Daniel C. Kammer. Associate Professor ... in which w is a diagonal matrix of modal frequencies, ... However, due to the Cayley-Hamilton theorem [IO], there are ...
ESTIMATION OF STRUCTURAL RESPONSE USING REMOTE SENSOR LOCATIONS Daniel C. Kammer Associate Professor Department of Nuclear Engineering and Engineering Physics University of Wisconsin Madison, WI 53706 (608) 262.5724
ABSTRACT A method is presented for estimating the response of a structure during its operation at discrete locations which are inaccessible for measurement using sensors. The prediction is based upon measuring response at other locations on the structure and transforming it into the response at the desired locations using a transformation matrix. The transformation is computed “sing the system Markov parameters determined from a vibration test in which the response is measured at both the locations which will possess sens”rs during structure operation and at the desired locations which will not possess sensors. Two different approaches are considered. The first requires as many sens”rs as there The second are modes responding in the data. approach, a generalization of the first, only requires as many sens”rs as the number of desired resp”“se locations. A numerical example is considered using the Controls-Structures Interaction Evolutionary Model (CEM) testbed at NASA LaRC. Acceleration response with 10% m~s noise at six sensor locations is used to predict the response at four force input locations. The proposed method is not computationally intensive, and combined with the fact that the process is causal, may allow real-time applications. 1.0
INTRODUCTION
Accurate knowledge of response produced at discrete locations within a flexible structure during its operation is vital to many engineering applications such as control, structural monitoring, damage detection, and parameter identification. In the case of flexible structure control, for many situations, the response of interest is at the actuator locations. Collocation of inputs and outputs, in the absence of sensor/actuator dynamics, is guaranteed to produce a stable closed loop system which is robust with respect to modeling err”rs [ 1, 21. In contrast, noncollocation of inputs and outputs produces a nonminimum phase system which possesses fundamental difficulties in both performance and stability such as the phenomena of control and observation spillover. Inverse filtering of signals [3-51 represents another instance in which collocation of inputs and outputs is desirable. A nonminimum phase
system by definition possesses unstable zeros. Inversion of such a system produces a filter which has unstable poles. The corresponding output of the filter becomes unbounded. In the case of structural monitoring and damage detection, response is required at identified critical areas within the structure. In many instances, sensors, such as accelerometers, can be easily placed at the locations of interest. However, situations can exist in which the desired locations within the structure are not accessible for measurement during the structure’s operation. For example, this may be the case when locations of interest are at interfaces between substructures of a larger structural system. In control and filtering problems, strict collocation of inputs and outputs is usually physically impossible. This paper presents a method by which the response at discrete inaccessible locations can be predicted using the response measured elsewhere on the structure. A transf”rmation matrix is produced based up”” the unit pulse response obtained from a vibration test of the structure or substructure prior t” it being placed in operation. Response measured during structural operation can then he transformed into the corresponding response at the desired locations. 2.0 DESCRIPTION OF STRUCTURAL SYSTEM The structural system to be analyzed has a discrete time representation which is governed b y t h e difference equation x(k + I) = Ax(k)+ Du(k)
(1)
in which x represents an n dimensional state vector, A is the nxn system matrix, D is the nxn, input influence matrix, and u is the n, dimensional force vector. Index k indicates the appropriate time step. The corresponding system output equation is given by y(k) = Cx(k) + Hu(k)
(2)
where y is an t~,~ dimensional sens”r output vector, C is an n,sxn output influence matrix, and H is an n,xn, direct feedthrough matrix. It is assumed throughout this analysis that
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accelerometers are used as sensors such that the direct feedthrough matrix is “““zero. In the case of a modal representation of the structure given by g+2g@+wzq=u
into sensor and desired locations to produce the two relations Y, = $3
(9)
Yd = 02
(10)
(3)
the matrices A and D will be time discretized versions of
in which subscripts s and d refer to sensor and desired location partitions, respectively. Equation (9) can be solved for the modal acceleration using a least-squares approach
while matrices C and H are. of the form c = [-9?Pi -W@]
H = $$,;
(5)
in which w is a diagonal matrix of modal frequencies, g is a diagonal matrix of modal damping coefficients, and @, and en are the mode shapes partitioned to the sensor and input locations, respectively. For zero initial conditions, equations (I) and (2) can be combined to produce the output at any time step in the form
y(k)= &(k-i)
(6)
,=n
Equation (6) represents a moving average model of the discrete system where the n,xn, weighting matrices H;, called Markov parameters, are given by
Ho= H
H;=
CA’-‘D
i=l,2,3-.
