Abstract
A method for identifying a piecewise-linear approximation to the nonlinear forces acting on a system is presented and demonstrated using response data from a micro-cantilever beam. It is based on the Restoring Force Surface (RFS) method by Masri and Caughey, which is very attractive when initially testing a nonlinear system because it does not require the user to postulate a form for the nonlinearity a priori. The piecewise-linear fitting method presented here assures that a continuous piecewise-linear surface is identified, is effective even when the data does not cover the phase plane uniformly, and is more computationally efficient than classical polynomial based methods. A strategy for applying the method in polar form to sinusoidally excited response data is also presented. The method is demonstrated on simulated response data from a cantilever beam with a nonlinear electrostatic force, which highlights some of the differences between the local, piecewise-linear model presented here and polynomial-based models. The proposed methods are then applied to identify the force-state relationship for a micro-cantilever beam, whose response to single frequency excitation, measured with a Laser Doppler Vibrometer, contains a multitude of harmonics. The measurements suggest that an oscillatory nonlinear force acts on the cantilever when its tip velocity is near maximum during each cycle.
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Abbreviations
- w L :
-
Relative tip deflection, [m]
- y :
-
Absolute tip displacement, [m]
- e b :
-
Base displacement, [m]
- y 0 :
-
Initial gap between beam and base, [m]
- ω :
-
Frequency, [rad/s]
- ω n :
-
Natural frequency, [rad/s]
- ζ :
-
Damping ratio, [unitless]
- f tot :
-
Total restoring forces, [N]
- f nl :
-
Nonlinear part of restoring forces, [N]
- m :
-
Effective mass, [kg]
- c :
-
Damping constant, [N s/m]
- k :
-
Stiffness, [N/m]
- ψ n :
-
nth basis function for piecewise-linear function, [unitless]
- q n :
-
Coefficient of nth basis function, [N or m/s2]
- N :
-
Number of basis functions in piecewise-linear approximation
- N f :
-
Number of time instants at which the acceleration, velocity and displacement are measured
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Sandia is a multi-program laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy’s National Nuclear Security Administration under Contract DE-AC04-94AL85000.
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Allen, M.S., Sumali, H. & Epp, D.S. Piecewise-linear restoring force surfaces for semi-nonparametric identification of nonlinear systems. Nonlinear Dyn 54, 123–135 (2008). https://doi.org/10.1007/s11071-007-9254-x
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DOI: https://doi.org/10.1007/s11071-007-9254-x