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Frequency response based sensor placement for the mid-frequency range
ARTICLE IN PRESS Mechanical Systems and Signal Processing 23 (2009) 1169–1179
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Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/jnlabr/ymssp
Frequency response based sensor placement for the mid-frequency range S. Nimityongskul, D.C. Kammer Department of Engineering Physics, University of Wisconsin, 1500 Engineering Drive, Madison, WI 53706, USA
a r t i c l e in fo
abstract
Article history: Received 31 October 2007 Received in revised form 29 August 2008 Accepted 15 November 2008 Available online 25 November 2008
Accurately characterizing mid-frequency vibrations is essential for structures that require ultra-quiet vibration environments. Selecting the proper sensor locations is an important step in the model verification and validation process. State-of-the-practice approaches to sensor placement are typically based on modes shapes of the pretest finite element model. However, these modal based techniques break down in the midfrequency range due to the high modal density. The purpose of this work was to develop a sensor placement technique based directly on a structure’s frequency response. The finite element model frequency response can be decomposed into principal directions and their corresponding singular values, which relate the principal directions to the system’s energy. A system’s response is usually dominated by a relatively small number of principal directions, even for frequency bands with high modal densities. Principal directions are always orthogonal, while mode shapes in general are not, which makes them more robust to modeling errors and experimental noise. The new method places sensors such that the independence and signal strength of the dominant principal directions are maintained. & 2008 Elsevier Ltd. All rights reserved.
Keywords: Sensor placement Effective independence Mid-frequency Frequency response
1. Introduction High-performance space vehicles being considered by the Air Force will require extremely low-level on-orbit vibration environments. Computer models of these high-performance vehicles must be valid to a higher frequency range than what is required for most other spacecraft. Physical vibration tests are performed, at least at the component level, to compare the performance of the computer model to that of the actual spacecraft. Sensor placement is a key element to ensure the success of any vibration test. Sensors should be placed in an optimal fashion to acquire all of the dynamically important data with minimum redundancy. Many techniques have been developed to pick sensor locations based on a variety of different criteria, such as modal kinetic energy [1], mass-to-stiffness ratio [2], maximizing an appropriate norm of the Fisher information matrix [3], etc. In practice, these techniques are paired with ‘‘engineering judgment’’ to determine the final sensor positions for vibration testing. The aerospace community has traditionally relied on modal methods and finite element analysis for dynamic predictions in the low frequency range. At low frequencies, a relatively small number of modes with widely spaced frequencies can be used to capture system behavior. Sensor placement and model validation are based on a relatively small set of target modes within the frequency range of interest. In contrast, precision spacecraft require models that are valid to a much higher frequency range for accurate predictions. This higher frequency band, lying between the low-frequency Corresponding author. Tel.: +1 608 262 5724; fax: +1 608 262 6707.
E-mail address: [email protected] (D.C. Kammer). 0888-3270/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ymssp.2008.11.006
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range of modal analysis and the high-frequency region of statistical energy analysis [4], is referred to as the mid-frequency range. The corresponding short wavelength vibration patterns require a fine spatial resolution in the finite element model (FEM), and result in a large number of densely packed modes in the structure. The drawback to using many of the current sensor placement methods in the mid-frequency range is that they are based on the mode shapes of the pretest FEM. Modal based techniques break down in the mid-frequency range due to the high modal density. As frequencies are increased, it becomes increasingly difficult to identify and select the dynamically important modes. Noise and model errors combined with the high modal density produce coupling sensitivity between test and analysis mode shapes. This sensitivity makes correlation metrics, such as orthogonality and cross-orthogonality [5], useless for matching test and analysis modes. As an alternative, a sensor placement technique based on the structure’s frequency response is presented in this paper. The new frequency response based technique is a generalization of the effective independence (Efi) approach [3,6,7]. Analytical frequency response can be calculated for FEMs based on specified frequency bands, damping, and input locations. The analytical frequency response can be decomposed into principal directions and singular values, which can be directly related to the system’s energy [8–10]. In general, principal directions do not coincide with the mode shapes, but it can be shown that the principal directions with non-zero singular values span the same subspace as the excited modes [11]. A system’s response is usually dominated by a relatively small number of principal directions, even for frequency bands with high modal density. Using principal directions thereby eliminates the difficult task of identifying the dynamically important mode shapes. The effects from input location and damping are automatically accounted for in the principal directions. Additionally, principal directions are always orthogonal, while mode shapes in general are not. This makes principal directions more robust to modeling errors and experimental noise [9]. These advantages make principal directions a more suitable basis for describing structural response in the mid-frequency range. 2. Background 2.1. Frequency response Modal based test/analysis correlation techniques break down in the mid-frequency range due to high modal density [12]. Therefore, modal based sensor placement is also inappropriate. Instead, the frequency domain approach to effective independence uses frequency response to directly select sensor locations without using the mode shapes. This eliminates the problem of identifying the dynamically important modes. Although the goal is to remove mode shapes from the sensor placement process, for large models, mode shapes still form and efficient basis for the computation of the analytical frequency response. The modal based acceleration frequency response can be calculated using the expression hpq ðoÞ ¼
r X j¼1
o2 fjp fjq
(1)
2 j
o o2 þ 2izj ooj
where o represents the forcing frequency in rad/s, oj is the undamped natural frequency of the jth mode, zj is the damping coefficient, fjp is the value of the jth mode at the pth output location, and fjq is the value of the jth mode at the qth input location [13]. Selecting all of the modes in the frequency band plus a sufficient amount of out-of-band modes on the high and low sides in Eq. (1) maintains the frequency response’s accuracy in the frequency band of interest. The frequency response data matrix, H, can then be defined as a collection of the individual frequency response matrices from the desired frequency band H ¼ ½ hðo1 Þ
hðo2 Þ
hðo3 Þ . . . hðof Þ
in which f is the number of data points, and hðoi Þ 2 C respectively.
(2) ns na
, where ns and na are the number of sensors and inputs,
2.2. Principal component analysis The frequency response data matrix can be decomposed by a technique called principal component analysis, which is similar to proper orthogonal decomposition, Karhunen–Loeve decomposition, and singular value decomposition. Using this approach, the data matrix can be written as (3)
H ¼ cSV
where c is an ns ns complex matrix whose orthonormal columns are the principal directions, S is a diagonal matrix of singular values, and V is a matrix with orthonormal rows containing the normalized frequency response of the principal directions. The principal directions of the frequency response data matrix are also the eigenvectors of the matrix HH ¼ cS 2 c
(4)
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in which * represents the complex conjugate transpose. Researchers have shown that the squares of the singular values are related to the total energy of the system contained in the corresponding principal directions [9,10,14]. The singular values are arranged from largest to smallest. Most of the system’s energy is usually concentrated in the first several singular values. The corresponding principal directions show how the energy is distributed in the structure [14]. There are a variety of methods for determining how many singular values to retain to properly characterize a system [15]. A common method is to keep the singular values and principal directions which correspond to the top 95–99% of the system’s total energy [8]. Like mode shapes, principal directions are the fundamental shapes that represent the system’s dynamics. However, principal directions automatically account for damping in the structure, effects of input locations, and out-of-band modes. Researchers have determined that principal directions converge to normal modes in symmetric linear systems if the mass matrix is diagonal and uniform, and the system is lightly damped [14,16]. In general, principal directions do not coincide with the mode shapes, but it can be shown that the principal directions with non-zero singular values span the same subspace as the excited modes [11]. While there may be many vibrational normal modes in a frequency band, the response is usually dominated by a relatively small number of principal directions. It is common practice to use only the real part of HH*, also known as the output covariance matrix, to determine principal directions and singular values for the system [10]. However, it can be determined that the sum of the k largest singular values for HH* will be greater than the sum of the k largest singular values for the covariance matrix [17], i.e. k X
li ð<ðHH ÞÞp
i¼1
k X
li ðHH Þ k ¼ 1; . . . ; n; with equality for k ¼ n.
