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close to an excitation frequency, then severe vibration and “beating” are likely. This usually necessitates alteration of the structure’s natural frequencies by resizing or by adding members or dampers. If frequencies of the structure and the excitation are well separated, the structure still vibrates, but the amplitude of the response is likely to be tolerable.
In static analysis, symmetry can be exploited, for example, by analyzing half of the entire structure. In vibration analysis, symmetry of structure and supports does not imply symmetry of all vibration modes. By imposing symmetry one would exclude all antisymmetric modes, which are probably as important as symmetric modes.
One should be aware that stabilization methods used to suppress mechanisms in underintegrated elements may be associated with relatively low stiffness. The associated nonphysical vibration modes may contaminate that portion of the vibration spectrum of greatest interest [13.70] .
13.6 TIME-HISTORY ANALYSIS. MODAL METHODS
In a time-history or dynamic response problem, we solve Eq. 13.2-12 for \{D\} , \{\dot{D}\} , and \{\ddot{D}\} as functions of time. When [M] , [C] , and [K] are known and are time-independent, the problem is linear. When initial values of \{D\} and \{\dot{D}\} are prescribed, Eq. 13.2-12 is called an initial value problem. If material behavior is nonlinear, then the internal force vector \{R^{int}\} replaces [K]\{D\} in Eq. 13.2-12, which is then called a nonlinear initial value problem (Eq. 13.2-13). For reasons that will become clear, in this section we consider only linear problems, for which \{R^{int}\} = [K]\{D\} .
The modal or mode superposition method of analysis transforms Eq. 13.2-12 so that \{D\} and its time derivatives are replaced by \{Z\} and its time derivatives, where \{Z\} is a vector of modal amplitudes, or, in other words, a vector of generalized d.o.f. If damping is orthogonal, as Rayleigh damping is, then the transformed equations are uncoupled and each can be solved independently of all others. Solutions of these modal equations are superposed to yield the solution of the original problem [13.1,13.18,13.20,13.21]. Other solution methods for dynamic response are considered in Section 13.9 et seq.
Modal analysis is effective because of special properties that eigenvectors possess. These properties are orthogonality and linear independence (see Appendix C). Eigenvectors \{\overline{\mathbf{D}}\} in Eq. 13.5-2 are orthogonal with respect to the (symmetric) mass and stiffness matrices. That is,
\{\overline {{{\mathbf {D}}}} \} _ {i} ^ {T} [ \mathbf {M} ] \{\overline {{{\mathbf {D}}}} \} _ {j} = 0 \quad \text { and } \quad \{\overline {{{\mathbf {D}}}} \} _ {i} ^ {T} [ \mathbf {K} ] \{\overline {{{\mathbf {D}}}} \} _ {j} = 0, \quad \text { where } \quad i \neq j \tag {13.6-1}
If eigenvectors are normalized with respect to the mass matrix, then the quadratic product of the stiffness matrix yields the undamped natural frequencies. That is, by Eq. 13.5-4 with \lambda = \omega^{2} ,
\text { if } \quad \{\overline {{\mathbf {D}}} \} _ {i} ^ {T} [ \mathbf {M} ] \{\overline {{\mathbf {D}}} \} _ {i} = 1 \quad \text { then } \quad \{\overline {{\mathbf {D}}} \} _ {i} ^ {T} [ \mathbf {K} ] \{\overline {{\mathbf {D}}} \} _ {i} = \omega_ {i} ^ {2} \tag {13.6-2}
(A vector is normalized by dividing each of its terms by the same constant.) If [\phi] is the modal matrix, that is, an n_{eq} by n_{eq} matrix whose columns are the normalized eigenvectors, then Eq. 13.6-2 yields
[ \phi ] ^ {T} [ \mathbf {M} ] [ \phi ] = [ \mathbf {I} ] \quad \text { and } \quad [ \phi ] ^ {T} [ \mathbf {K} ] [ \phi ] = [ \omega^ {2} ] \tag {13.6-3}
where \left[\omega^{2}\right] is a diagonal matrix of the squared natural frequencies. It is called the spectral matrix.
Using the property of linear independence of eigenvectors, we can express any vector \{\mathbf{D}\} as a linear combination of eigenvectors. Thus
\{\mathbf {D} \} = [ \phi ] \{\mathbf {Z} \} \tag {13.6-4}
where the Z_{i} in vector \{Z\} state the proportion of each eigenvector in the transformation. Equation 13.6-4 represents a convenient d.o.f. transformation in which the Z_{i} are modal amplitudes and, similar to \{D\} , are functions of time. No approximation is introduced by the transformation if all modes of the original system are retained in [\phi] .
