김경종
359 lines
32 KiB
Markdown
359 lines
32 KiB
Markdown
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and, as expected, are lower than the frequencies of Eqs. 13.7-9. With $\overline{w}_1 = 1$ , the amplitude of $\overline{\theta}_2$ in mode 1, from Eq. 13.7-8 and beam theory, respectively, is
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$$
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\bar {\theta} _ {2} = - 1. 5 7 0 4 / L (\text {approximate}) \quad \bar {\theta} _ {2} = - 1. 5 7 0 8 / L (\text {exact}) \tag {13.7-11}
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$$
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To condense the size of the eigenproblem of Eq. 13.7-8, we employ Guyan reduction and select $\overline{w}_{1}$ as master and $\overline{\theta}_{2}$ as slave. Using Eqs. 13.7-3 and 13.7-5, with $K_{ss} = 4EI/L$ and $K_{ms} = 6EI/L^{2}$ , we obtain
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$$
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[ \mathrm{T} ] = \left[ \begin{array}{c} 1 \\ - 3 / 2 L \end{array} \right] \quad \text { and } \quad \left(\frac {3 E I}{L ^ {3}} - \omega^ {2} \frac {2 0 4 m}{4 2 0}\right) \overline {{w}} _ {1} = 0 \tag {13.7-12}
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$$
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from which $\omega_{1}^{2}=6.1765EI/mL^{3}$ . This is the only eigenvalue obtainable. As expected, it is higher than $\omega_{1}^{2}$ in Eq. 13.7-9. With $\overline{w}_{1}=1$ , recovery of $\overline{\theta}_{2}$ from Eqs. 13.7-2 and 13.7-7, respectively, yields
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$$
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\overline {{{\theta}}} _ {2} = - 1. 5 0 0 0 / L \quad \text { and } \quad \overline {{{\theta}}} _ {2} = - 1. 5 7 0 9 / L \tag {13.7-13}
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$$
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An improved estimate of $\omega_{1}^{2}$ can be found by substituting $\{\overline{\mathbf{D}}\} = \lfloor 1 - 1.5709 / L \rfloor^{T}$ into the Rayleigh quotient (Eq. 13.5-4), with [K] and [M] taken from Eq. 13.7-8. The resulting eigenvalue is the same as $\omega_{1}^{2}$ in Eq. 13.7-9.
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A lumped-mass solution for the fundamental vibration frequency is easily obtained for this problem by placing a mass particle $m_1 = m / 2$ at node 1. From beam theory, $w_1 = P_1 L^3 / 3EI$ , so $K = P_1 / w_1 = 3EI / L^3$ . Then $\omega^2 = K / m_1 = 6EI / mL^3$ . Note that this value is not an upper bound.
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# 13.8 COMPONENT MODE SYNTHESIS
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General Remarks. In component mode synthesis, or simply modal synthesis, a structure is subdivided into components or substructures, each of which is analyzed independently for natural frequencies and, more importantly, for mode shapes. The component mode shapes are then “assembled” to give displacement shapes or load patterns (either interpretation is possible) of the original structure. These shapes or patterns are not eigenvectors of the original structure, but give rise to Ritz vectors that are subsequently used to transform the original displacement d.o.f. to generalized d.o.f. Thus the size of the system matrices is reduced. Eigenanalysis and/or time-history analysis of the reduced system equations can be performed much more economically and usually with surprising accuracy $[13.39]$ .
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Component mode synthesis can be regarded as an alternative to Guyan reduction. More importantly, modal synthesis has the managerial advantage of allowing different design groups to work on different parts of a large structure, as discussed in Section 8.14 with respect to static substructuring. There are many methods of component mode synthesis and an extensive literature $[13.2,13.40]$ . In what follows we present only fundamentals.
