Construction of frequency-energy plots for nonlinear dynamical systems from time-series data

Wang, Xingyuan

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https://hdl.handle.net/2142/16482 Copy

Description

Title

Construction of frequency-energy plots for nonlinear dynamical systems from time-series data

Author(s)

Wang, Xingyuan

Issue Date

2010-06-22T19:37:24Z

Director of Research (if dissertation) or Advisor (if thesis)

• McFarland, Donald M.
• Vakakis, Alexander F.
• Bergman, Lawrence A.

Department of Study

Mechanical Sci & Engineering

Discipline

Mechanical Engineering

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

M.S.

Degree Level

Thesis

Keyword(s)

• Frequency-energy plot (FEP)
• dynamical system
• nonlinear
• dynamics
• vibration
• vibrating
• system
• construction
• time-series
• experimental
• simulation
• results

Abstract

The frequency-energy plot (FEP) is a useful tool in analyzing nonlinear system response, but previous methods of generating the FEP were limited to analytical calculations or use of the wavelet transform. Although these are acceptable methods of constructing the FEP, the former requires significant computation, and the latter does not provide a sharp frequency estimate. This work presents the construction of the FEP with knowledge of the mass distribution and velocity time series, which is obtained either by direct measurement, using a laser vibrometer or equivalent; or through time differentiation or integration of measured displacement or acceleration, respectively. The FEP is generated through three intermediate steps. First, the total energy of the system is estimated by fitting a spline through the non-increasing peaks of the kinetic-energy time series. Use of the kinetic rather than the potential energy removes the need to know the nonlinear stiffnesses present in a system. Separately, the empirical mode decomposition (EMD) method is used to decompose the system response into intrinsic mode functions (IMFs) whose frequencies are then estimated with the Hilbert transform. The FEP is created by plotting these estimated frequencies against the total energy. The components of this FEP algorithm are first defined in greater detail with a variety of examples, and an FEP is plotted for a simulated system. The system has been previously shown to exhibit 1:3 transient resonance capture (TRC), and the resulting FEP reflects that internal resonance with a sharper frequency estimate than is possible with the wavelet transform. Next, the algorithm is used to show that it is in fact possible to track the impulsive manifold when imposing different initial conditions to the same Hamiltonian system. Finally, it is applied to experimental data and simulations of the parameters identified from experimental data. Additional observations on the application of the algorithm to experimental data are also discussed, with the key conclusion that processing challenges arise when short-duration signals are analyzed with this method.

Graduation Semester

2010-5

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http://hdl.handle.net/2142/16482

Copyright and License Information

(c) 2010 Xingyuan Wang

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