Model parameters are fit to empirical vibration data.

Two innovative mathematical models of nonlinear vibrations, and methods of applying them, have been conceived as byproducts of an effort to develop a Kalman filter for highly precise estimation of bending motions of a large truss structure deployed in outer space from a space-shuttle payload bay. These models are also applicable to modeling and analysis of vibrations in other engineering disciplines, on Earth as well as in
outer space.

The first model is denoted the amplitude- dependent stiffness (ADS) model to emphasize the difference between it and the classical linear harmonic-oscillator model, in which stiffness is a constant. The ADS model is embodied in the equation

where x is the instantaneous amplitude of the oscillating position or modal coordinate, ξ is a damping parameter, and K(x,x˙) is the ADS.

In the initial outer-space application, the ADS was represented by the following nonlinear function:

A Nonlinear Decaying Waveform is approximated with a best-fit sinusoidduring a moving window. The resulting sinusoidal amplitude and frequencydata are collected from fits for the entire sequence of window positionsand used to characterize the frequency versus amplitude of the nonlinearwaveform. These frequency versus amplitude data are then fit to an amplitude-dependent stiffness (ADS) representation.

The modal amplitude is given by

where K¯ (x,x˙) is any reasonable approximation of K(x,x˙). One can refine the approximation iteratively, starting from K(x,x˙)=a, then using the resulting value of A(x,x˙) in computing a value of K¯ (x,x˙) by use of the above equation for K(x,x˙) .

The second model, denoted the moment-expansion (ME) model, is embodied in the equation

where the function M(x,x˙) is a moment expansion that captures damping and stiffness effects. The moment expansion is given by

where both i and j range from 0 to 3, except that there is no (i,j) = (0,0) term. In the original outer-space application, the parameters p_ij are obtained from (1) modal position and velocity estimates obtained from Kalman-filter states and (2) derived accelerations.

In a test relevant to the original outer-space application, the ADS and ME models were compared with each other, with a linear model, and with a prior nonlinear model known as the Duffing model. The ADS model was found to yield the least error.

This work was done by Paul Brugarolas, David Bayard, John Spanos, and William Breckenridge of Caltech for NASA’s Jet Propulsion Laboratory. For further information, contact This e-mail address is being protected from spam bots, you need JavaScript enabled to view it