김경종
372 lines
32 KiB
Markdown
372 lines
32 KiB
Markdown
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Figure 9.3 The absolute value of the dynamic load factor
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(9.48) the response in the first $p$ modes; i.e., we use
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$$
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\mathbf {U} ^ {p} (t) = \sum_ {i = 1} ^ {p} \phi_ {i} x _ {i} (t) \tag {9.49}
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$$
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where $\mathbf{U}^p$ approximates the exact solution $\mathbf{U}$ of (9.1).
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The reason that only the lowest modes are considered in a practical finite element analysis lies in the complete modeling process for dynamic analysis. Namely, so far, we have been concerned only with the exact solution of the finite element system equilibrium equations in (9.1). However, what we really want to obtain is a good approximation to the actual exact response of the mathematical model under consideration. We showed in Section 4.3.3 that under certain conditions the finite element analysis can be understood to be a Ritz analysis. In such case, upper bounds to the exact frequencies of the mathematical model are obtained. Moreover, in general, even when the monotonic convergence conditions are not satisfied, the finite element analysis approximates the lowest exact frequencies best, and little or no accuracy can be expected in approximating the higher frequencies and mode shapes. Therefore, there is usually little justification for including the dynamic response in the mode shapes with the high frequencies in the analysis. In fact, the finite element mesh should be chosen such that all important exact frequencies and vibration mode shapes of the mathematical model are well approximated, and then the solution needs to be calculated including only the response in these modes. However, this can be achieved precisely using mode superposition analysis by considering only the important modes of the finite element system.
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It is primarily because of the fact that in a mode superposition analysis only a few modes may need to be considered that the mode superposition procedure can be much more effective than direct integration. However, it also follows that the effectiveness of mode superposition depends on the number of modes that must be included in the analysis. In general, the structure considered and the spatial distribution and frequency content of the
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loading determine the number of modes to be used. For earthquake loading, in some cases only the 10 lowest modes need to be considered, although the order of the system n may be quite large. On the other hand, for blast or shock loading, many more modes generally need to be included, and p may be as large as 2n/3. Finally, in vibration excitation analysis, only a few intermediate frequencies may be excited, such as all frequencies between the lower and upper frequency limits $\omega_{l}$ and $\omega_{u}$ , respectively.
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Considering the problem of selecting the number of modes to be included in the mode superposition analysis, it should always be kept in mind that an approximate solution to the dynamic equilibrium equations in (9.1) is sought. Therefore, if not enough modes are considered, the equations in (9.1) are not solved accurately enough. But this means, in effect, that equilibrium, including the inertia forces, is not satisfied for the approximate response calculated. Denoting by $U^{p}$ the response predicted by mode superposition when p modes are considered, an indication of the accuracy of analysis at any time t is obtained by calculating an error measure $\epsilon^{p}$ , such as
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$$
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\epsilon^ {p} (t) = \frac {\left\| \mathbf {R} (t) - \left[ \mathbf {M} \ddot {\mathbf {U}} ^ {p} (t) + \mathbf {K} \mathbf {U} ^ {p} (t) \right] \right\| _ {2}}{\left\| \mathbf {R} (t) \right\| _ {2}} \tag {9.50}
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$$
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where we assume that $\|\mathbf{R}(t)\|_{2}\neq0$ . If a good approximate solution of the system equilibrium equations in (9.1) has been obtained, $\epsilon^{p}(t)$ will be small at any time t. But $\mathbf{U}^{p}(t)$ must have been obtained by an accurate calculation of the response in each of the p modes considered because in this way the only error is due to not including enough modes in the analysis.
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The error measure $\epsilon^{p}$ calculated in (9.50) determines how well equilibrium including inertia forces is satisfied and is a measure of the nodal point loads not balanced by inertia and elastic nodal point forces [see (9.2)]. Alternatively, we may say that $\epsilon^{p}$ is a measure of that part of the external load vector that has not been included in the mode superposition analysis. Since we have $R = \sum_{i=1}^{n} r_{i} M \phi_{i}$ , we can evaluate
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$$
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\Delta \mathbf {R} = \mathbf {R} - \sum_ {i = 1} ^ {p} r _ {i} (\mathbf {M} \boldsymbol {\phi} _ {i}) \tag {9.51}
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$$
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For a properly modeled problem the response to $\Delta R$ should be at most a static response. Therefore, a good correction $\Delta U$ to the mode superposition solution $U^{p}$ can be obtained from
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$$
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\mathbf {K} \Delta \mathbf {U} (t) = \Delta \mathbf {R} (t) \tag {9.52}
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$$
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where the solution of (9.52) may be required only for certain times at which the maximum response is measured. We call $\Delta U$ calculated from (9.52) the static correction.
