Recursos y propósitos de la actividad

It is well known that a linear system has a directly proportional relationship between the excitation and the response (Equation 3-7). Houbolt (1955) applied this relationship to the taxiing of aeroplanes, making a further assumption that if the transmissibility is independent of the speed of the plane; then the speed will be linearly proportional to the response spectrum. Houbolt (1955) suggested the mean-squared response will therefore be linearly proportional to the velocity, and tests undertaken using a Boeing B-29 aeroplane at various constant speeds exhibited a linear relationship up to taxiing speeds of around 50 mph. Over time, this relationship has been examined by a number of authors in the field of vehicle dynamics, notably Gillespie (1985) and Sayers (1988). Dodds and Robson (1975) acknowledged that since the components of vehicles are generally nonlinear, calculating the output spectrum using Equation 3-7 will establish “an average linearised” spectrum which will approximate “that which would be measured in service.”

For in-service experimental testing, as the length of the test section (road) increases, the statistical reliability of the estimate also increases (Quinn & Zable 1966). It can be difficult to locate a road that is long enough without the presence of various factors that will disrupt the in- service measurement of constant speed testing. One common factor is that the flow of traffic may prevent the test vehicle from travelling at a constant speed. Other issues, such as steep ascents or descents, traffic lights and sharp turns, among other factors, may also prevent the vehicle from travelling at a constant speed. This highlights the benefits of using a purpose-built test track, where vehicles are able to travel at constant speed without interruption and can reduce the nonstationary component of the dynamic response.

Virchis and Robson (1971) developed a simple linear spring-mass-damper model with similar dynamic characteristics to a Sunbeam Imp vehicle and investigated the effect of the change in constant speed on the dynamic characteristics. The authors found that the deviations of the mean-square response values were “not large” which suggests that such nonstationary effects may be neglected. Quinn and Zable (1966) attempted to determine the dynamic tyre force PSD function using the differential air pressure. The vehicle was driven over a section of pavement at various constant operating speeds; while the nonlinear characteristics of the vehicle influenced the relationship, the results indicated that there was only a slight change in the response spectrum between operating speeds of 40, 50 and 60 mph (Quinn & Zable 1966).

Marcondes (1990) proposed an experimental approach to estimating the acceleration levels in truck-semitrailer beds using a combination of in-service response data and measured profile elevations. A truck-semitrailer combination (with 6 and 4 wheels, respectively) was used to undertake the in-service experiments, while the vibration acceleration was measured in the trailer bed above the axles. A profilometer was used to establish the pavement profiles for twenty highway sections in Michigan, USA. To calibrate the approach, Marcondes (1990) measured the pavement elevation profiles and computed the PSD function of each pavement; then measured the acceleration of the trailer bed while travelling at three constant speeds (45, 52 and 60 mph) to establish the transmissibility.

The transmissibilities established for the selected pavements illustrate the significant nonlinear behaviour of the vehicle; with variation between each speed travelled over a single pavement observed. The author then developed a model to predict the acceleration PSD functions of the truck bed at two constant speeds (55 and 60 mph) using the transmissibility obtained at an operating speed of 52 mph. Although the author states that the predicted acceleration PSD functions are “very close to the actual values” there is considerable variation between the predicted and actual estimates, including frequency shifting, particularly at low frequencies (Marcondes 1990). A sample of the predicted and obtained acceleration PSD functions for two pavement sections are shown in Figure 3-10.

Figure 3-10: The predicted and obtained acceleration PSDs for two different pavement sections, from Marcondes (1990).

Davis and Sack (2004) conducted a small number of in-service experiments by driving a semi- trailer over a number of suburban roads with a system to record the suspension airbag pressure. The instrument used in the study was originally designed for static applications as made evident by the slow response and poor resolution of the signal and is not suitable for measuring the dynamic response of vehicles. The vehicle speed during the in-service experiments is stated as

being up to 60 km/h, suggesting that the vehicle was not travelling at a constant speed during the experiments. The resulting frequency responses presented do not provide any useful information despite some optimistic conclusions from the authors.

