김경종
32 KiB
Examples. Consider the uniform bar element shown in Fig. 13.3-1a. Shape functions to be used in Eq. 13.2-5 are given by Eq. 3.8-4 with s replaced by x. Mass increment \rho \, dV in Eq. 13.2-5 can be written as (m/L) \, dx , where m = \rho AL is the total mass of the element. The consistent and lumped mass matrices are, respectively,
[ \mathbf {m} ] = \frac {m}{6} \left[ \begin{array}{l l l l} 2 & 0 & 1 & 0 \\ 0 & 2 & 0 & 1 \\ 1 & 0 & 2 & 0 \\ 0 & 1 & 0 & 2 \end{array} \right] \quad [ \mathbf {m} ] = \frac {m}{2} \left[ \begin{array}{l l l l} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right] \tag {13.3-1}
The lumped mass is obtained by placing half of the total element mass m as a particle at each node. The particle mass m/2 appears four times in [m] because four nodal acceleration vectors are resisted by inertia.
For the uniform beam element of Fig. 13.3-1b, shape functions are given in Fig. 3.13-2, and Eq. 13.2-5 yields the consistent mass matrix
[ \mathbf {m} ] = \frac {m}{4 2 0} \left[ \begin{array}{c c c c} 1 5 6 & 2 2 L & 5 4 & - 1 3 L \\ 2 2 L & 4 L ^ {2} & 1 3 L & - 3 L ^ {2} \\ 5 4 & 1 3 L & 1 5 6 & - 2 2 L \\ - 1 3 L & - 3 L ^ {2} & - 2 2 L & 4 L ^ {2} \end{array} \right] \tag {13.3-2}
where m = \rho AL is the total element mass. The (diagonal) lumped mass matrix of the beam is given by
[ \mathbf {m} ] = \frac {m}{2} \left[ \begin{array}{l l l l} 1 & \alpha L ^ {2} / 2 1 0 & 1 & \alpha L ^ {2} / 2 1 0 \end{array} \right] \tag {13.3-3}
where the second and fourth diagonal terms account for rotary inertia. Sometimes rotary inertia is neglected ( \alpha = 0 ). If included, it is often selected as the mass moment of inertia I of a uniform slender bar of length L/2 and mass m/2 = \rho AL/2 spinning about one end; that is, I = (m/2)(L/2)^{2}/3 , for which \alpha = 17.5 . If an element is tapered or has nonuniform density, its mass and stiffness matrices change.
A plane frame element has three d.o.f. per node. Its mass matrix is formed by expanding and then combining the bar and beam mass matrices. An element arbitrarily oriented in xy coordinates requires the coordinate transformation described in Section 7.4.
Further examples appear in Fig. 13.3-2.
Equations 13.3-1 and 13.3-3 illustrate ad hoc mass matrix lumping that is guided by intuition and physical insight. The lumped mass matrices obtained are effective
text_image
L 2 w₁ u₁ 1 x L w₂ u₂
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L 2 w₁ w₂ θ₂ 1 2 θ₁ x L
Figure 13.3-1. Uniform bar and beam elements with the respective \{d\} vectors \left[u_{1} \quad w_{1} \quad u_{2} \quad w_{2}\right]^{T} and \left[w_{1} \quad \theta_{1} \quad w_{2} \quad \theta_{2}\right]^{T} . Dashed lines show the conceptual shapes and tributary lengths used to obtain the lumped-mass coefficients associated with d.o.f. at node 1.
