Wave propagation, power flow, and resonance in a truss beam

Journal of Sound and Vibration

WAVE

(1988) 126( 1). 127-144

PROPAGATION,

POWER

FLOW,

IN A TRUSS

AND

RESONANCE

BEAM

J. SIGNORELLJ Department of Astronautics, United Slates Air Force Academy, Colorado Springs, Colorado, U.S.A. AND

A. H. VON

FLOTOW

Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, Massachusetts, U.S.A. (Received 18 August 1987, and in revisedform

19 April 1988)

Wave propagation irra periodic truss-work beam is investigated computationally. The analysis is based upon the transfer matrix of a single bay of the structure. The results,

with members modeled as rods with pinned joints, agree well with results obtained from an equivalent continuum model of the same structure. The inclusion of bending in member models shows that both the pinned rod model and the equivalent continuum models lose fidelity above the first resonant frequency of lateral motion of the members. Modeled with beam members, the truss exhibits complicated mechanical filtering properties, which are illuminated by investigation of wave-mode power flow. Boundary conditions are applied in wave-mode co-ordinates by reformulation in terms of reflection matrices. The phase closure principle is invoked to predict natural frequencies of a fixed-free portion of the truss. It is found that closely spaced resonant frequencies are not identified by this method. Computed results show other subtle erroneous characteristics which are attributed to computational inaccuracy.

1. INTRODUCTION

Many large spacecraft structures will be constructed, in part, of truss-work components. Truss structures generally consist of an assemblage of identical bays and are thus considered to be spatially periodic. Periodic structures have long been known to act as mechanical filters. In order to gain insight and understanding as to how such filtering properties can be exploited in the dynamics and control of large spacecraft structures, wave propagation in two mathematical models of a truss beam is examined in this paper. Any survey of the literature of wave propagation in periodic structures must mention the book by Brillouin [l]. Since Brillouin’s book, there have been many papers in which wave propagation has been treated in periodic structures, primarily mono-coupled systems (systems with one deflection co-ordinate linking neighboring bays). Models used include spring-mass [2], strings and rods [3], and periodically constrained beams [4]. The results of these works have all verified Brillouin’s dictum paraphrased here: “A one-dimensional periodic wave-guide supports as many travelling wave modes in each direction as the (minimum) number of coupling co-ordinates between bays. Each wave mode exhibits alternating (possibly overlapping) frequency ranges of pass-band and stop-band behavior. The number of pass bands is equal to the number of degrees of freedom within each bay”. Few works have dealt with multi-coupled periodic wave-guides. Mead has approached the problem mathematically, both for general situations [ 51, and for a specific 127 0022-460X/88/190127+ 18 $03.00/O 0 1988 Academic Press Limited

128

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H. VON

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model (Timoshenko beams with periodically attached inertias) [6]. Hodges, Powers, and Woodhouse have reported theoretical and confirming experimental work on wave propagation in periodic, rib-stiffened cylindrical shells [7]. Eatwell has considered wave propagation in periodic fluid loaded plates [8]. In each study, the introduction of multiple coupling co-ordinates between bays has permitted a new type of travelling wave mode, the complex mode, which both travels and is spatially attenuated. Such modes were also discovered in this work, in which four coupling co-ordinates link adjacent truss bays. A structural analysis is incomplete without consideration of boundary conditions. In this paper, conventional boundary conditions for truss beams (equations relating forces and deflections of boundary points) are converted to wave-mode co-ordinates. The result is a matrix of frequency dependent reflection coefficients at each boundary. The phaseclosure principle is then invoked to define and calculate natural frequencies: “Resonance occurs at those frequencies at which each propagation path closes on itself with a total phase change of 2kr (k = 1,2, . . . ) after one circumnavigation”. The truss is thus modeled as a multi-mode waveguide terminated by reflection matrices, rather than by system mass and stiffness matrices.

