US 20050072234 A1 Abstract A system and method for detecting structural damage is provided that utilizes a general order perturbation methodology involving multiple perturbation parameters. The perturbation methodology is used iteratively in conjunction with an optimization method to identify the stiffness parameters of structures using natural frequencies and/or mode shape information. The stiffness parameters are then used to determine the location and extent of damage in a structure. A novel stochastic model is developed to model the random impact series produced manually or to generate a random impact series in a random impact device. The random impact series method or the random impact device can be used to excite a structure and generate vibration information used to obtain the stiffness parameters of the structure. The method or the device can also just be used for modal testing purposes. The random impact device is a high energy, random, and high signal-to-noise ratio system.
Claims(46) 1. A system for determining stiffness parameters of a structure, comprising:
a sensor arranged to measure vibrations of said structure and output vibration information; and a stiffness parameter unit for receiving said vibration information, determining natural frequency data of said structure, and determining the stiffness parameters of said structure using said natural frequency data. 2. The system according to 3. The system according to 4. The system according to 5. The system according to 6. The system according to 7. A system for determining stiffness parameters of a structure, comprising:
a sensor arranged to measure vibrations of said structure and output vibration information; and a stiffness parameter unit for receiving said vibration information and determining said stiffness parameters with an iterative processing unit. 8. The system according to 9. The system according to 10. The system according to 11. A stiffness parameter unit for determining stiffness parameters for a structure, comprising:
an input for receiving vibration data related to the structure; an analyzer for converting said vibration data to spectral data; and an interative processing unit for receiving said spectral data and outputting said stiffness parameters using natural frequencies of the structure. 12. A stiffness parameter unit for determining stiffness parameters for a structure, comprising:
an input for receiving vibration data related to the structure; an analyzer for converting said vibration data to spectral data; and an interative processing unit for receiving said spectral data and outputting said stiffness parameters using a perturbation process. 13. The stiffness parameter unit according to 14. The stiffness parameter unit according to 15. A system for determining damage information of a structure, comprising:
a sensor arranged to measure vibrations of said structure and output vibration information; a stiffness parameter unit for receiving said vibration information, determining natural frequency data of said structure, and determining the stiffness parameters of said structure using said natural frequency data; and a damage information processor for receiving said stiffness parameters and outputting damage information. 16. The system according to 17. A system, comprising:
a structure; a sensor arranged to measure vibrations of said structure and output vibration information; and a stiffness parameter unit for receiving said vibration information, determining natural frequency data of said structure, and determining the stiffness parameters of said structure using said natural frequency data. 18. The system according to 19. The system according to 20. The system according to 21. The system according to 22. The system according to 23. The system according to 24. The system according to 25. The system according to 26. The system according to 27. The system according to 28. The system according to 29. The system according to 30. The system according to 31. The system according to 32. A device, comprising:
a random signal generating unit for generating first and second outputs; a random impact actuator for receiving said first and second outputs; and an impact applicator coupled to said random impact actuator and having an impact region; wherein said random impact actuator drives said impact applicator such that the force and arrival times of said impact applicator at said impact region are random. 33. The device of 34. The device of 35. The device of 36. A system, comprising:
a structure; a random impact device for inducing vibrations in said structure; a sensor arranged to measure vibrations of said structure and output vibration information; and a stiffness parameter unit for receiving said vibration information, determining natural frequency data of said structure, and determining the stiffness parameters of said structure using said natural frequency data. 37. The system of a random signal generating unit for generating first and second outputs; a random impact actuator for receiving said first and second outputs; and an impact applicator coupled to said random impact actuator and having an impact region; wherein said random impact actuator drives said impact applicator such that the force and arrival times of said impact applicator at said impact region are random. 38. The device of 39. The device of 40. The device of 41. A system for determining stiffness parameters of a structure, comprising:
a sensor arranged to measure vibrations of said structure and output vibration information; and a stiffness parameter unit for receiving said vibration information, determining mode shape information, and determining the stiffness parameters of said structure using said mode shape information. 42. The system according to 43. The system according to 44. The system according to 45. The system according to 46. The system according to Description This application claims priority to U.S. Provisional Patent Application Ser. No. 60/471,873 filed May 20, 2003 and U.S. Provisional Patent Application Ser. No. 60/512,656 filed Oct. 10, 2003, which are both incorporated herein by reference. 1. Field of the Invention The invention relates to a method and apparatus for detecting structural damage, and, more specifically, to a method and apparatus for detecting structural damage using changes in natural frequencies and/or mode shapes. 