Mass Templates for Bar2 Elements

Transcription

1 22 Mass Templates for Bar2 Elements 22 1

2 Chapter 22: MASS TEMPLATES FOR BAR2 ELEMENTS TABLE OF CONTENTS Page 22.1 The Two-Node Bar Element Bar2 Entry Weighted Template Bar2 One Parameter Mass Template Bar2 Alternative Parametrization Bar2 Angular Momentum Conservation Bar2 Fourier Analysis Bar2 Dispersion Diagrams Best µ By Low Frequency Fitting Folding Frequency Bar2 Test: Vibrations of a Fied-Free Bar Member Other Customization Options Bar2 Frequency Dependent Mass Bar2 Frequency Dependent Mass-Stiffness Pair Bar2 Frequency Dependent Mass Instances

3 22.1 THE TWO-NODE BAR ELEMENT The Two-Node Bar Element The template approach is best grasped through an eample that involves the simplest nontrivial structural finite element: a two-node prismatic bar of mass density ρ, area A and length l, that can only move along the longitudinal ais. See Figure 22.1(a). This element is often acronymed Bar2 for brevity s sake. The well known consistent and diagonally-lumped mass matri forms are M e C = me 6 [ ], M e L = me 2 [ 1 1 in which m e = ρ Al is the total element mass. These are derived in Bar2 Entry Weighted Template ]. (22.1) The most general mass matri form for Bar2 is the entry-weighted template [ ] [ ] [ ] M e M e = 11 M12 e µ11 µ M21 e M22 e = ρ A l 12 = m e µ11 µ 12. (22.2) µ 21 µ 22 µ 21 µ 22 The first form is merely a list of entries. Net the element mass m e = ρ A l is factored out. The emerging parameters µ 11 through µ 22 are numbers, which illustrates a general rule: template free parameters should be dimensionless. This simplifies analysis and implementation. To cut down on parameters one looks at configuration constraints. The most obvious ones are: Matri symmetry: M e = (M e ) T. For the epression (22.2) this requires µ 21 = µ 12. Physical symmetry: For a prismatic bar, M e in (22.2) must ehibit antidiagonal symmetry: µ 22 = µ 11. Conservation of total translational mass: same as conservation of linear momentum or of kinetic energy. Apply the uniform velocity field u = v to the bar. The associated nodal velocity vector is u e = v e = v [ 11] T. The kinetic energy is T e = 1 2 (ve ) T M e v e = 1 2 me v 2 (µ 11 + µ 12 + ν 2 + µ 22 ). This must equal 1 2 me v 2, whence µ 11 + µ 12 + µ 21 + µ 22 = 2(µ 11 + µ 12 ) = 1. Nonnegativity: M e should not be indefinite. [This is not an absolute must, and it is actually relaed in some elements discussed later.] Whether checked by computing eigenvalues or principal minors, this constraint is nonlinear and of inequality type. Consequently it is not often applied ab initio, unless the element is quite simple, as in this case, or can be stated through simple epressions Bar2 One Parameter Mass Template On applying the symmetry and conservation rules three parameters of (22.2) are eliminated. The remaining one, called µ, is taken for convenience to be µ 11 = µ 22 = (2 + µ)/6 and µ 12 = µ 21 = (1 µ)/6. This rearrangement gives [ ] M e µ = µ 1 µ 6 ρ A l = (1 µ)m e 1 µ 2 + µ C + µme L. (22.3) Epression (22.3) shows that the general Bar2 mass template can be recast as a linear combination of the CMM and DLMM instances listed in (22.1). Summarizing, we end up with a one-parameter, matri-weighted (MW) template that befits the LCD form (21.2). If µ = and µ = 1, (22.3) 22 3

