Journal of Vibration and Acoustics 125 (1) 129-131 (2003)
Proper Orthogonal MODEs of a Beam Sensed with strain gages
M. S. Riaz
Department of Mechanical Engineering
Michigan State University
2555 Engineering BuildingEast Lansing, MI 48824
Telephone: 517-355-2980, 517-353-1750 (fax)
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B. F. Feeny
Department of Mechanical EngineeringMichigan State University
2555 Engineering BuildingEast Lansing, MI 48824
Telephone: 517-353-9451, 517-353-1750 (fax)
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1.0 introduction
Proper orthogonal decomposition (POD) is a useful experimental tool in dynamical systems, for example in dimensionality studies [1,2] and reduced order modeling [3-6]. Application of POD in structural vibrations often involves sensed displacements, x1, x2, …, xM, at M locations on the structure. These displacements are sampled N times at a fixed sampling rate to form displacement arrays xj = [xj(t1),…,xj(tN)]T, j = 1,…, M. The means are often subtracted. An NxM ensemble matrix X = [x1,…, xM] is then built. The MxM correlation matrix is R = XTX/N. Since R is real and symmetric, its eigenvectors form an orthonormal basis. The eigenvectors are the proper orthogonal modes (POMs) and the eigenvalues are the proper orthogonal values (POVs).
The POMs in certain nonlinear structures have resembled the normal modes of the linearized system [1,7,8]. The POMs may converge to linear normal modes in multi-modal free responses of symmetric lightly damped lumped-mass linear systems, but only if the mass matrix has the form mI, which can be achieved by a coordinate transformation if the mass distribution is known [9,10] and equivalently [11]. Analysis and simulations show that this relationship is approximately true in distributed parameter systems if the mass and the sensor locations are uniformly distributed [12].
In this note, this application of POD for modal analysis is tested on a cantilevered beam sensed with strain gages. The objective is to verify the applicability of POD for modal analysis in a simple experiment, and to bring forth issues that arise from strain measurements. This parallels experimental work with accelerometers [13].
2.0 SYSTEM DESCRIPTION
An experiment was made to emulate a cantilevered Euler-Bernoulli beam. The 0.394× 0.012× 0.00079 m3 beam of mild steel had one end fixed in a steel clamp. The beam had Young’s modulus E = 128 ×109 N/m2 and a density = 7488 kg/m3.
Theoretical modal frequencies for this beam were computed based on the known solution [14] and were compared to experimental modal frequencies obtained from the fast Fourier transform of an impulse response. The six lowest theoretical frequencies were 4.52, 28.36, 79.38, 155.6, 257.2, and 384.2 Hz, and the experimental modal frequencies were 4.5, 27.2, 75.5, 147, 243, and 365 Hz.
The modal damping ratios were estimated as 0.0161, 0.016, 0.009, 0.0047, 0.0031, and 0.0022. These are rough estimates due to coarse resolution on the FFT analyzer, and the assumption of an ideal impulse input. Conventional modal analysis [14] would involve measuring frequency response functions between various input and output locations. Our beam was so floppy that we were unable to generate a meaningful signal on the impulse hammer. (This points to an advantage of the POD method, for which sensed inputs are not generally needed.)
The displacements at various points were measured with the help of twelve strain gages in six half wheat-stone bridges arranged for bending. The strain gages were mounted on the beam at locations from the clamped end specified as x1= 0.00527m, x2=0.0543m, x3=0.1033m, x4=0.1523m, x5=0.2013m, x6=0.2503m. The gages were biased towards the clamped end to improve strain sensitivity. The locations were “optimized” by performing numerical simulations and iterating the gage locations [15].
The displacements were estimated from the strain measurements by using assumed-mode basis functions, according to the following development.
By approximating
where Ms is the number of strain gage locations, and the i(x) form a basis satisfying the geometric conditions. Then we can write y = u, where yk = y(xk, t) are the elements of the Md-vector y, kj = j(xk) are the elements of Md Ms matrix , xk are the desired displacement locations, and uj(t) are the elements of Msmodal displacement vector u. Md is the number of desired displacement measurements on the beam.
The axial strain ε(x) on the surface of a symmetric beam in bending is related to the transverse displacement y(x) by
or
where c is half the width of the beam, i = (xi) are the elements of Ms strain vector , and ij = 2j(x)/x2|x=xi are the elements of MsMs matrix .
Strains were measured at Ms locations on the beam, and displacements were desired at Md locations. As such, the MsMs matrix is constructed based on the Ms actual strain-gage positions and the Ms basis functions, and the MdM matrix is built based on the Ms basis functions evaluated at Md desired displacement locations.
