencies of the significant modes span a wide range because the relative amplitudes of the different modes will be seriously distorted by inappropriate damping ratios.
|
These expressions show that for mass proportional damping, the damping ratio is inversely proportional to the frequency while for stiffness proportional damping it is directly in proportion with the frequency. In this regard it is important to note that the dynamic response generally will include contributions from all $N$ modes even though only a limited number of modes are included in the uncoupled equations of motion. Thus, neither of these types of damping matrix is suitable for use with an MDOF system in which the frequencies of the significant modes span a wide range because the relative amplitudes of the different modes will be seriously distorted by inappropriate damping ratios.
|
||||||
@ -553,14 +556,14 @@ These expressions show that for mass proportional damping, the damping ratio is
|
|||||||
An obvious improvement results if the damping is assumed to be proportional to a combination of the mass and the stiffness matrices as given by the sum of the two alternative expressions shown in Eq. (12-37a):
|
An obvious improvement results if the damping is assumed to be proportional to a combination of the mass and the stiffness matrices as given by the sum of the two alternative expressions shown in Eq. (12-37a):
|
||||||
假设阻尼与质量矩阵和刚度矩阵的组合成正比，如方程（12-37a）中所示的两种表达式之和，则显然可以获得显著的改进。
|
假设阻尼与质量矩阵和刚度矩阵的组合成正比，如方程（12-37a）中所示的两种表达式之和，则显然可以获得显著的改进。
|
||||||
$$
|
$$
|
||||||
\mathbf{c}=a_{0}\ \mathbf{m}+a_{1}\ \mathbf{k}
|
\mathbf{c}=a_{0}\ \mathbf{m}+a_{1}\ \mathbf{k}\tag{12-38a}
|
||||||
$$
|
$$
|
||||||
|
|
||||||
This is called Rayleigh damping, after Lord Rayleigh, who first suggested its use. By analogy with the development in Eqs. (12-37b) to (12-37d), it is evident that Rayleigh damping leads to the following relation between damping ratio and frequency
|
This is called Rayleigh damping, after Lord Rayleigh, who first suggested its use. By analogy with the development in Eqs. (12-37b) to (12-37d), it is evident that Rayleigh damping leads to the following relation between damping ratio and frequency
|
||||||
此方法称为 **瑞利阻尼**（Rayleigh damping），以瑞利勋爵（Lord Rayleigh）命名，他首次提出了这一概念。通过类比方程（12-37b）至（12-37d）的推导过程，可以得出瑞利阻尼下阻尼比与频率之间的关系如下：
|
此方法称为 **瑞利阻尼**（Rayleigh damping），以瑞利勋爵（Lord Rayleigh）命名，他首次提出了这一概念。通过类比方程（12-37b）至（12-37d）的推导过程，可以得出瑞利阻尼下阻尼比与频率之间的关系如下：
|
||||||
|
|
||||||
$$
|
$$
|
||||||
\xi_{n}=\frac{a_{0}}{2\omega_{n}}+\frac{a_{1}\omega_{n}}{2}
|
\xi_{n}=\frac{a_{0}}{2\omega_{n}}+\frac{a_{1}\omega_{n}}{2}\tag{12-38b}
|
||||||
$$
|
$$
|
||||||
|
|
||||||
The relationships between damping ratio and frequency expressed by Eqs. (12-37d) and (12-38b) are shown graphically in Fig. 12-2.
|
The relationships between damping ratio and frequency expressed by Eqs. (12-37d) and (12-38b) are shown graphically in Fig. 12-2.