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We will use a “reduced basis” [W] to replace d.o.f. $\{D\}$ of the entire (or assembled) structure by a vector of generalized coordinates $\{y\}$ that contains fewer d.o.f. than $\{D\}$ . The relation between $\{D\}$ and $\{y\}$ is stated by Eq. 13.6-13, here repeated:
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$$
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\left\{\mathbf {D} \right\} _ {n _ {\mathrm{eq}} \times 1} = \left[ \mathbf {W} \right] _ {n _ {\mathrm{eq}} \times m} \left\{\mathbf {y} \right\} _ {m \times 1} \tag {13.6-13}
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$$
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in which m is considerably less than $n_{eq}$ . Matrix [W] is an array of Ritz vectors. The question of how to establish [W] occupies much of our subsequent discussion. Equations 13.2-12 and 13.6-13 yield the reduced dynamic problem
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$$
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[ \mathbf {M} _ {r} ] \{\ddot {\mathbf {y}} \} + [ \mathbf {C} _ {r} ] \{\dot {\mathbf {y}} \} + [ \mathbf {K} _ {r} ] \{\mathbf {y} \} = \{\mathbf {R} _ {r} \} \tag {13.8-1}
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$$
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where $\{\mathbf{R}_r\} = [\mathbf{W}]^T\{\mathbf{R}^{\mathrm{ext}}\}$ and the reduced $m$ by $m$ mass, damping, and stiffness matrices are
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$$
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[ \mathbf {M} _ {r} ] = [ \mathbf {W} ] ^ {T} [ \mathbf {M} ] [ \mathbf {W} ] \quad [ \mathbf {C} _ {r} ] = [ \mathbf {W} ] ^ {T} [ \mathbf {C} ] [ \mathbf {W} ] \quad [ \mathbf {K} _ {r} ] = [ \mathbf {W} ] ^ {T} [ \mathbf {K} ] [ \mathbf {W} ] \tag {13.8-2}
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$$
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With $\{\mathbf{y}\} = \{\overline{\mathbf{y}}\}$ sin $\omega t$ and $\lambda = \omega^2$ , the reduced eigenproblem without damping is
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$$
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\left(\left[ \mathbf {K} _ {r} \right] - \lambda \left[ \mathbf {M} _ {r} \right]\right) \{\overline {{{\mathbf {y}}}} \} = \{\mathbf {0} \} \tag {13.8-3}
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$$
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Of the several methods of establishing an effective [W], we will summarize two. In the first, which we label CMS1 for short, [W] contains deflection vectors obtained by solving Eq. 13.6-14—that is, $[W] = [K]^{-1}[R]$ . Load patterns in [R] are assembled eigenvectors of the substructures—that is, the component modes. A modification is needed if [K] is singular, as for an unsupported structure; see [13.2]. In the second method we discuss, here labeled CMS2 for short, [W] contains the assembled component modes themselves. This method has no difficulty with an unsupported structure. In both of these methods, [W] also contains supplementary vectors related to the motion of d.o.f. shared by substructures. The supplementary vectors may be called rigid-body modes, constraint modes, or attachment modes, depending on how the shared d.o.f. are treated. In illustrations that follow we assume that shared d.o.f. are fixed for the substructure analyses that precede synthesis.
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Implementation. To illustrate method CMS1, consider a structure that is divided into $\ell$ substructures. Also assume that substructure k is attached to substructures k - 1 and $k + 1$ only; $k = 2, 3, \ldots, \ell - 1$ . Denote matrices for substructure k by $[K_k]$ and $[M_k]$ where all attachment d.o.f. (i.e., d.o.f. that are common to attaching substructures) are fixed. Then, for each substructure, the eigenproblem
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$$
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([ \mathbf {K} _ {k} ] - \lambda [ \mathbf {M} _ {k} ]) \{\overline {{{\mathbf {D}}}} _ {k} \} = \{\mathbf {0} \} \tag {13.8-4}
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$$
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is solved for normal modes of vibration $\{\overline{D}_{k}\}$ . These modes form the columns of the substructure modal matrix $[\phi_{k}]$ , which is $n_{k}$ by $n_{\phi}$ . Here $n_{k}$ is the number of interior d.o.f. of substructure k (i.e., the total number of d.o.f. of the substructure minus the number of attachment d.o.f.) and $n_{\phi}$ (usually $n_{\phi} \ll n_{k}$ ) is the number of normal nodes to be determined (the same for each substructure). Ritz vectors [W] for the entire structure can be obtained by solving Eq. 13.6-14 with
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$$
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\left[ \mathbf {R} \right] _ {n _ {\mathrm{eq}} \times m} = \left[ \begin{array}{c c c c} \phi_ {1} & \mathbf {0} & \mathbf {0} & \dots \\ \mathbf {0} & \mathbf {I} _ {1, 2} & \mathbf {0} & \dots \\ \phi_ {2} & \mathbf {0} & \mathbf {0} & \dots \\ \mathbf {0} & \mathbf {0} & \mathbf {I} _ {2, 3} & \dots \\ \vdots & \vdots & \vdots \\ \phi_ {\ell} & \mathbf {0} & \mathbf {0} & \dots \end{array} \right] \tag {13.8-5}
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$$
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where $\left[I_{k,k+1}\right]$ is a unit matrix with number of rows equal to the number of attachment d.o.f. between substructures k and $k+1$ , and $m=n_{\phi}+n_{a}$ , where $n_{a}$ is the total number of attachment d.o.f. in the synthesized model. The first $n_{\phi}$ columns of Eq. 13.8-5 represent assembled substructure normal modes (i.e., vertically stacked columns of substructure normal modes $[\phi_{k}]$ ) that are imposed as nodal loads in Eq. 13.6-14. However, since attachment d.o.f. for each substructure were fixed, the Ritz modes resulting from these load patterns are not capable of exciting attachment d.o.f. Thus, the first $n_{\phi}$ load patterns in Eq. 13.8-5 are supplemented by additional loads $\left[I_{k,k+1}\right]$ which physically correspond to successively applying a unit load to each attachment d.o.f. of the assembled structure while keeping all other d.o.f. load-free. The resulting Ritz vectors are called attachment modes and correspond to the displacements of all d.o.f. of the assembled structure due to the unit applied loads.
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Method CMS2 includes what is perhaps the most popular method of component mode synthesis, the Craig-Bampton method $[13.43]$ . Here the assembled substructure normal modes obtained by fixing attachment d.o.f. are used directly as Ritz vectors. These are then supplemented by constraint modes, which are deflection shapes of the assembled structure obtained by successively applying a unit displacement to each attachment d.o.f. while keeping all other attachment d.o.f. fixed.