Mode Displacement Method. In the mode displacement method of modal analysis, we premultiply Eq. 13.2-12 by [\phi]^{T} and combine the result with Eq. 13.6-4. Thus, in view of Eqs. 13.6-3, the coefficients of \{\ddot{Z}\} and \{Z\} become the respective diagonal matrices [I] and [\omega^{2}] . If [C] is given by Eq. 13.4-1, or by other orthogonal forms [13.1,13.18], then [\xi] = [\phi]^{T}[C][\phi] is also a diagonal matrix and the transformed equations are completely uncoupled. Using customary notation, we write the coefficients in a diagonal matrix [\xi] as 2\xi_{i}\omega_{i} , where \xi_{i} is the fraction of critical damping in mode i. Thus
\ddot {Z} _ {i} + 2 \xi_ {i} \omega_ {i} \dot {Z} _ {i} + \omega_ {i} ^ {2} Z _ {i} = p _ {i}, \quad \text { where } \quad p _ {i} = \{\boldsymbol {\phi} \} _ {i} ^ {T} \{\mathbf {R} ^ {\text { ext }} \} \tag {13.6-5}
in which \{\phi\}_{i} is the i th column of [\phi] . There are as many uncoupled ordinary differential equations as there are d.o.f. In each equation, p_{i} is a known function of time. Initial values for Eq. 13.6-5 are obtained from known initial values of \{\mathbf{D}\} and \{\dot{\mathbf{D}}\} as follows. Premultiply both sides of Eq. 13.6-4 by [\phi]^{T}[\mathbf{M}] and take note of Eqs. 13.6-3. Thus
\{\mathbf {Z} (t = 0) \} = [ \phi ] ^ {T} [ \mathbf {M} ] \{\mathbf {D} (t = 0) \} \quad \text { and } \quad \{\dot {\mathbf {Z}} (t = 0) \} = [ \phi ] ^ {T} [ \mathbf {M} ] \{\dot {\mathbf {D}} (t = 0) \} \tag {13.6-6}
When \{\mathbf{Z}\} has been determined as a function of time from Eq. 13.6-5, \{\mathbf{D}\} is obtained from Eq. 13.6-4.
There are many ways to time-integrate Eqs. 13.6-5. In fact, Eq. 13.6-5 can be solved exactly for Z_{i}(t) . If p_{i} is piecewise linear in time, then exact solutions can be written for Z_{i} and \dot{Z}_{i} in terms of e^{-\xi\omega t}\sin\omega t and e^{-\xi\omega t}\cos\omega t . However, for more general loading, an exact solution can be tedious and it is more effective to use a direct integration method, as discussed in Section 13.9 et seq.
At first glance, calculating [\phi] in Eq. 13.6-4 seems to be a prohibitive computational expense. But for many problems, the higher-frequency modes participate little in the structural response and therefore only a small number of low-frequency modes need be used. Thus, only the first m equations of Eq. 13.6-5, where typically m \ll n_{eq} , are solved and the transformation Eq. 13.6-4 is approximated by
\{\mathbf {D} \} \approx \sum_ {i = 1} ^ {m} \{\boldsymbol {\phi} \} _ {i} Z _ {i} \quad \text { where } \quad m < n _ {\mathrm{eq}} \tag {13.6-7}
in which \{\phi\}_{i} is the ith normalized eigenvector. A measure of the error at time t is denoted by e(t) . It can be quantified [13.18] by
e (t) \equiv \frac {\left\| \left\{\mathbf {R} ^ {\text {ext}} \right\} - [ \mathbf {M} ] \{\ddot {\mathbf {D}} \} - [ \mathbf {C} ] \{\dot {\mathbf {D}} \} - [ \mathbf {K} ] \{\mathbf {D} \} \right\|}{\left\| \left\{\mathbf {R} ^ {\text {ext}} \right\} \right\|} \tag {13.6-8}
where \{D\} and its time derivatives are obtained from Eq. 13.6-7 and \parallel denotes any vector norm. It is assumed in writing Eq. 13.6-8 that \{R^{ext}\} \neq \{0\} at the particular instant e(t) is computed. For an accurate analysis, e(t) should be small (1% or less, as a conjecture) for the entire duration of analysis.
In many structural dynamics problems, more modes participate in the quasi-static response than in the dynamic response. The mode displacement method may have difficulty in reproducing quasistatic deflection shapes of structures when m is small. A modification of the mode displacement method, discussed next, removes this deficiency. Additional comments appear in Section 13.14.
