Vibration confinement in beams

Journal of Sound and Vibration 399 (2017) 45–59

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Vibration confinement in beams Anthony A. Ruffa n, Michael A. Jandron, Raymond W. Roberts, Scott E. Hassan Naval Undersea Warfare Center, 1176 Howell Street, Newport, RI 02841, United States

a r t i c l e i n f o

abstract

Article history: Received 12 October 2016 Received in revised form 10 February 2017 Accepted 28 March 2017 Handling Editor: D.J Wagg Available online 2 April 2017

We develop approaches for vibration confinement to within an arbitrary region in both non-fluid-loaded beams and fluid-loaded beams. The approach for non-fluid-loaded beams makes use of a novel forward/backward substitution algorithm that generates an evanescent response that exhibits exponential growth from one end of the beam to a specified node, and then exponential decay from that node to the other end of the beam. A weighted sum of such solutions supports vibration confinement while requiring only one actuator at each edge of the confined region. We also demonstrate vibration confinement with just a single actuator term when one edge of the confined region extends to the end of the beam. For fluid-loaded beams, we use a weighted sum of solutions having compact support to achieve near-confinement. While the displacement vector is nonzero outside of the confined region (as a consequence of the acoustic loading on the beam), its amplitude is significantly reduced. Published by Elsevier Ltd.

Keywords: Beam Vibration Fluid-loaded Confinement Actuator Evanescent response

1. Introduction The control of structural vibrations and acoustic radiation has been an active research topic for over three decades. Many aspects of this ongoing research have been extensively documented in reference books [1,2]. In particular, there has been a considerable amount of work focusing on vibration confinement in flexible structures [3–9]. For example, Shelley and Clark [6] were able to localize all modes in a tridiagonal system with two control sensor/actuator pairs. However, the required number of pairs increases with the system bandwidth (i.e., the number of diagonals in the coefficient matrix). Other work [7] has either required a number of actuators that is equal to the dimension of the system of equations, or more recently, a limited number of actuators [8,9]. In the present paper, we focus on vibration confinement in beams. When there is negligible fluid loading (i.e., for beams in air), we use a finite difference approach and a novel solution algorithm that generates a response that involves exponential growth from one end of the beam to a specified node and exponential decay from that node to the other end of the beam. Furthermore, this response is associated with only three nonzero load terms, i.e., one at the specified node, and one at either adjacent node. We show that a weighted sum of such solutions (centered at different nodes) leads to vibration confinement to within an arbitrary region on the beam with only two actuators, i.e., one actuator at each end of the confined region. Outside the confining actuators, the vibration amplitude decays exponentially. We further show that only one actuator is required for vibration confinement when the region extends to one end of the beam. Our approach is a result of research into alternative solution approaches for banded linear systems [10–13]. One form involves a forward substitution approach that is initiated by assuming the values for the first q unknowns (where q is the n

Corresponding author. E-mail addresses: [email protected] (A.A. Ruffa), [email protected] (M.A. Jandron), [email protected] (R.W. Roberts), [email protected] (S.E. Hassan). http://dx.doi.org/10.1016/j.jsv.2017.03.038 0022-460X/Published by Elsevier Ltd.

