4.3 Experimental Methods
4.3.1 Impact testing
There are both analytic and numeric [9, 13, 98] approaches to finding the stability lobe diagram for a certain tooling structure, however both rely on knowledge of the dynamics of the free end of the structure, which is captured in the form the ’tool tip’ frequency response function. It should be noted that throughout the remainder of this thesis, the tool tip FRF is always used to refer to the FRF that is both measured and excited at the tool tip. The most common method to obtain the tool tip FRF is via impact testing [92]. By impacting the free end of the tool with an instrumented hammer, a force, f (t), is applied, and a wide range of frequencies can be excited simultaneously, then the resulting acceleration, ¨x(t), can be measured using a low mass accelerometer. By taking the Fourier transform of both the force and response, the FRF can be obtained. Assuming periodicity of the signals f (t) and ¨x(t) these are expressed as:
F (iω) = 1 tm Z tm 0 f (t)e−iωtdt (4.1) ¨ X(iω) = 1 tm Z tm 0 ¨ x(t)e−iωtdt (4.2)
where F (iω) is the Fourier transform of f (t), ¨X(iω) is the Fourier transform of ¨
x(t), ω is the frequency, and tm is the measurement time. The frequency response
function (Hacc(iω)) is then given by:
Hacc(iω) =
¨ X(iω)
F (iω) (4.3)
The above equation describes the accelerance FRF as it relates the the input force to the acceleration of the structure. For the rest of this thesis, however, we will be concerned with the receptance FRF, which relates the input force to the displacement of the structure. Since acceleration ¨x(t) and displacement x(t) are related (when the excitation is harmonic), by the relationship ¨x(t) = (iω)2Xeiωt = (iω)2x(t), the receptance FRF is calculated as
Hrec(iω) = X(iω) F (iω) = ¨ X(iω) (iω)2F (iω) (4.4)
H is used to refer to Hrec for the rest of this thesis.
In practice, the force and acceleration signals are measured at discrete time in- tervals (Ts) using a data acquisition (DAQ) board attached to a computer. If
a frequency resolution of ωs is required in the measurement data, measurements
must be collected for a total time of tm = 2π/ws, with N = tm/Ts data samples,
therefore
tm =
2π ωs
= N Ts (4.5)
Hence, the DAQ system works with Eqs. (4.1) and (4.2) in the discrete time form, Making the substitutions t = nTs, ω = kωs, and dt = Ts, the Fourier transforms
become F (ikωs) = 1 N N −1 X n=0 f (nTs)e−ik 2π Nn (4.6) ¨ X(ikωs) = 1 N N −1 X n=0 ¨ x(nTs)e−ik 2π Nn (4.7)
4.3. EXPERIMENTAL METHODS where F (ikωs) and ¨X(ikωs) are the discrete Fourier transforms or spectra of the
measured force and acceleration respectively. There are a number of additional steps that must be taken in practice to accurately calculate the FRF.
A necessary and sufficient condition for the correct reconstruction of a sampled signal, known as the Nyquist-Shannon sampling theorem, is that the sampling rate (1/Ts in Hz) is twice that of the highest frequency content in the signal (or
twice that of the highest frequency of interest). If this condition is not met, alias signals at higher frequencies will disrupt the reconstructed signal. To prevent such occurrences, data acquisition systems apply an anti-aliasing filter to the analogue signal, which attenuates the high frequency content above the Nyquist frequency. Such filters are known as low pass filters because frequency content below a certain frequency passes through the filter whilst higher frequency con- tent is removed. In the impact testing system used throughout this thesis an 8th order Bessel function filter is applied at 12500 Hz.
The contact time between the hammer and the tool is inherently very short and much shorter than the time for which the structure vibrates. Therefore a typical force signal contains a sharp peak followed by low amplitude noise. The effect of this noise on the quality of the FRF can be reduced by windowing the digital signals. Both the force and acceleration signals are multiplied by an exponential window function, which multiples the signals by unity during the period of con- tact and by an exponentially decreasing value thereafter. Once the signals have been windowed, the Fourier transforms can be calculated.
Despite the benefits of windowing the Fourier transforms will still contain some noise which results in Eq. (4.4) becoming inaccurate. The influence of noise can be attenuated by calculating the cross spectrum. The cross spectrum (Sxf¨ ) of
force and acceleration is found by multiplying the acceleration spectrum by the complex conjugate of the force spectrum, i.e.
◦ Anti-Aliasing Filter
Analog/Digital
Converter Windowing DFTX(ikω¨ s)
Auto- Spectrum (Sx¨¨x) Cross- Spectrum (Sxf¨) Auto- Spectrum (Sf f) DFTF (ikωs) Windowing Analog/Digital Converter Anti-Aliasing Filter ◦ + FRF1 (ikωs)2 Sxf¨ Sf f ¨ x(nTs) f (nTs)
Figure 4.2: Diagram showing the dimensions for both the unmodified and modi- fied holders
where∗ denotes the complex conjugate. The FRF is then found by dividing the
cross spectrum by the auto spectrum of the force signal such that
H(iω) = 1 (ikωs)2 Sxf¨ Sf f = 1 (ikωs)2 ¨ X(ikωs) · F∗(ikωs) F (ikωs) · F∗(ikωs) (4.9) It is also common to average a number of cross and auto spectra in order to further attenuate the noise in the signals, throughout this project five spectra were averaged before calculating the FRF.
The dual channel DAQ system which performs this signal processing can be ex- pressed as a block diagram as show in Fig. 2. There are many papers that discuss the accuracy of this form of impact testing, two noteworthy examples include Kim and Schmitz’s [99] paper, which discusses more physical problems in impact testing such as the effect of the mass of the accelerometer, and mis- alignment between the hammer and accelerometer, and Sims et al. [100], who list potential limitations with this type of impact test, such as the requirement of a skilled user, poor repeatability, and the ineffectiveness of the hammer to tap small tools.
One of the largest shortcomings of impact testing with a modal hammer is the inability to produce an impulse [99]. The frequency spectrum of an infinitely
4.3. EXPERIMENTAL METHODS narrow impulse signal is equal across all frequencies; meaning that, if a structure were excited by a true impulse all frequencies would be excited equally. Clearly, it is not possible to produce an impulse with a hammer, and instead the input signal usually resembles that of a half sine curve, the frequency spectrum of which decays with frequency. Therefore during modal testing, higher frequency modes may not be excited properly, causing error in the FRF. Moreover, as the input spectrum tends to zero, the FRF will tend to infinity, causing infinite flexibility around resonance, something which is not possible in reality. Therefore during the tests it is vital to monitor the frequency spectrum of the input and output sig- nals to assure the structure is properly excited over the whole measurement range. Visual inspections of the measured FRF data sets should also be made, either during or shortly after measurement. For instance, when the magnitude of an FRF is plotted on a logarithmic axes, the low frequency asymptote characterises the support conditions for the test. In the case of a milling machine, the bear- ings should produce results similar to those testing under cantilever conditions, therefore, the low frequency asymptote should tend to a stiffness line that corre- sponds to the static stiffness of the structure. Moreover, both the resonant and anti-resonant peaks should exhibit similar sharpness. This is an indication of ad- equate vibration levels and satisfactory frequency resolution. It is also important to examine the linearity of the structure, as the above theory is only applicable to linear structures. The point FRF of a linear structure should contain an an- tiresonance after each resonance, whilst the cross FRF (of two well spaced points) should contain more resonances than anitresonances.
Although an experimental modal analysis is not carried out in this chapter, the method will be used throughout the remainder of this thesis. Since it follows on directly from impact testing, it is sensible to discuss the theory of the method at this point.