Analysis Techniques

Various analytical methods are available to establish the transmissibility FRF (and dynamic characteristics) of an SDoF system when excited by either random or sinusoidal vibration and the excitation and response are simultaneously measured. The various methods used to compute the FRF from the measured excitation and response data are discussed, along with any considerations that must be made. As the FRF is the ratio of the response to the excitation, it is important to consider what is measured. Depending on the application, it may be of interest to measure the excitation force and the response acceleration, or the excitation displacement and the response acceleration may require measurement. For this thesis, the most relevant FRF to use is the transmissibility FRF which is obtained when the excitation and response are measured using the same units of motion.

This section describes the process to analyse the data obtained when exciting a system using random vibration. Establishing the FRF of the system depends on the method of excitation used, for example random vibration requires a different analysis approach compared to a sine (or swept-sine) excitation. Considering a constant-parameter linear system, the dynamic characteristics can be defined by an impulse response function, ℎ(𝜏𝜏), described in terms of the response of the system, 𝑦𝑦(𝑡𝑡), at any time due to a unit impulse excitation applied at a time 𝜏𝜏 before, shown in Equation 3-41 (Bendat & Piersol 2000, p. 29).

( )

∞

( ) (τ

τ τ)

−∞

=

∫

−

y t h x t d (3-41)

While the impulse response function is able to establish the dynamic characteristics in the time domain, it is also possible to establish them in the frequency domain via the FRF. To establish the FRF of a system, the FT of the impulse response function is computed, shown in Equation 3-42 (Bendat & Piersol 2000, p. 31).

( )₌

∞

( )_τ

− π τ

_τ

∫

2 0

e

j f

H f

h

d

(3-42) 55

If, instead, the FT is taken of both sides of Equation 3-41, the FRF can be established in terms of the excitation, 𝐸𝐸(𝑓𝑓), and response, 𝑌𝑌(𝑓𝑓), shown in Equation 3-43.

( )

=

( ) ( )

Y f H f E f (3-43)

Finally, rearranging Equation 3-43 yields the FRF in terms of the ratio of the excitation and response, presented in Equation 3-44.

( )=Y f( )_{( )}

H f

E f

(3-44)

In some cases, it may be beneficial to modify the FRF from the general relationship specified in Equation 3-44. Randall (1987, p. 240) describes two variations that can be implemented to calculate the FRF using convolution. The first FRF, denoted as 𝐻𝐻₁(𝑓𝑓), is obtained by multiplying both the numerator and denominator of Equation 3-44 by the complex conjugate of the excitation, presented in Equation 3-45.

( )=

( )_{( )}

( )_{( )}

1

*

.

*

Y f

E

f

H f

E f

E

f

(3-45)

Randall (1987, p.24) stated that the second variation, 𝐻𝐻₂(𝑓𝑓), is obtained by multiplying both the numerator and denominator in Equation 3-44 by the complex conjugate of the response, shown in Equation 3-46.

( )=

( )_{( )}

( )_{( )}

2

*

.

*

Y f

Y

f

H f

E f

Y

f

(3-46)

Randall (1987, p. 241-247) discussed the two variations of the FRF and stated that 𝐻𝐻₁(𝑓𝑓) is better for minimising the effect of extraneous noise in the measurements, while 𝐻𝐻₂(𝑓𝑓) is best used to minimise the effect of spectral leakage.

The ratio of these two FRFs is always equal to the coherence, shown in Equation 3-47 (Randall 1987, p. 240).

( )

( )

( )( )

( )

γ

= = 2 1 2 2 EY EE YY G f H f H f G f G f (3-47)

where 𝐺𝐺_{𝐸𝐸𝐸𝐸}(𝑓𝑓) is the single-sided cross spectrum and 𝐺𝐺_{𝐸𝐸𝐸𝐸}(𝑓𝑓) and 𝐺𝐺_{𝐸𝐸𝐸𝐸}(𝑓𝑓) are the single- sided autospectra of the excitation and response, respectively.

Various analysis parameters must be carefully considered when establishing the FRF of a system. As the FRF is established using the FFT (the DFT and FFT are discussed in detail in the previous section on response-only (transient) analysis techniques), a number of considerations must be made before the measurements are undertaken.