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In summary, therefore, assuming that the decoupled equations in (9.46) have been solved accurately, the errors in a mode superposition analysis using p < n are due to the fact that not enough modes have been used, whereas the errors in a direct integration analysis arise because too large a time step is employed.
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From the preceding discussion it may appear that the mode superposition procedure has an inherent advantage over direct integration in that the response corresponding to the higher, probably inaccurate frequencies of the finite element system is not included in the analysis. However, assuming that in the finite element analysis all important frequencies are represented accurately, meaning that negligible dynamic response is calculated in the modes
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that are not represented accurately, the inclusion of the finite element system dynamic response in these latter modes will not seriously affect the accuracy of the solution. In addition, we will discuss in Section 9.4 that in implicit direct integration, advantage can be taken also of integrating accurately only the first p equations in (9.46). This is achieved by using an unconditionally stable direct integration method and selecting an appropriate integration time step $\Delta t$ , which, in general, is much larger than the integration step used with a conditionally stable integration scheme.
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# 9.3.3 Analysis with Damping Included
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The general form of the equilibrium equations of the finite element system in the basis of the eigenvectors $\Phi_{i}, i = 1, \ldots, n$ was given in (9.43), which shows that provided damping effects are neglected, the equilibrium equations decouple and the time integration can be carried out individually for each equation. Considering the analysis of systems in which damping effects cannot be neglected, we still would like to deal with decoupled equilibrium equations in (9.43), and use essentially the same computational procedure whether damping effects are included or neglected. In general, the damping matrix C cannot be constructed from element damping matrices, such as the mass and stiffness matrices of the element assemblage, and it is introduced to approximate the overall energy dissipation during the system response (see, for example, R. W. Clough and J. Penzien [A]). The mode superposition analysis is particularly effective if it can be assumed that damping is proportional, in which case
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$$
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\boldsymbol {\phi} _ {i} ^ {T} \mathbf {C} \boldsymbol {\phi} _ {j} = 2 \omega_ {i} \xi_ {i} \delta_ {i j} \tag {9.53}
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$$
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where $\xi_{i}$ is a modal damping parameter and $\delta_{ij}$ is the Kronecker delta ( $\delta_{ij} = 1$ for $i = j$ , $\delta_{ij} = 0$ for $i \neq j$ ). Therefore, using (9.53), it is assumed that the eigenvectors $\Phi_{i}$ , $i = 1$ , $2, \ldots, n$ , are also C-orthogonal and the equations in (9.43) reduce to $n$ equations of the form
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$$
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\ddot {x} _ {i} (t) + 2 \omega_ {i} \xi_ {i} \dot {x} _ {i} (t) + \omega_ {i} ^ {2} x _ {i} (t) = r _ {i} (t) \tag {9.54}
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$$
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where $r_{i}(t)$ and the initial conditions on $x_{i}(t)$ have already been defined in (9.46). We note that (9.54) is the equilibrium equation governing motion of the single degree of freedom system considered in (9.46) when $\xi_{i}$ is the damping ratio.
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If the relation in $(9.53)$ is used to account for damping effects, the procedure of solution of the finite element equilibrium equations in $(9.43)$ is the same as in the case when damping is neglected (see Section 9.3.2), except that the response in each mode is obtained by solving $(9.54)$ . This response can be calculated using an integration scheme such as those given in Tables 9.1 to 9.4 or by evaluating the Duhamel integral to obtain
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$$
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x _ {i} (t) = \frac {1}{\bar {\omega} _ {i}} \int_ {0} ^ {t} r _ {i} (\tau) e ^ {- \xi_ {i} \omega_ {i} (t - \tau)} \sin \bar {\omega} _ {i} (t - \tau) d \tau + e ^ {- \xi_ {i} \omega_ {i} t} \left(\alpha_ {i} \sin \bar {\omega} _ {i} t + \beta_ {i} \cos \bar {\omega} _ {i} t\right) \tag {9.55}
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$$
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where $\bar{\omega}_i = \omega_i\sqrt{1 - \xi_i^2}$
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and $\alpha_{i}$ and $\beta_{i}$ are calculated using the initial conditions in (9.46).