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In all methods of component mode synthesis, the reduced eigenproblem, Eq. 13.8-3, is obtained by imposing constraints on the original large system. Thus, frequencies computed from Eq. 13.8-3 are upper bounds to those of the original system equations. Occasionally, it is possible to have all Ritz vectors orthogonal to one of the eigenvectors of the assembled structure. Then that eigenvector will be missing from the reduced eigenproblem, Eq. 13.8-3. In practice, the likelihood of missing lower-spectrum eigenvectors is reduced by using appropriate supplementary Ritz vectors such as attachment modes, constraint modes, and so on.
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In the following example problem, method CMS1 is more accurate than method CMS2. Method CMS2 is more prevalent in practice than method CMS1. However, when one considers that the original stiffness matrix $[K]$ must be factored to compute constraint modes (and/or other supplementary Ritz vectors), the additional cost required to obtain Ritz vectors by Eqs. 13.6-14 and 13.8-5 is small, and therefore method CMS1 merits consideration.
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Example. Consider axial vibrations of the structure shown in Fig. 13.8-1, whose stiffness and lumped mass matrices are
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$$
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[ \mathbf {K} ] = \frac {A E}{L} \left[ \begin{array}{r r r r r} 1 & - 1 & 0 & 0 & 0 \\ - 1 & 2 & - 1 & 0 & 0 \\ 0 & - 1 & 3 & - 2 & 0 \\ 0 & 0 & - 2 & 4 & - 2 \\ 0 & 0 & 0 & - 2 & 4 \end{array} \right] \quad [ \mathbf {M} ] = \frac {\rho A L}{2} \left[ \begin{array}{r r r r r} 1 & 0 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 & 0 \\ 0 & 0 & 3 & 0 & 0 \\ 0 & 0 & 0 & 4 & 0 \\ 0 & 0 & 0 & 0 & 4 \end{array} \right] \tag {13.8-6}
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$$
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Two substructures are created, one consisting of elements 1 and 2 and the other of elements 3, 4, and 5. With node 3 fixed, matrices for substructure 1 are
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$$
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[ \mathbf {K} _ {1} ] = \frac {A E}{L} \left[ \begin{array}{c c} 1 & - 1 \\ - 1 & 2 \end{array} \right] \quad \left[ \mathbf {M} _ {1} \right] = \frac {\rho A L}{2} \left[ \begin{array}{c c} 1 & 0 \\ 0 & 2 \end{array} \right] \tag {13.8-7}
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$$
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<details>
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<summary>text_image</summary>
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A, E
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2A, E
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① ② ③ ④ ⑤
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1 2 3 4 5
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L L L L L L
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</details>
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Figure 13.8-1. Bar with variable cross section modeled by five uniform elements of length L.
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and for substructure 2
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$$
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[ \mathbf {K} _ {2} ] = \frac {A E}{L} \left[ \begin{array}{c c} 4 & - 2 \\ - 2 & 4 \end{array} \right] \quad [ \mathbf {M} _ {2} ] = \frac {\rho A L}{2} \left[ \begin{array}{c c} 4 & 0 \\ 0 & 4 \end{array} \right] \tag {13.8-8}
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$$
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In the sequel, we assume that AE/L = 1 and $\rho AL/2 = 1$ . For substructure 1, the eigenproblem Eq. 13.8-4 is solved with the matrices of Eq. 13.8-7 to yield
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$$
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\lambda_ {1} = \frac {2 - \sqrt {2}}{2} \quad \lambda_ {2} = \frac {2 + \sqrt {2}}{2} \quad \{\overline {{\mathbf {D}}} _ {1} \} _ {1} = \left\{ \begin{array}{c} 1 \\ \sqrt {2} / 2 \end{array} \right\} \quad \{\overline {{\mathbf {D}}} _ {1} \} _ {2} = \left\{ \begin{array}{c} 1 \\ - \sqrt {2} / 2 \end{array} \right\} \tag {13.8-9}
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$$
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where eigenvectors are normalized so that the first coefficient has unit amplitude. Solving Eq. 13.8-4 using the second substructure's matrices, Eq. 13.8-8, we obtain
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$$
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\lambda_ {1} = \frac {1}{2} \quad \lambda_ {2} = \frac {3}{2} \quad \{\overline {{{\mathbf {D}}}} _ {2} \} _ {1} = \left\{ \begin{array}{l} 1 \\ 1 \end{array} \right\} \quad \{\overline {{{\mathbf {D}}}} _ {2} \} _ {2} = \left\{ \begin{array}{l} 1 \\ - 1 \end{array} \right\} \tag {13.8-10}
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$$
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For method CMS1, we must evaluate Eq. 13.8-5, which becomes
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$$
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[ \mathbf {R} ] = \left[ \begin{array}{c c c} 1 & 1 & 0 \\ \sqrt {2} / 2 & - \sqrt {2} / 2 & 0 \\ 0 & 0 & 1 \\ 1 & 1 & 0 \\ 1 & - 1 & 0 \end{array} \right] \tag {13.8-11}
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$$