Mode Acceleration Method. In the mode acceleration method, we perform modal transformation on only the inertial and viscous terms of Eq. 13.2-12 [13.2, 13.22, 13.23, 13.24]. This yields
[ \mathbf {M} ] [ \phi ] \{\ddot {\mathbf {Z}} \} + [ \mathbf {C} ] [ \phi ] \{\dot {\mathbf {Z}} \} + [ \mathbf {K} ] \{\mathbf {D} \} = \left\{\mathbf {R} ^ {\text {ext}} \right\} \tag {13.6-9}
If no rigid-body modes are possible and [K] is properly formed, then [K]^{-1} exists and Eq. 13.6-9 can be solved for \{D\} . From Eq. 13.6-9,
\{\mathbf {D} \} = [ \mathbf {K} ] ^ {- 1} \left\{\mathbf {R} ^ {\text {ext}} \right\} - [ \mathbf {K} ] ^ {- 1} ([ \mathbf {M} ] [ \phi ] \{\ddot {\mathbf {Z}} \} + [ \mathbf {C} ] [ \phi ] \{\dot {\mathbf {Z}} \}) \tag {13.6-10}
Equation 13.6-3 yields [K]^{-1}[\phi]^{-T} = [\phi]\left[\omega^{2}\right]^{-1} . Hence, if [C] is orthogonal, Eq. 13.6-10 can be written as
\{\mathbf {D} \} = [ \mathbf {K} ] ^ {- 1} \left\{\mathbf {R} ^ {\text { ext }} \right\} - [ \phi ] \left[ \omega^ {2} \right] ^ {- 1} \left(\left\{\ddot {\mathbf {Z}} \right\} + [ \xi ] \left\{\dot {\mathbf {Z}} \right\}\right) \tag {13.6-11}
where [\xi] is a diagonal matrix with ith diagonal coefficient 2\xi_{i}\omega_{i} (proof of Eq. 13.6-11 is left as an exercise). In Eq. 13.6-11, [K] is the original n_{eq} by n_{eq} stiffness matrix of Eq. 13.2-12. Here [\phi] is n_{eq} by n_{eq} , [\omega^{2}] and [\xi] are n_{eq} by n_{eq} diagonal matrices, and \{Z\} is an n_{eq} by 1 vector of modal amplitudes. As with mode displacement analysis, only a small number of modes, m, is usually required for accurate results. Hence, the matrix multiplication indicated in the second term of Eq. 13.6-11 is carried out over only the lowest m eigenvectors and Eq. 13.6-11 is approximated by
\{\mathbf {D} \} \approx [ \mathbf {K} ] ^ {- 1} \left\{\mathbf {R} ^ {\text {ext}} \right\} - \sum_ {i = 1} ^ {m} \left\{\phi \right\} _ {i} \left(\frac {1}{\omega_ {i} ^ {2}} \ddot {Z} _ {i} + \frac {2 \xi_ {i}}{\omega_ {i}} \dot {Z} _ {i}\right), \quad \text {where} \quad m < n _ {\mathrm{eq}} \tag {13.6-12}
The first term on the right-hand side of Eq. 13.6-12 represents the quasistatic response, and the second term represents the dynamic correction attributable to inertia and viscous effects. To implement the mode acceleration method, we solve Eq. 13.6-5 for \dot{Z}_{i} and \ddot{Z}_{i}, i = 1, 2, \ldots, m exactly as we would in a mode displacement analysis (note that we do not need Z_{i} ). Then Eq. 13.6-12 is used for superposition (rather than Eq. 13.6-7 as in the mode displacement method) to obtain \{D\} as a function of time.
Obviously, the mode acceleration algorithm is adept at reproducing the quasistatic structure deflection shape. If the magnitude of the external load (but not its distribution) varies with time, then \{R^{ext}\}=s(t)\{S\} , where \{S\} is time-independent and describes the distribution of external load, and s(t) is a time-dependent scalar that describes the amplitude of the external load. We call such loading proportional loading. Thus the quasistatic displacement [K]^{-1}\{S\} need only be scaled by s(t) at each instant in time. Usually, fewer modes are required in mode acceleration than in a mode displacement analysis of equivalent accuracy [13.24]. On the other hand, for nonproportional loading the mode acceleration method requires the solution of simultaneous equations at each step of the solution (only forward and back substitution after [K] has been factored). If \{R^{ext}\}=\{0\} , both methods yield identical results for the same m.
If rigid-body modes are possible, [K] is singular and the mode acceleration method cannot be employed in the straightforward manner indicated by Eqs. 13.6-11 and 13.6-12. Discussion of this matter appears in [13.2].
Superposition of Ritz Vectors. A disadvantage of the mode displacement and mode acceleration methods is that computation of eigenvectors is expensive. For large structures it is often the most costly part of a dynamic response analysis. Using Ritz vectors (which may also be called basis vectors), we make a transformation analogous to Eqs. 13.6-4 and 13.6-7. Ritz vectors do not have all of the desirable properties of eigenvectors but are much more economical to compute.