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In considering the implications of using $(9.53)$ to take account of damping effects, the following observations are made. First, the assumption in $(9.53)$ means that the total damping in the structure is the sum of individual damping in each mode. The damping in
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one mode could be observed, for example, by imposing initial conditions corresponding to that mode only (say ${}^{0}U = \phi_{i}$ for mode i) and measuring the amplitude decay during the free damped vibration. In fact, the ability to measure values for the damping ratios $\xi_{i}$ , and thus approximate in many cases in a realistic manner the damping behavior of the complete structural system, is an important consideration. A second observation relating to the mode superposition analysis is that for the numerical solution of the finite element equilibrium equations in (9.1) using the decoupled equations in (9.54), we do not calculate the damping matrix C but only the stiffness and mass matrices K and M.
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Damping effects can therefore readily be taken into account in mode superposition analysis provided that (9.53) is satisfied. However, assume that it would be numerically more effective to use direct step-by-step integration when realistic damping ratios $\xi_{i}, i = 1, \ldots, r$ are known. In that case, it is necessary to evaluate the matrix C explicitly, which when substituted into (9.53) yields the established damping ratios $\xi_{i}$ . If r = 2, Rayleigh damping can be assumed, which is of the form
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$$
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\mathbf {C} = \alpha \mathbf {M} + \beta \mathbf {K} \tag {9.56}
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$$
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where $\alpha$ and $\beta$ are constants to be determined from two given damping ratios that correspond to two unequal frequencies of vibration.
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EXAMPLE 9.9: Assume that for a multiple degree of freedom system $\omega_{1}=2$ and $\omega_{2}=3$ , and that in those two modes we require 2 percent and 10 percent critical damping, respectively; i.e., we require $\xi_{1}=0.02$ and $\xi_{2}=0.10$ . Establish the constants $\alpha$ and $\beta$ for Rayleigh damping in order that a direct step-by-step integration can be carried out.
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In Rayleigh damping we have
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$$
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\mathbf {C} = \alpha \mathbf {M} + \beta \mathbf {K} \tag {a}
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$$
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But using the relation in (9.53) we obtain, using (a),
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$$
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\boldsymbol {\phi} _ {i} ^ {T} (\alpha \mathbf {M} + \beta \mathbf {K}) \boldsymbol {\phi} _ {i} = 2 \omega_ {i} \xi_ {i}
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$$
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or $\alpha +\beta \omega_{i}^{2} = 2\omega_{i}\xi_{i}$ (b)
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Using this relation for $\omega_{1}$ , $\xi_{1}$ and $\omega_{2}$ , $\xi_{2}$ , we obtain two equations for $\alpha$ and $\beta$ ,
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$$
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\alpha + 4 \beta = 0. 0 8
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$$
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$$
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\alpha + 9 \beta = 0. 6 0 \tag {c}
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$$
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The solution of (c) is $\alpha = -0.336$ and $\beta = 0.104$ . Thus, the damping matrix to be used is
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$$
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\mathbf {C} = - 0. 3 3 6 \mathbf {M} + 0. 1 0 4 \mathbf {K} \tag {d}
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$$
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With the damping matrix given, we can now establish the damping ratio that is specified at any value of $\omega_{i}$ , when the Rayleigh damping matrix in (d) is used. Namely, the relation in (b) gives
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$$
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\xi_ {i} = \frac {- 0 . 3 3 6 + 0 . 1 0 4 \omega_ {i} ^ {2}}{2 \omega_ {i}}
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$$
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for all values of $\omega_{i}$ .
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In actual analysis it may well be that the damping ratios are known for many more than two frequencies. In that case two average values, say $\tilde{\xi}_{1}$ and $\tilde{\xi}_{2}$ , are used to evaluate $\alpha$ and $\beta$ . Consider the following example.