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The first two columns of Eq. 13.8-11 are the assembled component normal modes while the third column has the effect of “releasing” node 3. The resulting Ritz vectors, from Eqs. 13.6-14, 13.8-6, and 13.8-11, are
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$$
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[ \mathbf {W} ] = \left[ \begin{array}{l l l} 6. 7 6 8 & 2. 2 3 2 & 1. 5 \\ 5. 7 6 8 & 1. 2 3 2 & 1. 5 \\ 4. 0 6 1 & 0. 9 3 9 3 & 1. 5 \\ 3. 2 0 7 & 0. 7 9 2 9 & 1. 0 \\ 1. 8 5 4 & 0. 1 4 6 4 & 0. 5 \end{array} \right] \quad \text {(method CMS1)} \tag {13.8-12}
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$$
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Reduced stiffness and mass matrices, from Eqs. 13.8-2 and 13.8-12, are
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$$
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\left[ \mathbf {K} _ {r} \right] = \left[ \begin{array}{l l l} 1 5. 9 1 & 4. 0 4 3 & 4. 0 6 1 \\ 4. 0 4 3 & 2. 0 0 7 & 0. 9 3 9 3 \\ 4. 0 6 1 & 0. 9 3 9 3 & 1. 5 0 0 \end{array} \right] \quad \left[ \mathbf {M} _ {r} \right] = \left[ \begin{array}{l l l} 2 1 6. 7 & 5 2. 0 2 & 6 2. 2 6 \\ 5 2. 0 2 & 1 3. 2 7 & 1 4. 7 4 \\ 6 2. 2 6 & 1 4. 7 4 & 1 8. 5 0 \end{array} \right] \tag {13.8-13}
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$$
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TABLE 13.8-1. NATURAL FREQUENCIES FOR COMPONENT MODE ANALYSIS OF THE STRUCTURE SHOWN IN FIG. 13.8-1.
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<table><tr><td>Procedure</td><td> $\omega_1$ </td><td> $\omega_2$ </td><td> $\omega_3$ </td></tr><tr><td>Original structure (five d.o.f.)</td><td>0.2651</td><td>0.6156</td><td>1.000</td></tr><tr><td>Method CMS1 (three d.o.f.)</td><td>0.2656</td><td>0.8794</td><td>1.182</td></tr><tr><td>Method CMS2 (three d.o.f.)</td><td>0.2719</td><td>0.9927</td><td>1.268</td></tr></table>
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Natural frequencies of the reduced eigenproblem, Eq. 13.8-3, are reported in Table 13.8-1. Also reported are the first three frequencies of the original structure, whose matrices are given by Eq. 13.8-6.
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For method CMS2, we take as Ritz vectors the assembled component normal modes, plus constraint modes [13.43]. Thus
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$$
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[ \mathbf {W} ] = \left[ \begin{array}{c c c} 1 & 1 & 1 \\ \sqrt {2} / \dot {2} & - \sqrt {2} / 2 & 1 \\ 0 & 0 & 1 \\ 1 & 1 & 2 / 3 \\ 1 & - 1 & 1 / 3 \end{array} \right] \quad (\text { method CMS2 }) \tag {13.8-14}
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$$
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Here the third column of Eq. 13.8-14 (i.e., $\{\mathbf{w}_3\}$ ) is the deflection shape obtained by solving $[\mathbf{K}]\{\mathbf{w}_3\} = \{\mathbf{p}\}$ , where
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$$
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\left\{\mathrm{w} _ {3} \right\} = \left\lfloor u _ {1} \quad u _ {2} \quad 1 \quad u _ {4} \quad u _ {5} \right] ^ {T} \tag {13.8-15a}
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$$
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$$
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\{\mathbf {p} \} = \left[ \begin{array}{l l l l l} 0 & 0 & p _ {3} & 0 & 0 \end{array} \right] ^ {T} \tag {13.8-15b}
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$$
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and [K] is the original structure stiffness matrix of Eq. 13.8-6. As there is but one attachment d.o.f. in this simple problem, the third column of Eq. 13.8-12 and the third column of Eq. 13.8-14 describe the same displacement pattern. Natural frequencies obtained by solving the reduced eigenproblem, Eq. 13.8-3, are given in Table 13.8-1.
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As expected, both methods overestimate frequencies obtained by using all d.o.f. of the structure. The discrepancy is greater for higher frequencies. However, higher frequencies are usually of little interest, as structural response is dominated by lower frequencies. The several lowest frequencies are usually estimated accurately by a modest number of component modes.
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# 13.9 TIME-HISTORY ANALYSIS. DIRECT INTEGRATION METHODS
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In direct integration methods or step-by-step' methods, a finite difference approximation is used to replace the time derivatives appearing in Eq. 13.2-12 or 13.2-13 (i.e., $\{\ddot{\mathbf{D}}\}$ and $\{\dot{\mathbf{D}}\}$ ) by differences of displacement $\{\mathbf{D}\}$ at various instants of time. Finite difference methods for approximately solving initial value problems have been well studied and have a rich literature [13.47]. Over the past two decades, methods that are particularly effective for transient finite element equations have flourished and continue to be an active area of research [13.48,13.49]. Direct integration is an alternative to modal methods (Section 13.6). For many structural dynamics and wave propagation problems, including those with com-
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plicated nonlinearities, direct integration is more expedient. Many methods of direct integration are popular and the choice of method is strongly problem-dependent. In this section explicit and implicit methods are introduced. In Sections 13.10 and 13.11 these methods are discussed in detail.