|
||||||
@ -572,13 +575,13 @@ Now it is apparent that the two Rayleigh damping factors, $a_{0}$ and $a_{1}$ ,
|
|||||||
|
|
||||||
|
|
||||||
$$
|
$$
|
||||||
\left\{\begin{array}{c}{\xi_{m}}\\ {\xi_{n}}\end{array}\right\}=\frac{1}{2}\begin{array}{c}{\left[1/\omega_{m}\quad\omega_{m}\right]}\\ {\left[1/\omega_{n}\quad\omega_{n}\right]}\end{array}\left\{\begin{array}{c}{a_{0}}\\ {a_{1}}\end{array}\right\}
|
\left\{\begin{array}{c}{\xi_{m}}\\ {\xi_{n}}\end{array}\right\}=\frac{1}{2}\begin{array}{c}{\left[1/\omega_{m}\quad\omega_{m}\right]}\\ {\left[1/\omega_{n}\quad\omega_{n}\right]}\end{array}\left\{\begin{array}{c}{a_{0}}\\ {a_{1}}\end{array}\right\}\tag{12-39}
|
||||||
$$
|
$$
|
||||||
|
|
||||||
and the factors resulting from the simultaneous solution are
|
and the factors resulting from the simultaneous solution are
|
||||||
同时求解得到的各因子为：
|
同时求解得到的各因子为：
|
||||||
$$
|
$$
|
||||||
\left\{{\begin{array}{c}{a_{0}}\\ {a_{1}}\end{array}}\right\}=2\;{\frac{\omega_{m}\omega_{n}}{\omega_{n}^{2}-\omega_{m}^{2}}}\;\left[{\begin{array}{c c}{\omega_{n}}&{-\omega_{m}}\\ {-1/\omega_{n}}&{1/\omega_{m}}\end{array}}\right]\;{\left\{{\begin{array}{c}{\xi_{m}}\\ {\xi_{n}}\end{array}}\right\}}
|
\left\{{\begin{array}{c}{a_{0}}\\ {a_{1}}\end{array}}\right\}=2\;{\frac{\omega_{m}\omega_{n}}{\omega_{n}^{2}-\omega_{m}^{2}}}\;\left[{\begin{array}{c c}{\omega_{n}}&{-\omega_{m}}\\ {-1/\omega_{n}}&{1/\omega_{m}}\end{array}}\right]\;{\left\{{\begin{array}{c}{\xi_{m}}\\ {\xi_{n}}\end{array}}\right\}}\tag{12-40}
|
||||||
$$
|
$$
|
||||||
|
|
||||||
When these factors have been evaluated, the proportional damping matrix that will give the required values of damping ratio at the specified frequencies is given by the Rayleigh damping expression, Eq. (12-38a), as shown by Fig. 12-2.
|
When these factors have been evaluated, the proportional damping matrix that will give the required values of damping ratio at the specified frequencies is given by the Rayleigh damping expression, Eq. (12-38a), as shown by Fig. 12-2.
|
||||||
@ -587,7 +590,7 @@ When these factors have been evaluated, the proportional damping matrix that wil
|
|||||||
Because detailed information about the variation of damping ratio with frequency seldom is available, it usually is assumed that the same damping ratio applies to both control frequencies; i.e., $\xi_{m}=\xi_{n}\equiv\xi$ . In this case, the proportionality factors are given by a simplified version of Eq. (12-40):
|
Because detailed information about the variation of damping ratio with frequency seldom is available, it usually is assumed that the same damping ratio applies to both control frequencies; i.e., $\xi_{m}=\xi_{n}\equiv\xi$ . In this case, the proportionality factors are given by a simplified version of Eq. (12-40):
|
||||||
由于关于阻尼比随频率变化的详细信息鲜有可用，通常假设相同的阻尼比适用于两种控制频率，即：$\xi_{m} = \xi_{n} \equiv \xi$。此时，比例系数由 Eq. (12-40) 的简化形式给出：
|
由于关于阻尼比随频率变化的详细信息鲜有可用，通常假设相同的阻尼比适用于两种控制频率，即：$\xi_{m} = \xi_{n} \equiv \xi$。此时，比例系数由 Eq. (12-40) 的简化形式给出：
|
||||||
$$
|
$$
|
||||||
\left\{{\begin{array}{c}{a_{0}}\\ {a_{1}}\end{array}}\right\}={\frac{2\xi}{\omega_{m}+\omega_{n}}}\,\left\{{\begin{array}{c}{\omega_{m}\omega_{n}}\\ {1}\end{array}}\right\}
|
\left\{{\begin{array}{c}{a_{0}}\\ {a_{1}}\end{array}}\right\}={\frac{2\xi}{\omega_{m}+\omega_{n}}}\,\left\{{\begin{array}{c}{\omega_{m}\omega_{n}}\\ {1}\end{array}}\right\}\tag{12-41}
|
||||||
$$
|
$$
|
||||||
|
|
||||||
In applying this proportional damping matrix derivation procedure in practice, it is recommended that $\omega_{m}$ generally be taken as the fundamental frequency of the MDOF system and that $\omega_{n}$ be set among the higher frequencies of the modes that contribute significantly to the dynamic response. The derivation ensures that the desired damping ratio is obtained for these two modes (i.e., $\xi_{1}=\xi_{n}=\xi)$ ; then as shown by Fig. 12-2, modes with frequencies between these two specified frequencies will have somewhat lower values of damping ratio, while all modes with frequencies greater than $\omega_{n}$ will have damping ratios that increases above $\xi_{n}$ monotonically with frequency. The end result of this situation is that the responses of very high frequency modes are effectively eliminated by their high damping ratios.