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EXAMPLE 9.10: Assume that the approximate damping to be specified for a multiple degree of freedom system is as follows:
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$$
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\xi_ {1} = 0. 0 2; \quad \omega_ {1} = 2; \quad \xi_ {2} = 0. 0 3; \quad \omega_ {2} = 3
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$$
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$$
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\xi_ {3} = 0. 0 4; \quad \omega_ {3} = 7; \quad \xi_ {4} = 0. 1 0; \quad \omega_ {4} = 1 5
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$$
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$$
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\xi_ {5} = 0. 1 4; \quad \omega_ {5} = 1 9
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$$
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Choose appropriate Rayleigh damping parameters $\alpha$ and $\beta$ .
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As in Example 9.9, we determine $\alpha$ and $\beta$ from the relation
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$$
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\alpha + \beta \omega_ {i} ^ {2} = 2 \omega_ {i} \xi_ {i} \tag {a}
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$$
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However, only two pairs of values, $\bar{\xi}_{1}$ , $\bar{\omega}_{1}$ and $\bar{\xi}_{2}$ , $\bar{\omega}_{2}$ , determine $\alpha$ and $\beta$ . Considering the spacing of the frequencies, we use
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$$
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\bar {\xi} _ {1} = 0. 0 3; \quad \bar {\omega} _ {1} = 4 \tag {b}
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$$
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$$
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\overline {{{\xi}}} _ {2} = 0. 1 2; \quad \overline {{{\omega}}} _ {2} = 1 7
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$$
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For the values in (b) we obtain, using (a),
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$$
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\alpha + 1 6 \beta = 0. 2 4
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$$
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$$
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\alpha + 2 8 9 \beta = 4. 0 8
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$$
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Hence $\alpha = 0.01498$ , $\beta = 0.01405$ , and we obtain
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$$
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\mathbf {C} = 0. 0 1 4 9 8 \mathbf {M} + 0. 0 1 4 0 5 \mathbf {K} \tag {c}
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$$
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<details>
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<summary>line</summary>
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| ωi | ξi (Mass proportional damping) | ξi (Stiffness proportional damping) |
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|------|--------------------------------|-------------------------------------|
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| 0.025| 0.03 | 0.03 |
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| 1 | 0.015 | 0.015 |
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| 2 | 0.02 | 0.02 |
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| 3 | 0.025 | 0.025 |
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| 4 | 0.03 | 0.03 |
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| 5 | 0.035 | 0.035 |
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| 10 | 0.025 | 0.025 |
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| 20 | 0.02 | 0.02 |
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| 30 | 0.015 | 0.015 |
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</details>
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Figure E9.10 Damping as a function of frequency
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We can now calculate which actual damping ratios are employed when the damping matrix $\mathbf{C}$ in (c) is used. From (a) we obtain
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$$
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\xi_ {i} = \frac {0 . 0 1 4 9 8 + 0 . 0 1 4 0 5 \omega_ {i} ^ {2}}{2 \omega_ {i}}
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$$
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Figure E9.10 shows the relation of $\xi_{i}$ as a function of $\omega_{i}$ , where, based on the use of $\xi_{i}$ in (9.54), we also indicate the “mass proportional” and “stiffness proportional” damping regions.
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The procedure of calculating $\alpha$ and $\beta$ in Examples 9.9 and 9.10 may suggest the use of a more general damping matrix if more than only two damping ratios are used to establish C. Assume that the r damping ratios $\xi_{i}, i = 1, 2, \ldots, r$ are given to define C. Then a damping matrix that satisfies the relation in (9.53) is obtained using the Caughey series,
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$$
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\mathbf {C} = \mathbf {M} \sum_ {k = 0} ^ {r - 1} a _ {k} [ \mathbf {M} ^ {- 1} \mathbf {K} ] ^ {k} \tag {9.57}
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$$
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where the coefficients $a_k, k = 0, \ldots, r - 1$ , are calculated from the $r$ simultaneous equations
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$$
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\xi_ {i} = \frac {1}{2} \left(\frac {a _ {0}}{\omega_ {i}} + a _ {1} \omega_ {i} + a _ {2} \omega_ {i} ^ {3} + \dots + a _ {r - 1} \omega_ {i} ^ {2 r - 3}\right)
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$$
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We should note that with r = 2, (9.57) reduces to Rayleigh damping, as presented in (9.56). An important observation is that if r > 2, the damping matrix C in (9.57) is, in general, a full matrix. Since the cost of analysis is increased by a very significant amount if the damping matrix is not banded, in most practical analyses using direct integration, Rayleigh damping is assumed. A disadvantage of Rayleigh damping is that the higher modes are considerably more damped than the lower modes, for which the Rayleigh constants have been selected (see Example 9.10).