2. WAVE

MODE

DEFINITION

AND

DERIVATION

A wave mode on a one-dimensional wave-guide is described by both a wave-mode eigen-shape, and by an associated propagation coefficient. The eigen-shape is that unique mix of cross-sectional state variables which propagates with constant relative values and phases along the member. The associated propagation coefficient specifies the wavelength (or equivalently, phase speed), with which propagation occurs. A tensioned cable, for example, can support (in the classic approximation) three pairs of wave modes: one axial with a velocity of a, and the other two are lateral with velocity of m (EA is the axial stiffness, T is the tension, and p is the linear mass density). The cross-sectional state vector for such a cable would include three deflections and three corresponding forces. Note that, in agreement with Brillouin’s quote in the introduction, these three wave modes correspond to the three modeled deflection co-ordinates. Waves modes in periodic structures can be analogously defined. We select any reference cross-section in each bay, introduce kinematic assumptions, and assign a number of deflection variables to define the deformation state of that cross-section. (Mead has shown that a wise choice for the reference cross-section is the one that minimizes the number of deflection variables required [5].) 2.1. WAVE MODES IN A TRUSS BEAM Figure 1 is a sketch of the beam investigated in this paper, and of the chosen repeating element. Motion is constrained to the plane. The members are modeled as being pinned at the junctions; thus four deflections are required to define the deformation of the coupling cross-section. If one introduces the corresponding four coupling forces, and groups the coupling deflections and forces into a cross-sectional state vector, then the dynamics of the bay can be described by a transfer matrix relation YR = [T]Y,.

The bay transfer matrix is square, with dimension eight, and is frequency dependent. This transfer matrix can be obtained in several ways. In this analysis, for purposes of direct comparison with a conventional finite element analysis, we derive T( o) by numerical dynamic condensation of a finite element model of the bay.

WAVE

Figure

PROPAGATION

1. State vectors

associated

IN A TRUSS

129

BEAM

with one bay of a periodic

truss.

Two bay models are used, each based upon a particular finite element discretization. Each model yields a mass and stiffness matrix

which is then partitioned into left, U L; right, UR ; and internal, U,, degrees of freedom and manipulated to yield the transfer matrix. The dynamic stiffness matrix is partitioned,

])

(3)

and partially inverted to eliminate internal deflections U,,

E

DLL-DL,D~'D,L

: D

(4)

For compactness, these matrix partitions are relabeled:

(5) (Negative values of FL have been taken for compatibility between the force co-ordinate definitions of the transfer matrix and finite element analyses.) A further partial inversion leads to the bay transfer matrix,

I[

c I;;“, 1_-;;I UR = ____-----_i--_y_____i _---[ FR

I

(6)

or, compactly, YR = [T]Y,.

(7)

The first bay model investigated has four pinned-rod elements and yields 8 x 8 mass and stiffness matrices. Thus, no internal degrees of freedom need be eliminated. The

130

J. SIGNORELLI

A.

AND

H. VON

FLOTOW

second bay model includes member bending effects. Eight beam elements were used, as shown in Figure 2. Nodes 2, 4, 5, and 6 are clamped, while nodes 1, 3, 7,’and 8 are pinned. The resulting 28-degree-of-freedom finite element model thus includes 20 internal degrees of freedom. Note that only the linear deflections of nodes 1, 3, 7, and 8 are external degrees of freedom; the resulting transfer matrix is again 8 x 8. 2.2. WAVE MODE PROPERTIES INFERRED FROM THE TRANSFER MATRIX A wave mode propagating along a periodic structure can be characterized Yi+l

=

by

(8)

‘!fyi

indicating that the cross-sectional state vectors at station it- 1 and i are related by multiplication by a factor 5. This, together with the transfer matrix relation Yi+l= [TlYt

(9)

forms an eigenvalue problem for 5. The eigenvalues are generally complex and occur in 5 and l/t pairs. (Eigenvalues of a symplectic matrix occur in inverse pairs.) For each wave mode there are frequency regions in which the wave will propagate without attenuation, 151= 1 (pass bands), and regions in which the wave is attenuated, l,$i< 1 (stop bands). The relation between 5 and l/t can be seen by constructing a plot of the 5 plane (Figure 3). For a given frequency, the values of 161which lie on the unit circle are in a pass band. Those inside the unit circle are positive-going waves in a stop band while those reflected outside the circle are negative-going waves in a stop band. Values of 151 which lie in the interior (exterior) of the circle, but not on the real axes are termed complex wave modes. As a function of frequency, the eigenvalues move about the 5 plane, continually changing magnitude and phase. In the absence of damping, the transfer matrix, T, is real; thus its eigenvalues will be real or members of a complex conjugate pair. Complex wave modes thus occur only in groups of four. Undamped mono-coupled systems (with 2 x 2 transfer matrices) cannot support complex wave modes.

