The Art of Practical and Precise Strain Based ... - Webprofile.info

Strain-Based Measurement Introduction:Applications Note 1-7:The Fourier SeriesThe Fourier series is a mathematical expression that is used to define any timevaryingfunction as an infinite sum of the multiples of the fundamental frequencythat comprise the time function. The Fourier series is given by:qt ()=¥a---- 0+ a2 å ( k coskw 1 t + b k sinkw 1 t)k = 1(EQ 1-7)The Fourier series is not just the mathematical expression of a time function, itis the time function. Another way of regarding the Fourier Series is to considerthat every frequency component required to define a specific time function canbe regarded as a physically real subcomponent of the time function. The valueof k in the above expression is equal to the number of sinusoids, or harmonics,summed together to approximate the time function that is being modeled,where higher values of k yield closer approximations to the time function.The key issue concerning the Fourier Series expression of a time-varying signalis that the transfer function of the sensor and measurement system will acton each of the harmonic components as if these had been input separately. Theresult of this logic is that the resultant time-varying output is the summation, orsuperposition, of each of the outputs from the sensor or measurement systemcorresponding to each of the Fourier input components. Figure 1-21 shows thewaveform that will result if three simple Fourier components are summed. Itcan be seen intuitively that any complex waveform can be created by the successivesumming of the Fourier subcomponents. Correspondingly, each of theFourier components of the input will be phase-shifted by a different phaseangle depending upon the phase relationship of the measurement system transferfunction. The end result is that the output from the measurement system canbe both amplitude- and phase-distorted unless care is taken to assess the frequencyof the highest expected frequency component, and then to assess theCHAPTER 1-40 The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999