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number of superdiagonals and 2q + 1 is the system bandwidth). Every subsequent equation can then be solved for an additional unknown using the highest diagonal as a divisor. Since the approach assumes the value of the first q unknowns, it requires q additional unknowns that, once solved (via a q × q system), leads to the solution of the entire system. In the process, it decomposes the original system into q subsystems that can be solved in parallel. A second form involves both forward and backward substitution. This is necessary when the forward substitution process leads to a solution dominated by an exponential growth response, which prevents the solution approach in [10–12] from working. However, if the right hand side (RHS) vector has only a single nonzero term located at k ¼m, a modified approach will work when forward substitution is performed for 1 ≤ k ≤ m and backward substitution is performed for n ≥ k ≥ m , where n is the number of unknowns. The backward substitution process also leads to a solution that exhibits an exponential growth response. However, because it grows exponentially as the solution progresses in the backward direction, it is observed as exponential decay [12,13]. For a tridiagonal system, a single nonzero RHS term remains after completion of the forward and backward substitution processes. This term (and the solution vector) can then be appropriately scaled to obtain the required solution. For a pentadiagonal system (which is the focus of this paper), there will be three nonzero RHS terms remaining. This will require a more complicated procedure to arrive at an arbitrary RHS vector. The original intent was to use this approach as a parallelized solver for banded systems. However, it was found that such solutions provide insight into the systems themselves and can lead to novel solutions. For example, the finite difference form of the Euler-Bernoulli equation leads to a series of evanescent solutions having compact support that can be appropriately superimposed to obtain vibration confinement for any arbitrary load vector. Specifically, we found that this process could lead to vibration confinement with only one load term at each boundary of the confined region. Conventional approaches, on the other hand, require two terms at adjacent nodes at each boundary. A fluid-loaded beam (i.e., a beam in water) is more complex because of acoustic fields that impose loads on the entire beam. It is notable that past work has primarily focused on vibrating structures acting either in-vacuo or in-air. As a result, the fluid loading on the structure is often negligible and does not influence the mode shapes and structural dynamic response. Related work by Guiguo and Fuller [14] addresses the specific problem of active control of sound radiation from a semi-infinite beam with a clamped edge. Their approach is restricted to light fluid loading where the acoustic medium does not influence the beam structural dynamics. Other approaches for the light fluid loading case have been based on minimization of the local volume displacement into the fluid to control acoustic radiation [15]. In addition to the unique mathematical aspects of the present study, the control of sound radiation from a beam into a heavy fluid medium is noteworthy. Because of the fluid loading, it is not possible to use the solution approach developed for banded linear systems. It is also not possible to completely achieve vibration confinement with just two actuators. However, we have been able to form a weighted sum of solutions with compact support that achieves near-confinement. We can either set the displacement amplitude outside of the confined region to zero, which would result in a small amplitude load distribution outside of the confined region, or we can set the load vector to zero outside the confined region, which would result in a continuous (but small amplitude) displacement distribution outside of the confined region. This paper is organized as follows. We make use of a forward/backward solution approach in Section 2 to solve the beam vibration problem in air, obtaining an evanescent displacement vector and a load vector having compact support. We then develop a weighted sum of such solutions to achieve vibration confinement. We analyze the response to perturbations to that solution, and develop an analytical model that also demonstrates confinement with only two actuators on a uniform beam. In Section 3, we consider fluid-loaded beams. Here we generate solutions that emulate the evanescent displacement vector in air, and then consider other solutions having compact support, i.e., involving only one nonzero displacement term. We then perform a weighted sum of such solutions to generate near-confinement with only one actuator at either end of the confined region. Here the amplitude is small but nonzero outside the confined region as a consequence of the loads imposed by the acoustic field. Finally we present conclusions in Section 4.

2. Vibration confinement without fluid loading 2.1. Numerical algorithm As mentioned in the introduction, our approach for non-fluid-loaded beams is based on a numerical algorithm [12,13] for banded linear systems that we apply to the beam vibration problem. We use this algorithm in lieu of a conventional solver because it automatically generates an evanescent solution (exhibiting an exponential growth/decay response) and a load vector having compact support. We will show that such solutions can support vibration confinement with only a single load term (representing an actuator) at either end of the confined region. In contrast, the use of a conventional solver would lead to two actuators at either end of the confined region for a pentadiagonal system [6]. We model the Euler-Bernoulli equation, i.e.,

EI

∂ 4Ψ ∂x

4

+ ρA

∂ 2Ψ = W. ∂t 2

(1)

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47

1 0.9

Displacement (relative units)

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

200

400

600

800

1000 1200 Node number

1400

1600

1800

2000

Fig. 1. The ψ vector generated by the forward/backward substitution solution to the beam vibration problem for n¼2001 and m¼ 1001. Note that the exponential response dominates and the oscillatory response is not evident.