|
In applying this proportional damping matrix derivation procedure in practice, it is recommended that $\omega_{m}$ generally be taken as the fundamental frequency of the MDOF system and that $\omega_{n}$ be set among the higher frequencies of the modes that contribute significantly to the dynamic response. The derivation ensures that the desired damping ratio is obtained for these two modes (i.e., $\xi_{1}=\xi_{n}=\xi)$ ; then as shown by Fig. 12-2, modes with frequencies between these two specified frequencies will have somewhat lower values of damping ratio, while all modes with frequencies greater than $\omega_{n}$ will have damping ratios that increases above $\xi_{n}$ monotonically with frequency. The end result of this situation is that the responses of very high frequency modes are effectively eliminated by their high damping ratios.
|
||||||
@ -630,7 +633,7 @@ Hence, even though only the first and third damping ratios were specified, the r
|
|||||||
The mass and stiffness matrices used to formulate Rayleigh damping are not the only matrices to which the free-vibration mode-shape orthogonality conditions apply; in fact, it was shown earlier in Eq. (11-44) that an infinite number of matrices have this property. Therefore a proportional damping matrix can be made up of any combination of these matrices, as follows:
|
The mass and stiffness matrices used to formulate Rayleigh damping are not the only matrices to which the free-vibration mode-shape orthogonality conditions apply; in fact, it was shown earlier in Eq. (11-44) that an infinite number of matrices have this property. Therefore a proportional damping matrix can be made up of any combination of these matrices, as follows:
|
||||||
以下矩阵和刚度矩阵用于雷利阻尼（Rayleigh damping）的建模，并非唯一满足自由振动模态正交性条件的矩阵；实际上，如前所述（见式（11-44）），无限多个矩阵具有这一性质。因此，比例阻尼矩阵可以由这些矩阵的任意组合构成，具体如下：
|
以下矩阵和刚度矩阵用于雷利阻尼（Rayleigh damping）的建模，并非唯一满足自由振动模态正交性条件的矩阵；实际上，如前所述（见式（11-44）），无限多个矩阵具有这一性质。因此，比例阻尼矩阵可以由这些矩阵的任意组合构成，具体如下：
|
||||||
$$
|
$$
|
||||||
\mathbf{c}=m\,\sum_{b}a_{b}[m^{-1}\,k]^{b}\equiv\sum_{b}\,c_{b}
|
\mathbf{c}=m\,\sum_{b}a_{b}[m^{-1}\,k]^{b}\equiv\sum_{b}\,c_{b}\tag{12-42}
|
||||||
$$
|
$$
|
||||||
|
|
||||||
in which the coefficients $a_{b}$ are arbitrary. It is evident that Rayleigh damping is given by Eq. (12-42) if only the terms $b=0$ and $b=1$ are retained in the series. By retaining additional terms of the series a proportional damping matrix can be constructed that gives any desired damping ratio $\xi_{n}$ at a specified frequency $\omega_{n}$ for as many frequencies as there are terms in the series of Eq. (12-42).