Constant-strain triangle. With a linear displacement field in each direction, and \{d\} = \left[u_{1} \quad u_{2} \quad u_{3} \quad v_{1} \quad v_{2} \quad v_{3} \quad w_{1} \quad w_{2} \quad w_{3}\right]^{T} ,
[ \mathbf {m} ] _ {9 \times 9} = [ \mathbf {Q} \quad \mathbf {Q} \quad \mathbf {Q} ], \quad \text { where } \quad [ \mathbf {Q} ] = \frac {\rho A t}{1 2} \left[ \begin{array}{l l l} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2 \end{array} \right]
Bilinear rectangle. With a bilinear displacement field in each direction, and \{d\} = \left[u_{1} \quad u_{2} \quad u_{3} \quad u_{4} \quad v_{1} \quad v_{2} \quad v_{3} \quad v_{4} \quad w_{1} \quad w_{2} \quad w_{3} \quad w_{4}\right]^{T} ,
\left[ \begin{array}{l l} 4 & 3 \\ 1 & 2 \end{array} \right] \quad [ \mathrm{m} ] _ {1 2 \times 1 2} = [ \mathrm{Q} \quad \mathrm{Q} \quad \mathrm{Q} ], \quad \text { where } \quad [ \mathrm{Q} ] = \frac {\rho A t}{3 6} \left[ \begin{array}{l l l l} 4 & 2 & 1 & 2 \\ 2 & 4 & 2 & 1 \\ 1 & 2 & 4 & 2 \\ 2 & 1 & 2 & 4 \end{array} \right]
Figure 13.3-2. Consistent mass matrices for plane elements allowed to move in three dimensions. A = surface area, \rho = uniform mass density, and t = uniform thickness.
and widely used. However, for higher-order elements (e.g., quadratic-displacement plane elements) or elements of irregular shape, intuition can be risky. Accordingly, systematic schemes for lumping are necessary.
HRZ Lumping Scheme. The HRZ scheme [13.5,13.6] is an effective method for producing a diagonal mass matrix. It can be recommended for arbitrary elements. The idea is to use only the diagonal terms of the consistent mass matrix, but to scale them in such a way that the total mass of the element is preserved. Specifically, the procedural steps are as follows.
- Compute only the diagonal coefficients of the consistent mass matrix.
- Compute the total mass of the element, m.
- Compute a number
sby adding the diagonal coefficientsm_{ii}associated with translational d.o.f. (but not rotational d.o.f., if any) that are mutually parallel and in the same direction. - Scale all the diagonal coefficients by multiplying them by the ratio
m / s, thus preserving the total mass of the element.
As examples, for the bar and beam elements shown in Fig. 13.3-1, the preceding four steps yield, respectively, the diagonal matrices
[ \mathbf {m} ] = \frac {m}{2} \left[ \begin{array}{l l l l} 1 & 1 & 1 & 1 \end{array} \right] \tag {13.3-4}
[ \mathbf {m} ] = \frac {m}{7 8} \left[ \begin{array}{l l l l} 3 9 & L ^ {2} & 3 9 & L ^ {2} \end{array} \right] \tag {13.3-5}
Further examples appear in Fig. 13.3-3.
Test cases to date show that for flexural and low-order finite elements, the
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1/36 (3/76) 8/36 (16/76) Serendipity element (a)
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1/36 (1/36) • 16/36 (16/36) 4/36 (4/36) Lagrange element (b)
Figure 13.3-3. Diagonal mass matrices of plane rectangular elements obtained using the four-step HRZ scheme [13.5]. Each element has constant thickness. Numbers shown are fractions of the total element mass at each node. First number: based on 2 by 2 Gauss quadrature. Second number (in parentheses): based on 3 by 3 Gauss quadrature. (a) Serendipity element. (b) Lagrange element.
accuracy of this form of diagonal mass matrix is excellent, often surpassing that of the consistent mass matrix (see Table 13.3-1). For higher-order plane elements, such as the six-node quadratic triangle and the eight-node serendipity quadrilateral, this scheme may be less accurate for transient analysis than an optimally lumped mass matrix [13.7,13.8] .
Optimal Lumping. Mass lumping can be thought of as the result of applying an appropriate quadrature rule to evaluate \int\rho[N]^{T}[N]dV . If the integration points of a quadrature rule coincide with nodal locations of an element having translational d.o.f. only, then no off-diagonal terms are generated and the mass matrix is diagonal. If the element also has rotational d.o.f., then lumping by quadrature produces block-diagonal matrices that are of lesser practical usefulness because they are not diagonal. In the sequel, we shall be concerned with lumping by quadrature for elements with translational d.o.f. only.