2. Background of the Related Art Damage in a structure can be defined as a reduction in the structure's load bearing capability, which may result from a deterioration of the structure's components and connections. All load bearing structures continuously accumulate structural damage, and early detection, assessment and monitoring of this structural damage and appropriate removal from service is the key to avoiding catastrophic failures, which may otherwise result in extensive property damage and cost. A number of conventional non-destructive test (NDT) methods are used to inspect load bearing structures. Visual inspection of structural members is often unquantifiable and unreliable, especially in instances where access to damaged areas may be impeded or damage may be concealed by paint, rust, or other coverings. Penetrant testing (PT) requires that an entire surface of the structure be covered with a dye solution, and then inspected. PT reveals only surface cracks and imperfections, and can require a large amount of potentially hazardous dye be applied and disposed of. Similarly, magnetic particle testing (MT) requires that an entire surface of the structure be treated, can be applied only to ferrous materials, and detects only relatively shallow cracks. Further, due to the current required to generate a strong enough magnetic field to detect cracks, MT is not practically applied to large structures. Likewise, eddy current testing (ET) uses changes in the flow of eddy currents to detect flaws, and only works on materials that are electrically conductive. Ultrasonic testing (UT) uses transmission of high frequency sound waves into a material to detect imperfections. Results generated by all of these methods can be skewed due to surface conditions, and cannot easily isolate damage at joints and boundaries of the structure. Unless a general vicinity of a damage location is known prior to inspection, none of these methods are easily or practically applied to large structures which are already in place and operating. On the other hand, resonant inspection methods are not capable of determining the extent or location of damage, and is used only on a component rather than a assembled structure. None of the above NDT methods are easily or practically applied to large structures requiring a high degree of structural integrity. Because of these shortfalls in existing NDT methods when inspecting relatively large structures, structural damage detection using changes in vibration characteristics has received much attention in recent years. Vibration based health monitoring for rotating machinery is a relatively mature technology, using a non-model based approach to provide a qualitative comparison of current data to historical data. However, this type of vibration based damage testing does not work for most structures. Rather, vibration based damage detection for structures is model based, comparing test data to analytical data from finite element models to detect the location(s) and extent of damage. Vibration based damage detection methods fall into three basic categories. The first of these is direct methods such as optimal matrix updating algorithms, which identify damage location and extent in a single iteration. Because of the single iteration, these methods are not accurate in detecting a large level of damage. The second category is iterative methods. The methodology has only been for updating modeling, which determines modified structural parameters iteratively by minimizing differences between model and test data. The third category includes control-based eigenstructure assignment methods, which have the similar limitation to that of the direct methods indicated above and are not accurate in detecting a large level of damage. None of these current vibration based methods have been incorporated into an iterative algorithm that can detect small to large levels of damage, and the vibration based approach for structures remains an immature technology area which is not readily available on a commercial basis. An object of the invention is to solve at least the above problems and/or disadvantages and to provide at least the advantages described hereinafter. Another object of the invention is to provide a system and method for detecting structural damage based on changes in natural frequencies and/or mode shapes. An advantage of the system and method as embodied and broadly described herein is that it can be applied to a large operating structure with a large number (thousands or more) of degrees of freedom. Another advantage of the system and method as embodied and broadly described herein is that it can accurately detect the location(s) and extent of small to large levels of damage and is especially useful for detecting a large level of damage with severe mismatch between the eigenparameters of the damaged and undamaged structures. Another advantage of the system and method as embodied and broadly described herein is that it can work with a limited number of measured vibration modes. Another advantage of the system and method as embodied and broadly described herein is that it can use measurement at only a small number of locations compared to the degrees of freedom of the system. A modified eigenvector expansion method is used to deal with the incomplete eigenvector measurement problem arising from experimental measurement of a lesser number of degrees of freedom than that of the appropriate analytical model. Another advantage of the system and method as embodied and broadly described herein is that it can be applied to structures with slight nonlinearities such as opening and closing cracks. The random shaker test or the random impact series method can be used to average out slight nonlinearities and extract linearized natural frequencies and/or mode shapes of a structure. Another advantage of the system and method as embodied and broadly described herein is that it can handle structures with closely spaced vibration modes, where mode switching can occur in the damage detection process. Another advantage of the system and method as embodied and broadly described herein is that it can handle different levels of measurement noise with estimation errors within the noise levels. Another advantage of the system and method as embodied and broadly described herein is that the damage detection method and the vibration testing methods such as the random impact series method enables