Time-history analysis by superposition of Ritz vectors is a type of Rayleigh–Ritz analysis. A Ritz vector d.o.f. transformation
\{\mathbf {D} \} = [ \mathbf {W} ] \{\mathbf {y} \} \tag {13.6-13}
is used where [W] is an n_{eq} by m matrix whose columns are Ritz vectors \{w\}_{i} and \{y\} is a vector of generalized coordinates. Thus [W] = [w_{1} \quad w_{2} \quad \ldots \quad w_{m}] and \{y\} is an m by 1 vector whose terms y_{i} state the proportion of \{w\}_{i} in the transformation. The number m of Ritz vectors is chosen by the analyst. There are many ways to obtain the Ritz vectors in Eq. 13.6-13. Some of these procedures are rather arbitrary. If the Ritz vectors are the lowest m eigenvectors of Eq. 13.5-2 and are normalized according to Eq. 13.6-2, then Eq. 13.6-13 reduces to Eq. 13.6-7 of the mode displacement method. It is not necessary that Ritz vectors be close approximations of eigenvectors. Ideally, however, Ritz vectors are linear combinations of the lowest eigenvectors. A reliable method for generating good Ritz vectors (without solving an eigenproblem) is to determine [W] by solving
[ \mathbf {K} ] [ \mathbf {W} ] = [ \mathbf {R} ] \tag {13.6-14}
where [K] is the n_{eq} by n_{eq} stiffness matrix of the complete structure and [R] is an n_{eq} by m matrix whose columns are linearly independent load patterns selected by the analyst to excite important (i.e., lower) displacement modes of the structure. In writing Eq. 13.6-14, we assume that [K] is nonsingular. For treating problems with rigid-body modes, a modified method is required.
The use of Ritz vectors is not limited to problems of dynamics. Ritz vectors constitute a “reduced basis” that serves to reduce the cost of repetitive calculation cycles, which appear in modal methods of dynamics, in nonlinear static problems, and in design optimization.
Ritz Vectors in Time-History Analysis. Often, the load patterns used in Eq. 13.6-14 are obtained from actual load patterns applied to the structure. In dynamic analysis, a question that immediately arises is what load patterns are most effectively used in Eq. 13.6-14 when the loads that shake a structure, \{R^{ext}\} , are time-dependent. A method called superposition of Ritz vectors effectively addresses this question [13.25]. It is necessary that the external load be representable as a superposition of proportional loads—that is,
\{\mathbf {R} ^ {\mathrm{ext}} \} = \sum_ {j = 1} ^ {\ell} s _ {j} (t) \{\mathbf {S} \} _ {j} \tag {13.6-15}
where \{S\}_{j} describes the distribution of the jth load component, s_{j}(t) is a scalar that describes the time-dependent amplitude of \{S\}_{j} , and \ell is the number of \{S\}_{j} needed for superposition. For many practical problems \ell is small.
The dynamic response \{\mathbf{D}\}_{j} to load component s_j(t)\{\mathbf{S}\}_{j} is computed independently for each j . Then the total structural response is obtained by superposition
\{\mathbf {D} \} = \sum_ {j = 1} ^ {\ell} \{\mathbf {D} \} _ {j} \tag {13.6-16}
In the response analysis for each \{\mathbf{D}\}_{j} , a Ritz vector d.o.f. transformation
\{\mathbf {D} \} _ {j} = [ \mathbf {W} ] _ {j} \{\mathbf {y} \} _ {j} \tag {13.6-17}
is used where [W]_{j} is an n_{eq} by m matrix of Ritz vectors. An algorithm for generating [M]-orthogonal Ritz vectors, when given a load pattern \{S\}_{j} , is shown in Table 13.6-1. In this algorithm, the first Ritz vector \{w\}_{1} is proportional to the
TABLE 13.6-1. COMPUTATIONAL PROCEDURE FOR GENERATION OF [M]-ORTHOGONAL RITZ VECTORS [13.25].
| 1. Obtain $n_{eq}$ by $n_{eq}$ mass and stiffness matrices, [M] and [K]. | |
| 2. Factor the stiffness matrix; for example, [K] = [L][L] $^T$ . | |
| 3. For each proportional load component $\{S\}_{j}, j = 1, 2, \ldots, \ell$ , compute [W] $_j$ = [w $_1$ w $_2$ ... w $_m$ ] as follows: | |
| 3.1 Solve for the first Ritz vector, $\{w\}_{1}$ : | |
| $[K]\{w^{*}\}_{1} = \{S\}_{j}$ solve for $\{w^{*}\}_{1}$ $\{w\}_{i}^{T}[M]\{w\}_{1} = 1$ normalize $\{w^{*}\}_{1}$ to yield $\{w\}_{1}$ | |
| 3.2 Solve for additional Ritz vectors, $\{w\}_{i}$ ; $i = 2, 3, \ldots, m$ : | |
| $[K]\{w^{*}\}_{i} = [M]\{w\}_{i-1}$ solve for $\{w^{*}\}_{i}$ $\{w^{**}\}_{i} = \{w^{*}\}_{i} - \sum_{k=1}^{i-1} \{w\}_{k}^{T}[M]\{w^{*}\}_{i}\{w\}_{k}$ [M]-orthogonalize $\{w^{*}\}_{i}$ to yield $\{w^{**}\}_{i}$ $\{w\}_{i}^{T}[M]\{w\}_{i} = 1$ normalize $\{w^{**}\}_{i}$ to yield $\{w\}_{i}$ | |
| 3.3 Assemble Ritz vectors $\{w\}_{i}$ into $n_{eq}$ by $m$ matrix [W] $_j$ . | |
| 3.4 Next load component; $j \leftarrow j + 1$ , go to Step 3.1. |
quasistatic deflection shape of the structure under loads \{\mathbf{S}\}_{j} . To obtain the second Ritz vector, we impose \{\mathbf{w}\}_{1} as a nodal acceleration vector; hence, \{\mathbf{w}\}_{2} is proportional to the quasistatic deflection shape for inertial loads [\mathbf{M}]\{\mathbf{w}\}_{1} . To obtain the third Ritz vector, we impose \{\mathbf{w}\}_{2} as a nodal acceleration, and so on. If loads \{\mathbf{S}\}_{j} are zero, as for a structure that moves freely after initial velocities are prescribed, the procedure of Table 13.6-1 must be modified. A possible modification is to simply prescribe \{\mathbf{w}\}_{1} (e.g., as null except for unity corresponding to a d.o.f. expected to have significant displacement), then go to Step 3.2.