(5)
i¼1
This indicates that more of the system’s energy can be captured using fewer singular values and principal directions from the complex covariance matrix, as opposed to only its real part. For a given number, principal directions from the complex covariance matrix capture the information in the frequency response more accurately. The frequency resolution required in Eq. (2) is problem dependent, but should be high enough to capture the dynamics excited by the inputs. Increasing the resolution should not increase the number of principal directions required to maintain a desired energy level. 3. Frequency effective independence Effective independence is a sensor placement technique that selects sensor locations in a fashion that maximizes signal strength and renders the target mode shapes linearly independent. It was originally formulated to iteratively truncate sensor locations that have the smallest impact on the value of the determinant of the information matrix. Details can be found in Ref. [3]. Recently, work has been performed to improve and generalize the effective independence technique to create a sensor placement scheme for placing triaxial sensors as single units [6]. Additionally, the method has been modified to expand an initial sensor set, as opposed to truncating sensors from the candidate set [7]. The goal of the frequency response based effective independence (FEfI) technique is to place sensors to maintain the dynamically important information contained in the frequency response data within the desired frequency band. This is done by placing sensors such that the measured response is rich in the response of the dominant principal directions. As in the case of modal based EfI, the current sensor placement problem can also be cast in the form of a state estimation problem H ¼ cV þ N
(6)
where V¯ ¼ SV represents the frequency response of the dynamically important principal directions, and N is a matrix of Gaussian white noise. The sensors are placed properly if given the measured frequency response, the response of the dynamically important principal directions can be accurately estimated. The corresponding Fisher information matrix is
Qc ¼ cc W cc
(7)
where cc are the dominant principal directions partitioned to the candidate sensor set, and W is a weighting matrix, such as a mass matrix, or inverse of the noise covariance matrix. In this work, W is assumed to be an identity matrix. Maximizing an appropriate norm of the information matrix results in minimizing the error covariance matrix; this provides the best state estimate. As in EfI, the determinant of the information matrix will be selected as the appropriate norm for sensor placement. Note that the information matrix is Hermitian for this case, instead of being real and symmetric. This paper outlines the triaxial sensor set expansion technique for FEfI, while techniques for sensor set reduction would parallel work done in Refs. [3,6]. Sensor set expansion is useful for structures that have a very large candidate sensor set that needs to be reduced to a relatively small number of sensors. Individual sensors can be placed in the same manner. 3.1. Case 1: initial sensor set renders principal directions linearly independent For this case, it is assumed that the initial set of triaxial sensors renders the active principal directions linearly independent. The initial sensor set would typically include a few important sensor locations that were hand picked by the engineers, such as nodes around key instrumentation, or input locations. Linear independence requires that the initial sensor set has at least as many individual sensors as the number of principal directions. The goal of the triaxial sensor set
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expansion is to find the node from the candidate set that provides the greatest increase to the determinant of the current information matrix, and thereby maximize the contribution to the signal strength and independence of the principal directions. For expansion, the candidate node set consists of all of the possible triaxial sensor locations. Triaxial sensors, which measure the response in each of the translational degrees of freedom, are added one at a time to expand the initial set. Let Q+3i represent the Fisher information matrix with the ith triaxial sensor added to the initial set. This matrix can be written as
Qþ3i ¼ Qo þ c3i c3i ¼ Qo ½I þ Q1 o c3i c3i
(8)
in which c3i is the retained principal directions partitioned to the translational degrees of freedom corresponding to the ith triaxial sensor location, and Qo is the initial information matrix. Analogous to Ref. [7], the measure of goodness for triaxial FEfI was chosen to be FEfI3þ i ¼
detðQþ3i Þ detðQo Þ ¼ detðI 3 þ E 3i Þ 1 detðQo Þ
(9)
where
E 3i ¼ c3i Q1 o c3i .
(10)
This measure represents the fractional amount the determinant of the initial information matrix will increase if the ith candidate node is added to the initial set. All of the determinants are strictly real because each matrix in Eq. (9) is Hermitian, and therefore the FEfI3+ value is a real number. It can be shown that the value of FEfI3+ X0. The FEfI3+ value is computed for each candidate sensor, and the sensor with the largest value is added to the initial sensor set. The sensor set is iteratively expanded until the desired number of sensors is attained. 3.2. Case 2: initial sensor set renders principal directions linearly dependent In Case 2, it is assumed that the initial set of triaxial sensors does not render the dominant principal directions linearly independent. The initial information matrix, Qo, is singular so its inverse does not exist. The objective in Case 2 is to add sensors to make Qo full rank. Once Qo is full rank, the procedure outlined in Case 1 can be used to select additional triaxial sensors until the desired number is attained. For Case 2, with k retained principal directions, Qo is singular with rank lok. Let fo represent the set of orthonormal eigenvectors associated with the zero eigenvalues of Qo. These vectors span the kl dimensional space that is orthogonal to the information subspace of Qo. An orthogonal projector, P, can be formed from the eigenvectors using the expression
P ¼ fo fo .