damage detection and assessment to be automated, thus improving the reliability/integrity of results. Another advantage of the system and method as embodied and broadly described herein is that damage detection and assessment may be automated in the field so that structural health can be monitored at central location and useful service life may be optimized. Another advantage of the system and method as embodied and broadly described herein is that the random impact series method enables the modal parameters such as natural frequencies and/or mode shapes to be measured for a large structure or a structure in the field when there are noise effects such as those arising from the wind or other ambient excitation. Additional advantages, objects, and features of the invention will be set forth in part in the description which follows and in part will become apparent to those having ordinary skill in the art upon examination of the following or may be learned from practice of the invention. The objects and advantages of the invention may be realized and attained as particularly pointed out in the appended claims. The invention will be described in detail with reference to the following drawings in which like reference numerals refer to like elements wherein: Commonly measured modal parameters, such as natural frequencies and mode shapes, are functions of physical properties of a particular structure. Therefore, changes in these physical properties, such as reductions in stiffness resulting from the onset of cracks or a loosening of a connection, will cause detectable changes in these modal parameters. Thus, if the changes in these parameters are indicators of damage, vibration based damage detection may be, simplistically, reduced to a system identification problem. However, a number of factors have made vibration based damage detection difficult to implement in practice in the past. The system and method for detecting structural damage as embodied and broadly described herein is motivated by the observed advantages of vibration based damage detection over currently available technologies. It is well understood that this system and method may be effectively applied to damage detection and assessment for substantially all types and configurations of structures, including, but not limited to, simple beams, hollow tubes, trusses, frames, and the like. However, simply for ease of discussion, the system and method will first be discussed with respect to three examples—a mass-spring model, a beam, and a space frame—for conceptualization purposes. The system and method will later be applied lightning masts in electric substations. Stiffness parameter unit System The system and method may include a multiple-parameter, general order perturbation method, in which the changes in the stiffness parameters are used as the perturbation parameters. By equating the coefficients of like-order terms involving the same perturbation parameters in the normalization relations of eigenvectors and the eigenvalue problem, the perturbation problem solutions of all orders may be derived, and the sensitivities of all eigenparameters may be obtained. The perturbation method may be used in an iterative manner with an optimization method to identify the stiffness parameters of structures. Methodology This method presented below can simultaneously identify all the unknown stiffness parameters and is formulated as a damage detection problem. Since the effects of the changes in the inertial properties of a damaged structure are usually relatively small, only the changes in the stiffness properties due to structural damage are considered. Consider a N degree-of-freedom, linear, time-invariant, self-adjoint system with distinct eigenvalues. The stiffness parameters of the undamaged structure are denoted by G Let the k-th eigenvalue and mass-normalized eigenvector of the damaged structure be related to λ Using the normalization relations of the eigenvectors, φ Equating the coefficients of the δG Substituting (4)-(6) into (3) yields
Equating the coefficients of the δG Equating the coefficients of the δG We proceed now to derive the perturbation solutions for the general p-th order terms in (5) and (6). Equating the coefficients of the δG Equations (5) and (6) serve both the forward and inverse problems. In the former one determines the changes in the eigenparameters with changes in the stiffness parameters. Damage detection is treated as an inverse problem, in which one identifies iteratively the changes in the stiffness parameters from a selected set of measured eigenparameters of the damaged structure: λ Next, an optimization method may be used to find the changes in the stiffness parameters δG If the absolute values are not all less than ε, the process proceeds along an iterative path where the stiffness parameters are first updated (Block 7). The stiffness parameters are then bounded between 0 or ε Optimization Methods Neglecting the residuals of order p+1 in (5) and (6) yields a system of polynomial equations of order p. When n To minimize the objective function in (35) at the w-th iteration, one can use the algorithm
Quasi-Newton Methods Due to its successive linear approximations to the objective function, the gradient method can progress slowly when approaching a stationary point. The quasi-Newton methods can provide a remedy to the problem by using essentially quadratic approximations to the objective function near the stationary point. The iteration scheme of these methods is given by (36) with f Step Size Search Procedure The optimal step size for the gradient and quasi-Newton methods is determined in each nested iteration to minimize the function
Initial Bracketing. Choose the starting point x Golden Section Search. If |x The algorithm discussed above is used to identify the stiffness parameters of a N-degree-of-freedom system consisting of a serial chain of masses and springs, such as the system shown in We look at a forward problem first with N=3 and m=4. The stiffnesses of the damaged system are G
Consider now the damage detection problem with N=9, m=10, G 407 n=1; n=2; n=3; n=4; n=5; n=6; n=7 n=8; n=9. While the use of the second-order perturbations improves the accuracy of stiffness estimation in each iteration and reduces the number of iterations, it takes a much longer time to compute the higher-order perturbations and the associated optimal solutions.