4 Chapter 22: MASS TEMPLATES FOR BAR2 ELEMENTS (a) Total mass ρa 1 (e) 2 (b) = L e ρa ρa massless connector Figure The two-node prismatic bar element: (a) element configuration; (b) direct mass lumping to end nodes. reduces to MC e and Me L, respectively. This illustrates another requirement: the CMM and DLMM forms must be instances of the mass template. Finally we can apply the nonnegativity constraint. For the two principal minors of M e µ to be nonnegative, 2 + µ and (2 + µ) 2 (1 µ) 2 = 3 + 6µ. Both are satisfied if µ 1 2. Unlike the others, this constraint is of inequality type, and only limits the range of µ. The remaining task is to select the parameter. This is done by introducing an optimality criterion that fits the problem at hand. This is where customization comes in. Even for this simple case the answer is not unique. Thus the statement the best mass matri for Bar2 is so-and-so has to be qualified. Two specific optimization criteria are considered in and Bar2 Alternative Parametrization An alternative template epression that is useful in some investigations, such as those undertaken in Appendi V, is obtained by reparametrizing via χ = 1+2µ, the inverse of which is µ = (χ 1)/2. The resulting form is [ ] M e χ = χ 3 χ 12 ρ A l. (22.4) 3 χ 3 + χ This is called the χ form of the general Bar2 mass template. Observing that its determinant is ρ A lχ, M e χ is seen to be singular if χ =, and nonnegative if χ Bar2 Angular Momentum Conservation This criterion can only be applied in multiple dimensions, since angular rotations do not eist in 1D. Accordingly we allow the bar to move in the {, y} plane by epanding its nodal DOF to u e = [ u 1 u y1 u 2 u y2 ] T, whence (22.3) becomes a 4 4 matri 2 + µ 1 µ M e µ = 1 6 ρ A l 1 µ 2+ µ 2+ µ 1 µ (22.5) 1 µ 2+ µ Apply a uniform angular velocity θ about the midpoint. The associated node velocity vector at θ = is u e = 1 2 l θ [ 1 1] T. The discrete and continuum energies are T e µ = 1 2 ( ue ) T M e µ ue = 1 24 ρ Al3 (1 + 2µ), T e = l/2 l/2 ρ A ( θ ) 2 d = 1 24 ρ Al3. (22.6) Matching Tµ e = T e gives µ =. So according to this criterion the optimal mass matri is the consistent one (CMM). Note that if µ = 1, Tµ e = 3T e, whence the DLMM overestimates the element rotational (rotary) inertia by a factor or

5 22.1 THE TWO-NODE BAR ELEMENT (a) infinite continuum bar with ρ, E and A constant along u (,t) c CONTINUUM BAR (b) Over lattice, phase velocity and wavelength change to c and λ, respectively λ FEM discretization as infinite lattice (c) c j u (t) j d j λ d (d) d j j +d j Plane aial wave with phase velocity c and wavelength λ (aial displacement drawn transversally to bar for visualization conveniency) FEM-DISCRETIZED BAR j 1 j j+1 j 2-element, 3-node patch etracted at generic node j Figure Propagation of a harmonic plane wave over an infinite, prismatic, elastic bar: (a) propagation over a continuum bar; (b) FEM discretization as infinite regular lattice; (c) propagation of plane wave over Bar2-discretized lattice; (d) etraction of a typical two-element patch. For visualization convenience, the wave-profile aial displacement u(, t) is plotted normal to the bar Bar2 Fourier Analysis For longitudinal motions, a more useful customization criterion is to improve accuracy in the long wavelength, low-frequency limit; this is labeled LFCF in Table This is carried out by a well known tool: Fourier analysis. Physical interpretation: probe the fidelity with which planes waves are propagated over a FEM-discretized regular lattice, when compared to the propagation over a continuum bar. The essentials are illustrated in Figure The top half depicts the continuum bar whereas the bottom half shows stages of the Fourier analysis of its FEM-discretized counterpart. Symbols used for the analysis of plane wave propagation are collected in Table 22.1 for the reader s convenience. [The same notation is reused in later Sections.] Corresponding nomenclature for the FEM-discretized two-node bar lattice is collected in Table The continuum-versus-lattice notational rule is: corresponding quantities use the same symbol but the zero subscript is suppressed in the lattice. For eample, the continuum wavelength λ becomes the lattice wavelength λ. Plane wave propagation over a regular spring-mass lattice is governed by the semidiscrete linear equation of motion (EOM): M ü + Ku=, (22.7) 22 5