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In direct integration, the approach is to write the equation of motion, Eq. 13.2-12, at a specific instant of time,
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$$
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\boxed {[ \mathbf {M} ] \{\ddot {\mathbf {D}} \} _ {n} + [ \mathbf {C} ] \{\dot {\mathbf {D}} \} _ {n} + [ \mathbf {K} ] \{\mathbf {D} \} _ {n} = \{\mathbf {R} ^ {\text {ext}} \} _ {n}} \tag {13.9-1}
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$$
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where subscript n denotes time $n \Delta t$ and $\Delta t$ is the size of the time increment or time step. The absence of time step subscripts on matrices [M], [C], and [K] in Eq. 13.9-1 implies linearity. For problems with material nonlinearity, [K] is a function of displacements and therefore of time as well. Accordingly, from Eq. 13.2-13,
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$$
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[ \mathbf {M} ] \{\ddot {\mathbf {D}} \} _ {n} + [ \mathbf {C} ] \{\dot {\mathbf {D}} \} _ {n} + \left\{\mathbf {R} ^ {\text {int}} \right\} _ {n} = \left\{\mathbf {R} ^ {\text {ext}} \right\} _ {n} \tag {13.9-2}
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$$
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$\{R^{int}\}_{n}$ is the internal force vector at time $n \Delta t$ due to straining of material. It is obtained by assembling element internal force vectors, $\{r^{int}\}_{n}$ , given by Eq. 13.2-7 using $\{\sigma\}_{n}$ . For nonlinear problems, $\{R^{int}\}_{n}$ is a nonlinear function of $\{D\}_{n}$ and possibly time derivatives of $\{D\}_{n}$ . For linear problems, $\{R^{int}\}_{n} = [K]\{D\}_{n}$ . In Eq. 13.9-2, [M] and [C] are taken as time-independent, although for some problems these may be nonlinear also.
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In the following sections, [M] is assumed to be positive definite. Moreover, unless otherwise stated, [K] is positive semidefinite; that is, [K] may permit rigid-body motion.
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Difference methods for direct integration of Eqs. 13.9-1 and 13.9-2 can be categorized as explicit or implicit. Explicit methods have the form
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$$
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\{\mathbf {D} \} _ {n + 1} = f \left(\left\{\mathbf {D} \right\} _ {n}, \left\{\dot {\mathbf {D}} \right\} _ {n}, \left\{\ddot {\mathbf {D}} \right\} _ {n}, \left\{\mathbf {D} \right\} _ {n - 1}, \dots\right) \tag {13.9-3}
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$$
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and hence permit $\{D\}_{n+1}$ to be determined in terms of completely historical information consisting of displacements and time derivatives of displacements at time $n \Delta t$ and before. Implicit methods have the form
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$$
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\{\mathbf {D} \} _ {n + 1} = f (\{\dot {\mathbf {D}} \} _ {n + 1}, \{\ddot {\mathbf {D}} \} _ {n + 1}, \{\mathbf {D} \} _ {n}, \dots) \tag {13.9-4}
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$$
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and hence computation of $\{\mathbf{D}\}_{n+1}$ requires knowledge of the time derivatives of $\{\mathbf{D}\}_{n+1}$ , which are unknown. Explicit and implicit methods have markedly different properties. This has important practical implications.
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Methods that have the general form of Eqs. 13.9-3 and 13.9-4 are called multistep methods. When the right-hand sides of Eqs. 13.9-3 and 13.9-4 contain information dating back to time $n \Delta t$ only, the methods are called single-step methods. When the right-hand sides of Eqs. 13.9-3 and 13.9-4 contain information dating back to time $(n - 1) \Delta t$ , they are called two-step methods. Single-step methods are easy to start from initial conditions. Multistep methods require special starting procedures that may be awkward or may introduce inaccuracies. Poor starting procedures (not described in this book) may degrade the accuracy of the entire analysis [13.69].