When only the first few eigenvalues are used, for instance, n If the system has a large level of damage, i.e., G Finally, consider a large order system with a large level of damage: N=39, m=40, G The algorithm discussed above may be applied to detecting structural damage in an aluminum beam with fixed boundaries. The beam of length L Consider first the cases with N When only the translational degrees of freedom of an eigenvector are measured, a modified eigenvector expansion method is used to estimate the unmeasured rotational degrees of freedom. To this end, φ Finally, the effects of measurement noise on the performance of the algorithm are evaluated for the 10-element beam with the large level of damage. Simulated noise is included in the measured eigenparameters:
Space Frame Example The damage detection method can be applied to more complex structures, such as, for example, the modular, four bay space frame shown in In summary, the damage detection method identifies stiffness parameters in structures, which have a small, medium, and large level of damage if the maximum reduction in the stiffnesses is within 30%, between 30 and 70%, and over 70%, respectively: A large level of damage is studied in many examples because this poses the most challenging case, with sever mismatch between the eigenparameters of the damaged and undamaged structures. The damage detection method as embodied and broadly described herein can be applied to structures that can be modeled with beam elements. A beam element is an element that has one dimension that is much longer than the other two. This element is very good at modeling “I”-beams, rectangular beams, circular beams, “L”-angles, “C”-channels, pipes, and beams with varying cross sections. Structures that can be modeled with this element include, but are not limited to, lightning masts, light poles, traffic control poles, pillar type supports, bridges, pipelines, steel building frameworks, television, radio, and cellular towers, space structures, cranes, pipelines, railway tracks, and vehicle frames. Structures that can be modeled as beam elements are used simply for ease of discussion, and it is well understood that the damage detection method discussed above may be applied to structures that can be modeled by other elements using the finite element method or modeled by using other methods. Damage Detection Using Changes of Natural Frequencies: Simulation and Experimental Validation For structures such as beams and lightning masts in electric substations, using only the changes in the natural frequencies can relatively accurately detect the location(s) and extent of damage, even though the system equations are severely underdetermined in each iteration. This is an interesting finding as it is much easier to measure the natural frequencies than the mode shapes, and demonstrates the effectiveness of the iterative algorithm. Extensive numerical simulations on beams and lightning masts confirmed this finding. Experiments on the beam test specimens with different damage scenarios and a lightning mast in an electric substation validated the simulation results. The beam test specimens and the lightning mast are used as examples for demonstration purposes, and the method can be applied to other structures. Note that unlike the beam example shown earlier, where the Euler-Bernoulli beam finite element model is used, the Timoshenko beam finite element model is used in all the examples here. The Timoshenko beam theory is found to be more accurate in predicting the natural frequencies of the lightning masts and circular beams than the Euler-Bernoulli beam theory. For a cantilever beam, simulation results show that the damage located at a position within 0-35% and 50-95% of the length of the beam from the cantilevered end can be easily detected with less than 5 measured natural frequencies, and the damage located at a position within 35-50% of the length of the beam from the cantilevered end and at a position within 5% of the length of the beam from the free end can be relatively accurately detected with 10-15 measured natural frequencies. Numerical and Experimental Verification Cantilever Aluminum Beams Experimental damage