6 Chapter 22: MASS TEMPLATES FOR BAR2 ELEMENTS Table 22.1 Quantity Nomenclature for Harmonic Plane Wave Propagation over Continuum Bar Meaning (physical dimension in brackets) ρ, E, A Mass density, elastic modulus, and cross section area of bar (), () Abbreviations for derivatives with respect to space and time t, respectively ρ ü = Eu u (, t) Bar wave equation. Frequency domain forms: ω 2u = c2 u and u + k 2 u =. Plane wave function u = B ep ( i(k ω t) ) [length], in which i = 1 B Wave amplitude [length] λ Wavelength [length] k Wavenumber k = 2π/λ [1/length] κ Dimensionless wavenumber κ = k λ ω Circular (a.k.a. angular) frequency ω = k c = 2π f = 2πc /λ [radians/time] f Cyclic frequency f = ω /(2π) [cycles/time: Hz if time in seconds] T Period T = 1/f = 2π/ω = λ /c [time] c Dimensionless circular frequency = ω T = ω λ /c Group wave velocity c = ω /k = λ /T = E/ρ [length/time]. Often abbreviated to wavespeed. Physically, c is the longitudinal speed of sound. Unsubscripted counterpart symbols, such as k or c, pertain to a discrete FEM lattice; cf. Table 22.2 Table 22.2 Quantity Nomenclature for Harmonic Plane Wave Propagation over Bar2 Lattice Meaning (physical dimension in brackets) u(, t) Plane wave function (22.8) [length] u Node displacement vector, constructed by evaluating u(, t) at nodes [length] Mü + Ku = Semidiscrete lattice wave equation (22.7). K and M are infinite Toeplitz matrices B Wave amplitude [length] l Bar element length [length] λ Wavelength λ = 2π/k = 2πl/κ [length] k Wavenumber k = 2π/λ = κ/l [1/length] κ Dimensionless wavenumber κ = k l = 2πl/λ N eλ Number of elements per wavelength: λ/l : same as signal sampling rate ω Circular (a.k.a. angular) frequency ω = c /l [radians/time] f Cyclic frequency f = ω/(2π) [cycles/time: Hz if time in seconds] T Period T = 1/f = 2π/ω = λ/c [time] Dimensionless circular frequency = ωl/c c Group wave velocity over lattice: c = ω/ k = c ( / κ) [length/time] Wavespeed ratio c/c = / κ from discrete to continuum γ c Quantities unchanged from continuum to lattice, such as E, are not repeated in this Table. Note that the definition of uses the continuum wavespeed c = E/ρ; not the discrete wavespeed c. in which M and K are infinite, tridiagonal Toeplitz matrices. This EOM can be solved by Fourier methods. Figure 22.2(b) displays two characteristic lengths: λ and l. The element length-towavelength ratio is called ϒ = l/λ. The floor function of its inverse: N eλ = λ/l is the number of elements per wavelengths. Those ratios characterize the fineness of the discretization, as illustrated in Figure 22.2(b). Within constraints noted later the lattice can propagate real, travelling, harmonic plane waves of wavelength λ and grpup velocity c, as depicted in Figure 22.2(b,c). The wavenumber is k = 2π/λ 22 6

7 22.1 THE TWO-NODE BAR ELEMENT (a) Wavelength λ =, dimensionless wavenumber κ =. Sampling rate = elements per wavelength N = c> eλ (b) Wavelength λ =8, dimensionless wavenumber κ = π/4. Sampling rate = elements per wavelength N = 8 c> eλ λ (c) Wavelength λ =2, dimensionless wavenumber κ = π, Sampling rate = elements per wavelength N eλ = 2 λ λ c= (folding wavenumber) (d) Wavelength λ =, dimensionless wavenumber κ = 2π, Sampling rate = elements per wavelength N eλ = 1 c< Figure Selected plane waves of various wavelengths, illustrating the physical meaning of the dimensionless wavenumber (DWN) κ = kl = 2πl/λ. The number of elements per wavelength is N eλ = λ/l = 2π/κ, in which. denotes the floor function. (This is equivalent to the spatial sampling rate of filter technology.) The case λ = 2l pictured in (c) pertains to the folding or Nyquist frequency, at which κ = π, N eλ = 2, and the group velocity c vanishes. and the circular frequency ω = 2π/T = 2πc/λ = kc. The range of wavelengths that the lattice may transport is illustrated in Figure To study plane wave solutions it is sufficient to etract a two-element patch, a process depicted in Figure 22.2(d). A harmonic plane wave of amplitude B is described by the function u(, t) = B ep [ j (k ω t)] = B ep [ j ( κ c t )/ l], j = 1. (22.8) Here the dimensionless wavenumber κ and dimensionless circular frequency were introduced as κ = k l = 2πl/λ = 2πχ and = ωl/c, respectively, in which c = E/ρ is the elastic bar group velocity, which for the continuum is the same as the phase velocity. (In physical acoustics c is the sound speed of the material.) Using the well-known Bar2 static stiffness matri and the mass template (22.3) gives the patch equations ρ Al 6 [ ][ü ] 2 + µ 1 µ j 1 1 µ 4 + 2µ 1 µ ü j + EA 1 µ 2 + µ ü l j+1 [ 1 1 ][ ] u j u j =. (22.9) 1 1 u j+1 From this one takes the middle (node j) equation, which repeats in the infinite lattice: [ ] [ ] ρ Al ü j 1 6 [ 1 µ 4 + 2µ 1 µ ] ü j + EA u j 1 [ ] u j =. (22.1) ü l j+1 u j