We consider a beam of length L in which E = 2 × 1011 Pa is the Young's modulus, I is the moment of inertia, ρ = 8000 kg/m2 is the density, A is the cross-sectional area, W is the applied load, x and t are the independent spatial and temporal coordinates, respectively, and Ψ is the transverse displacement. We assume a harmonic time dependence, i.e., Ψ (x, t ) = ψ (x )eiωt (where i = −1 , ω = 2πf , and f is the frequency). A second order finite difference approximation leads to a pentadiagonal Toeplitz system (except for the first and last equations, which represent clamped/clamped boundary conditions) with row structure [1, −4 , 6 + γ , −4 , 1], where γ = − ω2ρAΔx 4E −1I −1, and Δx is the finite difference discretization length. We consider a square beam with a dimension of 0.1 m on each side with f = 100 Hz, Δx = λ /100 (where λ = 2π (EIρ−1A−1ω−2)1/4 is the wavelength), n ¼2001, and m ¼1001, so that L ≅ 20λ . We denote the system of equations as aψ = b , where b is the right hand side (RHS) vector. We implement a forward and backward substitution approach [12,13] by assuming that ψ1 = ψ2 = 1, and then using forward substitution to explicitly solve for ψ3 to ψm, where 3 < m < n − 2. Likewise, we assume that ψn = ψn − 1 = 1 and use backward substitution to explicitly solve for ψn  2 to ψm. We then scale the forward and backward substitution solutions so that ψm = 1 for both [12,13]. After scaling, ψm = 1, and both ψ1 and ψn are O(10−30), indicating that we can assume virtually any values for ψ1, ψ2, ψn  1, and ψn to initiate the forward and backward substitution processes except for ψ1 = ψ2 = 0 or ψn = ψn − 1 = 0. This process effectively satisfies the boundary conditions at both ends of the beam (i.e., ψ1 = dψ1/dx = 0 and ψn = dψn/dx = 0), even though they were not specified explicitly. The generated ψ vector is shown in Fig. 1. A consequence of assuming values of ψ to start the forward/backward substitution process is that the computation of the ψ vector does not make use of all of the equations in the system. Specifically, three equations (i.e., equations m − 1, m, and m + 1) are unused. We substitute appropriate values from the ψ vector into the unused equations and then solve for bm  1, bm, and bm + 1 to obtain a solution to (1). For this example, this leads to bm = 0.2510 and bm − 1 = bm + 1 = − 0.1257 (Fig. 2). The amplitude of the error vector, i.e., e = aψ − b (Fig. 3) is O(10−15). The exponential growth response shown in Fig. 1 arises from the forward/backward substitution algorithm. This response has been noted by previous researchers [16–20], i.e., a forward substitution process can exhibit an exponential growth response, depending on the roots of the difference equation corresponding to each equation in the Toeplitz system. Likewise, the backward substitution process can also lead to a solution exhibiting exponential growth behavior. However, because it grows exponentially as the solution progresses in the backward direction, it is observed as exponential decay [12,13]. The exponential growth response has sometimes been attributed to numerical stability issues. However, for this system, the error vector (Fig. 3) demonstrates that the solution is highly accurate, indicating that the observed exponential growth is not a result of the accumulation of roundoff errors [13]. We make use of this solution to achieve vibration confinement. 2.2. Vibration confinement A single evanescent solution as shown in Figs. 1 and 2 is by itself not useful. However, we can compute such solutions centered on each node on the beam and then superimpose them to obtain a solution that can satisfy any specified RHS vector. Furthermore, we can use that approach to achieve vibration confinement to within an arbitrary region with load

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0.3 0.25

Load (relative units)

0.2 0.15 0.1 0.05 0 −0.05 −0.1 −0.15 970

980

990

1000 Node number

1010

1020

1030

Fig. 2. The three nonzero RHS terms generated by the forward/backward substitution solution, i.e., bm  1, bm, and bm + 1, where m¼ 1001.