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In practice, reasonable Rayleigh coefficients in the analysis of a specific structure may often be selected using available information on the damping characteristics of a typical similar structure; i.e., approximately the same $\alpha$ and $\beta$ values are used in the analysis of similar structures. The magnitude of the Rayleigh coefficients is to a large extent determined by the energy dissipation characteristics of the construction, including the materials.
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In the above discussion we assumed that the damping characteristics of the structure can be represented appropriately using proportional damping, either in a mode superposition analysis or in a direct integration procedure. In many analyses, the assumption of proportional damping [i.e., that (9.53) is satisfied] is adequate. However, in the analysis of structures with widely varying material properties, nonproportional damping may need to be used. For example, in the analysis of foundation-structure interaction problems, significantly more damping may be observed in the foundation than in the surface structure. In this case it may be reasonable to assign in the construction of the damping matrix different Rayleigh coefficients $\alpha$ and $\beta$ to different parts of the structure, which results in a damping matrix that does not satisfy the relation in (9.53). Another case of nonproportional damping is encountered when concentrated dampers corresponding to specific degrees of freedom (e.g., at the support points of a structure) are specified.
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The solution of the finite element system equilibrium equations with nonproportional damping can be obtained using the direct integration algorithms in Tables 9.1 to 9.4 without
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modifications because the property of the damping matrix did not enter into the derivation of the solution procedures. On the other hand, considering mode superposition analysis using the free-vibration mode shapes with damping neglected as base vectors, we find that $\Phi^{T}C\Phi$ in (9.43) is in the case of nonproportional damping a full matrix. In other words, the equilibrium equations in the basis of mode shape vectors are no longer decoupled. But, if it can be assumed that the primary response of the system is still contained in the subspace spanned by $\phi_{1},\ldots,\phi_{p}$ , it is necessary to consider only the first p equations in (9.43). Assuming that the coupling in the damping matrix $\Phi^{T}C\Phi$ between $x_{i}, i=1,\ldots,p$ , and $x_{i}, i=p+1,\ldots,n$ , can be neglected, the first p equations in (9.43) decouple from the equations $p+1$ to n and can be solved by direct integration using the algorithms in Tables 9.1 to 9.4 (see Example 9.11). In an alternative analysis procedure, the decoupling of the finite element equilibrium equations is achieved by solving a quadratic eigenproblem, in which case complex frequencies and vibration mode shapes are calculated (see J. H. Wilkinson [A]).
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EXAMPLE 9.11: Consider the solution of the equilibrium equations
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$$
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\left[ \begin{array}{c c c} \frac {1}{2} & & \\ & 1 & \\ & & \frac {1}{2} \end{array} \right] \ddot {\mathbf {U}} + \left[ \begin{array}{c c c} 0. 1 & & \\ & 0 & \\ & & 0. 5 \end{array} \right] \dot {\mathbf {U}} + \left[ \begin{array}{c c c} 2 & - 1 & 0 \\ - 1 & 4 & - 1 \\ 0 & - 1 & 2 \end{array} \right] \mathbf {U} = \mathbf {R} (t) \tag {a}
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$$
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The free-vibration mode shapes with damping neglected and corresponding frequencies of vibration are calculated in Example 10.4 and are
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$$
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\boldsymbol {\Phi} = \left[ \begin{array}{c c c} \frac {1}{\sqrt {2}} & 1 & - \frac {1}{\sqrt {2}} \\ \frac {1}{\sqrt {2}} & 0 & \frac {1}{\sqrt {2}} \\ \frac {1}{\sqrt {2}} & - 1 & - \frac {1}{\sqrt {2}} \end{array} \right]; \quad \boldsymbol {\Omega} ^ {2} = \left[ \begin{array}{c c c} 2 & & \\ & 4 & \\ & & 6 \end{array} \right]
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$$
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Transform the equilibrium equations in (a) to equilibrium relations in the mode shape basis. Using $\mathbf{U} = \boldsymbol{\Phi}\mathbf{X}$ , we obtain corresponding to (9.43) the equilibrium relations
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$$