8 Chapter 22: MASS TEMPLATES FOR BAR2 ELEMENTS (a) Dimensionless frequency Ω=ω /c Continuum Bar CMM: µ = BLFM: µ = 1/2 DLMM: µ = Dimensionless wavenumber κ = k (b) Wavespeed ratio γ c = c/c DLMM: µ = 1 CMM: µ = Continuum Bar BLFM: µ = 1/ Dimensionless wavenumber κ = k Figure Results from Fourier analysis of Bar2 infinite regular lattice for three choices of µ, plus continuum: (a) dimensionless dispersion diagram (DDD); (b) dimensionless group velocity diagram (DGVD). Evaluate the wave motion (22.8) at = j 1 = j l, = j and = j+1 = j + l while keeping t continuous. Substitution into (22.1) gives the wave propagation condition ρ Ac 2 [ 6 (2 + µ) 2 ( 6 (1 µ) 2) ] ( cos κ cos c t i sin c ) t B =. (22.11) 3l l l If this is to be zero for any t and B, the epression in brackets, called the characteristic equation, must vanish. Solving gives the dimensionless frequency versus wavenumber relation 2 = Its inverse is [ 6 (2 + µ) 2 ] κ = arccos 6 + (1 µ) 2 6(1 cos κ) 2 + µ + (1 µ) cos κ = κ µ κ 4 + C 6 κ (22.12) 12 = 1 2µ µ + 2µ (22.13) 192 Transforming (22.12) to physical wavenumber k = κ/l and circular frequency ω = c /l gives ( 6c ω 2 2 ) ( = 1 cos(kl) l µ + (1 µ) cos(kl) = c2 k µ ) 12 k2 l 2 + C 6 k 4 l (22.14) in which C 6 = (1 1µ + 1µ 2 )/ Bar2 Dispersion Diagrams An equation that links frequency and wavenumber: = (κ) as in (22.12), or ω = ω(k), asin (22.14), is a dispersion relation. A plot of the dispersion relation with k and ω along horizontal and vertical aes, respectively, is called a dispersion diagram. When this is done in terms of dimensionless wavenumber κ and dimensionless frequency, the plot is called a dimensionless dispersion diagram, or DDD. Such diagrams ehibit a 2π period: (κ) = (κ + 2πn) for integer 22 8