1.5

x 10

Error (relative units)

1

0.5

0

−0.5

−1 500

600

700

800

900

1000 1100 Node number

1200

1300

1400

1500

Fig. 3. The error vector, i.e., e = aψ − f , associated with the forward/backward substitution solution.

terms at nodes α and β, where α < m < β . To do this, we generate β − α − 1 solutions centered at successive nodes from node α + 1 to node β − 1. We call these solutions ψ (α + 1) , ψ (α + 2) , ψ (α + 3) , …, ψ (β − 1). Each solution has a corresponding RHS vector, i.e., b(α + 1), b(α + 2), b(α + 3), …, b(β − 1). We then introduce weights wj so that β− α − 1

ψ=

∑

wjψ (α + j );

j=1

(2)

Likewise, β− α − 1

b=

∑ j=1

wjb (α + j).

(3)

We develop β − α − 1 equations that we use to solve for the weights corresponding to an arbitrary RHS vector within the confined region. For simplicity, the RHS vector in this example is zero except for bm ¼1. We can then develop equations as follows:

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1500

Displacement (relative units)

1000

500

0

−500

−1000

−1500 0

200

400

600

800

1000 1200 Node number

1400

1600

1800

2000

Fig. 4. The ψ vector showing confinement between nodes α = 700 and β = 1302 . Outside of these nodes, the amplitude decays exponentially, so that ψ1 and ψn are O(10−20) .

1 0.8

Load (relative units)

0.6 0.4 0.2 0 −0.2 −0.4 −0.6 0

200

400

600

800

1000 1200 Node number

1400

1600

1800

2000

Fig. 5. The RHS vector corresponding to the solution in Fig. 4 with α = 700 , m¼ 1001, and β = 1302. β− α − 1

bk = δkm =

∑ j=1

wjbk(α + j) ; α + 1 ≤ k ≤ β − 1;

(4)

Here δkm = 1 when k¼m, and δkm = 0 otherwise. This leads to a system with w as the unknown vector. We consider an example in which α = 700, m ¼1001, and β = 1302, so that the confined region contains 601 nodes (i.e., from node 701 to node 1301), or a length of 6λ . When we solve for w and substitute into (2), Fig. 4 is the result. Note that the terms bα and bβ are not included in (4). Instead, they are allowed to “float,” and as a result, they have nonzero values, as shown in Fig. 5. The weights (Fig. 6) exhibit an oscillatory nature having the same wavelength as that of ψ . It is also possible to use only one actuator to achieve confinement by locating the driving actuator between the confining actuator and one of the clamped ends, leading to the load vector shown in Fig. 8 and the displacement vector shown in Fig. 7. We can analyze the robustness of the numerical solution to small perturbations introduced into the load terms. For example, if the actuator term bα in Fig. 5 were perturbed, we can analyze the response by solving the system with a single

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80 60

Weight value (relative units)

40 20 0 −20 −40 −60 −80

0

100

200

300 Weight number

400

500

600

Fig. 6. The weights used to compute ψ shown in Fig. 4.

2500 2000

Displacement (relative units)

1500 1000 500 0 −500 −1000 −1500 −2000 −2500

0

200

400

600

800

1000 1200 Node number

1400

1600

1800

2000

Fig. 7. A simulation of an experiment with one confining actuator.

nonzero RHS term at node α = 700. The normalized solution is shown in Fig. 9. The actual response will be proportional to the perturbation amplitude introduced into bα . That response can then be superimposed with the response in Fig. 4 to obtain the total response. It indicates that the perturbed response will not affect the vibration confinement except for introducing a small amplitude oscillatory component that extends over the entire beam.