3

6

8

Figure 2. Finite element model of bay used to include member bending effects.

(a) Figure 3. Eigenvalues complex modes.

(b)

of the transfer matrix plotted

Cc) in the 5 plane.

(a) pass bands;

(b) stop bands;

(c)

WAVE

IN A TRUSS

PROPAGATION

131

BEAM

Particular results were calculated for a truss beam used in prior studies [9,10]. The bay members were assumed to have no structural damping, a bending stiffness El = 5.81482 x lo3 N m2, mass per unit length y =0*75948 kg/m, an axial stiffness EA = 1.93977 x lo7 N, and a longeron and batten length L = 1.397 m. Mills [9] developed continuum models for this truss, a Timoshenko beam model for bending, and a rod model for extension. Mills’ equivalent values for the truss are as follows: bending stiffness EI = 1.89284 x lo7 N m2, mass per unit length p = 3.35254 kg/m, axial stiffness EA = 3.87954~ lo7 N, shear stiffness GA =4-7562 x lo6 N, and inertia per length i, = 0.86460 N s’. Figures 4-l 1 present dispersion curves and wave mode shapes for the four right-going wave modes. Comparison with predictions from the continuum model is provided. Frequency steps of 0.2 Hz were taken. The spikes in the dispersion curves appear to be real and not the result of “noisy” data. Wave mode shapes were derived from the transfer matrix eigenvectors. One wavelength of each wave mode (if finite) is shown as a function of frequency. When member bending is modeled, the dispersion curves for the bending mode indicate complicated mechanical filtering as a function of frequency (Figure 4). At low frequencies the mode is in a pass band (propagation without attenuation) and the first wave mode shape exhibits a global sinusoidal response. Thus it is labeled as the bending mode (Figure 5). As the first resonant frequency of the bay diagonals is approached, the response becomes more localized in the bay diagonals. At 35 Hz, the diagonal members’ first pinned-pinned natural bending frequency, the mode becomes complex within a narrow

bondPass

1

20 (b)

- I20

0

I

I

I

I

20

40

60

80 Frequency

I

I

100

120

I

140

I

160

180

(Hz1

Figure 4. Dispersion curves for the bending mode. -, Beam-based finite element model; finite element model; - - -, Bernoulli-Euler beam model. (a) Amplitude; (b) phase.

-,

rod-based

132

J. SIGNORELLI

AND

A.

H. VON

FLOTOW

(b)

Cd)

(f)

Figure 5. Bending wave mode shapes, from the beam-based finite element model, as a function of frequency. (a) 10 Hz, 28 of 28 bays; (b) 30 Hz, 14 of 14 bays; (c) 50 Hz, 10 bays of evanescent mode; (d) 75 Hz, 18 of 18 bays; (e) 100 Hz, 28 of 44 bays; (f) 155 Hz, 19 of 19 bays.

frequency range, and both propagates and attenuates. At 40 Hz the mode enters a stop band, a region in which the wave mode does not propagate. The sharp spike at 70 Hz corresponds to the first pinned-pinned resonance of the bay longerons. The mode once again becomes complex after this resonant frequency. For higher frequencies, the truss response is no longer global, but becomes localized in the truss members. Predictions based on Bernoulli-Euler beam theory, and results of the analysis with pinned rod members compare favorably, but diverge from those of the more complete model at higher frequencies. Internal member resonances completely dominate the motion at these frequencies. Unlike the bending mode which is in a pass band at low frequencies, the second mode examined begins in a stop band (Figure 6). This mode initially has zero phase change per bay (non-propagating) and is attenuated. These properties are similar to those of the Timoshenko beam shear mode, which exhibits near field behavior below the cut-off frequency o = m = 360 Hz. Because of this initial similarity with the Timoshenko shear mode, this mode is labeled the shear mode. As with the bending mode, the shear mode alternately passes through stop and pass bands. The complex mode shapes (bandwidth 75-95 Hz) are identical to those of the bending mode since these modes couple to create the complex modes in this frequency range (Figure 7). Near 130 Hz the second pinned-pinned resonance of the diagonals appears in the mode shape plots.