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# 13.10 EXPLICIT DIRECT INTEGRATION METHODS
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A popular method, which is characteristic of explicit methods in general, is the central-difference method. It approximates velocity and acceleration by
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$$
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\{\dot {\mathbf {D}} \} _ {n} = \frac {1}{2 \Delta t} \left(\left\{\mathbf {D} \right\} _ {n + 1} - \left\{\mathbf {D} \right\} _ {n - 1}\right) \tag {13.10-1}
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$$
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$$
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\{\ddot {\mathbf {D}} \} _ {n} = \frac {1}{\Delta t ^ {2}} \left(\{\mathbf {D} \} _ {n + 1} - 2 \{\mathbf {D} \} _ {n} + \{\mathbf {D} \} _ {n - 1}\right) \tag {13.10-2}
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$$
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Equations 13.10-1 and 13.10-2 are obtained by expanding $\{\mathbf{D}\}_{n+1}$ and $\{\mathbf{D}\}_{n-1}$ in Taylor series about time $n\Delta t$ :
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$$
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\{\mathbf {D} \} _ {n + 1} = \{\mathbf {D} \} _ {n} + \Delta t \{\dot {\mathbf {D}} \} _ {n} + \frac {\Delta t ^ {2}}{2} \{\ddot {\mathbf {D}} \} _ {n} + \frac {\Delta t ^ {3}}{6} \{\dddot {\mathbf {D}} \} _ {n} + \dots \tag {13.10-3}
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$$
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$$
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\{\mathbf {D} \} _ {n - 1} = \{\mathbf {D} \} _ {n} - \Delta t \{\dot {\mathbf {D}} \} _ {n} + \frac {\Delta t ^ {2}}{2} \{\ddot {\mathbf {D}} \} _ {n} - \frac {\Delta t ^ {3}}{6} \{\dddot {\mathbf {D}} \} _ {n} + \dots \tag {13.10-4}
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$$
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Subtracting Eq. 13.10-4 from Eq. 13.10-3 yields Eq. 13.10-1 while adding Eqs. 13.10-3 and 13.10-4 yields Eq. 13.10-2. In both cases, terms containing $\Delta t^{2}$ and higher powers are omitted from Eqs. 13.10-1 and 13.10-2. Hence, the central-difference formulas, Eqs. 13.10-1 and 13.10-2, are said to be second-order accurate. In other words, the error is $O(\Delta t^{2})$ which implies that halving the time step should approximately quarter the error.
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Combining Eqs. 13.10-1 and 13.10-2 with Eq. 13.9-1 provides
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$$
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\begin{array}{l} \left[ \frac {1}{\Delta t ^ {2}} \mathbf {M} + \frac {1}{2 \Delta t} \mathbf {C} \right] \left\{\mathbf {D} \right\} _ {n + 1} \\ = \left\{\mathbf {R} ^ {\text {ext}} \right\} _ {n} - [ \mathbf {K} ] \left\{\mathbf {D} \right\} _ {n} + \frac {1}{\Delta t ^ {2}} [ \mathbf {M} ] \left(2 \left\{\mathbf {D} \right\} _ {n} - \left\{\mathbf {D} \right\} _ {n - 1}\right) + \frac {1}{2 \Delta t} [ \mathbf {C} ] \left\{\mathbf {D} \right\} _ {n - 1} \tag {13.10-5} \\ \end{array}
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$$
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# Remarks.
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1. Equation 13.10-5 is a system of linear algebraic equations. If [M] and [C] are diagonal, then the equations are uncoupled and $\{\mathbf{D}\}_{n+1}$ can be obtained without solving simultaneous equations.
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2. For small finite element models, [K] can be formed and stored in the computer's core memory and the internal force $\{\mathbf{R}^{\mathrm{int}}\}_{n} = [\mathbf{K}]\{\mathbf{D}\}_{n}$ at each time step can be obtained by matrix multiplication. However, it is more common, even for linear problems, to compute the internal force vector at each time step by summation of element contributions (i.e., element-by-element). Element contributions are given by Eq. 13.2-7. Because the element [k] need not be formed or stored, explicit methods can treat large three-dimensional models with comparatively modest computer storage requirements.
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3. Starting the method from $n = 0$ requires $\{\mathbf{D}\}_{-1}$ , which can be computed from known initial conditions $\{\mathbf{D}\}_0$ and $\{\dot{\mathbf{D}}\}_0$ and Eq. 13.10-4:
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$$
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\{\mathbf {D} \} _ {- 1} = \{\mathbf {D} \} _ {0} - \Delta t \{\dot {\mathbf {D}} \} _ {0} + \frac {\Delta t ^ {2}}{2} \{\ddot {\mathbf {D}} \} _ {0} \tag {13.10-6}
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$$
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where terms with $\Delta t^3$ and higher powers are omitted. $\{\ddot{\mathbf{D}}\}_0$ is obtained from the equation of motion, Eq. 13.9-1, at time zero:
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$$
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\{\ddot {\mathbf {D}} \} _ {0} = [ \mathbf {M} ] ^ {- 1} \left(\left\{\mathbf {R} ^ {\text {ext}} \right\} _ {0} - [ \mathbf {K} ] \{\mathbf {D} \} _ {0} - [ \mathbf {C} ] \{\dot {\mathbf {D}} \} _ {0}\right) \tag {13.10-7}
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$$
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4. To compute $\{\mathbf{D}\}_{n+1}$ requires $\{\mathbf{R}^{\mathrm{int}}\}_{n}$ . For nonlinear material constitutive laws that are functions of strain (but not of strain rate), $\{\mathbf{R}^{\mathrm{int}}\}_{n}$ is easy to evaluate because $\{\mathbf{D}\}_{n}$ , and hence the strain at time $n\Delta t$ , is known. For this reason, explicit methods are well suited to treatment of material nonlinearity.