(11)
The candidate sensor information matrix, Qc, can be filtered using the orthogonal projector to create Qc ¼ PQc P
(12)
¯ c represents the information associated with the candidate set that is orthogonal to the information contained in in which Q ¯ c is singular with rank kl, and can be decomposed in the form the initial sensor set. While Qc is full rank, the matrix Q
(13)
Qc ¼ f l f
where f represents the kl orthonormal eigenvectors associated with the non-zero eigenvalues l. In order to expand the ¯ c and place the sensor with initial sensor set, the objective is to rank the candidate sensors based on their contribution to Q the highest ranking into the initial sensor set. This adds sensors to the initial set that contain information that is orthogonal to the information that is already contained in the initial sensor set. The measure used to add a candidate sensor to the initial sensor set in the rank deficient case is given by FEfI3i ¼ 1 detðI 3 E 3i Þ
(14)
where þ
(15)
E 3i ¼ c3i Qc c3i ¯ c+ Q
represents the pseudo-inverse of the filtered candidate information matrix. The node with the largest in which FEfI3i value is added to the initial sensor set, and the process is repeated until the initial sensor set renders the principal directions linearly independent. This results in the information matrix being full rank. The method presented in Case 1 is then used to expand the sensor set further until the desired number of sensors is attained. 4. Effect of FEfI on observability The FEfI sensor placement approach is based on the identification of the dominant principal directions cc. Transforming the equations of motion to principal direction space produces
mq€ þ gq_ þ dq ¼ cca u
(16)
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in which q are the principal coordinates, m, g, and d are the generalized mass, damping, and stiffness matrices, respectively, cca are the principal directions row partitioned to the input locations, and u is the input vector. It is assumed that there is no rigid body motion within the frequency range of interest, such that m, g, and d are positive definite. Transforming to state space yields the state equation (17)
z_ ¼ Az þ Bu where " z ¼ f qT
q_T gT
A¼
0
I
m1 d
m1 g
(
# B¼
0
) (18)
cca
The general output equation is of the form (19)
ys ¼ Cz þ Nu in which ys is the physical response at the sensor locations. Realization methods used for system identification rely on a generalized observability matrix Vp given by 2 3 C 6 CA 7 6 7 7 Vp ¼ 6 . 6. 7 4. 5 p1 CA
(20)
In order to identify the k dimensional system in Eqs. (18) and (19), Vp must be full column rank, i.e., rk(Vp) ¼ 2k. In principal coordinates, Vp can be decomposed into the product of two matrices (21)
V p ¼ S p ðcc ÞZ p
where Sp is a (pns pk) matrix function of cc and Zp is a (pk 2k) matrix function of m, g, and d. An optimum sensor configuration will minimize the required size of the generalized observability matrix, but still maintain its rank. The smallest possible observability matrix that has rank 2k is given by C (22) V2 ¼ ¼ S2 Z2 CA while in practice, a much larger observability matrix would be used due to the presence of noise, etc., this minimum sized observability matrix can be used to demonstrate the effects of the FEfI sensor placement strategy. In the case of accelerometers, the output influence matrix is given by C ¼ ½ cc m1 d
cc m1 g
which results in the matrices " # " cc 0 m1 d S2 ¼ and Z 2 ¼ cc 0 m1 gm1 d
(23)
m1 g
# (24)
m1 d þ m1 gm1 g
According to Sylvester’s inequality [18] V2 will have rank 2k if S2 is full column rank and Z2 is nonsingular. The determinant of the partitioned matrix Z2 can be computed using simple matrix algebra, resulting in |Z2| ¼ |m1d|2. This indicates that Z2 is nonsingular regardless of the damping values and sensor placement. It is obvious from Eq. (24) that spatial matrix S2 will be full column rank, and thus the system will be observable, if and only if the principal direction partitions cc are full column rank. Therefore at least 2k sensors must be placed such that the principal direction partitions are linearly independent. This is precisely the objective of the FEfI sensor placement methodology described in this paper. The optimum sensor configuration should not only render the system observable, but it should also enhance the degree of observability. An observability grammian can be generated in the form W o ¼ V T2 V 2 ¼ Z T2 S T2 S 2 Z 2
(25) T
Wo1z
with an associated hyperellipsoid in observability space given by z ¼ 1. The volume of this hyperellipsoid is proportional to|Wo|1/2, therefore the determinant of the observability grammian is often used as a measure of the degree of observability of a system [19]. Sensors should be placed such that the determinant of the observability grammian is maximized. Using Eq. (25), the determinant can be written as jW o j ¼ jZ T2 j jS T2 S 2 j jZ 2 j ¼ jZ 2 j2 jS T2 S 2 j.
(26)
It was shown that sensor placement has no effect upon the matrix Z2 and thus also its determinant. However, the determinant of the matrix product ST2S2 is given by
jS T2 S 2 j ¼ jcc cc j2 ¼ jQc j2
(27)
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where Qc is the corresponding information matrix discussed earlier. Maximization of the determinant of the information matrix thus results in the maximization of the degree of system observability. Thus, the FEfI sensor placement technique not only results in a minimum sized generalized observability matrix, it also maximizes the observability of the system. 5. Applications 5.1. Simple beam To illustrate the usefulness of the FEfI technique, a sensor placement analysis was performed using a uniform unconstrained beam with 41 node points. Each node contains a single degree of freedom in the transverse direction. For this example, the mode shapes have well spaced natural frequencies, therefore modal based sensor placement and correlation techniques can be easily applied. This example is by no means a mid-frequency application, but it provides a simple illustration to show how FEfI picks sensor locations as compared to modal EfI. All 41 degrees of freedom were included in the candidate sensor set. Initial trials were performed for the frequency range of 5–205 Hz, which contains the first 6 elastic modes. The FEfI method was used on various input locations and damping values to show how the FEfI method is sensitive to different test conditions. The first set of examples consisted of a single input at the right hand end of the beam (node 41), and modal damping values of 1% and 10%. The input at the endpoint excited all 6 elastic modes in the frequency band. The frequency response was calculated using Eq. (1) with all 41 FEM mode shapes. The principal directions and singular values were extracted from the analytical frequency response. The singular values were truncated to the largest 6, which retained 95% of the system’s total energy. The principal directions corresponding to the retained singular values were used in the FEfI calculations. Modal based EfI using the first 6 elastic target modes provided a basis for comparison with the final sensor locations selected using FEfI. Due to the simplistic nature of this example, an iterative sensor reduction [3] was performed as opposed to the sensor set expansion that was outlined in this paper. Comparable results are expected. The frequency-based reduction mirrors its modal counterpart with principal directions replacing the mode shapes in the calculations. Each technique was used to reduce the candidate sensor set down to 10 final sensor locations. In the case with 1% modal damping, the final sensor sets and effective independence distributions are indistinguishable between the modal and frequency response based techniques. However, when the modal damping is increased to 10% the FEfI technique skews the sensor positions towards the input location at the right end, as illustrated in Fig. 1. This shows that FEfI is sensitive to changes in damping. Intuitively, as damping is increased the response further away from the input location would tend to decrease, and thus the final sensors are skewed more toward the input location. The next application consisted of changing the input location to vary which modes were excited. This example was run with 1% modal damping and the input at the mid-point of the beam. An input at the center only excites the symmetric modes (1, 3, and 5). If the target modes for modal EfI are chosen to be all of the modes in the frequency band, then half of the target modes will never be excited for this input. As for the FEfI technique, only 3 singular values are needed to make up 95% of the system’s total energy. This shows that FEfI automatically focuses resources on the 3 excited modes and ignores the unexcited modes in the frequency band. Fig. 2 shows the final 10 sensor locations and effective independence distributions for each technique when all of the modes in the frequency band are used in the modal EfI calculations. If only the symmetric modes are used in the modal EfI calculations, the FEfI and EfI sensor sets agree.
1 FEfI EfI
0.9 Effective Independence
0.8 10% Modal Damping
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0
5
10
15
20 25 30 Node Number
35
Fig. 1. Final EfI and FEfI distributions for 10% damping.
40
45
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0.5 FEfI EfI
0.45 Effective Independence
0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0
5
10
15
20 25 30 Node Number
35
40
45
Fig. 2. Final EfI and FEfI distributions—all modes.
