Method for spatially confining vibrational energy

A method for isolating vibrations from a source on a structure includes modeling the structure as a beam having a portion for isolation. A sensor is positioned proximate to the source, and at least one actuator is positioned on the structure between the source and the portion for isolation. A controller receives signals from the sensor and calculates vibrational inputs for each actuator that will isolate the structure portion. Driving signals are provided to each actuator by the controller in response to the calculated vibrational inputs, and each actuator is vibrated accordingly, isolating the structure portion from the source. This method can be implemented in multiple configurations to isolate the structure portion.

BACKGROUND OF THE INVENTION

(1) Field of the Invention

The present invention is directed to a method to confine vibrational energy within a physical structure and more particularly to a method to confine vibrational energy to a beam having various end conditions.

(2) Description of the Prior Art

There has been a considerable amount of work focusing on vibration confinement in flexible structures. It is known to localize individual longitudinal vibration modes utilizing 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). Vibration in a beam leads to a pentadiagonal matrix (i.e., having five diagonals) when modeled using the finite difference method.

FIG. 1illustrates a common vibration confinement problem. This involves isolation of a vibration source10on a beam12. Source10vibrates in a direction shown by arrow14. Beam12has simply supported end conditions16A and16B. The magnitude of vibrations is given by the x coordinate and the distance along the beam is given by the s coordinate. First end16A is at s=0 and has a vibration magnitude of x0. Second end16B is at s=S and has vibration magnitudes of xn+1, where n is the maximum finite difference node index. (Since x0=xn+1=0 for the cases considered here, the vibration amplitude at nodes 0 and n+1 are not modeled.) In this isolation problem, it is desirable to prevent vibrations created by source10from reaching a specified region between source10and end16A or16B.

FIG. 2is a graph of relative vibrations along beam12when subjected to vibrations from source10without modification. In the examples utilized herein, beam12is modeled utilizing finite difference modeling as a series of 3901 nodes. The node index is used as the s coordinate. Source is positioned at node 1951.

Thus, there is a need for a technique for computing isolating active vibration that only requires an actuator for each support. It is also desirable to have a technique for isolating broadband vibration from a specified region between the source and the end of a beam.

SUMMARY OF THE INVENTION

It is a first object of the present invention to provide a method for isolating portions of a structure from vibrations.

Another object is to provide the isolation method in real time utilizing sensor inputs.

Yet another object is to provide such isolation by using a minimum number of actuators.

Accordingly, there is provided a method for isolating vibrations from a source on a structure that includes modeling the structure as a beam having a portion for isolation. The source can represent a single load applied to a single finite difference node, or an arbitrary number of load terms applied to multiple finite difference nodes. In the latter case, the load distribution can be arbitrary in general. A sensor is positioned proximate to the source, and at least one actuator is positioned on the structure between the source and the portion for isolation. A controller receives signals from the sensor and calculates vibrational inputs for each actuator that will isolate the structure portion. Driving signals are provided to each actuator by the controller in response to the calculated vibrational inputs, and each actuator is vibrated accordingly, isolating the structure portion from the source. This method can be implemented in multiple configurations to isolate the structure portion, and the structure can have various end conditions.

DETAILED DESCRIPTION OF THE INVENTION

The vibration of a beam is governed by the Euler-Bernoulli equation, i.e.:

EI⁢∂4⁢x∂s4+ρ⁢⁢A⁢∂2⁢x∂t2=w.(1)
Here E is the Young's modulus for the beam, I is the moment of inertia, ρ is the density, A is the cross-sectional area, w is the applied load at the source, s and t are the independent spatial and temporal coordinates, respectively, and x is the transverse dis-placement. When the load w has a periodic time dependence at a known location s, a harmonic time dependence can be assumed, x(s,t)=x(s)eiωt. A finite difference approximation is used for the spatial derivatives:

∂4⁢x∂s4≈xn-2-4⁢⁢xn-1+6⁢xn-4⁢xn+1+xn+2Δ⁢⁢s4(2)
This leads to

xn-2-4⁢xn-1+6⁢xn+4⁢xn+1+xn+2Δ⁢⁢s4-ω2⁢ρ⁢⁢AEI⁢xn=0(3)
In the example ofFIG. 1, beam12is pinned on both ends16A and16B and cannot move vertically, this leads to the following boundary conditions:
x0=0   (4)

Equation (3) represents a banded Toeplitz system (except for the first and last equations) with a row structure of [1,−4,6+γ,−4,1], where γ=−ω2ρAΔs4/(EI). An analytical solution can be written as follows:
xk=Aeσk+Be−σk+Ccos(σk)+Dsin(σk)  (8)
Where σ=γ1/4. Denoting (3) as ax=b, the coefficient matrix a is:

Consider the case of a square beam with a length of 0.1 meter on each side and vibrating at 100 Hz. The finite difference discretization length is λ/100 (where λ is the wavelength of the vibration in the beam), and n=3901, so the beam effectively contains 39 wavelengths. The load vector is zero except for the term bm, where m=1951 (i.e., at the center of the beam). As described above,FIG. 2shows the resulting solution for this loading of the beam ofFIG. 1