Let p be the degree of the highest-degree complete polynomial in [N] and m the highest-order derivative in the strain energy expression (e.g., m = 1 for elasticity and m = 2 for bending). Fix [13.9] has shown that a quadrature rule
TABLE 13.3-1. PERCENTAGE ERRORS OF COMPUTED NATURAL FREQUENCIES OF A SIMPLY SUPPORTED THICK SQUARE PLATE [13.5]. HALF THE PLATE WAS MODELED BY A 4 BY 2 MESH OF EIGHT-NODE 24-D.O.F. ELEMENTS (TABLE 11.3-1). THE AD HOC LUMPED MASS MATRIX HAS EQUAL MASS PARTICLES AT EACH NODE. CONSISTENT MASS MATRIX RESULTS DO NOT GUARANTEE AN UPPER BOUND BECAUSE THE STIFFNESS MATRIX IS BASED ON REDUCED INTEGRATION.
| Mode | Type of Mass Matrix Used | |||
| m | n | Consistent (%) | HRZ Lumping (%) | Ad Hoc Lumping (%) |
| 1 | 1 | -0.11 | +0.32 | +0.32 |
| 2 | 1 | -0.40 | +0.45 | -0.45 |
| 2 | 2 | -0.35 | -2.75 | -4.12 |
| 3 | 1 | +5.18 | +0.05 | -5.75 |
| 3 | 2 | +4.68 | -2.96 | -10.15 |
| 3 | 3 | +13.78 | -5.18 | -19.42 |
| 4 | 2 | +16.88 | +1.53 | +31.70 |
with degree of precision at least 2(p - m) will yield comparable accuracy and cause no loss of convergence rate beyond what is already inherent in the finite element method with consistent mass matrices [13.68]. Diagonal mass matrices integrated in this way are said to be optimally lumped.
As an example of optimal lumping, consider the three-node quadratic-displacement bar element shown in Fig. 6.2-1, for which p = 2 and m = 1. Let the mass density \rho and cross-sectional area A be uniform and the nodes uniformly spaced. The minimum order of integration for optimal mass matrix lumping is n = 2(2 - 1) = 2 . An integration rule having at least this degree of precision and having integration points at nodal locations is the three-point Newton–Cotes formula [13.12], which is identical to Simpson's rule. (This integration rule will exactly integrate a cubic polynomial.)
\int_ {a} ^ {b} f (x) d x = (b - a) \left[ \frac {1}{6} f (x = a) + \frac {1}{6} f (x = (a + b) / 2) + \frac {1}{6} f (x = b) \right] \tag {13.3-6}
From Eq. 13.2-5, a single term of the mass matrix is
m _ {i j} = \rho A \int N _ {i} N _ {j} d x = \rho A \int_ {- 1} ^ {1} N _ {i} N _ {j} J d \xi \tag {13.3-7}
The Jacobian J is J = L / 2 , so Eq. 13.3-6 yields
\begin{array}{l} m _ {i j} = \rho A L [ \frac {1}{6} N _ {i} (\xi = - 1) N _ {j} (\xi = - 1) \\ + \frac {4}{6} N _ {i} (\xi = 0) N _ {j} (\xi = 0) + \frac {1}{6} N _ {i} (\xi = 1) N _ {j} (\xi = 1) ] \tag {13.3-8} \\ \end{array}
Thus m_{ij} = 0 for i \neq j , and, with node 3 at the middle of the bar,
[ \mathbf {m} ] = \frac {\rho A L}{6} \left[ \begin{array}{l l l} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{array} \right] \tag {13.3-9}
Equation 13.3-9 also agrees with the mass matrix obtained by HRZ lumping. In addition, the portion of element mass allocated to each d.o.f. is the same as the nodal allocation of a uniform traction load on the edge of a quadratic element as shown by Fig. 4.3-4.
