Page 126 - MODELING OF ASPHALT CONCRETE

104 Cha pte r F o u r

The quality of transfer function determines if the system output is totally caused by
the system input. Noise and/or nonlinear effects can cause large errors at various
frequencies, thus including errors when estimating the transfer function. Coherence
2
function g can be used to estimate the quality of system, where
γ = Response power caused by applied input (4-14)
2
d
Measured response power
2
G
γ = yx where 0 ≤ g ≤ 1 (4-15)
2
2
GG
xx yy
2
If g = 1 at any specific frequency, the system is said to have perfect causality at that
2
frequency. If g < 1, then extraneous noise is also contributing to the output power.
Differences in Employed Analysis Methods: Time Domain Methods
A study by Pellinen and Crockford (2003) compared three different filtering methods
and two different phase referencing methods of computing modulus and phase angle
from compressive dynamic modulus test data. The methods discussed were limited to
time domain techniques (other than FFT method) applied to cyclic loading in
compression. Study showed that the computed modulus values were less sensitive to
different analysis techniques than the phase angles.
Dynamic modulus test data of a dense graded asphalt mix obtained at five different
temperatures and six different frequencies were analyzed in this study. This type of a
data set is needed to construct a mastercurve of the mix required in the new pavement
design guide. Testing was conducted as a stress controlled test applying compressive
stress with a frequency sweep of 25, 10, 5, 1, 0.5, and 0.1 Hz.
The testing system capable of maximum sampling rates of 1 kHz and sampling rates
adequate to eliminate aliasing at the frequency of loading were used. Additional firmware
and sampling techniques were employed to minimize skewing of sequential samples and
noise in the incoming signal. In the presence of the noise remaining after firmware filtering,
the peaks of the waveforms generally exhibit the largest noise amplitude in a single cycle
and are therefore the worst locations to perform analyses that determine phase angles.
Greater noise at the peaks is due in part to the test machine actuator and the transducers
reversing their direction of movement at those times. It is assumed that it is much better to
determine phase angle from the “middle” of the waveform where the machine and
transducers are all moving in a relatively “steady state.” Using unfiltered peaks can easily
create phase shifts that are due to noise instead of fundamental signal peaks. The modulus
must be computed from the peaks, so additional software filtering is usually performed to
improve the peak measurement. This filtering must be carefully done so that time skewing
and alteration of the fundamental signal magnitude are minimized.
When any type of cyclic forcing function is applied to a material such as asphalt
concrete under load control, a strain response that mirrors the forcing function but with
different amplitude and a phase shift is expected. This is an oversimplification even in
the case of strictly compressive loading:
• Even if the forcing function’s wave shape is perfect, there is a nonzero average
stress level during the cyclic loading which causes the cyclic strain response
curve to be superimposed on a creep curve. For asphalt and polymers, this is
more apparent at high temperatures. Reducing the load may minimize the
creep, but typical load amplitude requirements do not eliminate the creep.