Once a mass-matrix-orthogonal [\mathbf{W}]_j is obtained, Eqs. 13.2-12, 13.6-16, and 13.6-17 are combined and premultiplied by [\mathbf{W}]_j^T to yield
\{\ddot {\mathbf {y}} \} _ {j} + [ \mathbf {C} ] _ {j} \{\dot {\mathbf {y}} \} _ {j} + [ \mathbf {K} ] _ {j} \{\mathbf {y} \} _ {j} = \{\mathbf {P} \} _ {j} \tag {13.6-18}
where the transformed stiffness matrix, damping matrix, and load vector are
[ \mathbf {K} ] _ {j} = [ \mathbf {W} ] _ {j} ^ {T} [ \mathbf {K} ] [ \mathbf {W} ] _ {j} \tag {13.6-19a}
[ \mathbf {C} ] _ {j} = [ \mathbf {W} ] _ {j} ^ {T} [ \mathbf {C} ] [ \mathbf {W} ] _ {j} \tag {13.6-19b}
\{\mathbf {P} \} _ {j} = s _ {j} (t) [ \mathbf {W} ] _ {j} ^ {T} \{\mathbf {S} \} _ {j} \tag {13.6-19c}
After the \{\mathbf{y}\}_{j} are known (as functions of time) for all j , Eqs. 13.6-16 and 13.6-17 yield \{\mathbf{D}\} = \{\mathbf{D}(t)\} . Note that we must generate a [\mathbf{W}]_j and also solve Eqs. 13.6-18 as many times as there are load components in Eq. 13.6-15.
Matrices [\mathbf{K}]_j and [\mathbf{C}]_j have dimension m by m . Therefore, if m \ll n_{\mathrm{eq}} , Eqs. 13.6-18 represent a much smaller system of simultaneous ordinary differential equations than the original system, Eq. 13.2-12. In general, matrices [\mathbf{K}]_j and [\mathbf{C}]_j are full (but [\mathbf{M}]_j is a unit matrix according to the procedure of Table 13.6-1). Equations 13.6-18 are usually solved for \{\mathbf{y}\}_j and its time derivatives by a direct integration method. Optionally, the Ritz vectors can be made to be also stiffness-matrix-orthogonal using the procedure of [13.25]. This entails greater expense in forming [\mathbf{W}]_j , but then [\mathbf{K}]_j is diagonal (as is [\mathbf{C}]_j if damping is orthogonal). Equations 13.6-18 are then completely uncoupled and can be solved independently, either exactly or approximately.
Several features of Ritz mode superposition are apparent from Table 13.6-1. Each Ritz vector is obtained by solving a system of simultaneous algebraic equations. Each solution is relatively inexpensive once [K] has been factored. As with the mode acceleration method, static structure deflection shapes are accurately modeled. However, since [K] must be nonsingular, the method cannot treat problems with rigid-body modes unless modified. When the number of loads \ell necessary for the superposition in Eq. 13.6-13 becomes large, the method loses its attractiveness since an independent set of Ritz vectors [\mathbf{W}]_j must be determined and an independent system of ordinary differential equations, Eqs. 13.6-18, must be solved for each load component. However, for many problems only a few load components are necessary. The reduction in the size of the original problem can be remarkable. For example, for earthquake shaking of structures, the number of Ritz vectors necessary for accurate response analysis is almost always less than 50, whereas equations in the original system may number several thousand [13.25].