9 22.1 THE TWO-NODE BAR ELEMENT n. Thus it is enough to plot (π) over either [ π, π] or[, 2π], a range called a Brillouin zone. All DDD in this paper use the [, 2π] range choice. Why is = atκ = 2π? The wavelenth λ = l pictured in Figure 22.3(d) has the same value at all nodes for each time t. This nodal sampling cannot be distinguished from the case λ = (that is, κ = ) shown in Figure 22.3(a). They must share the same frequency, which is zero; associated plane waves propagate with the same speed but in opposite directions. Similar arguments can be made to justify the dispersion curve symmetry about wavenumber κ = π, as well as the 2π periodicity Best µ By Low Frequency Fitting An oscillatory dynamical system is nondispersive if ω is linear in k, in which case c = ω/k is constant and the wavespeed (the group velocity) is the same for all frequencies. The physical dispersion relation for the continuum bar is c = ω /k = E/ρ. Hence all waves propagate with the same speed in this model. Group and phase velocities coalesce. The FEM-discretized lattice group velocity is c = ω/ k = c ( / κ), which differs from c ecept at ω = κ =. The Bar2 discrete model is dispersive for any fied µ, since from (22.14) we get γ c = c = c κ = 1 6(1 cos κ) κ 2 + µ + (1 µ) cos κ = µ κ 2 1 2µ + 2µ2 + κ (22.15) Plainly the best fit to the continuum for small wavenumbers κ = kl<<1 is obtained by taking µ = 1/2, which makes the second term of the series (22.12) or (22.15) vanish. So for LFCF customization the best mass matri is the average of the lumped and consistent ones: M e BLFM = Me µ = 1 2 Me C Me L = ρ Al [ µ= 1 2 ]. (22.16) This instance is labeled BLFM, for best low-frequency match. Figure 22.4(a) plots the dimensionless dispersion relation (22.12) for the CMM (µ = ), DLMM (µ = 1) and BLFM (µ = 1 2 ) mass matrices, along with the continuum-bar relation = κ. The superior small-κ fit provided by the BLFM is evident Folding Frequency The maimum lattice frequency occurs at the folding wavenumber κ = kl = π or λ = 2l, which is waveform (c) in Figure The sampling rate N eλ is then 2 values per element. This is called the folding or Nyquist frequency, and is denoted as af 2 = µ. (22.17) (The a in the subscript stands for acoustic branch; this notation is eplained in ). This varies from af = 12 = 2 3 for the CMM through af = 2 for the DLMM. Frequencies higher than af cannot be propagated over the lattice. As shown in Figure 22.4(b), the lattice wavespeed vanishes at the folding wavenumber κ = π, and is negative over the range (π, 2π]. Waveforms in that rage move with negative speed: c <. As discussed in , the waveform with l = λ, or κ = 2π, cannot be distinguished from a rigid motion such as that pictured in Figure 22.3(a), and the lattice frequency falls to zero. 22 9

10 Chapter 22: MASS TEMPLATES FOR BAR2 ELEMENTS (a) ρ=e=a =1 throughout (b) ρ=e=a =1 throughout L=π/2 L=π/2 Figure Fied-free homogeneous prismatic elastic bar member used in vibration test for Bar2 and Bar3 template instances. Both pictured discretizations display 4 elements. (a): member modeled with Bar2 elements; results reported in Table 22.3 and Figure (b): member modeled with Bar3 elements; results reported in Figure Correct digits in computed frequency 4 (a) (b) (c) 8 LFCF Bar2 Template Instances, Fi-Free Mode 1, 6 Eact Frequency ω 1= 1 BLFM DLMM Correct digits in computed frequency LFCF Bar2 Template Instances, Fi-Free Mode 2, Eact Frequency ω = 3 2 BLFM 2 CMM 2 2 DLMM DLMM CMM CMM Number of elements N e Number of elements N e Number of elements N e Correct digits in computed frequency LFCF Bar2 Template Instances, Fi-Free Mode 3, Eact Frequency ω = 5 3 BLFM Figure Performance of selected Bar2 template instances in predicting the first three natural frequencies ω i, i = 1, 2, 3 of the fied-free prismatic homogeneous bar shown in Figure 22.5(a). This is a graphical, log-log representation of the results of Table Horizontal ais shows number of elements while vertical ais displays correct digits of computed frequency. See tet for details of what is shown along each ais Bar2 Test: Vibrations of a Fied-Free Bar Member Natural frequency predictions of three Bar2 template instances are compared for predicting natural frequencies of longitudinal vibrations of the fied-free elastic bar member pictured in Figure The member is prismatic, with constant E = 1, A = 1, and ρ = 1. The total member length is taken as L = π/2 for convenience. With those numerical properties the continuum eigenfrequencies are (2i 1)π E ω i = = 2i 1, i = 1, 2, 3,... (22.18) 2L ρ The member is divided into N e identical elements, with N e = 1, 2, Figure 22.5(a) pictures the case N e = 4. Three template instances are compared: CMM (µ = ), DLMM (µ = 1) and BLFM (µ = 1/2). Numerical results obtained for the first three frequencies are collected in Table The O(κ 4 ) convergence of BLFM is obvious. For eample, 4 elements give ω 2 correct to 4 digits while both CMM and DLMM, which converge as O(κ 2 ), give only 2. As epected, CMM overestimates the continuum frequencies while DLMM underestimates them. The results of Table 22.3 are graphically reformatted in Figure 22.6, as accuracy versus elements log-log plots. The horizontal ais shows number of elements N e in log 2 scale. The vertical ais displays correct digits of computed frequency, computed as d = log 1 ω i, in which ω i = ω i ω i. (22.19). 22 1