These results demonstrate that the FEfI technique is automatically sensitive to input locations. Modal EfI requires the user to select the target modes based on engineering judgment, or additional techniques. While for this case, it is quite clear that only the symmetric modes will be excited, for a more complex structure, with a high modal density, the task of selecting the appropriate target modes becomes much more challenging. This example demonstrates that on a simple structure, for the case of light damping with inputs that excite all of the modes in the frequency band, the FEfI scheme picks sensors in a similar fashion to modal EfI. These examples also show how the FEfI technique automatically accounts for input location and damping effects. 5.2. Generic spacecraft The second example considers the Generic Spacecraft (GSC) to demonstrate how the FEfI technique places triaxial sensors on a more complex structure. The GSC is composed of a cubic core, two circular rib stiffened reflectors, and two rectangular photovoltaic (PV) arrays. The GSC’s FEM contains 1191 nodes and 1262 elements. Sensor placement was considered for two cases. The first trial is in the low frequency regime, so a comparison can be made to the modal based EfI technique. The second trial considers the mid-frequency range, where the number of modes present in the frequency band is prohibitive for modal based sensor placement approaches. 5.2.1. Low-frequency application The first case considers the frequency band from 0.1 to 10 Hz, which contains 16 undamped elastic mode shapes. The low frequency GSC example is similar to the beam example in that there are a relatively small number of modes contained in the frequency band of interest. However, unlike the beam, the modes of the GSC are closely spaced in frequency, which may make them more difficult to distinguish in a modal test. The GSC was excited with inputs in the z-direction (into the paper) located at six nodes, one on the end of each PV array and two on each reflector. Similar to what was seen in the beam example, the inputs do not guarantee that all of the mode shapes in the frequency band will be excited. One would expect that inputs only in the z-direction would not excite modes that have their primary displacements in the x–y plane. This makes the task of selecting appropriate target modes vital to the success of modal EfI, while FEfI automatically accounts for the active modes. The acceleration frequency response for the band from 0.1 to 10 Hz was calculated using Eq. (1), with an assumed 1% modal damping. The singular values were truncated to the top 95% of their total sum, resulting in 9 retained principal directions. This application places 100 triaxial accelerometers on the GSC in an attempt to capture all of the dynamically important data for the specified frequency band and input locations. The initial sensor set was created by selecting the highest ranked sensor location, based on the FEfI reduction technique [3], from the full set of nodes. All of the remaining node locations were then placed into the initial candidate sensor set. A total of 99 iterations were run to expand the initial sensor set to the final 100 sensor locations. The first 2 iterations used the method outlined in Case 2 to expand the initial sensor set until the information matrix was full rank. Then, 97 additional iterations, using Case 1, were performed to obtain the final sensor configuration. Fig. 3 shows the final triaxial sensor positions from the frequency response based technique. Next, modal based EfI [7] was run to obtain a sensor set for comparison. All 16 of the modes in the frequency band of interest were included in the target set. The initial sensor set was selected in the same fashion as for the frequency response based approach. The final 100 sensor locations based on modal EfI are shown in Fig. 4.
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Fig. 3. Sensor locations for FefI.
The sensor sets for FEfI and modal EfI share 68 of 100 common sensor locations. Each technique placed sensors around the edges of the circular reflectors and along the edges of the PV arrays. The frequency response based technique placed additional sensors near the input locations on the reflectors, while the modal-based technique placed more sensors around the edge of the PV arrays. The difference in sensor locations between the two techniques may be attributed to modal EfI placing sensors based on modes that were not excited by the inputs. Overall, the sensor sets for both techniques show a similar distribution. This shows that for the low-frequency range the frequency response based method is capable of selecting sensor locations in a similar fashion as compared to its modal counterpart.
5.2.2. Mid-frequency application The second case considers the GSC in a frequency range with high modal density. The frequency band of interest for this example is from 50 to 300 Hz, which contains 412 vibrational modes. It is important to note that the FEM being considered
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Fig. 4. Sensor locations for modal EfI.
in this example is not refined enough to make accurate predictions over this entire frequency band. However, the modes predicted are close in frequency, and very localized. Therefore, the model exhibits all the problems associated with the mid-frequency range, but is still easy to manipulate due to its relatively small size. The number of modes within the halfpower band at each frequency, also known as the mode overlap factor, peaks at 24 around 200 Hz. Closely spaced modes can be difficult to distinguish in a modal test, and when coupled with noise, create problems for correlation metrics, such as orthogonality and cross-orthogonality. The large number of modes present in the frequency band makes it impractical to try to identify each of the modes in a vibration test. For the mid-frequency range, modal based techniques fail because the number of sensors required for linear independence of the modes is prohibitive. Here, modal based EfI would require a minimum of 138 triaxial sensors to render the mode shapes linearly independent, and as many as 412 triaxial sensors to guarantee independence. This problem lends itself more naturally to the frequency response based techniques outlined in
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this paper, where the frequency response is dominated by a relatively small number of principal directions, as compared to the mode shapes. This case uses the same input locations and damping characteristics as used in the previous low-frequency example. The analytical frequency response for the band from 50 to 300 Hz was calculated using Eq. (1). To account for contributions from out-of-band modes, all mode shapes up to 450 Hz were used in the calculation. The singular values of the frequency response were computed and truncated to the top 95% of their total sum, resulting in 121 being retained. This shows that the bulk of the system’s energy is captured in 121 principle directions, which is roughly 70% fewer shapes than the number modes contained in the 50–300 Hz. frequency band. The FEfI triaxial sensor set expansion method was used to obtain 60 triaxial sensor locations. The initial sensor set was once again created by selecting the highest ranked sensor location based on FEfI for the full set of nodes. A total of 59 iterations were run to expand the initial sensor set to the final 60 sensor locations. The first 40 iterations used the method
Fig. 5. Sensor locations for FEfI in mid-frequency range.
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outlined in Case 2 to expand the initial sensor set until the information matrix was full rank. Then, 19 additional iterations, using Case 1, were performed to obtain the final sensor configuration. Fig. 5 shows the final triaxial sensor positions on the GSC. It is apparent that all of the sensors for this mid-frequency example are clustered on the circular reflectors. This can be attributed to the fact that the majority of the modes in this frequency range have displacements that are highly localized to the reflector surfaces. 6. Conclusion A frequency domain sensor placement scheme was presented based on principal component analysis of the frequency response data matrix. The method is a generalization of effective independence [3,6,7]. Sensors are placed to maintain the dynamically important information in the frequency response and the overall system energy within the frequency range of interest. The frequency response data automatically accounts for input location and damping, therefore this technique eliminates the difficult task of identifying target modes. Two examples were presented that showed that frequency and modal based EfI methods provide comparable sensor configurations for systems with low damping and well separated modal frequencies. The beam example showed that as damping levels increased, the frequency-based EfI approach automatically skewed the sensor locations toward the input location. Both the beam and the Generic Spacecraft examples demonstrated that the frequency-based EfI technique automatically accounts for how the selected inputs excite the structure, and thereby eliminate the need for the user to identify the target modes. In the high modal density example, where the modal density was very high, it was not only impossible to select and distinguish the appropriate target modes, there were so many of them that there were not enough sensors available to render them linearly independent. This makes modal EfI impossible to use, and frequency EfI, based on principal directions, a valuable tool for sensor placement.