Nonlinear Problems. In material-nonlinear problems, [K] and [C] are time-dependent. Usually [M] does not change with time. Modal techniques employ superposition and hence are often assumed to be inapplicable to nonlinear problems. However, nonlinear problems can be accommodated if all nonlinearities are treated as pseudoloads and incorporated with the external load \{R^{ext}\} . This approach requires that modes be superposed at each time step to obtain \{D\} (and if necessary \{\dot{D}\} ) so that the material constitutive law can be evaluated. Pseudoloads are then calculated and transformed back to modal equations, the solution of which is then incremented in time by one step. Strictly speaking, when response is nonlinear [\phi] does not represent normal modes of vibration. Equation 13.6-4 should then be viewed as simply a coordinate transformation [13.26,13.27]. For structural dynamics problems with nonlinearities that are mild and localized in the mesh, mode displacement superposition has been used, sometimes effectively. For severe nonlinearities, convergence of the pseudoload approach is poor. For most nonlinear problems, direct time integration methods are preferable to superposition methods.
13.7 MASS CONDENSATION. GUYAN REDUCTION
In static analyses, problems with 10,000 d.o.f. or more are common. In dynamic analyses in which we ask for natural frequencies and mode shapes, problems with only 1000 d.o.f. can be difficult. The major difficulty is the expense of computing eigenvalues and eigenvectors.
However, condensation can be employed to reduce the number of d.o.f., which reduces the expense of computing eigenvalues and eigenvectors and the expense of subsequent calculations. Condensation is detrimental to accuracy, but negligibly so if properly used. However, one may not wish to use condensation if [M] has been obtained by optimal lumping, as the use of optimal lumping implies that particular effort is being made to ensure accuracy.
The literature is vast so we present only important ideas [13.1,13.18,13.28] . In Section 13.8, the component mode synthesis method is described, which has significant condensation features.
We present a condensation algorithm known as Guyan reduction, mass condensation, or eigenvalue economization [13.29]. Equation 13.5-2 is partitioned and written as
\left(\left[ \begin{array}{l l} \mathbf {K} _ {m m} & \mathbf {K} _ {m s} \\ \mathbf {K} _ {m s} ^ {T} & \mathbf {K} _ {s s} \end{array} \right] - \lambda \left[ \begin{array}{l l} \mathbf {M} _ {m m} & \mathbf {M} _ {m s} \\ \mathbf {M} _ {m s} ^ {T} & \mathbf {M} _ {s s} \end{array} \right]\right) \left\{ \begin{array}{l} \overline {{\mathbf {D}}} _ {m} \\ \overline {{\mathbf {D}}} _ {s} \end{array} \right\} = \left\{ \begin{array}{l} 0 \\ 0 \end{array} \right\} \tag {13.7-1}
where the m “master” d.o.f. \{\overline{D}_{m}\} are to be retained and the s “slave” d.o.f. \{\overline{D}_{s}\} are to be removed by condensation. The principal assumption in Guyan reduction is that for the lowest-frequency modes, inertia forces on slave d.o.f. are much less important than elastic forces transmitted by the master d.o.f. (subsequently, this assumption will be used as a guideline for selecting whether a d.o.f. should be a master or a slave). In other words, slave d.o.f. are assumed to move quasi-statically in response to the motion of master d.o.f. Thus, in order to obtain a relation between \{D_{s}\} and \{D_{m}\} , we temporarily ignore all mass but [M_{mm}] . From the lower partition of Eq. 13.7-1,
\left\{\overline {{\mathbf {D}}} _ {s} \right\} _ {s \times 1} = - \left[ \mathbf {K} _ {s s} \right] _ {s \times s} ^ {- 1} \left[ \mathbf {K} _ {m s} \right] _ {s \times m} ^ {T} \left\{\overline {{\mathbf {D}}} _ {m} \right\} _ {m \times 1} \tag {13.7-2}
Accordingly, with n_{\mathrm{eq}} = m + s and [\mathbf{I}] an m by m identity matrix,
\left\{ \begin{array}{l} \overline {{\mathbf {D}}} _ {m} \\ \overline {{\mathbf {D}}} _ {s} \end{array} \right\} = \underset {n _ {\mathbf {e q}} \times m} {\left[ \mathbf {T} \right]} \left\{ \begin{array}{l} \overline {{\mathbf {D}}} _ {m} \\ m \times 1 \end{array} \right\}, \quad \text {where} \quad [ \mathbf {T} ] = \left[ \begin{array}{c} \mathbf {I} \\ - \mathbf {K} _ {s s} ^ {- 1} \mathbf {K} _ {m s} ^ {T} \end{array} \right] \tag {13.7-3}
Substitution of Eq. 13.7-3 into Eq. 13.7-1 and premultiplication by [T]^{T} yields the condensed eigenproblem
\left(\left[ \mathbf {K} _ {r} \right] - \lambda \left[ \mathbf {M} _ {r} \right]\right) \left\{\overline {{\mathbf {D}}} _ {m} \right\} = \{\mathbf {0} \} \tag {13.7-4}
where the reduced matrices are symmetric and are given by
\left[ \mathbf {K} _ {r} \right] _ {m \times m} = [ \mathbf {T} ] ^ {T} [ \mathbf {K} ] [ \mathbf {T} ] \quad \text { and } \quad \left[ \mathbf {M} _ {r} \right] _ {m \times m} = [ \mathbf {T} ] ^ {T} [ \mathbf {M} ] [ \mathbf {T} ] \tag {13.7-5}
Note that even if [K] is banded and [M] is diagonal, [K_{r}] and [M_{r}] are in general full and [M_{r}] is a combination of both mass and stiffness coefficients. If damping matrix [C] and external loads \{R^{ext}\} appear in the equation of motion, then condensed damping matrix [C_{r}] = [T]^{T}[C][T] and condensed external loads \{R_{r}^{ext}\} = [T]^{T}\{R^{ext}\} appear in the reduced equation of motion
[ \mathbf {M} _ {r} ] \{\ddot {\mathbf {D}} _ {m} \} + [ \mathbf {C} _ {r} ] \{\dot {\mathbf {D}} _ {m} \} + [ \mathbf {K} _ {r} ] \{\mathbf {D} _ {m} \} = \left\{\mathbf {R} _ {r} ^ {\text {ext}} \right\} \tag {13.7-6}
where \{\mathbf{D}_m\} represents the displacements of master d.o.f.
