Nonlinear Dynamics

, Volume 54, Issue 1–2, pp 123–135 | Cite as

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

Original Paper


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.


Nonlinear system identification Restoring force surface Micro electro-mechanical system MEMS Nonlinear vibration Harmonic distortion Force state mapping 



Relative tip deflection, [m]


Absolute tip displacement, [m]


Base displacement, [m]


Initial gap between beam and base, [m]


Frequency, [rad/s]


Natural frequency, [rad/s]


Damping ratio, [unitless]


Total restoring forces, [N]


Nonlinear part of restoring forces, [N]


Effective mass, [kg]


Damping constant, [N s/m]


Stiffness, [N/m]


nth basis function for piecewise-linear function, [unitless]


Coefficient of nth basis function, [N or m/s2]


Number of basis functions in piecewise-linear approximation


Number of time instants at which the acceleration, velocity and displacement are measured


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Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  • Matthew S. Allen
    • 1
  • Hartono Sumali
    • 2
  • David S. Epp
    • 2
  1. 1.Department of Engineering PhysicsUniversity of Wisconsin-MadisonMadisonUSA
  2. 2.Sandia National LaboratoriesAlbuquerqueUSA

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