WAVE

c 0.4 -

PROPAGATION

stop band

IN

Pass band Complex

A TRUSS

Complex

133

BEAM

Pass band-

Stop band

0.2 -

I 60

(b)

40

0

-40

-6C I 0

I

I

I

I

20

40

60

60 Frequency

Figure 6. Dispersion curves for the shear mode. -, finite element model. (a) Amplitude; (b) phase.

I

1

100

120

I

140

I

160

I

160

(Hz)

Beam-based

finite element

model;

--,

rod-based

(a)

(b)

(d)

Figure 7. Shear wave mode shapes, from the beam-based finite model, as a function of frequency. (a) 10 Hz, 10 bays of evanescent mode; (b) 30 Hz, 30 bays of evanescent mode; (c) 50 Hz, 28 of 53 bays; (d) 130 Hz, 10 bays of evanescent mode; (e) 165 Hz, 20 of 20 bays.

134

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A. H. VON

FLOTOW

The low-frequency behavior of the third wave mode is essentially compression/extension and all models predict similar response. This compression mode is characterized by quite large pass bands separated by narrow stop bands (Figure 8). Complex mode formation occurs between 70.8 and 71.3 Hz. Near 50 Hz, the compression/extension response gives way to longeron and diagonal response (Figure 9). Only the horizontal longerons remain in bending by 65 Hz. Near 150 Hz the response is confined to the second pinned-pinned resonance of the diagonals. The magnitude and phase of the fourth, or evanescent mode, over the frequency range investigated is essentially zero (Figure 10). This indicates that response “dies out” so quickly that it can be considered to be confined to a single bay. With the exception of a very narrow pass band at 72 Hz (the first pinned-pinned resonance of the diagonal members), this mode is always in a stop band. The response of the first bay as a function of frequency is shown in Figure 11. For most frequencies, the response of the structure dies out before reaching the next bay. Between 71 and 72 Hz the evanescent mode is complex. The continuum (Timoshenko) model supports no analogous evanescent wave mode. After the first appearance of the complex modes, the dispersion curves cross over each other and become intertwined (Figure 12). At this point it becomes difficult, if not impossible, to identify which mode is which. An attempt was made to track the phase of the wave mode through the complex mode regions. Because of possible errors in this tracking, the mode nomenclature may be valid only for low frequencies.

Pass band band

Cw -

80

(b)

-60 0

1 20

I

I

I

40

60

60 Freguency

I

I

100

Ix)

1

140

1

160

180

(Hz)

Figure 8. Dispersion curves for the compression mode. -, Beam-based finite element model; --, rod-based finite element model; - - -, axial compression wave from continuum model. (a) Amplitude; (b) phase.

WAVE

PROPAGATION

IN

A TRUSS

BEAM

135

(a)

(b)

(f)

Figure 9. Compression wave mode shapes, from the beam-based finite element model, as a function of frequency. (a) 20 Hz, 28 of 119 bays; (b) 50 Hz, 12 of 12 bays; (c) 65 Hz, 8 of 8 bays; (d) 90 Hz, 28 of 35 bays; (e) 120 Hz, 9 of 9 bays; (f) 150 Hz, 7 of 7 bays.

2.3.

COMPLEX

MODES

wave modes have not received much attention in the structural dynamics literature, and appear to have been mentioned in only a few published papers [5-81. The pinned joint truss mode1 investigated reveals two frequency bands (over the range investigated) in which wave modes are complex-from 35 to 40 Hz, and from 72 to 95 Hz. If the dispersion curves of the bending and shear modes are plotted together, some interesting observations can be made (Figure 12). Both modes are complex throughout the same frequency ranges. In addition, the magnitudes of the eigenvalues are exactly the same. The two wave modes couple throughout these regions, producing complex modes. Complex mode formation appears to be initiated at a member resonant frequency (a joining point). The first bending/shear complex mode pair forms at the first pinned-pinned frequency of the diagonals while the second complex mode pair forms at the first pinned-pinned frequency of the longerons. At break-away points the modes decouple and once again take on separate character. The frequency range 70-73 Hz contains many complex modes (Figure 13). Within this range there are three right-going (and three left-going) complex mode pairs. The bending and compression modes couple in a very short band centered at 71.2 Hz. At 71 Hz even the evanescent mode forms a complex mode pair with the shear mode. The longest region of coupling is between the bending and shear modes (72.5 to 95 Hz). Complex

2.4.