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\begin{array}{l} \ddot {\mathbf {X}} (t) + \left[ \begin{array}{c c c} 0. 3 & - 0. 2 \sqrt {2} & - 0. 3 \\ - 0. 2 \sqrt {2} & 0. 6 & 0. 2 \sqrt {2} \\ - 0. 3 & 0. 2 \sqrt {2} & 0. 3 \end{array} \right] \dot {\mathbf {X}} (t) + \left[ \begin{array}{c c c} 2 & & \\ & 4 & \\ & & 6 \end{array} \right] \mathbf {X} (t) \\ = \left[ \begin{array}{c c c} \frac {1}{\sqrt {2}} & \frac {1}{\sqrt {2}} & \frac {1}{\sqrt {2}} \\ 1 & 0 & - 1 \\ - \frac {1}{\sqrt {2}} & \frac {1}{\sqrt {2}} & - \frac {1}{\sqrt {2}} \end{array} \right] \mathbf {R} (t) \tag {b} \\ \end{array}
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$$
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If it were now known that because of the specific loading applied, the primary response lies only in the first mode, we could obtain an approximate response by solving only
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$$
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\ddot {x} _ {1} (t) + 0. 3 \dot {x} _ {1} (t) + 2 x _ {1} (t) = \left[ \frac {1}{\sqrt {2}} \quad \frac {1}{\sqrt {2}} \quad \frac {1}{\sqrt {2}} \right] \mathbf {R} (t) \tag {c}
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$$
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and then calculating
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$$
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\mathbf {U} (t) = \left[ \begin{array}{c} \frac {1}{\sqrt {2}} \\ \frac {1}{\sqrt {2}} \\ \frac {1}{\sqrt {2}} \end{array} \right] x _ {1} (t)
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$$
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However, it should be noted that because $\Phi^{T}C\Phi$ in (b) is full, the solution of $x_{1}(t)$ from (c) does not give the actual response in the first mode because the damping coupling has been neglected.
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# 9.3.4 Exercises
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9.7. Obtain the solution of the finite element equations in Exercise 9.1 by mode superposition using all modes of the system (see Example 9.6).
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9.8. Consider the finite element system in Exercise 9.1.
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(a) Establish $a$ load vector which will excite only the second mode of the system.
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(b) Assume that $\mathbf{R} = \mathbf{0}$ and ${}^0\mathbf{U} = \mathbf{0}$ but ${}^0\dot{\mathbf{U}} \neq \mathbf{0}$ . Establish $a$ value of ${}^0\dot{\mathbf{U}}$ which will make the system vibrate only in its first mode.
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9.9. Calculate the curve corresponding to $\xi = 0.2$ given in Fig. 9.3.
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9.10. Perform a mode superposition solution of the equations given in Exercise 9.1 but using only the lowest mode. Also, evaluate the static correction [i.e., in (9.51) we have $p = 1$ ].
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9.11. Establish a damping matrix C for the system considered in Exercise 9.1, which gives the modal damping parameters $\xi_{1}=0.02$ , $\xi_{2}=0.08$ .
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9.12. A finite element system has the following frequencies: $\omega_{1}=1.2$ , $\omega_{2}=2.3$ , $\omega_{3}=2.9$ , $\omega_{4}=3.1$ , $\omega_{5}=4.9$ , $\omega_{6}=10.1$ . The modal damping parameters at $\omega_{1}$ and $\omega_{4}$ shall be $\xi_{1}=0.04$ , $\xi_{4}=0.10$ , respectively. Calculate a Rayleigh damping matrix and evaluate the damping ratios used at the other frequencies.
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# 9.4 ANALYSIS OF DIRECT INTEGRATION METHODS
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In the preceding sections we presented the two principal procedures used for the solution of the dynamic equilibrium equations
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$$
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\mathbf {M} \ddot {\mathbf {U}} (t) + \mathbf {C} \dot {\mathbf {U}} (t) + \mathbf {K} \mathbf {U} (t) = \mathbf {R} (t) \tag {9.58}
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$$
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where the matrices and vectors have been defined in Section 9.1. The two procedures were mode superposition and direct integration. The integration schemes considered were the central difference method, the Houbolt method, the Newmark integration procedure (see Tables 9.1 to 9.4), and the Bathe method. We stated that using the central difference scheme, a time step $\Delta t$ smaller than a critical time step $\Delta t_{cr}$ has to be used; but when employing the other three integration schemes, a similar time step limitation is not applicable.