Assuming is invertible (as it is made up of evaluations of linearly independent basis functions), we obtain y in terms of , such that
y
3.0 RESULTS
The beam was excited with an impulse, and the resulting multi-modal free vibration was monitored. The six strain histories were simultaneously sampled and recorded in 0.2 second windows which were then pasted together in time. The data was taken for 10 seconds with a sampling rate of 800 samples per second in all the tests, which was enough time to capture the modal characteristics of the cantilever beam.
3.1 Theoretical Modes as a Basis
The first set of tests involved basis functions i(x) consisting of the linear normal modes of the cantilevered beam model. This may seem pointless, as the linear normal modes are ultimately to be estimated by the POD. But the linear normal modes provide the best basis for the strain-to-displacement conversion, resulting in the best displacement estimates, and thus yield an illustrative example for the method. Figure 1 shows the POMs, with Md = 6, compared with LNMs. Continuous lines represent the LNMs and circles show the POMs. The POVs corresponding to this plot were 2.88, 0.00374, 1.19e-5, 1.19e-6, 2.22e-8, and 1.53e-9.
Tests were done at various sampling rates and impulse locations, and results were consistent for the lower three modes. In all the cases we saw the maximum energy with the first mode. When the input was applied near a nodal point, the corresponding mode was excited less.
3.2 Use of Redundant Sensors
We generated a large number of “pseudo sensors” on the beam by evaluating the continuous functions at Md > 6 locations. The pseudo sensors are redundant sensors derived by interpolating the measurements through the usage of the basis functions, which in this case were the linear normal modes of the cantilevered beam model. The motive for using the pseudo sensors was to improve the resolution associated with the rectangular rule integration that effectively underlies the relationship between orthogonality of linear normal modal functions, orthogonality between discrete POMs and the uniformly discretized modal vector [12]. However, as we used only first six strain measurements and six LNMs, the system measurement only contains six independent displacements. The pseudo sensors apparently performed an equivalent task to Gramm-Schmidt orthonomalization used in converting discrete POMs to continuous orthogonal functions [15]. Either approach depends on the interpolating functions and how well they depict the physics of the problem.
Figure 2 shows an example when 21 pseudo sensors were used. The use of pseudo sensors provides some improvement, and also helps in visualizing the highly undulatory modes.
3.3 On the Choice of Basis Functions
The LNMs are typically unknown, and not available as a basis for a strain-to-modal-coordinate conversion. As such, we made example computations using various bases. Figure 3 shows the results for basis functions of the form i(x)=1-cos(ix), where i=(2i+1)/2L, i=0,1,…,5. These basis functions are not orthogonal, but are linearly independent, and meet the geometric boundary conditions and the zero-strain condition at x=L. The POMs are not as good as the those with a basis of LNMs, but are distinct from the basis functions.
Not shown are results for basis functions of the form i(x) = sin(ix), where i = (4i+1)/4L, i=0,1,…,5. This basis only satisfies the displacement boundary condition at x=0. It imposes a false constraint of zero slope at x=L. The resulting POMs nearly (not exactly) accommodated the other geometric boundary condition, as responses obey the boundary conditions [16]. But the POMs exhibited the erroneous zero slope at x=L. Otherwise, the POMs were qualitatively representative of the LNMs.
We also used orthogonal polynomials of degrees two through seven as basis functions [17]. The results are not shown, but the POMs were distorted from the true LNMs toward the free end of the beam. This is probably because the high-degree polynomials, evaluated at several values of x, are not well conditioned for the computation.
4.0 CONCLUSION
POD was applied to a beam sensed with strain gages. Basis functions were used to convert strains to displacements. Results were best when the basis functions were trivially chosen as the LNMs. When the basis functions differed from the LNMs, the POMs were still a reasonable approximation of the LNMs, and were distinct from the basis itself, even though the basis was used to interpolate redundant sensors. The choice of the basis functions is thought to influence the POD by its influence on the quality of the displacement estimations.
Acknowledgments
This research was supported by the National Science Foundation (CMS-9624347 and CMS-0099603).
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Figure 1: Proper orthogonal modes with the impulse applied between 4th and 5th strain gage from the clamp. Circles depict the POMs, and lines show the theoretical LNMs.
Figure 2: Proper orthogonal modes with the impulse applied between 4th and 5th strain gage and using 21 pseudo sensors. Circles: POMs. Lines: LNMs.
Figure 3: Proper orthogonal modes using a basis made of cosine functions. Circles: POMs. Lines: LNMs.
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