WAVE

MODE

POWER

FLOW

The preceding figures show that each wave mode has frequency bands in which there is propagation, bands in which there is no propagation, and bands in which there is both propagation and attenuation (the complex modes). Intuition tells us that when a wave propagates, it transmits energy along the structure, and when it does not propagate,

136

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SIGNORELLI

AND

A.

H.

VON

FLOTOW

,_(b)

-100

1

0

20

I

I

I

I

I

I

/

40

60

80

100

I20

140

I60

Frequency

180

(Hz)

Figure 10. Dispersion curves for the evanescent mode as obtained by using a beam-based finite element model. (a) Amplitude; (b) phase.

5___-__ (c)

(f)

(e)

(d)

Figure 11. First bay of the evanescent mode wave mode shapes, from the beam-based finite element model, as a function of frequency. (a) 10 Hz, (b) 35 Hz, (c) 50 Hz, (d) 70 Hz, (e) 120 Hz, (f) 155 Hz.

energy cannot move along the structure. Mead in 1973 [5] found theoretically that there is no net power flow complex modes. Instantaneous power is the product of the instantaneous velocity and force: P(t)=Re[i~(U~+iU,)(cos~f+isinwt)]

* Re[(F,+iF,)(coswr+isinwt)].

(10)

Averaging over one period gives P ovg= $o(U~. F, -U,

.

FR).

Substitution of the deflection and force of the wave mode eigenvector average power flow for each wave mode were it alone in the structure.

(II) then gives the

WAVE

PROPAGATION 1

I

1

IN

I

A TRUSS I

I

137

BEAM I

I

I

I-0

I

I

\

4

0.8-

b_#

L

\

\ \

\

\

: ----.__A

0.6 -

k P

0.4 -

0.2 -

I’ 9 I 20

0

I 40

I 100

I 80

I 60

Frequency

I 140

I I20

I 160

Figure 12. Complex mode coupling between the bending and shear modes. --, mode.

I

I

‘I I.0

I

I

160

(Hz)

I

I

Bending mode; - -, shear

I

I

---

0.8

0.6

P

----

1

o-4 ! ,‘I

o-2 t

Complex

mode +++

1

I \

1

71 Frequency

/’

_ 72

(Hz)

Figure 13. Complex mode coupling between 68 and 75 Hz. -, compression mode; ., evanescent mode.

bending mode; --,

shear mode; - - -,

Power flow was calculated for the eight wave modes present in this truss, with members free to bend. The eigenvector components used in (11) were normalized so that the axial displacement of node one (Figure 1) was 2.54 cm. Figure 14 present plots of the logarithm of the magnitude of power flow in the four right-going wave modes. As expected, the bending, shear, and compression modes show power flow in the pass bands-the power flows in each left and right-going “brother” wave pair being equal and opposite. Stop band regions show up as “noisy” data in these plots, with complex modes appearing to propagate many orders of magnitude greater than pure stop bands. This residual power flow in stop bands is believed to be an artifact of finite-precision arithmetic, specifically suggesting that the calculated eigenvectors are impure. Power flow calculated in the evanescent mode also exhibits behavior suggesting a problem with numerical accuracy. A small pass band at 70 Hz shows up clearly in the

138

J. SIGNORELLI

AND

A.

H.

VON

FLOTOW

IE2

stop band

IE-2

IE-4

0

20

40

60

60

100

I20

140

160

Frequency

Figure 14. Logarithm frequency. (a) Bending

0

20

40

60

80

loo

120

140

I60

(Hz)

of the absolute value of the power flow in the left-going wave modes as a function mode; (b) shear mode; (c) compression mode; (d) evanescent mode.

of

plot of Figure 14(d). But outside this pass band, power flow is also evident, once again, probably due to poorly calculated eigenvectors. Interaction between wave modes, creating other forms of power flow, is also possible. Miller [ 1l] has shown that the two evanescent wave modes supported by the BernoulliEuler beam model can interact to create power flow. Miller showed that power flow can be expressed as a complete quadratic function of all wave modes present, and proposed a matrix formulation, P au*

=

WPIW,

(12)

where W is a vector of complex wave-mode amplitudes, and P is a Hermitian matrix. The superscript H (Hermitian) denotes complex conjugate transpose. The model under investigation in this paper would require an 8 x 8 power flow matrix P. Figure 14 shows the frequency dependence of the diagonal entries of P. The off-diagonal terms have not been investigated.