If [M] is diagonal and slave d.o.f. carry no mass, then [K_{r}] is the same matrix as produced by “static condensation,” [M_{r}] contains the nonzero M_{ii} of [M], and condensation produces no loss of accuracy. (Static condensation is discussed in Section 8.1.)
In vibration problems, when eigenvalues \lambda_{i} and eigenvectors \{\overline{D}_{m}\}_{i} of the reduced system are known, slave modes \{\overline{D}_{s}\}_{i} can be recovered by use of Eq. 13.7-2. However, it is more accurate to recover \{\overline{D}_{s}\}_{i} from the lower partition of Eq. 13.7-1 in which advantage is taken of the previously neglected slave node masses. Thus
\{\overline {{{\mathbf {D}}}} _ {s} \} _ {i} = - \left[ \mathbf {K} _ {s s} - \lambda_ {i} \mathbf {M} _ {s s} \right] ^ {- 1} \left[ \mathbf {K} _ {m s} ^ {T} - \lambda_ {i} \mathbf {M} _ {m s} ^ {T} \right] \{\overline {{{\mathbf {D}}}} _ {m} \} _ {i} \tag {13.7-7}
Reference 13.34 includes another form of this expression that is more computationally efficient.
Remarks. Why create a detailed finite element model if we subsequently intend to discard many d.o.f. by condensation? Because coarse finite element discretizations usually do not have sufficient detail for accurate stiffness and mass representations. Furthermore, accurate stress computations usually require fine discretizations [13.30].
Because reduction destroys any preexisting band structure and sparsity, m must be considerably smaller than n_{eq} for Guyan reduction to be cost-effective. If original matrices [M] and [K] have unusually small bandwidths, it may be prudent
to avoid condensation because a gain in efficiency can only be obtained by taking m \ll n_{eq} , in which case accuracy may suffer.
If applied only to [K], Eq. 13.7-3 becomes the static condensation algorithm, Eq. 8.1-3. There are computational advantages to generating [T] in terms of flexibility instead of stiffness [13.28,13.31,13.32].
We see that [T] plays the same role as the matrix [W]_{j} of Ritz vectors in Eqs. 13.6-19. Indeed, [T] can be regarded as a matrix of Ritz vectors and Guyan reduction as a Rayleigh–Ritz method. However, [T] and [W]_{j} are not the same: [T] is obtained more efficiently, without reference to the applied loading, but is not mass-matrix-orthogonal. [W]_{j} is more adept at reproducing quasistatic structure deflection shapes, and since it is obtained from the applied loading, it can usually have fewer columns than [T] for a given level of accuracy.
If [M] is lumped so that \left[M_{mm}\right] contains all the nonzero masses, and [M_{ss}] is null, then Eq. 13.7-2 follows without the necessity of our principal assumption. This suggests another approach to condensation: the analyst can lump mass at only the d.o.f. to be retained as masters. However, this approach requires experience and is less accurate than condensation with a more populated mass matrix.
In going from Eq. 13.7-1 to Eq. 13.7-4, eigenvalues are raised because constraints are imposed [13.33]. This behavior, and the considerable accuracy possible when m \ll n_{\mathrm{eq}} , are shown in Fig. 13.7-1. The higher eigenvalues of the original system are absent from the reduced system because several d.o.f. have been discarded. After \{\overline{\mathbf{D}}_s\} has been recovered by use of Eq. 13.7-7, an eigenvalue can be improved by substituting the eigenvector \{\mathbf{D}\} = \{\overline{\mathbf{D}}_m, \overline{\mathbf{D}}_s\} into the Rayleigh quotient of the full system, Eq. 13.5-4.