Nodes of Lagrangian elements coincide with integration points of the Lobatto quadrature rule [13.10]. Lobatto integration weights are always positive, hence optimal lumping for Lagrangian elements results in positive definite diagonal mass matrices; that is, each node in the element has a positive mass lump associated with it. Results for the quadratic Lagrange element are shown in Fig. 13.3-4. Additional results appear in [13.7]. In cubic and higher-order Lagrange elements, nodal masses are positive but nodes in the reference element must be at special positions. In general it is not possible to construct an integration rule of a required accuracy that simultaneously permits arbitrary specification of integration point locations and has positive weighting. Thus optimally lumped diagonal mass matrices for triangular and serendipity quadrilateral elements (particularly the quadratic and higher-order elements) frequently have some zero or negative nodal masses as shown in Fig. 13.3-4.
For low-order elements, such as the linear-displacement bar, the constant-strain
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0 A/3 -A/12 A/3 4A/9 A/36 A/9
Figure 13.3-4. Optimal mass matrix lumping for some common two-dimensional elements [13.7]. The triangular element is an equilateral reference element in area coordinates. Rectangular elements are isoparametric reference elements. Elements are uniform and have mass proportional to element area A.
triangle, and the bilinear quadrilateral, ad hoc lumping usually gives the same result as optimal lumping. Also, for the quadratic Lagrange element, HRZ lumping and optimal lumping produce the same diagonal mass matrix. For cubic and higher-order Lagrange elements, HRZ lumping and optimal lumping may be different; however, these elements are rarely used in dynamics. For quadratic and higher-order triangular and serendipity quadrilateral elements, HRZ lumping and optimal lumping are markedly different: numerical tests show that the displacement, velocity, and acceleration time-history results for HRZ lumped models can be less accurate than for optimally lumped models in some problems [13.7, 13.8] .
Remarks. With any mass matrix, the product [m]\{\ddot{d}\} or [m]\{\ddot{d}\} must yield the correct total force on an element according to Newton's law F = ma when \{\ddot{d}\} represents a rigid-body translational acceleration. The rationale is that for convergence to correct results, this kind of motion must be correctly represented because it is the only motion experienced by an element when a mesh is indefinitely refined.
Consistent mass matrices [m] and [M] are positive definite (i.e., the kinetic energy \frac{1}{2}\{\dot{d}\}^{T}[m]\{\dot{d}\}>0 for all \{\dot{d}\}\neq\{\mathbf{0}\} ). A lumped mass matrix is positive semi-definite or indefinite if zero or negative masses, respectively, appear on the diagonal. The zeros may or may not make some operations awkward, depending on the algorithm, and negative masses usually (but not always [13.7]) require some special treatment.
If the mesh layout correctly represents the structure volume, elements are compatible and not softened by low-order integration rules, and mass matrices are consistent, then computed natural frequencies are upper bounds to the exact values. If any of these restrictions is violated, such as by the use of any lumping scheme, a bound cannot be guaranteed [13.11]. The upper bound is computed with an error of order h^{2(q-1)} , where h and q are defined in Section 18.6 [13.13]. The upper-bound property is illustrated by the numerical example in Section 13.5.
We cannot say that either lumped or consistent mass matrices are best for all problems. Consistent matrices are more accurate for flexural problems, such as beams and shells, but negligibly so if the wavelength of the mode spans more than about four elements [13.14]. Lumped matrices usually yield natural frequencies that are less than the exact values. For a bar element, MacNeal [13.15] finds that [m] and [m] of Eq. 13.3-1 yield natural frequency errors of order h^{2} in opposite directions and that an [m] that is the average of the two yields a natural frequency error of only order h^{4} . Similar improvement is possible with beam elements [13.46].
In wave propagation problems using linear-displacement field elements, lumped masses give greater accuracy because of fewer spurious oscillations. For higher-order elements, diagonal mass matrices obtained by ad hoc lumping may be deficient in accuracy compared to optimal lumping.