3. WAVE MODE BOUNDARY CONDITIONS Up to this point in the analysis no reference has been made to truss boundary conditions. In order to consider wave mode propagation in a finite length truss, boundary conditions must be taken into account. The concept of a scattering matrix will be used to give the infinite truss closure.

WAVE The cross-sectional transformation

PROPAGATION

IN

A TRUSS

state vector, Y, may be transformed

139

BEAM

into wave co-ordinates by the

Y = V(O)W,

(13)

where W is the cross-sectional state vector in wave mode co-ordinates and v is a matrix with columns consisting of the eigenvectors of the transfer matrix T. The cross-sectional state vector W can be partitioned into components which represent right-going waves, W+, and left-going waves, W-: w=

KY_ [ w- 1 .

(14)

We also label wave modes which arrive at a member boundary, a, and those which depart a boundary, d. The relationship between the arriving and departing wave modes at beam boundaries is depicted in Figure 15. The boundary conditions at the ends of the truss may be written as

(15)

DHw)lY=Fe.x,(~).

where the boundary conditions, B, and external forces, F, may be functions of frequency. In wave mode co-ordinates this becomes

A partial inversion yields the boundary condition in causal form, d = -B;‘(to)B,(u)a+B~‘F,,,,

(17)

or, by introducing the scattering matrix, S(w), d=S(w)a+Bi’F,,,.

(18)

Without external forcing this becomes d = [S(w)]a.

(19)

Components of the scattering matrix are complex, frequency dependent reflection coefficients. The second term of equation (18), the wuue mode generating matrix, indicates how external forces at the boundary generate outgoing wave modes [3]. 3.1.

DERIVATION

OF

THE

SCATTERING

MATRICES

FOR

A PINNED-FREE

TRUSS

BEAM

The boundary conditions for the pinned end of a truss (taken to be the left end) are [I ; O]

1FL1 L

=o.

(20)

Following the preceding derivation, one obtains the scattering matrix.

+-Figure

15. Representation

of arriving,

QL

-+-

a, and departing,

6 d, wave modes

at beam boundaries.

140

J. SIGNORELLI

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A.

H. VON

FLOTOW

Application of the free boundary conditions at the right end of the truss requires a bit more care. These boundary conditions must include the effect of the member which completes the truss end. This member has a force-deflection relation

(21) where [E F; G H] can be obtained from dynamic condensation of a finite element model of that member. In this analysis, the member was modeled by two beam elements; thus the nine-dimensional finite element model must be reduced to the form (21) by dynamic condensation of five internal degrees of freedom. Following the procedure defined by equations (15), (16), and (17) leads to a scattering matrix with 16 entries each of which is a frequency dependent reflection coefficient. Plots of these entries reveal them to be active functions of frequency. Detailed plots of the individual entries are available in reference [ 121. Checks can made on the validity of these reflection coefficients by examining limiting cases of these values. At low frequencies the bending mode resembles a beam in bending, while the compression mode resembles a rod in tension/compression. The detlectionnormalized reflection coefficient of a compression wave in an elastic rod is well known to be unity for a free end, and negative unity for a fixed end. The S,(4,4) plot of the right scattering matrix (Figure 16) indicates that the reflection coefficient for the compression mode at the free boundary tends to unity while the SL(4, 4) plot at the fixed end tends to negative unity [12]. The low-frequency limiting behavior of these two terms is thus correct.

“O_,,,,,,,,,,,,,,,,“l,,l,,,‘,,,,,~ 1

(b)

0.8 -

Figure 16. The S,(4,4) Real(S); . . ., Imag(S).