The forward and backward substitution technique for solution of tridiagonal systems can be expanded for use with the pentadiagonal system used here. This is best illustrated with the exemplary tridiagonal Toeplitz system for a longitudinal vibrational system having row structure as given below:

[-2100……………01-2100⋱⋱⋱⋱⋮01-2100⋱⋱⋱⋮001-2100⋱⋱⋮⋮001-2100⋱⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋱⋱⋱01-2100⋮⋱⋱⋱001-210⋮⋱⋱⋱⋱001-210⋯…………001-2].(10)
Consider equation (10) with n=3901 and a single nonzero load term, bm=1, where m=1951. Such a system represents the coefficient matrix for the finite difference approximation to Laplace's equation in one dimension.

The forward substitution process applied to the system having the coefficient matrix in equation (10) begins by assuming x1=1 and solves for x2using the first equation:
−2x1+x2=0  (11)
Next, x3is determined with the second equation:
x1−2x2+x3=0  (12)

The forward substitution process is continued using each successive equation to find xkfor 1≤k≤1951 until x1951is determined with the equation:
x1949−2x1950+x1951=0.  (13)

Similarly, backward substitution begins by assuming x3901=1 and then using each successive equation to find xkfor 3901≥k≥1951 until x1951is determined with the equation:
x1951−2x1952+x1953=0.  (14)

Next, the solutions are scaled so that x1951found by forward and backward substitution are equal. This can be done since the load vector is zero for 1≤k≤m−2 and m+2≤k≤n. This leaves the equation:
x1950−2x1951+x1952=Bm.  (15)
The term Bmis found from (15), and then the entire solution is scaled by the factor bm/Bmto obtain the solution.

Extending the forward and backward substitution approach to the pentadiagonal beam problem leads to the result shown inFIG. 3A. The eσkand e−σkterms in equation (8) dominate the solution for large systems. There are three remaining equations:
x1948−4x1949+(6+γ)x1950−4x1951+x1952=B1950;  (16)
x1949−4x1950(6+γ)x1951−4x1952+x1953=B1951; and  (17)
x1950−4x1951+(6+γ)x1952−4x1953+x1954=B1952.  (18)
Equations (16)-(18) lead to B1951=0.2510 and B1950=B1952=−0.1257. This load is shown onFIG. 3B. Note that the node reference label has been changed to focus in on the added terms. The calculated error indicates that this is an accurate solution. B1950=B1952≠0 leads to the evanescent response, and in any case actuation at these points would be difficult to implement, since the terms are only spaced by a hundredth of a wavelength; therefore, a different approach is needed for the beam problem to effectively contain vibrations within a region of the beam.

It is possible to create a solution so that xk≠0 in general for a limited range of values of k, and xk=0 otherwise. For example, introducing dynamic load terms at k=1751 and k=2151 leads to xk≠0 for 1751≤k≤2151, and xk=0 for all other values of k. These load terms represent the load terms required at k=1751 and k=2151 required to provide a confined vibrational energy state in the structure. Consider the following equation:
x1749−4x1750+(6+γ)x1751−4x1752+x1753=B1751.  (19)

The presence of b1751supports setting xk=0 for k≤1752 by setting x1753=b1751. Likewise, x2149=b2151, so that xk=0 for k≥2150 using the following equation:
x2149−4x2150+(6+γ)x2151−4x2152+x2153=B2151.  (20)

Forward substitution is then performed for 1751≤k≤1951 and backward substitution is performed for 1951≤k≤2151. These two solutions are then matched at k=m=1951. However, two independent solutions are needed for both 1751≤k≤m and m≤k≤2151. The second solution is developed with a forward solution for 1≤k≤m generated with a zero load vector. This is an exponentially growing solution; however, it can be scaled and subtracted with the forward confined solution to produce a second forward solution that is independent of the first. This can be done since the load vector corresponding to the second solution is zero. A similar procedure can develop an independent back solution.

The solutions represented byFIGS. 3A and 3B(obtained by forward and backward substitution of the system having a coefficient matrix represented by equation (9) and load terms that are the subsequent solution of equations (16)-(18)) are superimposed with these two solutions to obtain two independent forward and back solution vectors, i.e., f1, f2, and b1and b2. A set of four equations are then developed to match the forward and back solutions. The four unknowns are α1, α2, β1, and β2, which govern the contributions of the forward and back solutions of f1, f2, and b1and b2. The equations are as follows:
α1f19511+α2f19512=β1b19511+β2b19512;  (21)
α1f19481+α2f19482−4(α1f19491+α2f19492)+(6+γ)(α1f19501+α2f19502)−4(β1b19511+β2b19512)+β1b19521+β2b19522=0;  (22)
α1f19491+α2f19492−4(α1f19501+α2f19502)+(6+γ)(α1f19511+α2f19512)−4(β1b19521+β2b19522)+β1b19531+β2b19532=b1951; and  (23)
α1f19501+α2f19502−4(α1f19511+α2f19512)+(6+γ)(α1f19521+α2f19522)−4(β1b19531+β2b19532)+β1b19541+β2b19542=0.  (24)

This leads to the solutionFIG. 4Aand the associated source termsFIG. 4B. As can be seen fromFIG. 4A, displacements are contained between node 1751 and 2151.FIG. 4Bshows relative magnitude and positioning of the original load and two additional loads at these nodes.