detection results for four different scenarios are shown first, followed by various simulation results. Scenario 1: Evenly-Distributed Damage Machined from the Top and the Bottom Surfaces of the Beam Test Specimen. The aluminum beam test specimen shown in Scenario 2: The Same Aluminum Beam Test Specimen as in Scenario 1 Clamped at the Other End. The same beam was tested in the clamped-free configuration with the clamped end reversed, which placed the damage from 25 cm to 30 cm (from the 23 Scenario 3: Undamaged Cantilever Aluminum Beam Test Specimen with the Same Dimensions as Above. An undamaged aluminum beam test specimen was clamped at one end with the same configuration as shown in Scenario 4: A Cut of Small Width on a Cantilever Aluminum Beam Test Specimen, Shown in The beam shown below has a cut that is 0.4191 cm deep and 0.1016 cm wide, which corresponds to a 96% reduction in bending stiffness at the cut. The beam is divided into 45 elements and the cut is located in the middle of the 23 To examine the effectiveness and robustness of the damage detection algorithm, various simulations with different damage scenarios were carried out. In this way, we can gain more insight concerning the accuracy of the finite element model, convergence of the estimated bending stiffnesses of all the elements of the beam with the increased numbers of measured natural frequencies and/or mode shapes, and region of the beam within which the damage can be detected with few measured frequencies. With the cantilever aluminum beam divided into 40 elements, the following two simulations have the similar damage location and extent to those in Scenarios 1 and 2 in the experiments: Simulation 1: Uniform Damage Between 9.0 cm and 15.75 cm from the Cantilevered End of the Beam with the Same Dimensions as Discussed Above. Simulation 2: Uniform Damage Between 29.25 cm and 36 cm from the Cantilevered End of the Beam with the Same Dimensions as Discussed Above. Simulation results in With the cantilever aluminum beam divided into the same number of elements, two more simulations are presented here: one for a multiple damage scenario—70% of damage at the 3 Simulation 3: Multiple Damage of the Beam with the Same Dimensions as Discussed Above: 70% of Damage at the 3 Simulation results in Simulation 4: Uniform Damage from the 16 Simulation results in Lightning Masts The lighting mast shown in
Damage detection was then performed using only the first 4 to 7 measured natural frequencies, as shown in Table 2. The mast is modeled by 40 elements; the lower section has 18 elements, the upper section has 15 elements, and the spike has 7 elements. The experimental damage detection results are shown in Simulations for the same lightning mast as shown in Methods to Handle Some Ill-Conditioned System Equations When the first order perturbation approch is used, one needs to solve in each iteration a system of linear algebraic equations
Ill-conditioning problems do not occur in all the examples described above. Sometimes they can occur. Consider, for example, a cantilever aluminum beam of the same dimensions as those of the beam shown in -
- Method 1. Estimate A
^{+}from (A^{T}A+ηI)^{−1 }A^{T}, where η is a small positive constant that can be searched in several ways. One way is set η=η×1.618×min(**10,∥δG∥**_{∞}), where μ·μ_{∞}denotes the infinity norm, with the initial value η=∥A^{T}A∥_{∞}, until ∥δG∥_{∞}is less than 0.8, for example. - Method 2. To constrain the magnitude of δG, we include it in the objective function to be minimized. For example, we can minimize the following objective function
$\begin{array}{cc}f\left(\delta \text{\hspace{1em}}G\right)=\sum _{i=1}^{{N}_{o}}\text{\hspace{1em}}\sum _{j=1}^{m}\text{\hspace{1em}}{\left({F}_{i}-{A}_{\mathrm{ij}}\delta \text{\hspace{1em}}{G}_{j}\right)}^{2}+\sum _{j=1}^{m}{\left(\delta \text{\hspace{1em}}{G}_{i}\right)}^{2}& \left(48\right)\end{array}$ instead of (35), where N_{o}=n_{λ}+n_{φ}N_{m }is the number of equations in (5) and (6). The optimal solution can be obtained by using the generalized inverse method for the expanded system A*=[A;I] and the expanded vector F*=[F;0], where I is the m×m identity matrix and 0 is the m×1 zero vector.