2.3. Analytical solution We can develop an analytical solution that also demonstrates the use of just two actuators to achieve confinement. If we assume confining actuators at x = xα and x = xβ with a load term at x = x m , we can develop four solutions as follows (assuming a harmonic time dependence):

ψ (1) = C1(1)e σ (x − xα) ; 0 ≤ x ≤ xα ;

(5)

ψ (2) = C1(2)e σ (x − xm) + C2(2)e−σ (x − xα) + C3(2)cosσx + C4(2)sinσx; xα ≤ x ≤ x m ;

(6)

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1

Load (relative units)

0.5

0

−0.5

−1

−1.5

0

200

400

600

800

1000 1200 Node number

1400

1600

1800

2000

Fig. 8. The amplitudes of the driving and confining actuators leading to the displacements in Fig. 7.

1

Normalized perturbation amplitude

0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8

0

200

400

600

800

1000 1200 Node number

1400

1600

1800

2000

Fig. 9. The normalized response for a single nonzero load term bα for α = 700 . This response can be scaled by the perturbation amplitude and then superimposed with the confined solution to estimate the effect of small perturbations introduced into the confining actuators.

ψ (3) = C1(3)e σ (x − xβ ) + C2(3)e−σ (x − xm) + C3(3)cosσx + C4(3)sinσx; x m ≤ x ≤ xβ ;

(7)

ψ (4) = C2(4)e−σ (x − xβ ) ; xβ ≤ x ≤ L.

(8)

Here σ = (ω2ρAE −1I −1)1/4 . At x = xα , the following equations hold:

ψ (1) = ψ (2); dψ (1) dψ (2) = ; dx dx d2ψ (1) d2ψ (2) = ; dx2 dx2 3 (1) 3 (2) W dψ dψ = − α; EI dx 3 dx 3 Similarly, at x = x m , the following equations hold:

(9) (10) (11) (12)

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8

x 10

Displacement (relative units)

6 4 2 0 −2 −4 −6 −8 0

2000

4000

6000

8000 10000 Node number

12000

14000

16000

Fig. 10. A comparison of ψ (1) , ψ (2) , and ψ (3) (i.e., the analytical solution), each term scaled by 1/Δx 3 , with ψ (the numerical solution). Here Δx = λ /800, which leads to 16,001 nodes for L ≅ 20λ .

ψ (2) = ψ (3); dψ (2) dψ (3) = ; dx dx 2 (2) 2 (3) dψ dψ = ; dx2 dx2 W d3ψ (2) d3ψ (3) = − m; 3 EI dx dx 3

(13) (14) (15) (16)

Finally, at x = xβ , the following equations hold:

ψ (3) = ψ (4); dψ (3) dψ (4) = ; dx dx d2ψ (3) d2ψ (4) = ; 2 dx dx2 Wβ d3ψ (3) d3ψ (4) = − ; 3 3 EI dx dx

(17) (18) (19) (20)

Eqs. (9)–(20) can be solved for twelve unknowns, i.e., C1(1) , C1(2) , C2(2) , C3(2), C4(2) , C1(3) , C2(3) , C3(3), C4(3) , C2(4) , Wα, and Wβ. The solution closely approximates the numerical solution when Δx ≈ λ /800 and the analytical displacement is scaled by 1/Δx 3 (see Figs. 10–12). These conditions are necessary because each load in the analytical solution approximates a Dirac delta function, while each load has a width of Δx in the numerical solution. Note also that the boundary conditions are not satisfied explicitly; however, the solution decays sufficiently so that the amplitude at the boundaries is negligible.

3. Vibration confinement with fluid loading The approach for fluid-loaded beams is more complicated, since the acoustic field imposes a load on the entire beam. The use of conventional approaches can lead to an infinite set of coupled equations [21]. As previously mentioned in the introduction, past work has primarily focused on vibrating structures acting either invacuo or in-air so that the fluid loading on the structure does not influence the mode shapes and structural dynamic response [14,15]. If we had restricted the fluid loading so that the beam structural dynamics was unaffected, we would have been able to continue using the algorithms from the last section. However, we consider here the case involving heavy fluid loading, which complicates the coefficient matrix so we cannot use the forward/backward substitution algorithm. Instead, we will develop solutions with compact support with the weighted sum process that we used for non-fluid-loaded beams.