Davis et al. (2007) performed a similar series of experiments as those by Davis and Sack (2004), except in this case an accelerometer was mounted onto the chassis of the vehicle to measure the vibration response. The authors conducted a series of in-service experiments over “typical, uneven road sections,” with various vehicles at speeds ranging between 40 – 90 km/h. The application of Fourier analysis to the random vibration responses appears to be flawed making it impossible to draw meaningful conclusions about the location of peaks (natural frequencies), let alone the damping ratios. In the papers, the authors do not provide the specific location or accurate description of the roads, merely an elementary indication of the pavement roughness. Another issue with the experiments is the lack of repeatability; the operating speeds of the vehicles vary during the experiments and it is therefore impossible to attempt to replicate them. Furthermore, the authors do not disclose a sound scientific basis for these in-service road tests; only the assumption that by “driving the combination on some normal, uneven suburban roads” it was “hoped that this would apply a random signal” to the vehicle (Davis & Sack 2004).

Gonzalez et al. (2008) proposed that the PSD of a longitudinal pavement profile could be obtained using only the vertical response acceleration of a vehicle, based on Equation 3-7. The outcome the authors were seeking was to determine the classification of the road as described in ISO standard 8608 (1995). A numerical model of a half car (side car) was developed and the vertical acceleration was measured at the front and rear axles and at the body. The transfer function was obtained by combining the measured profile with the acceleration measurements (for each of the front and rear axles and the body). The effect of travelling at different constant speeds was also investigated by Gonzalez et al. (2008), finding the speed to have an influence on the estimated PSD – the authors recommend that transfer functions be established for a range of speeds (in steps of 5 km/h) to provide “sufficient accuracy.”

The profiles were then artificially contaminated with noise, with the results indicating that as the Signal to Noise (S/N) ratio increases, the accuracy decreases at lower spatial frequencies for estimating the road profile. The authors further commented that “the influence of S/N on the high spatial frequency components of the spectra is not so important, and once the frequencies are above 0.1 c/m, the road class is generally accurately predicted regardless of the noise level.” When randomly generated noise was introduced into the acceleration measurements, it was

found that the low spatial frequency components were overestimated. Also, the results found that the body accelerations are “as accurate as axle acceleration data” (Gonzalez et al. 2008). Rouillard and Sek (2010b) proposed that an estimate of a vehicle’s operational FRF can be established using only in-service vibration response data, provided the road profile can be represented by an appropriate spectral function. The authors assumed that, over sufficient distances, the average spectral shape of the road profile approaches a brown spectrum in displacement as defined in ISO standard 8608 (1995). If a vehicle travelled over the same road at two different nominally constant speeds, provided the vehicle is linear during operation, the dynamic characteristics and the spectral exponent of the road may be determined. Rouillard and Sek (2010b) proposed this relationship, based on Equation 3-7, in Equation 3-55.

( )

( )

( )

( )

=

R f

R f

G f

G f

where the acceleration response PSD function at constant speed 𝑣𝑣 is 𝑅𝑅_𝑥𝑥̈(𝑓𝑓)_𝑣𝑣.

Multiple experiments using a small transport vehicle travelling at various constant speeds over a long country road (approximately 40 km) showed that the use of a standard value of 2.0 for the spectral exponent, 𝑤𝑤, did not yield the best agreement between the predicted vehicle FRF and the FRF measured using broad-band excitation on a vibration table. One of the issues encountered by the authors was the presence of an “unidentified response spectral peak” within 25 – 30 Hz. Rouillard and Sek (2010b) concluded that the approach is promising, however further work is required to establish the technique for a variety of experimental vehicles and roads, to quantify the effects of vehicle speed and suspension type, to investigate the influence of the record length on the response PSD function and to assess whether the spectral exponent actually tends towards two as the length of the road increases.

As with the analysis techniques for excitation-response and response-only (transient) data, the PP method can also be used to estimate the damped natural frequency of the system from the autospectrum of the in-service response data. From the response autospectrum, the occurrence of peaks can generally be assumed to be due to either the excitation spectrum or the response of the system (Bendat & Piersol 1980, p. 181). For broad-band random excitation, it is generally assumed that the excitation spectrum will not include any significant peaks or troughs within the relevant frequency bandwidth. The major limitation of these approaches is that estimating the

damping ratio is “uncertain or impossible” using only in-service response data (Alam et al. 2009).