Acknowledgement This material is based on work supported by the Air Force Office of Scientific Research under Grant FA9550-06-1-0171. This funding is gratefully acknowledged. References [1] M.R. Salma, Rose, J. Garba, Optimal placement of excitations and sensors for verification of large dynamical systems. In: The 28th Structures, Structural Dynamics, and Materials Conference, 1987. [2] R.D. Henshell, J.H. Ong, Automatic masters for eigenvalue economization, Earthquake Engineering Structural Dynamics 3 (1975) 375–383. [3] D.C. Kammer, Sensor placement for on-orbit modal identification and correlation of large space structures, Journal of Guidance, Control, and Dynamics 14 (1991) 251–259. [4] M. Castanier, C. Pierre, Mid-Frequency Vibration Analysis of Complex Structures: State of the Art and Future Directions, Air Force Office of Scientific Research, University of Michigan, 2002. [5] T.K.Hasselman, R.N. Coppolino, D.C. Zimmerman, Criteria for Modeling Accuracy: A State-of-the-Practice Survey, In: IMAC 18. 2000, IMAC: San Antonio, TX. [6] D.C. Kammer, M.L. Tinker, Optimal placement of triaxial accelerometers for modal vibration tests, Mechanical Systems and Signal Processing 18 (2004) 29–41. [7] D.C. Kammer, Sensor set expansion for modal vibration testing, Mechanical Systems and Signal Processing 19 (2005) 700–713. [8] B.F. Feeny, On proper orthogonal coordinates as indicators of modal activity, Journal of Sound and Vibration 255 (2002) 805–817. [9] T.K. Hasselman, Principal components analysis for nonlinear model correlation updating, and uncertainty evaluation. In: The 16th International Modal Analysis Conference, 1998. [10] A. Sarkar, R. Ghanem, Mid-frequency structural dynamics with parameter uncertainty, Computer Methods in Applied Mechanics and Engineering 191 (2002) 5499–5513. [11] G. Kerschen, P. De Boe, J.C. Golinval, K. Worden, Sensor validation using principal component analysis, Smart Materials and Structures 14 (2005) 36–42. [12] P. Blelloch, Cross-Orthogonality of Closely Spaced Modes, In: IMAC XXIV. 2006, IMAC: St. Louis. [13] M.I. Friswell, J.E. Motershead, Finite Element Model Updating in Structural Dynamics, Kluwer Academic Publishers, 1995. [14] B.F. Feeny, R. Kappagantu, On the physical interpretation of proper orthogonal modes in vibrations, Journal of Sound and Vibration 211 (4) (1998) 607–616. [15] J.E. Jackson, A User’s Guide to Principal Components, John Wiley & Sons Inc., New York, USA, 1991. [16] B.F. Feeny, Y. Liang, Interpreting proper orthogonal modes of randomly excited vibration systems, Journal of Sound and Vibration 265 (2003) 953–966. [17] R.A. Horn, C.R. Johnson, Topics in Matrix Analysis, Cambridge University Press, New York, USA, 1994. [18] R.A. Horn, C.R. Johnson, Matrix Analysis, Cambridge University Press, New York, USA, 1992. [19] X. Guangqian, P.M. Bainum, Actuator placement using degree of controllability for discrete-time systems, Journal of Dynamic Systems, Measurement, and Control 114 (3) (1992) 508–516.
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Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/jnlabr/ymssp
Frequency response based sensor placement for the mid-frequency range S. Nimityongskul, D.C. Kammer Department of Engineering Physics, University of Wisconsin, 1500 Engineering Drive, Madison, WI 53706, USA
a r t i c l e in fo
abstract
Article history: Received 31 October 2007 Received in revised form 29 August 2008 Accepted 15 November 2008 Available online 25 November 2008
Accurately characterizing mid-frequency vibrations is essential for structures that require ultra-quiet vibration environments. Selecting the proper sensor locations is an important step in the model verification and validation process. State-of-the-practice approaches to sensor placement are typically based on modes shapes of the pretest finite element model. However, these modal based techniques break down in the midfrequency range due to the high modal density. The purpose of this work was to develop a sensor placement technique based directly on a structure’s frequency response. The finite element model frequency response can be decomposed into principal directions and their corresponding singular values, which relate the principal directions to the system’s energy. A system’s response is usually dominated by a relatively small number of principal directions, even for frequency bands with high modal densities. Principal directions are always orthogonal, while mode shapes in general are not, which makes them more robust to modeling errors and experimental noise. The new method places sensors such that the independence and signal strength of the dominant principal directions are maintained. & 2008 Elsevier Ltd. All rights reserved.
Keywords: Sensor placement Effective independence Mid-frequency Frequency response
1. Introduction High-performance space vehicles being considered by the Air Force will require extremely low-level on-orbit vibration environments. Computer models of these high-performance vehicles must be valid to a higher frequency range than what is required for most other spacecraft. Physical vibration tests are performed, at least at the component level, to compare the performance of the computer model to that of the actual spacecraft. Sensor placement is a key element to ensure the success of any vibration test. Sensors should be placed in an optimal fashion to acquire all of the dynamically important data with minimum redundancy. Many techniques have been developed to pick sensor locations based on a variety of different criteria, such as modal kinetic energy [1], mass-to-stiffness ratio [2], maximizing an appropriate norm of the Fisher information matrix [3], etc. In practice, these techniques are paired with ‘‘engineering judgment’’ to determine the final sensor positions for vibration testing. The aerospace community has traditionally relied on modal methods and finite element analysis for dynamic predictions in the low frequency range. At low frequencies, a relatively small number of modes with widely spaced frequencies can be used to capture system behavior. Sensor placement and model validation are based on a relatively small set of target modes within the frequency range of interest. In contrast, precision spacecraft require models that are valid to a much higher frequency range for accurate predictions. This higher frequency band, lying between the low-frequency Corresponding author. Tel.: +1 608 262 5724; fax: +1 608 262 6707.
E-mail address: [email protected] (D.C. Kammer). 0888-3270/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ymssp.2008.11.006
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range of modal analysis and the high-frequency region of statistical energy analysis [4], is referred to as the mid-frequency range. The corresponding short wavelength vibration patterns require a fine spatial resolution in the finite element model (FEM), and result in a large number of densely packed modes in the structure. The drawback to using many of the current sensor placement methods in the mid-frequency range is that they are based on the mode shapes of the pretest FEM. Modal based techniques break down in the mid-frequency range due to the high modal density. As frequencies are increased, it becomes increasingly difficult to identify and select the dynamically important modes. Noise and model errors combined with the high modal density produce coupling sensitivity between test and analysis mode shapes. This sensitivity makes correlation metrics, such as orthogonality and cross-orthogonality [5], useless for matching test and analysis modes. As an alternative, a sensor placement technique based on the structure’s frequency response is presented in this paper. The new frequency response based technique is a generalization of the effective independence (Efi) approach [3,6,7]. Analytical frequency response can be calculated for FEMs based on specified frequency bands, damping, and input locations. The analytical frequency response can be decomposed into principal directions and singular values, which can be directly related to the system’s energy [8–10]. In general, principal directions do not coincide with the mode shapes, but it can be shown that the principal directions with non-zero singular values span the same subspace as the excited modes [11]. A system’s response is usually dominated by a relatively small number of principal directions, even for frequency bands with high modal density. Using principal directions thereby eliminates the difficult task of identifying the dynamically important mode shapes. The effects from input location and damping are automatically accounted for in the principal directions. Additionally, principal directions are always orthogonal, while mode shapes in general are not. This makes principal directions more robust to modeling errors and experimental noise [9]. These advantages make principal directions a more suitable basis for describing structural response in the mid-frequency range. 2. Background 2.1. Frequency response Modal based test/analysis correlation techniques break down in the mid-frequency range due to high modal density [12]. Therefore, modal based sensor placement is also inappropriate. Instead, the frequency domain approach to effective independence uses frequency response to directly select sensor locations without using the mode shapes. This eliminates the problem of identifying the dynamically important modes. Although the goal is to remove mode shapes from the sensor placement process, for large models, mode shapes still form and efficient basis for the computation of the analytical frequency response. The modal based acceleration frequency response can be calculated using the expression hpq ðoÞ ¼
r X j¼1
o2 fjp fjq
(1)
2 j
o o2 þ 2izj ooj
where o represents the forcing frequency in rad/s, oj is the undamped natural frequency of the jth mode, zj is the damping coefficient, fjp is the value of the jth mode at the pth output location, and fjq is the value of the jth mode at the qth input location [13]. Selecting all of the modes in the frequency band plus a sufficient amount of out-of-band modes on the high and low sides in Eq. (1) maintains the frequency response’s accuracy in the frequency band of interest. The frequency response data matrix, H, can then be defined as a collection of the individual frequency response matrices from the desired frequency band H ¼ ½ hðo1 Þ
hðo2 Þ
hðo3 Þ . . . hðof Þ
in which f is the number of data points, and hðoi Þ 2 C respectively.