As for efficiency, lumped mass matrices are simpler to form, occupy less storage space, and require less computational effort. Indeed, some methods of dynamic analysis are practicable only with lumped mass matrices. Usually it is more important to have a diagonal mass matrix in time-history analysis than in vibration analysis, as time-history analysis is usually much more expensive.
A practitioner contemplating the use of lumped matrices for higher-order elements having translational d.o.f. only has basically four choices: (1) use lower-order elements instead; (2) use Lagrangian elements with optimal lumping, which always results in positive nodal masses; (3) use non-Lagrangian elements with optimal lumping, which may result in a nonpositive definite mass matrix with diagonal coefficients that can be positive, zero, or negative; or (4) use ad hoc lumping which sacrifices optimal accuracy but maintains positive nodal masses. While option 3 is an accurate and workable computational alternative, the most expedient analyses will result from options 1 and 2. Option 4 may give inaccurate results and should be avoided.
13.4 DAMPING
Damping in structures is not viscous; rather, it is due to mechanisms such as hysteresis in the material and slip in connections. These mechanisms are not well understood. Moreover, they are awkward to incorporate into the equations of structural dynamics, or they make the equations computationally difficult. Therefore, the actual damping mechanism is usually approximated by viscous damping. Comparisons of theory and experiment show that this approach is sufficiently accurate in most cases.
The treatment of damping in computational analyses can be categorized as (1) phenomenological damping methods, in which the actual physical dissipative mechanisms such as elastic–plastic hysteresis loss, structural joint friction, or material microcracking are modeled, or (2) spectral damping methods, in which viscous damping is introduced by means of specified fractions of critical damping [13.16] . (Critical damping, for which the damping ratio is \xi = 1 , marks the transition between oscillatory and nonoscillatory response.) Phenomenological methods require detailed models for the dissipative mechanisms and almost always result in nonlinear analyses; hence, they are seldom used. With spectral damping approaches, experimental observations of the vibratory response of structures are used to assign a fraction of critical damping as a function of frequency, or more commonly, a single damping fraction for the entire frequency range of a structure [13.16] . The damping ratio \xi depends on the material and the stress level. In steel piping, \xi ranges from about 0.5% at low stress levels to about 5% at high stress levels. In bolted or riveted steel structures, and in reinforced or prestressed concrete, \xi has the approximate range 2% to 15%.
A popular spectral damping scheme, called Rayleigh or proportional damping, is to form damping matrix [C] as a linear combination of the stiffness and mass matrices, that is,
[ \mathbf {C} ] = \alpha [ \mathbf {K} ] + \beta [ \mathbf {M} ] \tag {13.4-1}
where \alpha and \beta are called, respectively, the stiffness and mass proportional damping constants. Matrix [C] given by Eq. 13.4-1 is an orthogonal damping matrix because it permits modes to be uncoupled by eigenvectors associated with the undamped eigenproblem (Section 13.6). The relationship between \alpha , \beta , and the fraction of critical damping \xi at frequency \omega is given by the following equation, proof of which is left as an exercise:
\xi = \frac {1}{2} \left(\alpha \omega + \frac {\beta}{\omega}\right) \tag {13.4-2}
Damping constants \alpha and \beta are determined by choosing the fractions of critical damping ( \xi_{1} and \xi_{2} ) at two different frequencies ( \omega_{1} and \omega_{2} ) and solving simultaneous equations for \alpha and \beta . Thus
\alpha = 2 \left(\xi_ {2} \omega_ {2} - \xi_ {1} \omega_ {1}\right) / \left(\omega_ {2} ^ {2} - \omega_ {1} ^ {2}\right) \tag {13.4-3a}
\beta = 2 \omega_ {1} \omega_ {2} (\xi_ {1} \omega_ {2} - \xi_ {2} \omega_ {1}) / (\omega_ {2} ^ {2} - \omega_ {1} ^ {2}) \tag {13.4-3b}
Shown in Fig. 13.4-1 is the fraction of critical damping versus frequency. Damping attributable to \alpha[K] increases with increasing frequency, whereas damping attributable to \beta[M] increases with decreasing frequency. For structures that may have rigid-body motion, it is important that the mass-proportional damping not be excessive. Positive values of \beta less than about 0.1 per time unit are usually acceptable [13.17].