Spectral averaging is often implemented when establishing the FRF in order to reduce the influence of noise and other effects and so, when dealing with random excitation, the excitation and response of a system should be measured for an extended period of time. Spectral averaging is used to minimise the statistical uncertainty inherent in random data. The spectral uncertainty or error, 𝜀𝜀_𝑟𝑟, is equal to the inverse of the square root of the number of distinct spectral averages, 𝑁𝑁𝑠𝑠, provided in Equation 3-48 (Bendat & Piersol 2000, pp. 306-308).

ε

_r

=

1

d

N

(3-48)

The original relationship between the temporal and frequency resolutions, previously outlined in Equation 3-24, does not take into account spectral averaging. With the introduction of spectral averaging, the temporal resolution is now dependent upon the duration of each average, also known as the sub-record length, 𝑇𝑇_{𝑠𝑠𝑟𝑟}, and the number of independent averages, given in Equation 3-49.

=

r d sr

T N T (3-49)

Substituting the sub-record length into Equation 3-49, the following relationship for the frequency resolution with spectral averaging is obtained, shown in Equation 3-50.

ε

∆ =

d

=

1

₂ r r r

N

f

T

T

(3-50)

As discussed briefly in the response-only (transient) analysis techniques section, the use of windowing is important to minimise spectral leakage. Spectral leakage is a consequence of taking a finite section of a time-history coupled with the assumption of periodicity. A “practical solution” to this issue is the use of windowing (Ewins 1995, p. 137). Using the DFT and applying no windowing function to the signal prior to analysis is akin to applying a rectangular window which possesses a value of one (unity) across the duration (Randall 1987, p. 157). This rectangular window possess large side lobes in the DFT and allows for power to leak at frequencies well separated from the main lobe into adjacent bands (Randall 1987, p. 155). This leakage of power may cause significant distortions in the estimated frequency spectrum. The role of windowing (also known as time-history tapering) is to eliminate any discontinuities at the ends of each sub-record and reduce side lobe leakage. While various windows are available for use, the Hanning window is the most common and Randall (1987, p. 157) considered it to be “an excellent general purpose window.” The process of windowing reduces the overall magnitude in each spectrum and requires a multiplication factor to restore this loss.

Two additional enhancement techniques can also be implemented to improve the FRF estimates obtained; zero-padding the data and overlapping of spectral averages. The requirement to maintain a fine temporal resolution by increasing the sub-record length will result in a decreased number of averages, increasing the spectral uncertainty of the estimates. Overlapping of averages seeks to provide an increased number of averages without increasing the duration required for analysis. Each data record is divided into equal segments that overlap one another, rather than dividing the records into independent segments. The role of overlapped averages, in a statistical sense, retrieves most of the information lost due to windowing (Randall 1987, p. 175; Bendat & Piersol 2000, p. 430). Randall (1987, p. 175) stated that “overlaps greater than 50 % generally give a more uniform overall weighting function, but no appreciable improvement in statistical error.” A numerical and experimental study undertaken by Lamb et al. (2010) found that the use of zero-padding and overlapped spectral averages improves the modal parameter extraction results.

The following section describes the various analysis procedures and considerations required for establishing the FRF of a system when excited by a sine-based excitation. While the same relationship is used (Equation 3-44), it should be noted that as sine-based excitations will satisfy the periodicity requirement of the FFT, no spectral leakage will occur as long as complete cycles are used and there is no need to apply a windowing function to the excitation and response data. An alternative approach that can be used is to determine the ratio of the peak acceleration (or rms) of the excitation and response at each frequency. This involves subjecting the system to a series of discrete sinusoidal excitations at various frequencies. The peak or rms ratio at each discrete frequency is combined to construct the transmissibility FRF of the vehicle at all the selected excitation frequencies during the test. This technique can be rather time- consuming and cumbersome, particularly if the controller is manually operated and a fine frequency resolution is required. Sweatman et al. (2000) suggested that when a swept-sine excitation is applied to a vehicle the phase spectrum can be analysed to determine the level of damping in the suspension system whereas the adhesion spectrum (the adhesion is the ratio of the minimum dynamic tyre force at a given frequency to the static weight of the vehicle) can be used to determine the frequencies of the sprung and unsprung masses.