(a) and S,(4,4)

4. NATURAL

(b) reflection coefficients for the free end of the truss. - - - - -,

FREQUENCIES

BY PHASE

CLOSURE

The phase closure principle states that natural frequencies occur when all wave modes complete a circumnavigation of the structure with a total phase change of 2kn. For the case of an n-bay truss, the wave modes arriving and departing the two ends (Figure 15) are related by aR = g”dL,

aL = kndR,

(22)

WAVE

PROPAGATION

IN

A TRUSS

141

BEAM

where 5 is a diagonal matrix of eigenvalues associated with the right-going wave modes. Equation (18) can be written for both boundaries as dL =

PA,

dR=[SRbR.

(23)

By repeated substitutions of (22) and (23), one obtains aR after one circumnavigation the beam: aR+g%&*SRaR.

of

(24)

Resonance occurs when this relation is an equality [g”SLg”SR -T]a,

=o.

(25)

Therefore, det [e”S,g”SR -I] =O

(26)

is satisfied at a truss resonance. Notice that by replacing only one value in the formulation (the number of bays, n), the natural frequencies for a truss with an arbitrary number of bays can be rapidly determined. The order of the problem does not increase with increasing number of bays, but remains that of the scattering matrix. Figure 17 is a plot of the determinant of equation (26) for a seven bay truss (member bending modeled) with one free and one pinned end. The natural frequencies of the truss can be identified whenever equation (26) tends to being satisfied. Resonant frequencies determined by the phase closure method are listed in Table 1. These frequencies reproduce those determined by a finite element analysis of the same truss [9] except whenever the frequencies are closely spaced. The finite element analysis obtains five modes within the 35.13-35.31 Hz bandwidth while the phase closure method locates only two. This remains true even when frequency steps of 0.001 Hz are used in equation (26). The same results also occur about 70 and 154 Hz. Because the isolated modes are so accurately determined,

Frequency (Hz) Figure 17. Det [s”S,&“S, -I] end, as a function of frequency.

for a seven bay truss (member bending modeled)

with one free and one pinned

142

J. SlGNORELLl AND A. H. VON FLOTOW TABLE

1

Natural frequencies for a seven bay pinned-free truss obtained by the phase closure principle compared to naturalfrequencies as obtained by a jinite element analysis; n/i, not identljied; ?, possible identijcation Present analysis

Finite elements

12.10 33.97 m n/i n/i n/i 37.16 51.84 64.51 m ? n/i 70.22 n/i n/i n/i n/i n/i n/i

12.10 33.97 3513 35.25 35.29 3531 37.16 51.84 64.51

n/i n/i n/i n/i n/i n/i n/i n/i 89.23 103.83 121.14 147.91 153.45

6700

69.50 70.04 70.22 70.39 70.40 70.43 70.47 70.47 70.49

n/i n/i 155.28 157.07

one may be tempted to attribute the phase closure method’s during eigenvalue/eigenvector

behavior

EFFECTS

An actual spacecraft truss may exhibit significant

has been assumed.

non-linearities, particularly if it is deployable The effects of such joint non-linearities It seems plausible

failure to error introduced

calculation.

5. UNMODELED Linear

70.51 70.52 70.54 70.54 70.57 70.59 70.59 70.59 89.23 103.83 121.12 147.91 153.45 154.32 154.57 154.69 155.28 157.07

(and thus could have relatively loose joints).

upon the results presented here are not known.

that the situation would

become

even more complex,

and that the

pattern of stop and pass bands, at any given response amplitude, would suffer some sort of blurring. Dissipation

also has the potential

for strongly modifying

the dynamics.

Dissipation

will result in spatial attenuation of travelling wave modes, and will attenuate the localized response mentioned in the following paragraph. Even if an actual truss were linear, it would not be perfectly periodic. Small, unintentional variations from perfect periodicity

would be present. The statistical effect on wave

propagation

in bay properties

of such random

variations

is the subject of references

[13, 141. Although in these studies only mono-coupled systems have been considered, they show that the first order effect of slight disorder is that all wave modes at all frequencies

will be spatially attenuated. The degree of attenuation is proportional to the proportional to the coupling strength between bays. (Suitable non-dimensional measures of randomness and coupling strength must be introduced.) The physical explanation for such localization is that the coherent wave is scattered into incoherence; the vibrational energy is transferred into a spatially localized response. We