Usually, \omega_{1} and \omega_{2} are chosen to bound the design spectrum. Thus \omega_{1} is taken as the lowest natural frequency of the structure, and \omega_{2} is the maximum frequency of interest in the loading or response. For example, in seismic analyses, 30 Hz is often used as the upper frequency because the spectral content of seismic design spectra are insignificant above that frequency.
More general proportional damping schemes are possible in which the fraction
line
| Frequency | Stiffness-proportional damping: ξ = αω/2, β = 0 | Mass-proportional damping: ξ = β/2ω, α = 0 | | --------- | ----------------------------------------------- | ------------------------------------------ | | ω₁ | ξ₁ | ξ₂ | | ω₂ | ξ₂ | ξ₂ |Figure 13.4-1. Fraction of critical damping versus frequency for Rayleigh damping. Contribution of stiffness and mass proportional damping to total damping is also shown.
of critical damping at any desired number of frequencies can be specified [13.18]. However, these schemes usually produce fully populated damping matrices, and hence are rarely used in practice.
13.5 NATURAL FREQUENCIES AND MODE SHAPES
The Eigenvalue Problem. An undamped structure, with no external loads applied to unrestrained d.o.f., undergoes harmonic motion (caused perhaps by initial conditions) in which each d.o.f. moves in phase with all other d.o.f. Thus
\{\mathbf {D} \} = \{\overline {{\mathbf {D}}} \} \sin \omega t \quad \text { and } \quad \{\ddot {\mathbf {D}} \} = - \omega^ {2} \{\overline {{\mathbf {D}}} \} \sin \omega t \tag {13.5-1}
where \{\overline{D}\}= amplitudes of nodal d.o.f. vibration and \omega=circular\ frequency (radians per second). The cyclic frequency (in Hertz) is f=\omega/2\pi and the period is T=1/f (seconds). Both \omega and f are called simply “frequency” and pertain to undamped motion unless stated otherwise.
Combining Eqs. 13.5-1 with Eq. 13.2-12, and with [C] and \{\mathbf{R}^{\mathrm{ext}}\} both zero, we obtain
([ \mathbf {K} ] - \lambda [ \mathbf {M} ]) \{\overline {{\mathbf {D}}} \} = \{\mathbf {0} \}, \quad \text { where } \quad \lambda = \omega^ {2} \tag {13.5-2}
This is the basic statement of the vibration problem. Equation 13.5-2 is called a generalized eigenproblem or simply an eigenproblem. When the matrix [K] - \lambda[M] is nonsingular, Eq. 13.5-2 has only the trivial solution \{\overline{D}\} = \{0\} . We are interested in nontrivial solutions and hence wish to determine the eigenvalues (or characteristic numbers, or latent roots) \lambda that satisfy
\det ([ \mathbf {K} ] - \lambda [ \mathbf {M} ]) = 0 \tag {13.5-3}
Associated with each eigenvalue \lambda_{i} is an eigenvector \{\overline{D}\}_{i} , which is sometimes called a normal (or natural, or characteristic, or principal) mode. The lowest nonzero \omega_{i} is called the fundamental vibration frequency. Appendix C lists some important properties of eigenvalues and eigenvectors such as orthogonality and linear independence.