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5. Equation 13.10-5 is conditionally stable and requires $\Delta t$ such that
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$$
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\Delta t \leq 2 / \omega_ {\max} \tag {13.10-8}
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$$
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where $\omega_{max}$ is the highest natural frequency of $\det([K] - \omega^{2}[M]) = 0$ . If Eq. 13.10-8 is not satisfied, computations will be unstable. This is indicated by an obviously erroneous time-history solution that grows unbounded, perhaps by orders of magnitude per time step. $^{2}$ (In nonlinear problems, instabilities may be more difficult to detect.)
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6. A feature of Eq. 13.10-5 is that stability (i.e., maximum allowable time step size) is not affected by damping. For the central difference method to be economically competitive with implicit methods, both [M] and [C] must be diagonal. For reasons discussed in Section 13.13, explicit integration is usually more accurate with lumped mass matrices than with consistent mass matrices. However, it is difficult to model damping by spectral methods if [C] must be diagonal.
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7. It is conceivable that we may know the displacement and velocity of a structure at a given instant (i.e., initial conditions) and wish to integrate backward in time (i.e., use a negative $\Delta t$ ) to determine the configuration of the structure at some instant in the past. The central-difference method is conditionally stable for such applications and requires $-2/\omega_{max} \leq \Delta t \leq 2/\omega_{max}$ . Henceforth, we will be concerned with positive $\Delta t$ only.
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Alternative Form for Nondiagonal [C]. A form of the central-difference method that does not require diagonal [C] is obtained by approximating the velocity and acceleration by
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$$
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\{\dot {\mathbf {D}} \} _ {n - 1 / 2} = \frac {1}{\Delta t} \left(\left\{\mathbf {D} \right\} _ {n} - \left\{\mathbf {D} \right\} _ {n - 1}\right) \tag {13.10-9}
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$$
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$^{2}$ The case $\Delta t = 2/\omega_{max}$ may be called “limiting stability.” If $\Delta t = 2/\omega_{max}$ , the numerical solution may diverge in certain cases, but only in arithmetic fashion. If $\Delta t > 2/\omega_{max}$ , divergence is exponential. Further discussion of these matters appears in Section 13.13.
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$$
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\begin{array}{l} \{\ddot {\mathbf {D}} \} _ {n} = \frac {1}{\Delta t} \left(\{\dot {\mathbf {D}} \} _ {n + 1 / 2} - \{\dot {\mathbf {D}} \} _ {n - 1 / 2}\right) \\ = \frac {1}{\Delta t ^ {2}} \left(\{\mathbf {D} \} _ {n + 1} - 2 \{\mathbf {D} \} _ {n} + \{\mathbf {D} \} _ {n - 1}\right) \tag {13.10-10} \\ \end{array}
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$$
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The equation of motion, Eq. 13.9-1, is modified by lagging the velocity by one-half time step:
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$$
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[ \mathbf {M} ] \{\ddot {\mathbf {D}} \} _ {n} + [ \mathbf {C} ] \{\dot {\mathbf {D}} \} _ {n - 1 / 2} + [ \mathbf {K} ] \{\mathbf {D} \} _ {n} = \left\{\mathbf {R} ^ {\text {ext}} \right\} _ {n} \tag {13.10-11}
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$$
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Combination of Eqs. 13.10-9 through 13.10-11 yields
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$$
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\frac {1}{\Delta t ^ {2}} [ \mathbf {M} ] \{\mathbf {D} \} _ {n + 1} = \left\{\mathbf {R} ^ {\text {ext}} \right\} _ {n} - [ \mathbf {K} ] \{\mathbf {D} \} _ {n} + \frac {1}{\Delta t ^ {2}} [ \mathbf {M} ] \left(\left\{\mathbf {D} \right\} _ {n} + \Delta t \left\{\dot {\mathbf {D}} \right\} _ {n - 1 / 2}\right) - [ \mathbf {C} ] \left\{\dot {\mathbf {D}} \right\} _ {n - 1 / 2} \tag {13.10-12}
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$$
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If [M] is lumped, then computation of $\{D\}_{n+1}$ does not require the solution of simultaneous equations. There are no restrictions on the form of [C]. The method can be started using the initial displacement $\{D\}_{0}$ and the approximation $\{\dot{D}\}_{-1/2} \approx \{\dot{D}\}_{0}$ . Alternatively, $\{\dot{D}\}_{-1/2}$ can be approximated by the forward difference formula with negative $\Delta t$ —that is,
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$$
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\{\dot {\mathbf {D}} \} _ {- 1 / 2} = \{\dot {\mathbf {D}} \} _ {0} - \frac {\Delta t}{2} \{\ddot {\mathbf {D}} \} _ {0} \tag {13.10-13}
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$$
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where $\{\ddot{\mathbf{D}}\}_{0}$ is obtained from Eq. 13.10-7. Although the central-difference formulas, Eqs. 13.10-9 and 13.10-10, are second-order accurate, we can only guarantee first-order accuracy in the time integration of Eq. 13.10-11 when $[\mathbf{C}] \neq [\mathbf{0}]$ because viscous forces $[\mathbf{C}]\{\dot{\mathbf{D}}\}_{n-1/2}$ lag by half a time step. However, for practical structures (which are not heavily damped), Eqs. 13.10-5 and 13.10-12 have almost the same accuracy.