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1 0.8

Load (relative units)

0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8

0

2000

4000

6000

8000 10000 Node number

12000

14000

16000

Fig. 11. The load vectors corresponding to the analytical and numerical displacements in Fig. 10. The numerical load terms have a width of Δx .

−5

8

x 10

6

Relative Error

4

2

0

−2

−4

−6

0

2000

4000

6000

8000 10000 Node number

12000

14000

16000

Fig. 12. The relative error between the analytical and numerical displacements in Fig. 10.

3.1. Numerical model For this problem, we model both the fluid and the beam with a finite difference approximation and couple them via coincident nodes (i.e., pairs of nodes in the fluid and beam with the same coordinates) to connect the two systems. In the fluid, we model the Helmholtz equation, i.e., ∇2ϕ + k 2ϕ = 0, where ϕ is the velocity potential. The finite difference grid is shown in Fig. 13. We impose a rigid section at each end of the beam with a length of one acoustic wavelength. The fluid domain extends vertically for a distance of two acoustic wavelengths above the beam. We impose plane wave boundary conditions on the upper boundary (i.e., p = ρw cw ∂ϕ/∂y , where y is the vertical coordinate), and on the left and right boundaries (i.e., p = ± ρw cw ∂ϕ/∂x ), where p is the acoustic pressure and ρw is the fluid density. At the lower boundary, we impose a load on each node on the beam due to the acoustic pressure so that (1) becomes the following (assuming a harmonic time dependence):

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Fig. 13. The computational grid.

1.8

x 10

Displacement amplitude (relative units)

1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0

0

20

40

60

80

100 Node number

120

140

160

180

200

Fig. 14. The displacement amplitude on a fluid-loaded beam for a single nonzero RHS term.

d 4ψ dx 4

−

ω2ρA pΔz ψ=W+ . EI EI

(21)

We compute the acoustic pressure from the velocity potential, i.e., p = − iωρw ϕ , and Δz is the width of the beam along the z coordinate that is perpendicular to the x and y coordinates. At the fluid boundary, iωψ = ∂ϕ/∂y provides the required boundary condition. For the beam, E = 2 × 1011 Pa is the Young's modulus, ρ = 8000 kg /m2 is the density, and its cross section is a square with a dimension of 0.1 m on each side, and f = 100 Hz. The fluid domain has a density and sound speed of ρw = 1000 kg/m2 and cw ¼1500 m/s, respectively. The acoustic wavelength in the fluid is λ w = cw /f = 15 m, and λ w ≈ 5λ , where λ is the wavelength governing propagating transverse waves in the beam. The finite difference spacing is Δx = Δy = λ /10 ≈ λ w /50 . The rigid section at either end of the beam spans 50 nodes, or approximately one acoustic wavelength, or approximately 5λ . 3.2. Development of a solution with compact support Fig. 14 shows the displacement amplitude on the beam for a single nonzero harmonic load term imposed on a single node on the beam (Fig. 15). The acoustic pressure field is shown in Fig. 16. Although we cannot perform the forward/ backward substitution process, we can use a weighted sum approach to develop a useful solution with compact support. We obtain such solutions centered on each node on the beam outside of the rigid sections (i.e., from node 51 to node

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1 0.9

Load amplitude (relative units)

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

20

40

60

80

100 Node number

120

140

160

180

200

Fig. 15. The RHS vector leading to the response in Fig. 14. 30

25

Distance (m)

20

15

10

5

10

20

30 Distance (m)

40

50

60

Fig. 16. The amplitude of the acoustic pressure associated with the beam displacement vector in Fig. 14.

