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Piecewise-linear restoring force surfaces for semi-nonparametric identification of nonlinear systems

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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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Authors and Affiliations

  1. Department of Engineering Physics, University of Wisconsin-Madison, 535 ERB, 1500 Engineering Drive, Madison, WI, 53706, USA

    Matthew S. Allen

  2. Sandia National Laboratories, P.O. Box 5800, Albuquerque, NM, 87185, USA

    Hartono Sumali & David S. Epp

Authors
  1. Matthew S. Allen
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  2. Hartono Sumali
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  3. David S. Epp
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Correspondence to Matthew S. Allen.

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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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