randomness and inversely

WAVE

PROPAGATION

IN A TRUSS BEAM

143

thus anticipate that an actual truss (as compared to its idealized mathematical model) will exhibit the characteristics described by this paper only approximately. Real-world effects (non-linearity, disorder, and others) will tend to modify this response (especially at higher frequencies) to be more of an ill-defined rattle that slowly appears and disappears in local portions of the structure. 6. CONCLUSIONS This research computationally investigated wave mode propagation in a periodic truss beam. Some conclusions based on this research follow. (1) The transfer matrix technique proved useful in that the dynamics of a complete truss beam were determined by analyzing only one of the periodic elements. Conventional analysis tools such as the finite element analysis become computationally cumbersome as the number of degrees of freedom needed to model the structure increases. In the transfer matrix method the order of the problem depends only on the order of the cross-sectional state vector. (2) The results obtained by examining a pinned rod truss by transfer matrices closely match the results obtained by continuum models of the same structure. (3) As with continuum models of the truss structure, the pinned rod truss loses fidelity at the first resonant frequency of the truss members. The rod modelling masks all lateral member dynamics that would be present if member bending were modeled. (4) The pinned beam truss exhibits complicated mechanical filtering properties. (5) Complex wave modes must form in pairs. Thus, in a periodic structure, there must be at least four complex modes present for any to exist at all. Mono-coupled systems cannot, therefore, support complex wave modes. (6) Complex mode formation is initiated at member resonant frequencies. No explanation could be found for termination of complex mode coupling. (7) Results obtained indicate there may be numerical round-off errors in this formulation. This may have given rise to the erroneous apparent power flow in the evanescent modes throughout much of the bandwidth examined. Also, closely spaced natural frequencies of the truss are not detected when analyzed by phase closure. ACKNOWLEDGMENT This work was supported in part by the United States Air Force Office of Scientific Research, with Dr Anthony Amos as contract monitor. REFERENCES 1. L. BRILLOUIN 1946 Wave Propagation in Periodic Structures. New York: Dover. 2. R. A, JOHNSON 1983 Mechanical Filters in Electronics. New York: John Wiley. 3. A. H. VON FLOTOW 1986Journal ofSound and Vibration 106,433-450. Disturbance propagation in structural networks. 4. J. W. MILES 1956 American Society of Civil Engineering Journal Jan, 1-9. Vibrations of beams on many supports. 5. D. J. MEAD 1973 Journal of Sound and Vibration 27, 235-260. A general theory of harmonic wave propagation in linear periodic systems with multiple coupling. 6. D. J. MEAD 1986 Journal of Sound and Vibration 104,9-27. A new method of analyzing wave propagation in periodic structures: applications to periodic Timoshenko beams and stiffened plates. 1985 JournalofSound and Vibration 101, 7. C. H. HODGES,J. POWERS and J. WOODHOUSE 219-256. The low frequency vibration of a ribbed cylinder.

144

J. SIGNORELLI AND

A. H. VON

FLOTOW

1983 Journal of Sound and Vibration 88, 507-522. Free-wave propagation in an irregularly stiffened, fluid-loaded plate. 9. R. A. MILLS 1985 S.M. Thesis, Department of Aeronauks and Asrronauks, Massachusetts Institute of Technology. Natural vibrations of beam-like trusses. 10. A. H. VON FLOTOW 1986 American Institute of Aeronautics and Astronautics, Proceedings of rhe Guidance and Control Conference, Williamsburg, Virginia 1, 622-628. Control motivated dynamic tailoring of spacecraft truss structures. 11. D. MILLER, A. H. VON FLOTOW and S. R. HALL 1987 Proceedings ofrhe American Conrrols Conference, Minneapolis, Minnesota, Volume 2, 1318-1324. Active modification of wave reflection and transmission in flexible structures. 12. J. SIGNORELLI 1986 S.M. Thesis, Department of Aeronautics and Astronautics, Massachusetts Institute of Technology. Wave propagation in periodic truss structures. 13. C. H. HODGES and J. WOODHOUSE 1983 Journal of the Acoustical Society of America 74, 894-905. Vibration isolation from irregularity in a periodic structure: Theory and measurements. 28th Structures, Srructual 14. G. KISSEL 1987 Proceddings of rhe AZAAIASMEIASCEIAHS Dynamics andMaterials Conference, Monterey, California, 1046- 1055. Localization in disordered periodic structures. 8. G. P. EATWELL