If [K] and [M] are n_{\mathrm{eq}} by n_{\mathrm{eq}} matrices, then, under conditions usually satisfied in structural analysis, Eq. 13.5-2 has n_{\mathrm{eq}} eigenvalues and n_{\mathrm{eq}} eigenvectors (see Appendix C). All eigenvalues are positive if [K] and [M] are both positive definite, as is usually the case when [K] has all rigid-body modes constrained and when [M] is either a consistent mass matrix or a lumped mass matrix with strictly positive diagonal coefficients. A partly or completely unsupported structure has positive semidefinite [K] and has one zero eigenvalue associated with each possible rigid-body motion. When [M] is lumped—that is, a diagonal matrix—some of its coefficients M_{ii} may be zero (which is commonly the case if rotational d.o.f. are present in \{\overline{\mathbf{D}}\} but not associated with rotary inertia) or negative (which is a rare situation that is due to optimal lumping). Typically each zero M_{ii} is associated with an infinite eigenvalue, and each negative M_{ii} is associated with a negative eigenvalue, which gives rise to an imaginary frequency. Some algorithms for eigenvalue extraction require positive definite mass matrices and hence will not
work for lumped matrices having some zero or negative diagonal coefficients. When there are negative diagonal coefficients, the reader is referred to algorithms described in Appendix C and in [13.7]. When a diagonal coefficient M_{ii} is zero, the associated d.o.f. can be removed from \{\overline{D}\} by condensation prior to eigenanalysis.
Rayleigh Quotient. Let [K] be symmetric and [M] be positive definite and symmetric. If we premultiply Eq. 13.5-2 by \{\overline{D}\}^{T} and solve for \lambda , we obtain the Rayleigh quotient
\lambda = \frac {\{\overline {{{\mathbf {D}}}} \} ^ {T} [ \mathbf {K} ] \{\overline {{{\mathbf {D}}}} \}}{\{\overline {{{\mathbf {D}}}} \} ^ {T} [ \mathbf {M} ] \{\overline {{{\mathbf {D}}}} \}} \tag {13.5-4}
If \{\overline{D}\} approximates the ith eigenvector with first-order error, then \lambda approximates the corresponding eigenvalue with second-order error. Thus a casual estimate of \{\overline{D}\} may result in an accurate estimate of \lambda . The Rayleigh quotient is, in fact, an extreme value when \{\overline{D}\} varies in the neighborhood of an exact eigenvector [13.18,13.19]; accordingly, the extraction of an eigenvalue can be approached as an optimization problem. Values of the Rayleigh quotient are bounded by the largest and smallest eigenvalues of the mesh. That is, for any vector \{v\} ,
\lambda_ {\min} \leq \frac {\{\mathbf {v} \} ^ {T} [ \mathbf {K} ] \{\mathbf {v} \}}{\{\mathbf {v} \} ^ {T} [ \mathbf {M} ] \{\mathbf {v} \}} \leq \lambda_ {\max} \tag {13.5-5}
where \lambda_{min} and \lambda_{max} are the smallest and largest eigenvalues of Eq. 13.5-2. Equation 13.5-5 presumes that [M] is positive definite. If [M] is indefinite, as may result from optimal lumping or lumping with zero rotary inertia, the Rayleigh quotient may be positive or negative infinity for some choices of \{v\} .
In certain direct integration algorithms, it is necessary to have prior knowledge of the largest eigenvalue of the mesh, \lambda_{max} . An upper bound to \lambda_{max} can be obtained by considering the eigenproblem for a single, unsupported element
\det ([ \mathbf {k} ] - \lambda^ {\prime} [ \mathbf {m} ]) = 0 \tag {13.5-6}
The largest eigenvalue of Eq. 13.5-6 can often be obtained by hand calculation. If we denote the largest eigenvalue of Eq. 13.5-6 among all elements by \lambda_{\mathrm{max}}^{\prime} , then
\lambda_ {\max} \leq \lambda_ {\max} ^ {\prime} \tag {13.5-7}
The reason is that the largest eigenvalue is a maximum of the Rayleigh quotient, Eq. 13.5-5. Suppose we consider an alternative Rayleigh quotient instead, in which no constraints of nodal-value sharing are imposed. Then the resulting maximum eigenvalue would be \lambda_{max}^{\prime} of Eq. 13.5-6. The actual \lambda_{max} could be obtained from the alternative Rayleigh quotient by imposing a large number of linear constraints that specify the nodal-value sharing of the original mesh. Optimization theory shows that imposing such constraints can only lower a maximum, and thus \lambda_{max} \leq \lambda_{max}^{\prime} . Applying the same argument to \lambda_{min} shows that \lambda_{min} \geq \lambda_{min}^{\prime} , which is consistent with our physical intuition that imposing constraints raises the fundamental frequency.