(2) ns na
, where ns and na are the number of sensors and inputs,
2.2. Principal component analysis The frequency response data matrix can be decomposed by a technique called principal component analysis, which is similar to proper orthogonal decomposition, Karhunen–Loeve decomposition, and singular value decomposition. Using this approach, the data matrix can be written as (3)
H ¼ cSV
where c is an ns ns complex matrix whose orthonormal columns are the principal directions, S is a diagonal matrix of singular values, and V is a matrix with orthonormal rows containing the normalized frequency response of the principal directions. The principal directions of the frequency response data matrix are also the eigenvectors of the matrix HH ¼ cS 2 c
(4)
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in which * represents the complex conjugate transpose. Researchers have shown that the squares of the singular values are related to the total energy of the system contained in the corresponding principal directions [9,10,14]. The singular values are arranged from largest to smallest. Most of the system’s energy is usually concentrated in the first several singular values. The corresponding principal directions show how the energy is distributed in the structure [14]. There are a variety of methods for determining how many singular values to retain to properly characterize a system [15]. A common method is to keep the singular values and principal directions which correspond to the top 95–99% of the system’s total energy [8]. Like mode shapes, principal directions are the fundamental shapes that represent the system’s dynamics. However, principal directions automatically account for damping in the structure, effects of input locations, and out-of-band modes. Researchers have determined that principal directions converge to normal modes in symmetric linear systems if the mass matrix is diagonal and uniform, and the system is lightly damped [14,16]. In general, principal directions do not coincide with the mode shapes, but it can be shown that the principal directions with non-zero singular values span the same subspace as the excited modes [11]. While there may be many vibrational normal modes in a frequency band, the response is usually dominated by a relatively small number of principal directions. It is common practice to use only the real part of HH*, also known as the output covariance matrix, to determine principal directions and singular values for the system [10]. However, it can be determined that the sum of the k largest singular values for HH* will be greater than the sum of the k largest singular values for the covariance matrix [17], i.e. k X
li ð<ðHH ÞÞp
i¼1
k X
li ðHH Þ k ¼ 1; . . . ; n; with equality for k ¼ n.
(5)
i¼1
This indicates that more of the system’s energy can be captured using fewer singular values and principal directions from the complex covariance matrix, as opposed to only its real part. For a given number, principal directions from the complex covariance matrix capture the information in the frequency response more accurately. The frequency resolution required in Eq. (2) is problem dependent, but should be high enough to capture the dynamics excited by the inputs. Increasing the resolution should not increase the number of principal directions required to maintain a desired energy level. 3. Frequency effective independence Effective independence is a sensor placement technique that selects sensor locations in a fashion that maximizes signal strength and renders the target mode shapes linearly independent. It was originally formulated to iteratively truncate sensor locations that have the smallest impact on the value of the determinant of the information matrix. Details can be found in Ref. [3]. Recently, work has been performed to improve and generalize the effective independence technique to create a sensor placement scheme for placing triaxial sensors as single units [6]. Additionally, the method has been modified to expand an initial sensor set, as opposed to truncating sensors from the candidate set [7]. The goal of the frequency response based effective independence (FEfI) technique is to place sensors to maintain the dynamically important information contained in the frequency response data within the desired frequency band. This is done by placing sensors such that the measured response is rich in the response of the dominant principal directions. As in the case of modal based EfI, the current sensor placement problem can also be cast in the form of a state estimation problem H ¼ cV þ N
(6)
where V¯ ¼ SV represents the frequency response of the dynamically important principal directions, and N is a matrix of Gaussian white noise. The sensors are placed properly if given the measured frequency response, the response of the dynamically important principal directions can be accurately estimated. The corresponding Fisher information matrix is
Qc ¼ cc W cc
(7)
where cc are the dominant principal directions partitioned to the candidate sensor set, and W is a weighting matrix, such as a mass matrix, or inverse of the noise covariance matrix. In this work, W is assumed to be an identity matrix. Maximizing an appropriate norm of the information matrix results in minimizing the error covariance matrix; this provides the best state estimate. As in EfI, the determinant of the information matrix will be selected as the appropriate norm for sensor placement. Note that the information matrix is Hermitian for this case, instead of being real and symmetric. This paper outlines the triaxial sensor set expansion technique for FEfI, while techniques for sensor set reduction would parallel work done in Refs. [3,6]. Sensor set expansion is useful for structures that have a very large candidate sensor set that needs to be reduced to a relatively small number of sensors. Individual sensors can be placed in the same manner. 3.1. Case 1: initial sensor set renders principal directions linearly independent For this case, it is assumed that the initial set of triaxial sensors renders the active principal directions linearly independent. The initial sensor set would typically include a few important sensor locations that were hand picked by the engineers, such as nodes around key instrumentation, or input locations. Linear independence requires that the initial sensor set has at least as many individual sensors as the number of principal directions. The goal of the triaxial sensor set
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expansion is to find the node from the candidate set that provides the greatest increase to the determinant of the current information matrix, and thereby maximize the contribution to the signal strength and independence of the principal directions. For expansion, the candidate node set consists of all of the possible triaxial sensor locations. Triaxial sensors, which measure the response in each of the translational degrees of freedom, are added one at a time to expand the initial set. Let Q+3i represent the Fisher information matrix with the ith triaxial sensor added to the initial set. This matrix can be written as
Qþ3i ¼ Qo þ c3i c3i ¼ Qo ½I þ Q1 o c3i c3i
(8)
in which c3i is the retained principal directions partitioned to the translational degrees of freedom corresponding to the ith triaxial sensor location, and Qo is the initial information matrix. Analogous to Ref. [7], the measure of goodness for triaxial FEfI was chosen to be FEfI3þ i ¼
detðQþ3i Þ detðQo Þ ¼ detðI 3 þ E 3i Þ 1 detðQo Þ
(9)
where
E 3i ¼ c3i Q1 o c3i .
(10)
This measure represents the fractional amount the determinant of the initial information matrix will increase if the ith candidate node is added to the initial set. All of the determinants are strictly real because each matrix in Eq. (9) is Hermitian, and therefore the FEfI3+ value is a real number. It can be shown that the value of FEfI3+ X0. The FEfI3+ value is computed for each candidate sensor, and the sensor with the largest value is added to the initial sensor set. The sensor set is iteratively expanded until the desired number of sensors is attained. 3.2. Case 2: initial sensor set renders principal directions linearly dependent In Case 2, it is assumed that the initial set of triaxial sensors does not render the dominant principal directions linearly independent. The initial information matrix, Qo, is singular so its inverse does not exist. The objective in Case 2 is to add sensors to make Qo full rank. Once Qo is full rank, the procedure outlined in Case 1 can be used to select additional triaxial sensors until the desired number is attained. For Case 2, with k retained principal directions, Qo is singular with rank lok. Let fo represent the set of orthonormal eigenvectors associated with the zero eigenvalues of Qo. These vectors span the kl dimensional space that is orthogonal to the information subspace of Qo. An orthogonal projector, P, can be formed from the eigenvectors using the expression
P ¼ fo fo .