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The stability condition for Eq. 13.10-12 is
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$$
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\Delta t \leq \frac {2}{\omega_ {\max}} \left(\sqrt {1 + \xi^ {2}} - \xi\right) \tag {13.10-14}
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$$
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where $\xi$ is the fraction of critical damping at the highest undamped natural frequency, $\omega_{max}$ . For proportional damping, $\xi$ at frequency $\omega_{max}$ can be computed from Eq. 13.4-2. Equation 13.10-14 is more restrictive than Eq. 13.10-8.
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Implementation. A computational procedure for central-difference integration of undamped equations of motion with possible material nonlinearity is given in Table 13.10-1 (modification of this scheme to include damping is left as an exercise). In the element internal force evaluation, $\int [B]^{T}\{\sigma\}_{n} dV$ requires the same order of quadrature as used for the element stiffness matrix $\int [B]^{T}[E][B] dV$ . Therefore, guidelines given in Section 6.11 for stiffness matrix calculation are also applicable to internal force calculation. Note that internal forces must be computed
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TABLE 13.10-1. COMPUTATIONAL PROCEDURE FOR DIRECT INTEGRATION BY THE CENTRAL-DIFFERENCE METHOD, USING THE FORM GIVEN IN Eq. 13.10-12 BUT WITHOUT DAMPING.
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1. Set initial conditions $\{\mathbf{D}\}_{0} = \{\mathbf{D}(t = 0)\}$ and $\{\dot{\mathbf{D}}\}_{-1/2} = \{\dot{\mathbf{D}}(t = 0)\}$ , $n = 0$ .
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2. Assemble [M].
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3. Compute internal force $\{\mathbf{R}^{\mathrm{int}}\}_n$ as follows:
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3.1 Loop over elements $e = 1, 2, \ldots$ ;
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3.2 Compute element internal force $\{\mathbf{r}_e^{\mathrm{int}}\}_{n} = \int_{V_e} [\mathbf{B}]^T \{\boldsymbol{\sigma}\}_{n} dV$ ;
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3.3 Assemble element internal force, $\{\mathbf{r}_e^{\mathrm{int}}\}_n$ , into global internal force $\{\mathbf{R}^{\mathrm{int}}\}_n$ ;
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3.4 Next element; go to Step 3.1.
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4. Update displacement by Eq. 13.10-12:
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$$
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\{\mathbf {D} \} _ {n + 1} = \Delta t ^ {2} [ \mathbf {M} ] ^ {- 1} (\{\mathbf {R} ^ {\text { ext }} \} _ {n} - \{\mathbf {R} ^ {\text { int }} \} _ {n}) + \{\mathbf {D} \} _ {n} + \Delta t \{\dot {\mathbf {D}} \} _ {n - 1 / 2}
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$$
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5. Update velocity [at time $(n + \frac{1}{2})\Delta t]$ by Eq. 13.10-9:
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$$
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\{\dot {\mathbf {D}} \} _ {n + 1 / 2} = \frac {1}{\Delta t} (\{\mathbf {D} \} _ {n + 1} - \{\mathbf {D} \} _ {n})
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$$
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6. Output if desired; $n \leftarrow n + 1$ , go to Step 3.
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at each time step. This is the most expensive part of the per-time-step cost of an explicit method. Hence, there is considerable motivation to use reduced quadrature to evaluate internal forces. For example, explicit transient analysis with the four-node bilinear quadrilateral element with one-point quadrature will be roughly one-fourth as expensive as analysis using four-point quadrature. In three dimensions, the savings are even more dramatic. However, when reduced integration is used, additional precautions must be taken to prevent mesh instabilities (discussed in Section 6.12). Flanagan and Belytschko [13.52] present an effective scheme using hourglass control, or a stabilization matrix, that adds artificial stiffness to the element to suppress the zero-energy modes (see also [13.49, 13.53, 13.54]).
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Stability: Estimation of $\omega_{max}$ . The central-difference method, as well as explicit methods in general, is conditionally stable. If $\Delta t$ is too large, the method fails. If $\Delta t$ is much smaller than necessary, computations are too expensive. Therefore, it is necessary to determine, or accurately bound, $\omega_{max}$ in Eq. 13.10-8 or Eq. 13.10-14. Equation 13.5-7 demonstrates that $\omega_{max}$ for the assembled finite element model is bounded by the maximum frequency of the constituent unassembled and unsupported elements. This frequency can often be computed by hand calculation.
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Consider the uniform linear-displacement bar element shown in Fig. 13.5-1. With lumped masses, the highest frequency of this element is given in the Example of Section 13.5 as
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$$
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(\omega_ {\max}) _ {e} = \frac {2 c}{L} \tag {13.10-15}
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$$
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where the dilatational wave speed, acoustic wave speed, or simply wave speed, $c = \sqrt{E/\rho}$ is the speed at which information travels in the bar. From Eq. 13.10-8, we must use $\Delta t \leq 2/\omega_{max}$ . Therefore, if Eq. 13.10-15 represents the maximum element frequency among all elements in a mesh, then stable integration
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