|
in which the coefficients $a_{b}$ are arbitrary. It is evident that Rayleigh damping is given by Eq. (12-42) if only the terms $b=0$ and $b=1$ are retained in the series. By retaining additional terms of the series a proportional damping matrix can be constructed that gives any desired damping ratio $\xi_{n}$ at a specified frequency $\omega_{n}$ for as many frequencies as there are terms in the series of Eq. (12-42).
|
||||||
@ -639,13 +642,13 @@ in which the coefficients $a_{b}$ are arbitrary. It is evident that Rayleigh dam
|
|||||||
To understand the procedure, consider the generalized damping value $C_{n}$ for any normal mode "n" see Eqs. (12-37b) and (12-37c):
|
To understand the procedure, consider the generalized damping value $C_{n}$ for any normal mode "n" see Eqs. (12-37b) and (12-37c):
|
||||||
要理解该程序，请考虑任意简正模态“n”的广义阻尼系数 $C_{n}$（见方程（12-37b）和（12-37c））：
|
要理解该程序，请考虑任意简正模态“n”的广义阻尼系数 $C_{n}$（见方程（12-37b）和（12-37c））：
|
||||||
$$
|
$$
|
||||||
C_{n}=\pmb\phi_{n}^{T}\,\mathbf c\,\pmb\phi_{n}=2\xi_{n}\,\omega_{n}\,M_{n}
|
C_{n}=\pmb\phi_{n}^{T}\,\mathbf c\,\pmb\phi_{n}=2\xi_{n}\,\omega_{n}\,M_{n}\tag{12-43}
|
||||||
$$
|
$$
|
||||||
|
|
||||||
If c in this expression is given by Eq. (12-42), the contribution of term $b$ to the generalized damping value is
|
If c in this expression is given by Eq. (12-42), the contribution of term $b$ to the generalized damping value is
|
||||||
如果表达式中的 $c$ 由式（12-42）给出，则项 $b$ 对广义阻尼值的贡献为：
|
如果表达式中的 $c$ 由式（12-42）给出，则项 $b$ 对广义阻尼值的贡献为：
|
||||||
$$
|
$$
|
||||||
C_{n b}=\phi_{n}^{T}\,\mathbf{c}_{b}\,\phi_{n}=a_{b}\,\mathbf{m}\,[\mathbf{m}^{-1}\,\mathbf{k}]^{b}\,\phi_{n}
|
C_{n b}=\phi_{n}^{T}\,\mathbf{c}_{b}\,\phi_{n}=a_{b}\,\mathbf{m}\,[\mathbf{m}^{-1}\,\mathbf{k}]^{b}\,\phi_{n}\tag{12-44a}
|
||||||
$$
|
$$
|
||||||
|
|
||||||
Now if Eq. (11-39) $(k\,\phi_{n}=\omega_{n}^{2}\,m\,\phi_{n})$ is premultiplied on both sides by $\phi_{n}^{T}\,\mathbf{k}\,\mathbf{m}^{-1}$ , the result is
|
Now if Eq. (11-39) $(k\,\phi_{n}=\omega_{n}^{2}\,m\,\phi_{n})$ is premultiplied on both sides by $\phi_{n}^{T}\,\mathbf{k}\,\mathbf{m}^{-1}$ , the result is
|
||||||
@ -657,13 +660,13 @@ $$
|
|||||||
By operations equivalent to this it can be shown that
|
By operations equivalent to this it can be shown that
|
||||||
通过与上述操作等效的方法可以证明：
|
通过与上述操作等效的方法可以证明：
|
||||||
$$
|
$$
|
||||||
\pmb{\phi}_{n}^{T}\,\mathbf{m}\left[\mathbf{m}^{-1}\,\mathbf{k}\right]^{b}\,\pmb{\phi}_{n}=\omega_{n}^{2b}\,M_{n}
|
\pmb{\phi}_{n}^{T}\,\mathbf{m}\left[\mathbf{m}^{-1}\,\mathbf{k}\right]^{b}\,\pmb{\phi}_{n}=\omega_{n}^{2b}\,M_{n}\tag{12-45}
|
||||||
$$
|
$$
|
||||||
|
|
||||||
and consequently
|
and consequently
|
||||||
|
|
||||||
$$
|
$$
|
||||||
C_{n b}=a_{b}\,\omega_{n}^{2b}\,M_{n}
|
C_{n b}=a_{b}\,\omega_{n}^{2b}\,M_{n}\tag{12-44b}
|
||||||
$$
|
$$
|
||||||
|
|
||||||
On this basis, the generalized damping value associated with any mode $n$ is
|
On this basis, the generalized damping value associated with any mode $n$ is
|
||||||
@ -675,13 +678,13 @@ $$
|
|||||||
from which
|
from which
|
||||||
|
|
||||||
$$
|
$$
|
||||||
\xi_{n}=\frac{1}{2\omega_{n}}\sum_{b}a_{b}\;\omega_{n}^{2b}
|
\xi_{n}=\frac{1}{2\omega_{n}}\sum_{b}a_{b}\;\omega_{n}^{2b}\tag{12-46}
|
||||||
$$
|
$$
|
||||||
|
|
||||||