Example. We consider an elementary example of a matrix eigenproblem and its solution. Consider the uniform one-dimensional unsupported bar with mass density \rho , elastic modulus E, and cross-sectional area A shown in Fig. 13.5-1. With consistent [m], Eq. 13.5-2 becomes
\left(\frac {A E}{L} \left[ \begin{array}{c c} 1 & - 1 \\ - 1 & 1 \end{array} \right] - \omega^ {2} \frac {\rho A L}{6} \left[ \begin{array}{c c} 2 & 1 \\ 1 & 2 \end{array} \right]\right) \left\{ \begin{array}{l} \overline {{u}} _ {1} \\ \overline {{u}} _ {2} \end{array} \right\} = \left\{ \begin{array}{l} 0 \\ 0 \end{array} \right\} \tag {13.5-8}
For nontrivial amplitudes \{\overline{d}\} = \left[\overline{u}_{1}, \overline{u}_{2}\right]^{T} to exist, the determinant of the expression in parentheses must vanish. Thus
\omega^ {2} (\omega^ {2} \rho A L - 1 2 A E / L) = 0 \tag {13.5-9}
from which \omega_{1}=0 and \omega_{2}=(2/L)\sqrt{3E/\rho}=(3.464/L)\sqrt{E/\rho} . The easiest way to determine the eigenvector associated with any eigenvalue \lambda_{i} is to set one d.o.f. in the eigenvector \{\overline{d}\}_{i} to an arbitrary nonzero number (\overline{d}_{1}=1 , for example), substitute the known value of \lambda_{i} , and solve for the remaining amplitudes in \{\overline{d}\}_{i} . (If, by coincidence, the d.o.f. amplitude that was assumed to be nonzero is in fact zero, then the resulting system of equations will be singular and no solution will exist for the remaining amplitudes. Then the eigenvector can usually be found by assuming a nonzero value for one of the other amplitudes.) Thus, from Eq. 13.5-8,
\text { for } \omega_ {1} = 0 \quad \{\overline {{\mathbf {d}}} \} _ {1} = \left\lfloor 1 - 1 \right\rfloor^ {T} \tag {13.5-10a}
\text { for } \omega_ {2} = (3. 4 6 4 / L) \sqrt {E / \rho} \quad \{\overline {{{\mathbf {d}}}} \} _ {2} = \left\lfloor 1 - 1 \right\rfloor^ {T} \tag {13.5-10b}
The first eigenvector describes a rigid-body translation in the x direction. The second describes an axial straining mode (for which the exact fundamental frequency of a continuous unsupported bar of length L is (\pi/L)\sqrt{E/\rho} ). Hence, the single-element consistent-mass model overpredicts the exact fundamental frequency by about 10%, thus illustrating the upper-bound property noted in Section 13.3.
If, instead of the consistent mass matrix, the lumped mass matrix [m] = (\rho AL/2)[1 \_ 1] is used in Eq. 13.5-4, the computed frequencies are \omega_{1} = 0 and \omega_{2} = (2/L)\sqrt{E/\rho} . Thus we see that the upper-bound property is destroyed by lumping. Eigenvectors for the lumped-mass case are the same as those of Eqs. 13.5-10.
In many design situations, we wish to know if severe dynamic excitation of a structure is likely. Therefore we compare the frequency spectrum of the structure with that of the time-dependent loading. If a natural frequency of the structure is
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u₁ 1 x L (a) 2 u₂
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Exact and one element 1.0 ω₁ = 0 Exact 0 L x One element ω₂ > 0 (b)
Figure 13.5-1. (a) Unsupported two-d.o.f. uniform bar. (b) Vibration modes \omega_{1}=0 (rigid-body translation) and \omega_{2}>0 (axial straining mode).