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An important observation was that the cost of a direct integration analysis (i.e., the number of operations required) is directly proportional to the number of time steps required for solution. It follows that the selection of an appropriate time step in direct integration is
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of much importance. On one hand, the time step must be small enough to obtain accuracy in the solution; but, on the other hand, the time step must not be smaller than necessary because with such time step the solution is more costly than actually required. The aim in this section is to discuss in detail the problem of selecting an appropriate time step $\Delta t$ for direct integration. The two fundamental concepts to be considered are those of stability and accuracy of the integration schemes. The analysis of the stability and accuracy characteristics of the integration methods results in guidelines for the selection of an appropriate time step.
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A first fundamental observation for the analysis of a direct integration method is the relation between mode superposition and direct integration. We pointed out in Section 9.3 that, in essence, using either procedure the solution is obtained by numerical integration. However, in the mode superposition analysis, a change of basis from the finite element nodal displacements to the basis of eigenvectors of the generalized eigenproblem,
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$$
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\mathbf {K} \boldsymbol {\phi} = \omega^ {2} \mathbf {M} \boldsymbol {\phi} \tag {9.59}
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$$
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is performed prior to the time integration. Writing
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$$
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\mathbf {U} (t) = \boldsymbol {\Phi} \mathbf {X} (t) \tag {9.60}
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$$
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where the columns in $\Phi$ are the M-orthonormalized eigenvectors (free-vibration modes) $\phi_{1}, \ldots, \phi_{n}$ , and substituting for $\mathbf{U}(t)$ into (9.58) we obtain
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$$
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\ddot {\mathbf {X}} (t) + \boldsymbol {\Delta} \dot {\mathbf {X}} (t) + \boldsymbol {\Omega} ^ {2} \mathbf {X} (t) = \boldsymbol {\Phi} ^ {T} \mathbf {R} (t) \tag {9.61}
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$$
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where $\Omega^2$ is a diagonal matrix listing the eigenvalues of (9.59) (free-vibration frequencies squared) $\omega_1^2, \ldots, \omega_n^2$ . Assuming that the damping is proportional, $\Delta$ is a diagonal matrix, $\Delta = \mathrm{diag}(2\omega_i\xi_i)$ , where $\xi_i$ is the damping ratio in the $i$ th mode.
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The equation in (9.61) consists of n uncoupled equations, which can be solved, for example, using the Duhamel integral. Alternatively, one of the numerical integration schemes discussed as direct integration procedures may be used. Because the periods of vibration $T_{i}, i = 1, \ldots, n$ , are known, where $T_{i} = 2\pi/\omega_{i}$ , we can choose in the numerical integration of each equation in (9.61) an appropriate time step that ensures a required level of accuracy. On the other hand, if all n equations in (9.61) are integrated using the same time step $\Delta t$ , then the mode superposition analysis is completely equivalent to a direct integration analysis in which the same integration scheme and the same time step $\Delta t$ are used. In other words, the solution of the finite element system equilibrium equations would be identical using either procedure. Therefore, to study the accuracy of direct integration, we may focus attention on the integration of the equations in (9.61) with a common time step $\Delta t$ instead of considering (9.58). In this way, the variables to be considered in the stability and accuracy analysis of the direct integration method are only $\Delta t$ , $\omega_{i}$ , and $\xi_{i}, i = 1, \ldots, n$ , and not all elements of the stiffness, mass, and damping matrices. Furthermore, because all n equations in (9.61) are similar, we only need to study the integration of one typical row in (9.61), which may be written
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$$
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\ddot {x} + 2 \xi \omega \dot {x} + \omega^ {2} x = r \tag {9.62}
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$$
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and is the equilibrium equation governing motion of a single degree of freedom system with free-vibration period $T$ , damping ratio $\xi$ , and applied load $r$ .
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It may be mentioned here that this procedure of changing basis, i.e., using the transformation in (9.60), is also used in the convergence analysis of eigenvalue and eigenvector
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solution methods (see Section 11.2). The reason for carrying out the transformation in Section 11.2 is the same, namely, many fewer variables need to be considered in the analysis.
