Earthquake Engineering Research - HKU Libraries - The University ...

249
WOLFE, MASRI, CAFFREY
Synthetic data was developed utilizing a nonlinear single-degree-of-freedom (sdof)
Duffing oscillator. Manipulating the equation of motion yields the oscillator
restoring force as
f(x,x)=m[2£aK + G) 2 (x + £x 3 )] (1)
Unit mass and system period were prescribed for this oscillator. The natural
frequency was set at In. These values yielded the system stiffness, damping
coefficient and nonlinear term of 39.48, 1.26 and 1.23, respectively. Noise pollution
was incorporated into the simulated data by adding stationary, zero-mean, 0.10
standard deviation noise. Potential noise sources in real-world applications include
instrumentation susceptibility, cabling interference, acquisition hardware, etc. The
system response to a stationary random excitation is depicted in Figure 2.1.
2.2. Identification Procedure
The restoring force method [1] was utilized to identify the system parameters from the
system response. This method expresses the estimated restoring force in terms of
two-dimensional orthogonal polynomials as
where the T's are defined as Chebyshev polynomials, taking advantage of their equalerror
approximation within the interval of interest. Chebyshev polynomials are
defined as
(3)
satisfying the weighted orthogonality property
0
(4)
The normalized displacement and velocity values are defined as
-x_)/2]