Master d.o.f. should be those for which inertia is most important. Such d.o.f. have a large mass-to-stiffness ratio. Thus rotational d.o.f. rarely appear as masters. A master d.o.f. should be retained at each node that carries a time-varying applied load or has a time-varying prescribed displacement. Master d.o.f. should not be clustered in one area of the mesh. If they are, some vibration modes may be almost linearly dependent. This is not of major concern, but, if ignored, can lead to the disconcerting appearance of negative eigenvalues in the highest modes of the reduced eigenproblem [13.35] .
The selection of master and slave d.o.f. can be automated as follows [13.36]. Diagonal coefficients of [K] and [M] are scanned, and the d.o.f. i for which K_{ii} / M_{ii}
natural_image
Grid pattern with diagonal lines and circular markers, no text or symbols present
Full system, 90 d.o.f. \omega_{1} = 3.469
(one displacement \omega_{2} = 8.535
and two rotations \omega_{3} = 21.450
at each node) \omega_{4} = 27.059
Reduced system, 6 mas- \omega_{1} = 3.473
ter d.o.f. (lateral \omega_{2} = 8.604
displacements at \omega_{3} = 22.690
nodes circled) \omega_{4} = 29.490
Figure 13.7-1. First four vibration frequencies of a thin, square cantilever plate [13.37]. The analysis uses triangular plate elements and consistent mass matrices.
is largest is selected as the first slave. In case of a tie, the first d.o.f. encountered is taken as a slave. Then [K] and [M] are condensed (by one order). The condensed matrices are now scanned, the largest K_{rii}/M_{rii} is selected as the next slave, and another condensation is performed. This process repeats until a user-specified number of d.o.f. remain. These are the masters, chosen in a near-optimal way.
The number of masters can be chosen automatically by specifying a cut-off frequency, \omega_{c} [13.44]. This frequency is taken to be about three times the highest frequency of interest in the excitation and/or structural response. Modes of the structure having frequencies greater than \omega_{c} are quasistatic with respect to the excitation and can therefore be neglected in the dynamic response, although they may participate in the quasistatic response. Automatic selection of masters and slaves proceeds as described above except that condensation is terminated when the largest ratio K_{rii}/M_{rii} is less than \omega_{c}^{2} . Another procedure that may be more efficient is given in [13.38].
One may combine manual selection of some master d.o.f. with automatic selection of the rest. A motivation would be to retain as masters those d.o.f. subjected to time-varying loads. It would be very inconvenient to eliminate these d.o.f. as slaves.
The number of master d.o.f., m, may be as low as 1/10 or 1/20 the total number of d.o.f. If there are two or three times as many master d.o.f. as eigenvalues to be computed, the highest computed eigenvalue may err by less than 10%. These estimates are rough and strongly problem-dependent [13.36,13.37].
Mass condensation yields good results if the choice of master d.o.f. is good and if the ratio of master d.o.f. to total d.o.f. (m/n_{\mathrm{eq}}) is not too small. As alternatives, the subspace iteration and Lanczos methods of eigenproblem solution employ condensation techniques that may be more reliable than Eq. 13.7-3.
Example: Beam Vibration. Nonzero d.o.f. in the problem of Fig. 13.7-2 are w_{1} and \theta_{2} . Accordingly, using the stiffness and mass matrices of Eqs. 4.2-5 and 13.3-2, respectively, we obtain the eigenproblem
\left(\frac {E I}{L ^ {3}} \left[ \begin{array}{l l} 1 2 & 6 L \\ 6 L & 4 L ^ {2} \end{array} \right] - \frac {\omega^ {2} m}{4 2 0} \left[ \begin{array}{c c} 1 5 6 & - 1 3 L \\ - 1 3 L & 4 L ^ {2} \end{array} \right]\right) \left\{ \begin{array}{l} \overline {{w}} _ {1} \\ \overline {{\theta}} _ {2} \end{array} \right\} = \left\{ \begin{array}{l} 0 \\ 0 \end{array} \right\} \tag {13.7-8}
where m = \rho AL is the mass of the beam. The eigenvalues of this two-d.o.f. system are
\omega_ {1} ^ {2} = 6. 1 3 6 2 \frac {E I}{m L ^ {3}} \quad \text { and } \quad \omega_ {2} ^ {2} = 7 5 8. 1 7 \frac {E I}{m L ^ {3}} \tag {13.7-9}
From continuous beam theory, the two lowest frequencies are
\omega_ {1} ^ {2} = 6. 0 8 8 1 \frac {E I}{m L ^ {3}} \quad \text { and } \quad \omega_ {2} ^ {2} = 4 9 3. 1 3 \frac {E I}{m L ^ {3}} \tag {13.7-10}
text_image
w₁ θ₂ 1 2 L
Figure 13.7-2. A uniform beam. The left end is allowed to displace but not to rotate. The right end is simply supported.