- Method 1. Estimate A
Note that the solutions from the regularization methods are not strictly the optimal solutions for the original objective function in (35). With accurate and sufficient measurement information and proper handling of the ill-conditioning problem, the system equations can become well conditioned in the last few iterations and regulation does not need to be applied. Consequently the stiffness parameters can be more accurately determined. Sometimes regulation may over constrain the magnitude of δG, and the termination criterion ∥δG∥ Using only the translational degrees of freedom of the first eigenvector and the first regulation method, the estimated bending stiffnesses of all the elements of the beam are shown in Similarly, using the translational degrees of freedom of the first eigenvector and the second regulation method, the estimated bending stiffnesses of all the elements of the beam are shown in Conclusions Thus, the sensitivities of eigenparameters of all orders may, for the first time, be derived using a multiple-parameter, general-order perturbation method. The higher-order solutions may be used to estimate the changes in the eigenparameters with large changes in the stiffness parameters. The perturbation method may be combined with an optimization method to form a robust iterative damage detection algorithm. The gradient and quasi-Newton methods can be used for the first or higher order system equations, and the generalized inverse method can be used efficiently with the first order system equations because it does not involve nested iterations. Including the higher-order perturbations can significantly reduce the number of iterations when there is a large level of damage. A modified eigenvector expansion method is used to estimate the unmeasured component of the measured mode shape. For many cases, the location(s) and extent of damage can be relatively accurately detected using only measured natural frequencies. Methods to handle ill-conditioned system equations that may occasionally arise are developed and shown to be effective. Numerical simulations on different structures including spring-mass systems, beams, lightning masts, and frames show that with a small number of measured eigenparameters, the stiffness parameters of the damaged system may be accurately identified in all the cases considered. Experiments on the different beam test specimens and the lightning mast in an electric substation validated the theoretical predictions. The methodology can be readily applied to various operation structures of different sizes by incorporating their finite element models or other mathematical models. One example of a practical application of this damage detection method is shown in The structure In accordance with the method as described above, upon receipt of the response signal from the sensor Although the convergence of the system to resolution is not dependent on where the impact F is applied to the structure The structure Different methods have been employed in conventional vibration testing in order to excite a test specimen. Shaker testing, in which a specimen is, simplistically, shaken in order to impart a high level of energy, can produce a high signal to noise ratio, and can induce random excitation, which can average out slight nonlinearities and extract linearized eigenparameter parameters. However, shaker testing is not practically employed in the field on relatively large structures, and can be cost prohibitive to conduct. Single impact hammer testing addresses the shortfalls of shaker testing, in that it is portable and inexpensive to conduct. However, single impact hammer testing falls short where shaker testing is strong, in that low energy input of single impact hammer testing produces a low energy input, a low signal to noise ratio with no randomization. To address the need for a system which combines the advantages of shaker testing and single impact hammer testing, a Random Impact Series method for hammer testing is presented which yields a high energy, high signal to noise ratio, random system. A novel stochastic model is developed to simulate the random impact series produced manually and to generate a random impact series for a specially designed random impact device. Output Impact applicator Although random signal generating unit As discussed above, vibration information is obtained by sensor Experiments conducted on lightning masts by the applicant confirm that multiple impact testing performs better than single impact testing when there is wind excitation to the masts. Results are shown for a 65 foot tall mast with a 5 foot spike, as shown in A stochastic model will now be discussed which describes the random impact series F Stochastic Model of a Random Impact Series A random impact series is modeled here as a sum of force pulses with the same shape and random amplitudes and arrival times:
Since a finite time record is used for the force signal in modal testing, we consider only the pulses that arrive during the time interval (0,T] of length T, as shown in For the Poisson process N(t) with stationary increments, the probability of the event {N(t)=n}, where n is an integer, is
The force spectrum is the Fourier transform of the force signal in (50):
The average power density of the force signal in (50) is defined as
The first-order cumulant function of x(t) in (50), κ _{xx}(t _{1} ,t _{2})=λE[ψ _{1} ^{2}]∫_{0} ^{T} y(t _{1}−α)y(t _{2}−α)dα (78)
where tε[0,T+Δτ]. Let t−α=u in (75), then dα=−du. We have from (77) Let W(t)=∫_{0} ^{t} y(u)du (80)
Noting that y(u)=0 when u<0 and u>Δτ, we have from (79) where T>Δτ is assumed. Let t _{1}−α=u and t_{2}−t_{1}=k in (78), then dα=−du and t_{2}−α=u+k. We have from (78)
When |k|>Δτ, y(u)y(u+k)=0 and hence κ _{xx}(t_{1}, t_{2})=0. When 0≦k≦Δτ, we have from (82) for different t_{1 }
When −Δτ≦k<0, we have from (82) for different t _{1 }
Let u+k=ν in (84), then u=ν−k and du=dν. We have from (84) after changing ν back to u Combining the second equations in (83) and (85), we have for t _{1 }and t_{2 }in [Δτ,T]
Since by the second equation in Eq. (81), E[x(t)] is a constant for t ε[Δτ, T], and by (86), κ Applying the Fourier transform to E[x(t)] in Eq. (81) yields
We will show that F{E[x(t)]} in (92) is equivalent to E[X(jω)] in (91). By (80), we have
Equation (92) consists of two parts: the first part,
Since x(t) is stationary in [Δτ,T], the average power density of x(t) in [Δτ,T] is defined as
The power spectral density of x(t) can be obtained from (100) by increasing T to infinity
S _{x}(ω)=lim/T→∞E[S_{2}(ω)]=∫_{−∞} ^{∞} R _{xx}(k)e ^{jωk} dk=∫_{−∞} ^{∞} R _{xx}(k)e ^{−jωk} dk=F[R _{xx}(k)] (103)
where R _{xx}(−k)=R_{xx}(k) has been used. Equation (103) is the well-known Wiener-Khintchine theorem, which states the power spectral density is the Fourier transform of the autocorrelation function. Substituting (87) into (83), and noting that I(u)I(u+|k|)=0 when |k|>Δτ and the Fourier transform of 1 is 2πδ(ω), yields (101). Note that the power spectral density is only defined for a wide-sense stationary process with an infinite time record. When the mean amplitude of each pulse E[ψ_{1}] is not equal to zero, there is an associated delta function in the power spectral density.
Comparison of E[S _{1}(ω))] and E[S_{2}(ω)]
By (100), E[S _{1}(ω)] in (75) consists of two parts, the first part,
depends also on the shape function y() as the shape function is used in calculating the power associated with the nonstationary parts of x(t) in [0,Δτ] and [T, T+Δτ]. When the shape function is a delta function (i.e., Δτ→0), the nonstationary parts of x(t) vanish and E[S When the shape function of the pulses is represented by a half sine wave, i.e.,
Consider next the normalized shape function y(t) shown in _{1}]=0.8239N, and E[ψ_{1} ^{2}]=0.7163N^{2}. The curve for E[x(t)] in the time interval from 0 to 8.15625 s, shown as a solid line in _{1}(jω))]} in the same frequency range, shown as a solid line in _{1}(jω)]} with λ=1/s and the other parameters unchanged is shown as a dashed line in _{1}(jω)] increases by 4.14 to 15.6247 times in the frequency range shown when λ is increased from λ_{2}=1/s to λ_{1}=4.14/s. This result can be shown by using (75)
This shows that a larger arrival rate λ would increase the energy input to the structure over the entire frequency domain. Numerical simulation is undertaken next to validate the analytical predictions. The random number N(T) satisfying the Poisson distribution in (51) with λ=4.14/s and T=8 s is generated using MATLAB. Similarly, the random numbers corresponding to the random variables τ The stochastic model was experimentally validated for an experimenter conducting manually a random series of impacts on the four bay space frame as shown in Thus, the system and method for detecting structural damage and the random impact series method as embodied and broadly described herein can be applied to an unlimited number and type of structures to provide automated, reliable damage detection and assessment and to conduct modal testing. This system could be further automated to conduct periodic tests and provide results to a centralized monitoring section. Regular health monitoring of these types of structures could provide additional protection against potential failure, as well as a characterization of usage and wear over time in particular environmental conditions for predicting useful service life. The foregoing embodiments and advantages are merely exemplary and are not to be construed as limiting the present invention. The present teaching can be readily applied to other types of systems. The description of the present invention is intended to be illustrative, and not to limit the scope of the claims. Many alternatives, modifications, and variations will be apparent to those skilled in the art. In the claims, means-plus-function clauses are intended to cover the structures described herein as performing the recited function and not only structural equivalents but also equivalent structures. Referenced by
Classifications
Legal Events
Rotate |