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Considering the solution characteristics of a direct integration method, the problem is, therefore, to estimate the integration errors in the solution of (9.62) as a function of $\Delta t/T$ , $\xi$ , and r. For such investigations, see, for example, L. Collatz [A] and R. D. Richtmyer and K. W. Morton [A]. In the following discussion we employ a relatively simple procedure in which the first step is to evaluate an approximation and load operator that relates explicitly the unknown required variables at time $t + \Delta t$ to previously calculated quantities (see K. J. Bathe and E. L. Wilson [A]).
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# 9.4.1 Direct Integration Approximation and Load Operators
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As discussed in the derivation of the direct integration methods (see Section 9.2), assume that we have obtained the required solution for the discrete times 0, $\Delta t$ , $2\Delta t$ , $3\Delta t$ , $\ldots$ , $t - \Delta t$ , t and that the solution for time $t + \Delta t$ is required next. Then for the specific integration method considered, we aim to establish the following recursive relationship:
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$$
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^ {t + \Delta t} \hat {\mathbf {X}} = \mathbf {A} ^ {t} \hat {\mathbf {X}} + \mathbf {L} (^ {t + \nu} r) \tag {9.63}
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$$
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where $t^{+\Delta t}\hat{X}$ and $t^{\hat{X}}$ are vectors storing the solution quantities (e.g., displacements, velocities) and $t^{+\nu}r$ is the load at time $t + \nu$ . We will see that $\nu$ may be 0, $\Delta t$ , or $\theta \Delta t$ for the integration methods considered. The matrix A and vector L are the integration approximation and load operators, respectively. Each quantity in (9.63) depends on the specific integration scheme employed. However, before deriving the matrices and vectors corresponding to the different integration procedures, we note that (9.63) can be used to calculate the solution at any time $t + n\Delta t$ , namely, applying (9.63) recursively, we obtain
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$$
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\begin{array}{l} { } ^ { t + n \Delta t } \hat { \mathbf { X } } = \mathbf { A } ^ { n } { } ^ { t } \hat { \mathbf { X } } + \mathbf { A } ^ { n - 1 } \mathbf { L } ( { } ^ { t + \nu } r ) + \mathbf { A } ^ { n - 2 } \mathbf { L } ( { } ^ { t + \Delta t + \nu } r ) + \dots \\ + \mathbf {A} \mathbf {L} \left(^ {t + (n - 2) \Delta t + \nu} r\right) + \mathbf {L} \left(^ {t + (n - 1) \Delta t + \nu} r\right) \tag {9.64} \\ \end{array}
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$$
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It is this relation that we will use for the study of the stability and accuracy of the integration methods. In the following sections we derive the operators A and L for the different integration methods considered, where we refer to the presentations in Sections 9.2.1 to 9.2.4.
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The central difference method. In the central difference integration scheme we use (9.3) and (9.4) to approximate the acceleration and velocity at time t, respectively. The equilibrium equation (9.62) is considered at time t; i.e., we use
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$$
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^ {\prime} \ddot {x} + 2 \xi \omega^ {\prime} \dot {x} + \omega^ {2} {} ^ {\prime} x = ^ {\prime} r \tag {9.65}
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$$
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$$
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^ {\prime} \ddot {x} = \frac {1}{\Delta t ^ {2}} \left(^ {t - \Delta t} x - 2 ^ {\prime} x + ^ {t + \Delta t} x\right) \tag {9.66}
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$$
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$$
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{ } ^ { t } \dot { x } = \frac { 1 } { 2 \Delta t } \left( - { } ^ { t - \Delta t } x + { } ^ { t + \Delta t } x \right) \tag {9.67}
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$$
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Substituting (9.66) and (9.67) into (9.65) and solving for $t^{+\Delta t}x$ , we obtain
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$$
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{ } ^ { t + \Delta t } x = \frac { 2 - \omega ^ { 2 } \Delta t ^ { 2 } } { 1 + \xi \omega \Delta t } { } ^ { t } x - \frac { 1 - \xi \omega \Delta t } { 1 + \xi \omega \Delta t } { } ^ { t - \Delta t } x + \frac { \Delta t ^ { 2 } } { 1 + \xi \omega \Delta t } { } ^ { t } r \tag {9.68}
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$$
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