Since the load source terms can be introduced anywhere in the system, it follows that vibrational energy can be confined to a chosen region on the beam by this approach. This approach is general in nature and can be extended to two dimensional and three dimensional problems. The number of introduced sources will be N=(BW−1)/2, where BW is the bandwidth of the coefficient matrix. There are some physical limitations on this. It is noted that by applying these new load terms, energy in the beam is being confined to a shorter length of beam, resulting in greater deflections in the shorter beam length. If the vibration amplitude becomes great enough so that it becomes nonlinear, then this technique cannot be fully effective because it is based on solutions to the Euler-Bernoulli equation, which assumes linear vibrations.

FIG. 5shows an embodiment for confining a vibration to a region in the center of a beam. Of course, this is applicable to other structures that can be modeled as beams. To build an apparatus based on these principles, a sensor18on beam12is used to detect vibrations14of noise source10. Sensor18can be an accelerometer or any other sensor capable of measuring vibrations14. Sensor18provides data to a controller20. Controller20is joined to actuators22A and22B. Actuators22A and22B are joined to beam12to provide vibrational loading at the actuator's mounting position on beam12. To simulate the finite difference model, actuators22A and22B should apply loads to the beam with a small spatial extent. For the case presented here, each finite difference node is associated with a spatial extent of λ/100 where λ is an estimated wavelength of the applied load. The width of each confining actuator probe in contact with the beam should not exceed this width. Controller20receives vibration signal14from sensor18and utilizes signal14with beam model as described above for calculating output signals to actuators22A and22B. Actuators22A and22B provide vibrational loading in combination with that provided by source10that will result in vibrations being contained in beam12between the location of actuator22A and that of actuator22B.

FIG. 6provides an alternate embodiment having multiple sources10′. Each source10′ is at a known position s and has an associated vibrational loading14′. A sensor18is associated with each source10′. Controller20′ utilizes a beam model to calculate signals for actuators22A and22B that will result in vibrations being contained in the region of beam12between actuator22A and actuator22B.

FIG. 7provides a third embodiment having a single source10but only one actuator22A. In this embodiment, actuator22A blocks vibrations from propagating to end16A while vibrations are free to propagate to end16B. Controller20receives a vibrational signal14from sensor18positioned at source10. Controller20utilizes this signal with a predetermined beam model to calculate a blocking vibrational load signal which is provided to actuator22A. Vibrations14are blocked from the region of beam12between end16A and actuator22A location. End16B is subjected to the resulting vibrations from source10and actuator22A. The solution is given inFIG. 8Afor the load terms shown inFIG. 8B.

In the example the noise sources all had the same phase, so all of the load terms were real. This will not be true in general, which will mean that the introduced load terms will be complex, each will have a computed amplitude and phase.

Note that each set of noise sources can be isolated individually or all of them can be isolated with just two introduced sources, as long as the vibrations are linear (so that superposition holds). Although just using two introduced sources is simpler, multiple sets of introduced sources may lead to better results (i.e., by isolating multiple regions individually rather than isolating all the sources as a single larger region) when the noise sources that need to be suppressed are distributed over most of the beam.

An extension of the procedure can suppress noise sources on a shell, which is a two dimensional extension of a beam. In this case, the noise sources to be isolated are then measured with accelerometers and the results are put into a model. Sources are then introduced in the banded matrix that will surround the noise sources. The amplitude and phase of the introduced sources will again be computed that will isolate the noise sources.

The principal advantage is the ability to confine vibrational energy to any region on a beam by introducing two source terms whose values are calculated by this approach. This can be useful in sonar and other acoustic applications to isolate and remove self-noise due to machinery and other sources.

Although the example here focused on beam vibration, this approach will work on any system that leads to a banded coefficient matrix. For example, it will work for one, two, and three dimensional vibrational problems, as well as any other system that can be modeled with a partial differential equation via a numerical approach (e.g., finite differences or finite elements) that leads to a banded coefficient matrix.

FIG. 9shows application of this technique to a cantilevered beam30having a cantilevered end32A and a distal end32B. A vibration source10is positioned proximate distal end32B. Vibration source10provides a vibration loading14. A sensor18detects vibrational loading14and provides a sensor signal to controller20. Controller20calculates a blocking vibration signal from the received vibration loading signal and a predetermined model of the cantilevered beam30, as described above. Blocking vibration signal is provided to actuator34in communication with controller20. Vibrational loading from actuator34blocks vibrations from source10from reaching cantilevered end32A of beam30.