Equation (12-46) provides the means for evaluating the constants $a_{b}$ to give the desired damping ratios at any specified number of modal frequencies. As many terms must be included in the series as there are specified modal damping ratios; then the constants are given by the solution of the set of equations, one written for each damping ratio. In principle, the values of $b$ can lie anywhere in the range $-\infty<b<\infty$ , but in practice it is desirable to select values of these exponents as close to zero as possible. For example, to evaluate the coefficients that will provide specified damping ratios in any four modes having the frequencies $\omega_{m}$ $m,\omega_{n},\omega_{o},$ $\omega_{p}$ , the equations resulting from Eq. (12-46) using the terms for $b=-1,0,+1$ , and $+2$ are
|
Equation (12-46) provides the means for evaluating the constants $a_{b}$ to give the desired damping ratios at any specified number of modal frequencies. As many terms must be included in the series as there are specified modal damping ratios; then the constants are given by the solution of the set of equations, one written for each damping ratio. In principle, the values of $b$ can lie anywhere in the range $-\infty<b<\infty$ , but in practice it is desirable to select values of these exponents as close to zero as possible. For example, to evaluate the coefficients that will provide specified damping ratios in any four modes having the frequencies $\omega_{m}$ $m,\omega_{n},\omega_{o},$ $\omega_{p}$ , the equations resulting from Eq. (12-46) using the terms for $b=-1,0,+1$ , and $+2$ are
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方程（12-46）提供了评估常数 $a_{b}$ 的方法，以实现任意指定模态频率下所需的阻尼比。在级数展开中，必须包含与指定阻尼比数量相等的项；然后，这些常数由一组方程的解给出，每个方程对应一个阻尼比。理论上，指数 $b$ 的取值范围为 $-\infty < b < \infty$，但在实际应用中，应尽可能选择接近零的指数值。例如，为了评估在频率分别为 $\omega_{m}$、$\omega_{n}$、$\omega_{o}$ 和 $\omega_{p}$ 的任意四个模态中提供指定阻尼比的系数，根据方程（12-46）并采用 $b = -1, 0, +1$ 和 $+2$ 的项所得到的方程为：
|
方程（12-46）提供了评估常数 $a_{b}$ 的方法，以实现任意指定模态频率下所需的阻尼比。在级数展开中，必须包含与指定阻尼比数量相等的项；然后，这些常数由一组方程的解给出，每个方程对应一个阻尼比。理论上，指数 $b$ 的取值范围为 $-\infty < b < \infty$，但在实际应用中，应尽可能选择接近零的指数值。例如，为了评估在频率分别为 $\omega_{m}$、$\omega_{n}$、$\omega_{o}$ 和 $\omega_{p}$ 的任意四个模态中提供指定阻尼比的系数，根据方程（12-46）并采用 $b = -1, 0, +1$ 和 $+2$ 的项所得到的方程为：
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$$
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$$
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\left\{\begin{array}{c}{\displaystyle\xi_{m}}\\ {\displaystyle\xi_{n}}\\ {\displaystyle\xi_{o}}\\ {\displaystyle\xi_{p}}\end{array}\right\}=\frac{1}{2}\left[\begin{array}{c c c c}{1/\omega_{m}^{2}}&{1/\omega_{m}}&{\omega_{m}}&{\omega_{m}^{3}}\\ {1/\omega_{n}^{2}}&{1/\omega_{n}}&{\omega_{n}}&{\omega_{n}^{3}}\\ {1/\omega_{o}^{2}}&{1/\omega_{o}}&{\omega_{o}}&{\omega_{o}^{3}}\\ {1/\omega_{p}^{2}}&{1/\omega_{p}}&{\omega_{p}}&{\omega_{p}^{3}}\end{array}\right]\left\{\begin{array}{c}{\displaystyle a_{-1}}\\ {\displaystyle a_{0}}\\ {\displaystyle a_{1}}\\ {\displaystyle a_{2}}\end{array}\right\}
|
\left\{\begin{array}{c}{\displaystyle\xi_{m}}\\ {\displaystyle\xi_{n}}\\ {\displaystyle\xi_{o}}\\ {\displaystyle\xi_{p}}\end{array}\right\}=\frac{1}{2}\left[\begin{array}{c c c c}{1/\omega_{m}^{2}}&{1/\omega_{m}}&{\omega_{m}}&{\omega_{m}^{3}}\\ {1/\omega_{n}^{2}}&{1/\omega_{n}}&{\omega_{n}}&{\omega_{n}^{3}}\\ {1/\omega_{o}^{2}}&{1/\omega_{o}}&{\omega_{o}}&{\omega_{o}^{3}}\\ {1/\omega_{p}^{2}}&{1/\omega_{p}}&{\omega_{p}}&{\omega_{p}^{3}}\end{array}\right]\left\{\begin{array}{c}{\displaystyle a_{-1}}\\ {\displaystyle a_{0}}\\ {\displaystyle a_{1}}\\ {\displaystyle a_{2}}\end{array}\right\}\tag{12-47}