There are three common analysers that are used to establish the magnitude and phase from the excitation and response from a sine-based test. The first analyser is a tracking filter, which is a (generally analogue) device used to generate the FRF of a system subjected to a slow sine- sweep excitation. The oscillator (or controller) used to generate the sine-sweep is coupled with a set of narrowband filters, through which the transducer signals are passed, and the centre frequency of the filters are continuously adjusted in order to match the current frequency of the excitation (Ewins 1995, p. 128). The tracking filter is able to establish both the magnitude and phase of the component of each of the transducer signals related to the actual excitation source (Ewins 1995, p. 129).

The second analyser, the Frequency Response Analyser (FRA), is different to tracking filters in that the processing is performed digitally. The command signal (sine wave at the desired frequency) is digitally generated within the FRA and output through a digital-to-analogue converter (Ewins 1995, p. 129). Within the FRA, the excitation and response signals are digitised and correlated numerically with the outgoing signal (digitally generated) to eliminate all the components in the incoming signals that are not at the frequency of the sine wave (Ewins 1995, p. 129). The advantage of using an FRA over a tracking filter is that both channels are treated in exactly the same way; there is no need to tune the two tracking filters to obtain accurate data (Ewins 1995, p. 129).

As with many similar instruments, the measurement of the FRAs can be improved by performing the correlation over a longer duration (Ewins 1995, p. 130). The third analyser that may be implemented is a spectrum analyser, which is significantly different than the FRA (Ewins 1995, p. 130). Unlike the FRA, the spectrum analyser simultaneously measures all the frequency components in a complex time-varying signal (Ewins 1995, p. 130). The spectrum analyser, which can be digital or analogue, will produce a spectrum with the relative magnitudes of the range of frequencies present in the signal (Ewins 1995, p. 130). The digital spectrum analyser is commonly in use today and uses the DFT to obtain many important properties of signals, including establishing the FRF of the system.

Once the FRF of the system has been established, numerous techniques are available for the extraction of the dynamic characteristics of the system, namely the natural frequency and damping ratio of the first mode. Two common approaches to estimating the dynamic characteristics are discussed. The simplest approach is to use the PP method to estimate the natural frequency of the system by identifying the location of the maximum amplitude of the FRF (He & Fu 2001, p. 163). To estimate the damping ratio, the half-power bandwidth method can be used. This method uses the half-power points of the FRF, located at either side of the resonance (𝑓𝑓_{𝑟𝑟𝑅𝑅𝑠𝑠}) amplitude at frequencies 𝑓𝑓₁ and 𝑓𝑓₂, divided by the square-root of two, shown in Equation 3-51 (Bendat & Piersol 2000, p. 37).

( )

2

=

( )

2

=

( )

2

1 2

1

2

res

H f

H f

H f

(3-51)

Once the half-power point frequencies have been established, the damping ratio can be approximated via Equation 3-52 (Bendat & Piersol 2000, p. 37). It is important to recognise that this is only an approximation and is limited to systems with “light damping” (Bendat & Piersol 2000, p. 37). Once the damping ratio has been established, the natural frequency can be found using Equation 3-6.

ζ

≈

2

−

1

2

_res

f

f

f

(3-52)

where the relationship is valid for 𝜁𝜁≤ 0.1.

The second approach to establish the dynamic characteristics is to apply a curve-fit to the transmissibility FRF. The magnitude and phase of the transmissibility FRF of an SDoF system is presented in Equation 3-53 and Equation 3-54, respectively (Rao 2005, p. 240).

( )

(

)

(

)

(

)

ζ

ζ

+ = − + 2 2 2 2 1 2 1 2 r T f r r (3-53)

( )

₍

ζ

₎

φ

ζ





=

_

_

+

−





3 2

2

arctan

1

4

1

r

f

r

(3-54)

where the frequency ratio is =𝑓𝑓 𝑓𝑓⁄ _𝑛𝑛 .

The magnitude of the transmissibility FRF (Equation 3-53) is of significant interest, as it may be used to provide a curve-fit (via least-squares regression) to extract the sprung mass mode natural frequency and damping ratio. The data that is used for the curve-fit must be selected by the user beforehand, as not all of the data provided by the FRF is relevant for modal parameter estimation. The region of resonance is almost always selected, whether by manual selection or by using a set number of points either side of the maximum magnitude of the transmissibility FRF (Ewins 1995, p. 190). For the estimation of the modal parameters of the system via curve- fitting, it is recommended that no fewer than six data points be used (Ewins 1995, p. 190).

In document Estimating the Dynamic Characteristics of Road Vehicles Using Vibration Response Data (Page 84-90)