150), each with a single RHS term centered on each finite difference node on the beam, i.e., 100

ψ=

∑ wjψ ( j+ 50); j=1

(22)

and 100

b=

∑ wjb ( j+ 50). j=1

(23)

We then perform a weighted sum to obtain a solution (Fig. 17) that closely approximates the displacement vector of the evanescent solution of a beam in air. The imposed load (represented by the RHS vector) is shown in Fig. 18. Unlike the solution in air, the load vector is nonzero over the entire beam, reflecting the effects of acoustic coupling. Although we could have used a weighted sum of such solutions, we chose instead another solution having even more compact support. Fig. 19 shows a solution with a displacement vector having a single nonzero term, and the corresponding RHS terms (Fig. 20). We will use this result to demonstrate the confinement in fluid-loaded beams. 3.3. Vibration confinement To demonstrate confinement in beams, we first generate the solution shown in Figs. 19 and 20 centered on every node on the beam. We then perform another weighted sum of these solutions to attempt confinement with a single nonzero load

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1

Displacement amplitude (relative units)

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

20

40

60

80

100 Node number

120

140

160

180

200

Fig. 17. The displacement vector for a fluid-loaded beam closely approximating that associated with the evanescent response of a beam in air.

5

x 10

4.5

Load amplitude (relative units)

4 3.5 3 2.5 2 1.5 1 0.5 0 0

20

40

60

80

100 Node number

120

140

160

180

200

Fig. 18. The imposed load vector that would be required to obtain the response in Fig. 17. 1

Displacement amplitude (relative units)

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

20

40

60

80

100 Node number

120

140

160

180

Fig. 19. The displacement amplitude on a fluid-loaded beam consisting of a single nonzero term.

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57

x 10

14

Load amplitude (relative units)

12

10

8

6

4

2

0

0

20

40

60

80

100 Node number

120

140

160

180

200

Fig. 20. The imposed load vector that would be required to obtain the response in Fig. 19.

1 0.9

Load amplitude (relative units)

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

20

40

60

80

100

120

140

160

180

200

Node number

Fig. 21. The load vector required to achieve vibration confinement in a fluid-loaded beam.

term at node 109 with confining actuators at nodes 89 and 129. This will require the superposition of 39 solutions centered at nodes 90 to 128, i.e., 39

ψ=

∑ wjψ ( j+ 89); j=1

(24)

39

b=

∑ wjb ( j+ 89). j=1

(25)

The results are shown in Figs. 21 and 22. Note that the central load term generates a displacement vector that is zero outside of the two terms at either border of the confined region. Because of the fluid loading, however, the load vector is nonzero outside of this region on either side. Finally, we take the load vector and further limit it to just three nonzero terms representing one driving actuator and two confining actuators (Fig. 23). We then use this RHS vector and solve the new system to obtain the displacement vector shown in Fig. 24. This should be more practical (from the standpoint of performing an experiment) than attempting to implement a continuous load distribution. However, it leads to a nonzero displacement vector outside of the confining actuators (Fig. 24).

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x 10

1.8

Displacement amplitude (relative units)

1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0

0

20

40

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Fig. 22. The displacement vector corresponding to the RHS vector in Fig. 21. 1 0.9

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0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

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Fig. 23. A load vector derived from the RHS vector in Fig. 21 containing only three nonzero actuator terms.

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1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0

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Fig. 24. The displacement vector corresponding to the load vector in Fig. 23.

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A.A. Ruffa et al. / Journal of Sound and Vibration 399 (2017) 45–59

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4. Conclusions We have demonstrated vibration confinement on a non-fluid-loaded beam with only two actuators, i.e., one at each end of the confined region. This is done with finite difference approximation and a novel forward/backward solution approach that generates an evanescent solution that exhibits exponential growth of the displacement vector from one end of the beam to a specified node, and then exponential decay from that node to the other end of the beam. We also shown that near-confinement can be achieved on a fluid-loaded beam with only two actuators. The displacement vector is nonzero outside the confined region, with a significantly reduced amplitude. Acknowledgements This work was completed with funding from the Naval Undersea Warfare Center Section 219 internal research program under the supervision of Mr. Neil Dubois.