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$$
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$$
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When the coefficients $a_{-1},\,a_{0},\,a_{1}$ , and $a_{2}$ have been evaluated by the simultaneous solution of Eq. (12-47), the viscous damping matrix that provides the four required damping ratios at the four specified frequencies is obtained by superposing four matrices (one for each value of $b$ ) in accordance with Eq. (12-42). Figure $12{-}3a$ illustrates the relation between damping ratio and frequency that would result from this matrix. To simplify the figure it has been assumed here that the same damping ratio, $\xi_{x}$ , was specified for all four frequencies; however, each of the damping ratios could have been specified arbitrarily. Also, $\omega_{m}$ has been taken as the fundamental mode frequency, $\omega_{1}$ , and $\omega_{p}$ is intended to approximate the frequency of the highest mode that contributes significantly to the response, while $\omega_{n}$ and $\omega_{0}$ are spaced about equally within the frequency range. It is evident in Fig. 12-3a that the damping ratio remains close to the desired value $\xi_{x}$ throughout the frequency range, being exact at the four specified frequencies and ranging slightly above or below at other frequencies in the range. It is important to note, however, that the damping increases monotonically with frequency for frequencies increasing above $\omega_{p}$ . This has the effect of excluding any significant contribution from any modes with frequencies much greater than $\omega_{p}$ , thus such modes need not be included in the response superposition.
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When the coefficients $a_{-1},\,a_{0},\,a_{1}$ , and $a_{2}$ have been evaluated by the simultaneous solution of Eq. (12-47), the viscous damping matrix that provides the four required damping ratios at the four specified frequencies is obtained by superposing four matrices (one for each value of $b$ ) in accordance with Eq. (12-42). Figure $12{-}3a$ illustrates the relation between damping ratio and frequency that would result from this matrix. To simplify the figure it has been assumed here that the same damping ratio, $\xi_{x}$ , was specified for all four frequencies; however, each of the damping ratios could have been specified arbitrarily. Also, $\omega_{m}$ has been taken as the fundamental mode frequency, $\omega_{1}$ , and $\omega_{p}$ is intended to approximate the frequency of the highest mode that contributes significantly to the response, while $\omega_{n}$ and $\omega_{0}$ are spaced about equally within the frequency range. It is evident in Fig. 12-3a that the damping ratio remains close to the desired value $\xi_{x}$ throughout the frequency range, being exact at the four specified frequencies and ranging slightly above or below at other frequencies in the range. It is important to note, however, that the damping increases monotonically with frequency for frequencies increasing above $\omega_{p}$ . This has the effect of excluding any significant contribution from any modes with frequencies much greater than $\omega_{p}$ , thus such modes need not be included in the response superposition.