(7)
The Markov parameters represent the response of the discrete system to unit force pulses at the input locations and thus contain tbe dynamic properties of the structure. They can be obtained by experimentally measuring the output of the system due to a known input and computing the corresponding frequency response functions. The Markov parameters are then derived by computing the inverse discrete Fourier transform. 3.0 PREDICTION OF RESPONSE USING AS MANY SENSORS AS MODES
Assume that y represents the acceleration response due to input forces, u, at both the sensor and desired response locations and that it can be written in the form y=f#q
Substitution into Eq. (IO) produces (12) where j,, represents the estimate of the acceleration response at the desired locations. Using the transformation matrix P, the desired location response can be estimated from the acceleration response measured at other locations on the structure. Note that for this approach to be valid, the number of sensors, nS, mutt be greater than or equal to the number of modes, n,,,, that are excited by the inputs, and the modal partitions, I$,, must be linearly independent. The author has developed a method called Effective Independence which can be used to optimally place sensors to satisfy the above conditions [6-81. It is proposed that mode shapes derived from a pretest finite element model can be used to optimally place sensors on the structure. In order to compute the transformation matrix P using Eq. (l2), the modal coefficients at both the desired and sensor locations must be known. The test based modal coefficients can be computed from the same test data that must be obtained to compute the Markov parameters. Several methods have been developed for this purpose [9]. Note that the response of the structure must be measured during the test at both the desired response locations and the locations that will possess a sensor when the structure is in operation. An alternative approach can be taken for computing the transformation P directly from the test data without estimating the mode shapes. Assuming that there are n, points in the test data, there will in general be n, Markov parameters, Hi, which can also be partitioned according to sensor and desired locations as
[ 1d,
(8)
H;= ; in which @ is the matrix of mode shapes which respond in the output and q is the vector of corresponding modal accelerations. Equation (8) can be partitioned
i = 0, n, - 1
(13)
From previous discussions, it is known that the r e s p o n s e a t a n y l o c a t i o n can be computed by
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convolving the corresponding forward system Markov parameters with the input forces. Therefore, Y,(k) = $ HJ4 - 4
(14)
y,(k)=iH,u(k-i) i-0
(15)
but to increase the row rank of 6, such that Hd will always lie in its row space. The proper form for F?, can be found by expanding Eq. (14) for k time steps into the matrix expression H,,,
Equation (12) then implies i = 0, n, - I
H, = PK.
0 0 :“.
(16)
However, due to the Cayley-Hamilton theorem [IO], there are at most 2n,,, + I independent forward Markov parameters. Therefore, only N, t 2n,,, + 1 equations from (16) must be considered. Assembling the sensor and desired location Markov parameters into the matrices
Hq, H,,,
0
:
0
...
H,,
...
H,m, ;
...
H,t-,s,
!
u(k) u (k-1)
: 4,
(21)
while for the kth time step, Eq. (15) is given by and using Eq. (16) results in PHr = H,,
u(k) u(k - I)
(18)
= rn,!, and @, is full column rank.
40)
u(k)
(23)
u(k -I) =H,
4.0 PREDICTION OF RESPONSE USING AS MANY SENSORS AS DESIRED LOCATIONS
P& = H,
40)
Premultiplying Eq. (21) by the transformation P and utilizing Eqs. (20) through (22) produces
[I I]. Using p, a n y r e s p o n s e m e a s u r e d o n t h e substructure at the sensor locutions during vehicle operation can be used to estimate the corresponding response at the desired locations.
In some cases, there may be many modes responding in the sensor output, making the proposed method of predicting the response at the input locations cumbersome due to the large number of required Se”SO*S. This section focuses on modifying the previous method such that fewer than nm sensors are required. It is now assumed that nn 5 n, 2 n,,, The idea is to still use an equation analogous to (18) such as
(22)
i
= y,,(k)
_ 40) _
If transformation P can be found, the response at the desired locations at time k, y,,(k), can be predicted using the current sensor response, yT(k), and the response from the past N, -I time steps. The corresponding matrix equation that must be solved for P is given by
(20)
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Hr, 0 po I 0
H,,, H,,
... ... 0
... ... :
H,+ H,,+, f ."
4,
= [ Hz,,
...
Hnv-,.
H