References [1] C.R. Fuller, S.J. Elliott, P.A. Nelson, Active Control of Vibration, Academic Press, London, 1997. [2] C. Hansen, S. Snyder, X. Qiu, L. Brooks, D. Moreau, Active Control of Noise and Vibration, 2nd ed. CRC Press Boca Raton, FL, 2013. [3] A.S. Yigit, Vibration confinement and suppression in flexible structures by distributed feedback and time delay, in: Proceedings of the 33rd IEEE Conference on Decision and Control, 1994, vol. 2, IEEE, 1994, pp. 1244–1245. [4] A.S. Yigit, S. Choura, Vibration confinement in flexible structures via alteration of mode shapes by using feedback, J. Sound Vib. 179 (4) (1995) 553–567. [5] S. Zhang, Y. Luo, X. Zhang, C. Ma, Mode localization in a satellite system due to parameters detuning, in: Proceedings of the 21st International Congress on Sound and Vibration, Beijing, China, 13–17 July 2014. [6] F.J. Shelley, W.W. Clark, Eigenvector scaling for mode localization in vibrating systems, J. Guid. Control Dyn. 19 (6) (1996) 1342–1348. [7] S. Choura, Control of flexible structures with the confinement of vibrations, J. Dyn. Syst. Meas. Control 117 (2) (1995) 155–164. [8] S. Choura, A.S. Yigit, Confinement and suppression of structural vibrations, J. Vib. Acoust. 123 (4) (2001) 496–501. [9] S. Choura, S. Bouaziz, A.S. Yigit, Control of linear time-varying structures by vibration confinement using limited number of actuators, J. Vib. Control 16 (11) (2010) 1685–1699. [10] A.A. Ruffa, A Solution Approach for Lower Hessenberg Linear Systems, ISRN Applied Mathematics 2011, 2011, 236727. [11] M.A. Jandron, A.A. Ruffa, J. Baglama, A new asynchronous solver for banded linear systems, in: Proceedings of the SIAM Conference on Applied Linear Algebra, Atlanta, GA, 26–30 October 2015. [12] M.A. Jandron, A.A. Ruffa, J. Baglama, An asynchronous direct solver for banded linear systems, Numer. Algorithms (2017) 1–25, http://dx.doi.org/10.1007/s11075-016-0251-3. [13] A.A. Ruffa, M.A. Jandron, B. Toni, Parallelized solution of banded linear systems with an introduction to p-adic computation, in: B. Toni (Ed.), Mathematical Sciences with Multidisciplinary Applications, Springer, 2016, pp. 431–464. [14] C. Guiguo, C.R. Fuller, Active control of sound radiation from a semi-infinite elastic beam with a clamped edge, J. Sound Vib. 168 (3) (1993) 507–523. [15] M.B. Zahui, K. Naghshineh, J.W. Kamman, Narrow band active control of sound radiated from a baffled beam using local volume displacement minimization, Appl. Acoust. 62 (1) (2001) 47–64. [16] N. Levinson, The Wiener RMS (root mean square) error criterion in filter design and prediction, in: N. Wiener (Ed.), Extrapolation, Interpolation, and Smoothing of Stationary Time Series with Engineering Applications, Wiley, New York, 1949, pp. 129–148. [17] W.F. Trench, An algorithm for the inversion of finite Toeplitz matrices, J. Soc. Ind. Appl. Math. 12 (3) (1964) 515–522. [18] S. Zohar, The solution of a Toeplitz set of linear equations, J. ACM 21 (2) (1974) 272–276. [19] D.T. Gavel, Solution to the problem of instability in banded Toeplitz solvers, IEEE Trans. Signal Process. 40 (2) (1992) 464–466. [20] A.J. MacLeod, Instability in the solution of banded Toeplitz systems, IEEE Trans. Acoust. Speech Signal Process. 37 (9) (1989) 1449–1450. [21] B.E. Sandman, Fluid-loaded vibration of an elastic plate carrying a concentrated mass, J. Acoust. Soc. Am. 61 (6) (1977) 1503–1510.