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@ -698,14 +701,14 @@ The general implication of this observation is that extended Rayleigh damping ma
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A second method is available for evaluating the damping matrix associated with any given set of modal damping ratios. In principle, the procedure can be explained by considering the complete diagonal matrix of generalized damping coefficients, given by pre- and postmultiplying the damping matrix by the mode-shape matrix:
|
A second method is available for evaluating the damping matrix associated with any given set of modal damping ratios. In principle, the procedure can be explained by considering the complete diagonal matrix of generalized damping coefficients, given by pre- and postmultiplying the damping matrix by the mode-shape matrix:
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||||||
第二种方法可用于评估与任意给定模态阻尼比相关的阻尼矩阵。从原理上讲，该程序可以通过考虑广义阻尼系数的完整对角矩阵来解释，该矩阵由阻尼矩阵与模态形状矩阵的**前乘**和**后乘**得到：
|
第二种方法可用于评估与任意给定模态阻尼比相关的阻尼矩阵。从原理上讲，该程序可以通过考虑广义阻尼系数的完整对角矩阵来解释，该矩阵由阻尼矩阵与模态形状矩阵的**前乘**和**后乘**得到：
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$$
|
$$
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||||||
C=\Phi^{T}\mathrm{\bf~c}\;\Phi=2\;\left[\begin{array}{c c c c}{\xi_{1}\omega_{1}M_{1}}&{0}&{0}&{\cdot\cdot\cdot}\\ {0}&{\xi_{2}\omega_{2}M_{2}}&{0}&{\cdot\cdot}\\ {0}&{0}&{\xi_{3}\omega_{3}M_{3}}&{\vdots}\\ {\vdots}&{\vdots}&{\vdots}&{\vdots}\end{array}\right]\;
|
C=\Phi^{T}\mathrm{\bf~c}\;\Phi=2\;\left[\begin{array}{c c c c}{\xi_{1}\omega_{1}M_{1}}&{0}&{0}&{\cdot\cdot\cdot}\\ {0}&{\xi_{2}\omega_{2}M_{2}}&{0}&{\cdot\cdot}\\ {0}&{0}&{\xi_{3}\omega_{3}M_{3}}&{\vdots}\\ {\vdots}&{\vdots}&{\vdots}&{\vdots}\end{array}\right]\;\tag{12-48}
|
||||||
$$
|
$$
|
||||||
|
|
||||||
It is evident from this equation that the damping matrix can be obtained by pre- and postmultiplying matrix $c$ by the inverse of the transposed mode-shape matrix and the inverse of the mode-shape matrix, respectively, yielding
|
It is evident from this equation that the damping matrix can be obtained by pre- and postmultiplying matrix $c$ by the inverse of the transposed mode-shape matrix and the inverse of the mode-shape matrix, respectively, yielding
|
||||||
由此方程可以看出，阻尼矩阵可以通过将矩阵 $c$ 左乘以模态形状矩阵的转置矩阵的逆矩阵，右乘以模态形状矩阵的逆矩阵，从而得到，即：
|
由此方程可以看出，阻尼矩阵可以通过将矩阵 $c$ 左乘以模态形状矩阵的转置矩阵的逆矩阵，右乘以模态形状矩阵的逆矩阵，从而得到，即：
|
||||||
|
|
||||||
$$
|
$$
|
||||||
\left[\Phi^{T}\right]^{-1}C\;\Phi^{-1}=\left[\Phi^{T}\right]^{-1}\Phi^{T}\;\mathbf{c}\;\Phi\;\Phi^{-1}=\mathbf{c}
|
\left[\Phi^{T}\right]^{-1}C\;\Phi^{-1}=\left[\Phi^{T}\right]^{-1}\Phi^{T}\;\mathbf{c}\;\Phi\;\Phi^{-1}=\mathbf{c}\tag{12-49}
|
||||||
$$
|
$$
|
||||||
Since for any specified set of modal damping ratios $\xi_{n}$ , the generalized damping coefficients in matrix $c$ can be evaluated, as indicated in Eq. (12-43), the damping matrix c can be evaluated using Eq. (12-49).