(11)
The candidate sensor information matrix, Qc, can be filtered using the orthogonal projector to create Qc ¼ PQc P
(12)
¯ c represents the information associated with the candidate set that is orthogonal to the information contained in in which Q ¯ c is singular with rank kl, and can be decomposed in the form the initial sensor set. While Qc is full rank, the matrix Q
(13)
Qc ¼ f l f
where f represents the kl orthonormal eigenvectors associated with the non-zero eigenvalues l. In order to expand the ¯ c and place the sensor with initial sensor set, the objective is to rank the candidate sensors based on their contribution to Q the highest ranking into the initial sensor set. This adds sensors to the initial set that contain information that is orthogonal to the information that is already contained in the initial sensor set. The measure used to add a candidate sensor to the initial sensor set in the rank deficient case is given by FEfI3i ¼ 1 detðI 3 E 3i Þ
(14)
where þ
(15)
E 3i ¼ c3i Qc c3i ¯ c+ Q
represents the pseudo-inverse of the filtered candidate information matrix. The node with the largest in which FEfI3i value is added to the initial sensor set, and the process is repeated until the initial sensor set renders the principal directions linearly independent. This results in the information matrix being full rank. The method presented in Case 1 is then used to expand the sensor set further until the desired number of sensors is attained. 4. Effect of FEfI on observability The FEfI sensor placement approach is based on the identification of the dominant principal directions cc. Transforming the equations of motion to principal direction space produces
mq€ þ gq_ þ dq ¼ cca u
(16)
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in which q are the principal coordinates, m, g, and d are the generalized mass, damping, and stiffness matrices, respectively, cca are the principal directions row partitioned to the input locations, and u is the input vector. It is assumed that there is no rigid body motion within the frequency range of interest, such that m, g, and d are positive definite. Transforming to state space yields the state equation (17)
z_ ¼ Az þ Bu where " z ¼ f qT
q_T gT
A¼
0
I
m1 d
m1 g
(
# B¼
0
) (18)
cca
The general output equation is of the form (19)
ys ¼ Cz þ Nu in which ys is the physical response at the sensor locations. Realization methods used for system identification rely on a generalized observability matrix Vp given by 2 3 C 6 CA 7 6 7 7 Vp ¼ 6 . 6. 7 4. 5 p1 CA
(20)
In order to identify the k dimensional system in Eqs. (18) and (19), Vp must be full column rank, i.e., rk(Vp) ¼ 2k. In principal coordinates, Vp can be decomposed into the product of two matrices (21)
V p ¼ S p ðcc ÞZ p
where Sp is a (pns pk) matrix function of cc and Zp is a (pk 2k) matrix function of m, g, and d. An optimum sensor configuration will minimize the required size of the generalized observability matrix, but still maintain its rank. The smallest possible observability matrix that has rank 2k is given by C (22) V2 ¼ ¼ S2 Z2 CA while in practice, a much larger observability matrix would be used due to the presence of noise, etc., this minimum sized observability matrix can be used to demonstrate the effects of the FEfI sensor placement strategy. In the case of accelerometers, the output influence matrix is given by C ¼ ½ cc m1 d
cc m1 g
which results in the matrices " # " cc 0 m1 d S2 ¼ and Z 2 ¼ cc 0 m1 gm1 d
(23)
m1 g
# (24)
m1 d þ m1 gm1 g
According to Sylvester’s inequality [18] V2 will have rank 2k if S2 is full column rank and Z2 is nonsingular. The determinant of the partitioned matrix Z2 can be computed using simple matrix algebra, resulting in |Z2| ¼ |m1d|2. This indicates that Z2 is nonsingular regardless of the damping values and sensor placement. It is obvious from Eq. (24) that spatial matrix S2 will be full column rank, and thus the system will be observable, if and only if the principal direction partitions cc are full column rank. Therefore at least 2k sensors must be placed such that the principal direction partitions are linearly independent. This is precisely the objective of the FEfI sensor placement methodology described in this paper. The optimum sensor configuration should not only render the system observable, but it should also enhance the degree of observability. An observability grammian can be generated in the form W o ¼ V T2 V 2 ¼ Z T2 S T2 S 2 Z 2
(25) T
Wo1z
with an associated hyperellipsoid in observability space given by z ¼ 1. The volume of this hyperellipsoid is proportional to|Wo|1/2, therefore the determinant of the observability grammian is often used as a measure of the degree of observability of a system [19]. Sensors should be placed such that the determinant of the observability grammian is maximized. Using Eq. (25), the determinant can be written as jW o j ¼ jZ T2 j jS T2 S 2 j jZ 2 j ¼ jZ 2 j2 jS T2 S 2 j.
(26)
It was shown that sensor placement has no effect upon the matrix Z2 and thus also its determinant. However, the determinant of the matrix product ST2S2 is given by
jS T2 S 2 j ¼ jcc cc j2 ¼ jQc j2
(27)
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where Qc is the corresponding information matrix discussed earlier. Maximization of the determinant of the information matrix thus results in the maximization of the degree of system observability. Thus, the FEfI sensor placement technique not only results in a minimum sized generalized observability matrix, it also maximizes the observability of the system. 5. Applications 5.1. Simple beam To illustrate the usefulness of the FEfI technique, a sensor placement analysis was performed using a uniform unconstrained beam with 41 node points. Each node contains a single degree of freedom in the transverse direction. For this example, the mode shapes have well spaced natural frequencies, therefore modal based sensor placement and correlation techniques can be easily applied. This example is by no means a mid-frequency application, but it provides a simple illustration to show how FEfI picks sensor locations as compared to modal EfI. All 41 degrees of freedom were included in the candidate sensor set. Initial trials were performed for the frequency range of 5–205 Hz, which contains the first 6 elastic modes. The FEfI method was used on various input locations and damping values to show how the FEfI method is sensitive to different test conditions. The first set of examples consisted of a single input at the right hand end of the beam (node 41), and modal damping values of 1% and 10%. The input at the endpoint excited all 6 elastic modes in the frequency band. The frequency response was calculated using Eq. (1) with all 41 FEM mode shapes. The principal directions and singular values were extracted from the analytical frequency response. The singular values were truncated to the largest 6, which retained 95% of the system’s total energy. The principal directions corresponding to the retained singular values were used in the FEfI calculations. Modal based EfI using the first 6 elastic target modes provided a basis for comparison with the final sensor locations selected using FEfI. Due to the simplistic nature of this example, an iterative sensor reduction [3] was performed as opposed to the sensor set expansion that was outlined in this paper. Comparable results are expected. The frequency-based reduction mirrors its modal counterpart with principal directions replacing the mode shapes in the calculations. Each technique was used to reduce the candidate sensor set down to 10 final sensor locations. In the case with 1% modal damping, the final sensor sets and effective independence distributions are indistinguishable between the modal and frequency response based techniques. However, when the modal damping is increased to 10% the FEfI technique skews the sensor positions towards the input location at the right end, as illustrated in Fig. 1. This shows that FEfI is sensitive to changes in damping. Intuitively, as damping is increased the response further away from the input location would tend to decrease, and thus the final sensors are skewed more toward the input location. The next application consisted of changing the input location to vary which modes were excited. This example was run with 1% modal damping and the input at the mid-point of the beam. An input at the center only excites the symmetric modes (1, 3, and 5). If the target modes for modal EfI are chosen to be all of the modes in the frequency band, then half of the target modes will never be excited for this input. As for the FEfI technique, only 3 singular values are needed to make up 95% of the system’s total energy. This shows that FEfI automatically focuses resources on the 3 excited modes and ignores the unexcited modes in the frequency band. Fig. 2 shows the final 10 sensor locations and effective independence distributions for each technique when all of the modes in the frequency band are used in the modal EfI calculations. If only the symmetric modes are used in the modal EfI calculations, the FEfI and EfI sensor sets agree.
1 FEfI EfI
0.9 Effective Independence
0.8 10% Modal Damping
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0
5
10
15
20 25 30 Node Number
35
Fig. 1. Final EfI and FEfI distributions for 10% damping.