11 22.1 THE TWO-NODE BAR ELEMENT Table 3. Bar2 Instance Results for Vibrations of a Fied-Free Bar Member Instance N e ω 1 ω 2 ω 3 CMM DLMM BLFM * frequency not provided by discrete FEM model Here ω i is the frequency error of computed values with respect to continuum frequencies ω i = 2 i 1, given by (22.18). The plots clearly show at a glance that, for the same N e, BLFM roughly doubles the number of correct digits provided by the other two instances. It also illustrates that CMM and DLMM give the same error magnitude (within plot accuracy) although of different signs. Thus log-log plots such as those in Figure 22.6 are unable to show whether the convergence is from above or below, because of the taking of absolute values in (22.19). That visualization deficiency should be kept in mind should error signs be important Other Customization Options The last three customization options listed in Table 21.1 are not relevant to this element. RHFP is unnecessary because the dispersion diagram does not have an optical branch. MSTS is pointless because the DLMM in (22.1) is unique. Finally, RDAW does not apply to 1D elements Bar2 Frequency Dependent Mass As noted in , it is occasionally useful to make the mass and/or stiffness matri frequency dependent. The goal is to eactly match the continuum dispersion relation = κ for all frequencies, or at least a finite range that includes = κ =. Such an eact fit allows for coarser discretizations. The cost paid is that matri entries become trigonometric functions of frequency. Both the EOM and associated eigenproblems become trascendental. Unless the frequency is specified beforehand (for eample, in pure harmonic ecitation) an iterative process is unavoidable. Therefore eactness gains might be illusory: the dog chases its own tail

12 Chapter 22: MASS TEMPLATES FOR BAR2 ELEMENTS Early publications that follow this approach are cited in H.6. For reasons indicated there, those formulations are not necessarily instances of the general template derived in The simplest way to introduce frequency dependency is to allow the mass template parameter µ in (22.3) to be frequency dependent, while the stiffness matri is held fied. To find the epression of µ, set κ in the characteristic equation etracted from (22.11): 6 (2 + µ ω ) 2 ( 6 (1 µ ω ) 2) cos =, (22.2) in which µ has been renamed µ ω. Solving for it gives Since κ = for the continuum, µ ω = cos = (22.21) 288 µ ω = κ cos κ = 1 2 κ2 4 κ4 18 κ6... (22.22) 288 As orκ both (22.21) and (22.22) approach /. The indeterminacy is removed by the Taylor epansions given above, which show that the limit is µ ω 1, as may be epected. As 2 or κ grows, µ ω decreases so the template gradually favors the CMM more. Two interesting values should be noted. If κ = , µ ω =, which makes the CMM frequency eact; this occurs at the intersection of the continuum and CMM dispersion curves in Figure 22.4(a). If κ = κ lim = , µ ω = 1 2, which makes Me singular. If κ>κ lim, M e becomes indefinite. It follows that the match (22.21) or (22.22) is practically limited to the DWN range κ< Bar2 Frequency Dependent Mass-Stiffness Pair The most general FDMS template for Bar2 has 8 free parameters. These are chosen as deviations from the optimal frequency-independent matrices: ([ ] [ ]) ([ ] [ ]) M e 5 1 µ ω = C M + 11 µ ω µ ω 21 µ ω, K e 1 1 β ω = C K + 11 β12 ω β21 ω β22 ω, (22.23) in which C M = ρ Al/12 and C K = EA/l. All parameters may be frequency dependent. For brevity that dependency will not be eplicitly shown unless necessary. If all µ ij ω vanish, Me reduces to (22.16), which is BLFM optimal. If all βij ω vanish, K e reduces to the well known stiffness of a 2-node prismatic bar. Thus in the zero-frequency (static) limit all parameters must vanish, which provides useful checks. To cut down on parameters, we impose diagonal and antidiagonal symmetry conditions a priori: µ ω 21 = µω 12, µω 22 = µω 11, βω 21 = βω 12, and βω 22 = βω 11. In addition setting β12 ω = βω 21 = βω 11 avoids singularities in the static limit, as noted later. Thus (22.23) reduces to ([ ] [ ]) ([ ] [ ]) M e 5 1 µ ω = C M + 11 µ ω µ ω, K e 1 1 β ω = C 12 µ K + 11 β11 ω β11 ω β11 ω. (22.24) 22 12