|
Since for any specified set of modal damping ratios $\xi_{n}$ , the generalized damping coefficients in matrix $c$ can be evaluated, as indicated in Eq. (12-43), the damping matrix c can be evaluated using Eq. (12-49).
|
||||||
对于任意指定的模态阻尼比 $\xi_{n}$ 集合，可根据式（12-43）评估矩阵 $c$ 中的广义阻尼系数，从而利用式（12-49）求解阻尼矩阵 $c$。
|
对于任意指定的模态阻尼比 $\xi_{n}$ 集合，可根据式（12-43）评估矩阵 $c$ 中的广义阻尼系数，从而利用式（12-49）求解阻尼矩阵 $c$。
|
||||||
@ -714,25 +717,25 @@ In practice, however, this is not a convenient procedure because inversion of th
|
|||||||
在实际应用中，该方法并不方便，因为模态形状矩阵的求逆需耗费大量计算资源。相反，更有效的方法是利用模态形状相对于质量矩阵的正交性。系统的对角广义质量矩阵可通过以下关系式获得：
|
在实际应用中，该方法并不方便，因为模态形状矩阵的求逆需耗费大量计算资源。相反，更有效的方法是利用模态形状相对于质量矩阵的正交性。系统的对角广义质量矩阵可通过以下关系式获得：
|
||||||
|
|
||||||
$$
|
$$
|
||||||
M=\Phi^{T}\textbf{m}\Phi
|
M=\Phi^{T}\textbf{m}\Phi\tag{12-50}
|
||||||
$$
|
$$
|
||||||
|
|
||||||
Premultiplying this equation by the inverse of the generalized-mass matrix then gives
|
Premultiplying this equation by the inverse of the generalized-mass matrix then gives
|
||||||
将该方程两边左乘广义质量矩阵的逆矩阵，得到：
|
将该方程两边左乘广义质量矩阵的逆矩阵，得到：
|
||||||
$$
|
$$
|
||||||
\mathbf{I}=M^{-1}\,M=\left[M^{-1}\,\,\Phi^{T}\,\mathbf{\Phi}\mathbf{m}\right]\,\Phi=\Phi^{-1}\,\Phi
|
\mathbf{I}=M^{-1}\,M=\left[M^{-1}\,\,\Phi^{T}\,\mathbf{\Phi}\mathbf{m}\right]\,\Phi=\Phi^{-1}\,\Phi\tag{12-51}
|
||||||
$$
|
$$
|
||||||
|
|
||||||
from which it is evident that the mode-shape-matrix inverse is
|
from which it is evident that the mode-shape-matrix inverse is
|
||||||
|
|
||||||
$$
|
$$
|
||||||
\Phi^{-1}=M^{-1}\;\Phi^{T}\;{\bf m}
|
\Phi^{-1}=M^{-1}\;\Phi^{T}\;{\bf m}\tag{12-52}
|
||||||
$$
|
$$
|
||||||
|
|
||||||
Operating on this expression, one can obtain
|
Operating on this expression, one can obtain
|
||||||
|
|
||||||
$$
|
$$
|
||||||
\left[\Phi^{T}\right]^{-1}=\mathbf{m}\;\Phi\;M^{-1}
|
\left[\Phi^{T}\right]^{-1}=\mathbf{m}\;\Phi\;M^{-1}\tag{12-53}
|
||||||
$$
|
$$
|
||||||
|
|
||||||
Substituting Eqs. (12-52) and (12-53) into Eq. (12-49) yields
|
Substituting Eqs. (12-52) and (12-53) into Eq. (12-49) yields
|
||||||
|
|||||||
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Reference in New Issue
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