40
45
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0.5 FEfI EfI
0.45 Effective Independence
0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0
5
10
15
20 25 30 Node Number
35
40
45
Fig. 2. Final EfI and FEfI distributions—all modes.
These results demonstrate that the FEfI technique is automatically sensitive to input locations. Modal EfI requires the user to select the target modes based on engineering judgment, or additional techniques. While for this case, it is quite clear that only the symmetric modes will be excited, for a more complex structure, with a high modal density, the task of selecting the appropriate target modes becomes much more challenging. This example demonstrates that on a simple structure, for the case of light damping with inputs that excite all of the modes in the frequency band, the FEfI scheme picks sensors in a similar fashion to modal EfI. These examples also show how the FEfI technique automatically accounts for input location and damping effects. 5.2. Generic spacecraft The second example considers the Generic Spacecraft (GSC) to demonstrate how the FEfI technique places triaxial sensors on a more complex structure. The GSC is composed of a cubic core, two circular rib stiffened reflectors, and two rectangular photovoltaic (PV) arrays. The GSC’s FEM contains 1191 nodes and 1262 elements. Sensor placement was considered for two cases. The first trial is in the low frequency regime, so a comparison can be made to the modal based EfI technique. The second trial considers the mid-frequency range, where the number of modes present in the frequency band is prohibitive for modal based sensor placement approaches. 5.2.1. Low-frequency application The first case considers the frequency band from 0.1 to 10 Hz, which contains 16 undamped elastic mode shapes. The low frequency GSC example is similar to the beam example in that there are a relatively small number of modes contained in the frequency band of interest. However, unlike the beam, the modes of the GSC are closely spaced in frequency, which may make them more difficult to distinguish in a modal test. The GSC was excited with inputs in the z-direction (into the paper) located at six nodes, one on the end of each PV array and two on each reflector. Similar to what was seen in the beam example, the inputs do not guarantee that all of the mode shapes in the frequency band will be excited. One would expect that inputs only in the z-direction would not excite modes that have their primary displacements in the x–y plane. This makes the task of selecting appropriate target modes vital to the success of modal EfI, while FEfI automatically accounts for the active modes. The acceleration frequency response for the band from 0.1 to 10 Hz was calculated using Eq. (1), with an assumed 1% modal damping. The singular values were truncated to the top 95% of their total sum, resulting in 9 retained principal directions. This application places 100 triaxial accelerometers on the GSC in an attempt to capture all of the dynamically important data for the specified frequency band and input locations. The initial sensor set was created by selecting the highest ranked sensor location, based on the FEfI reduction technique [3], from the full set of nodes. All of the remaining node locations were then placed into the initial candidate sensor set. A total of 99 iterations were run to expand the initial sensor set to the final 100 sensor locations. The first 2 iterations used the method outlined in Case 2 to expand the initial sensor set until the information matrix was full rank. Then, 97 additional iterations, using Case 1, were performed to obtain the final sensor configuration. Fig. 3 shows the final triaxial sensor positions from the frequency response based technique. Next, modal based EfI [7] was run to obtain a sensor set for comparison. All 16 of the modes in the frequency band of interest were included in the target set. The initial sensor set was selected in the same fashion as for the frequency response based approach. The final 100 sensor locations based on modal EfI are shown in Fig. 4.
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Fig. 3. Sensor locations for FefI.
The sensor sets for FEfI and modal EfI share 68 of 100 common sensor locations. Each technique placed sensors around the edges of the circular reflectors and along the edges of the PV arrays. The frequency response based technique placed additional sensors near the input locations on the reflectors, while the modal-based technique placed more sensors around the edge of the PV arrays. The difference in sensor locations between the two techniques may be attributed to modal EfI placing sensors based on modes that were not excited by the inputs. Overall, the sensor sets for both techniques show a similar distribution. This shows that for the low-frequency range the frequency response based method is capable of selecting sensor locations in a similar fashion as compared to its modal counterpart.
5.2.2. Mid-frequency application The second case considers the GSC in a frequency range with high modal density. The frequency band of interest for this example is from 50 to 300 Hz, which contains 412 vibrational modes. It is important to note that the FEM being considered
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Fig. 4. Sensor locations for modal EfI.
in this example is not refined enough to make accurate predictions over this entire frequency band. However, the modes predicted are close in frequency, and very localized. Therefore, the model exhibits all the problems associated with the mid-frequency range, but is still easy to manipulate due to its relatively small size. The number of modes within the halfpower band at each frequency, also known as the mode overlap factor, peaks at 24 around 200 Hz. Closely spaced modes can be difficult to distinguish in a modal test, and when coupled with noise, create problems for correlation metrics, such as orthogonality and cross-orthogonality. The large number of modes present in the frequency band makes it impractical to try to identify each of the modes in a vibration test. For the mid-frequency range, modal based techniques fail because the number of sensors required for linear independence of the modes is prohibitive. Here, modal based EfI would require a minimum of 138 triaxial sensors to render the mode shapes linearly independent, and as many as 412 triaxial sensors to guarantee independence. This problem lends itself more naturally to the frequency response based techniques outlined in
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this paper, where the frequency response is dominated by a relatively small number of principal directions, as compared to the mode shapes. This case uses the same input locations and damping characteristics as used in the previous low-frequency example. The analytical frequency response for the band from 50 to 300 Hz was calculated using Eq. (1). To account for contributions from out-of-band modes, all mode shapes up to 450 Hz were used in the calculation. The singular values of the frequency response were computed and truncated to the top 95% of their total sum, resulting in 121 being retained. This shows that the bulk of the system’s energy is captured in 121 principle directions, which is roughly 70% fewer shapes than the number modes contained in the 50–300 Hz. frequency band. The FEfI triaxial sensor set expansion method was used to obtain 60 triaxial sensor locations. The initial sensor set was once again created by selecting the highest ranked sensor location based on FEfI for the full set of nodes. A total of 59 iterations were run to expand the initial sensor set to the final 60 sensor locations. The first 40 iterations used the method
Fig. 5. Sensor locations for FEfI in mid-frequency range.
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outlined in Case 2 to expand the initial sensor set until the information matrix was full rank. Then, 19 additional iterations, using Case 1, were performed to obtain the final sensor configuration. Fig. 5 shows the final triaxial sensor positions on the GSC. It is apparent that all of the sensors for this mid-frequency example are clustered on the circular reflectors. This can be attributed to the fact that the majority of the modes in this frequency range have displacements that are highly localized to the reflector surfaces. 6. Conclusion A frequency domain sensor placement scheme was presented based on principal component analysis of the frequency response data matrix. The method is a generalization of effective independence [3,6,7]. Sensors are placed to maintain the dynamically important information in the frequency response and the overall system energy within the frequency range of interest. The frequency response data automatically accounts for input location and damping, therefore this technique eliminates the difficult task of identifying target modes. Two examples were presented that showed that frequency and modal based EfI methods provide comparable sensor configurations for systems with low damping and well separated modal frequencies. The beam example showed that as damping levels increased, the frequency-based EfI approach automatically skewed the sensor locations toward the input location. Both the beam and the Generic Spacecraft examples demonstrated that the frequency-based EfI technique automatically accounts for how the selected inputs excite the structure, and thereby eliminate the need for the user to identify the target modes. In the high modal density example, where the modal density was very high, it was not only impossible to select and distinguish the appropriate target modes, there were so many of them that there were not enough sensors available to render them linearly independent. This makes modal EfI impossible to use, and frequency EfI, based on principal directions, a valuable tool for sensor placement.
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