13 22.1 THE TWO-NODE BAR ELEMENT Table 4. General FDMS Template For Bar2 Free parameters Linkage equation (top line); Taylor series at ω = κ = (bottom line) µ ω 11,µω 12 β11 ω = (κ2 (5 + µ ω 11 ) + (12 + κ2 (1 µ ω 12 )) cos κ 12)/(12(1 cos κ)) β11 ω κ = [µ ω 11 µ ω 12 ] κ 2 + O(κ 3 ) β ω 11,µω 12 µ ω 11 = ( βω 11 5κ2 ( β11 ω + κ2 (1 µ ω 12 )) cos κ)/κ2 µ ω 11 κ = [µ ω 11 µ ω 12 ] κ 2 + O(κ 3 ) β ω 11,µω 11 µ ω 12 = ( βω 11 + κ2 ( β11 ω κ2 (5 + µ ω 11 )) sec κ)/κ2 µ ω 12 κ = [µ ω 11 µ ω 12 ] κ 2 + O(κ 3 ) Template parameters are generally functions of κ or ; e.g., µ ω 11 = µω 11 (κ) = µω 11 ( ), etc. Parameter arguments are usually omitted to reduce clutter unless necessary. In the bottom-line series, β ω 11, µ ω 11 and µ ω 12 denote parameter values at κ = =. which has 3 free parameters: µ ω 11, µω 12 and βω 11. These matrices are nonnegative if 4 + µ ω 11 + µω 12, 6 + µω 11 µω 12, 1 + βω 11. (22.25) Imposing the plane wave motion (22.8) on a two-element patch, etracting the middle node equation and dropping etraneous factors yields the comple characteristic equation [ 12(1+β11 ) (5+µ ω 11 ) 2 ( β 11 + (1 µ ω 12 ) 2) cos κ ] ep( jκ) =. (22.26) Since the comple eponential never vanishes, it may be dropped and (22.26) reduces to the real equation 12 (1+β ω 11 ) (5+µω 11 ) 2 ( β ω 11 + (1 µω 12 ) 2) cos κ =. (22.27) To match the continuum, is replaced by κ, whence f cm = 12 (1+β ω 11 ) (5+µω 11 )κ2 ( β ω 11 + (1 µω 12 )κ2 ) cos κ =. (22.28) This establishes a linear constraint among the 3 parameters. Consider these as functions of κ: β11 ω = βω 11 (κ), µω 11 = µω 11 (κ), and µω 12 = µω 12 (κ). Epanding in Taylor series about κ = yields f cm = ( ( ) 6 β11 ω µ ω 11 + µ ω ) 12 κ βω 11 κ µω 11 κ + µω 12 κ κ =, (22.29) This shows that in the static limit = κ = the continuum equation is identically satisfied. If β 12 = β11 ω, however, a term in κ 2 appears in (22.29); this is the reason for presetting β12 ω = βω 11. Further developments depend on which parameter pair is kept. Table 22.4 lists three possibilities: (β11 ω,µω 11 ), (βω 11,µω 12 ), and (µω 11,µω 12 )

14 Chapter 22: MASS TEMPLATES FOR BAR2 ELEMENTS Bar2 Frequency Dependent Mass Instances Some relatively simple FDM instances can be obtained by setting β11 ω µ ω 12 = µω 11 and solving for the latter gives = in (22.24). Taking µ ω 11 = µω 12 = κ cos κ = κ2 2 κ4 54 κ6... (22.3) 144 The resulting M e is indefinite if κ> This is the equivalent of the FDM instance considered in The difference between (22.3) and (22.22) lies in the choice of baseline matri for null free parameters. On the other hand, setting µ ω 12 = along with βω 11 = yields µ ω 11 = κ2 + (12 + κ 2 ) cos κ κ 2 = κ4 4 11κ6..., (22.31) 144 This correction is smaller than (22.3) if κ<π/2. The resulting M e is indefinite if κ>