MÓDULO FORMATIVO 2

Supposing that the second car gets onto the plate structure at a time gap of Δ after the first car does. The time instants of two cars have a relationship of

% # $5 (6.9)

where _$ and _% are the time instants of the first car and the second car when they are arriving at the plate, respectively. If the time gap is short, eight wheels of the two cars can be acting on the plate structure at the same time for some time. However, when the time gap is long enough so that the second car arrives after the first car leaves the plate structure, the second car encounters a plate undergoing free vibration caused by the departure of the first car. Another special case is six wheels acting on the plate structure (rear wheel-set of the first car and two wheel-sets of the second car), which is not easy to achieve in experiments and not studied in this thesis.

(a) Situation one: two separate cars with short time gap

The first car carries a mass marked as No. 1 and they weigh 4.335 kg in total (mass ratio of 0.41), and the second car also carries a mass marked as No. 2 and they weigh 4.207 kg in total (mass ratio of 0.40). As the Laser Vibrometer can only measure the speed of one car at a time, the speed of the first car is measured first for a while. When the second car reaches the plate structure, its speed is measured until it leaves the plate structure. The measured speeds of the two cars are given in Figure 6.5. The initial speed of the first car is estimated to be 1.21 m/s and its acceleration is -0.0647 m/s% _{from the fitted straight line. After the first car travels for Δ # 2.385 s}

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(estimated from Figure 6.6), the second car reaches the plate structure with an initial speed of 1.59 m/s and an acceleration of -0.0461 m/s%.

(a) (b)

Figure 6.5. Measured speeds of moving cars: (a) car 1, (b) car 2, with linear fitting

curves

The simulated dynamic response of the plate structure agrees well with the measured response, as shown in Figure 6.6. There are high frequency oscillations in the measured results, especially during the period when the second car is travelling on the plate structure, which is mainly due to the second car travelling on a vibrating plate structure at a higher speed which causes a stronger car-plate interaction. The frequency of the high oscillation is around 14.01 Hz, seen from Figure 6.7 (b). A sudden change of the dynamic response of the plate structure can be seen at the grey points in Figure 6.6 when the second car gets onto the plate structure. The response of the plate structure subjected to two cars can be seen as the sum of the responses of the plate structure caused by two cars individually, as the vibration of the plate structure behaves linearly in the tests. The spectrum of the measured results in Figure 6.7 (a) shows two driving frequencies caused by two cars individually in the spectrum of the car-excited response of the plate structure and only the first two plate frequencies are excited largely in the free response of the plate structure shown in Figure 6.8. A new frequency of 1.25 Hz is found in the dynamic response of the plate structure, which is because the loading of the second car changes the original displacement trace of the plate structure.

The frequency of 21.95 Hz in Figure 6.7 (b) is close to the 1st structural frequency of 21.5 Hz in Figure 6.8, and the frequency of 23.0 Hz in the former figure is next to the 2nd structural frequency of 23.5 Hz in the latter figure. The amplitudes of the above frequencies in the former figure are increased in the latter figure.

C ar 1 s p ee d ( m /s )

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Figure 6.6. Displacements of the plate at measured points by simulation (blue curve)

compared with experimental results (red curve) for two separate moving cars with short time gap

(a) (b)

Figure 6.7. Spectrum of measured car-excited displacement of the plate structure at

1st measured point: (a) whole amplitude view, (b) part amplitude view at initial car speeds of 1.21 m/s and 1.59 m/s with short time gap

0 1 2 3 4 5 -1.5 -1 -0.5 0 0.5 1st point 0 1 2 3 4 5 -1.5 -1 -0.5 0 0.5 2nd point 0 1 2 3 4 5 Time (s) -1.5 -1 -0.5 0 0.5 D is p la ce m en ts ( m m ) 3rd point 0 1 2 3 4 5 -1.5 -1 -0.5 0 0.5 4th point

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Figure 6.8. Spectrum of measured free vibration of the plate structure at 1st measured

point at initial car speeds of 1.21 m/s and 1.59 m/s with short time gap

(b) Situation two: two separate cars with long time gap

The first car travels on the plate structure with an estimated initial speed of about 1.11 m/s and an acceleration of around -0.0628 m/s%, as shown in Figure 6.9 (a). After a time gap of Δ # 4.298 s, the second car starts to move on the plate structure with an estimated initial speed of about 1.79 m/s and an acceleration of around - 0.0644 m/s% as shown in Figure 6.9 (b). An obvious change occurs at the grey points when the second car arrives at the plate structure as seen in Figure 6.10. A time period of free vibration of the structure can be seen in the figure after the first car leaves the plate structure and before the second car arrives, but the free vibration is small and dies out quickly, which does not affect the vibration of the second car-plate system much. Larger high frequency oscillations can be seen in the car-excited response and free vibration response of the plate structure when the second car is travelling on the plate structure at a higher car speed, compared with those during the period before the arrival of the second car. The spectrum of the responses of the plate structure caused by two cars separately is similar to that by one car.

The spectrum of free vibration of the plate structure after the second car leaves the structure is depicted in Figure 6.11. It can be seen in the figure that the first two structural frequencies are excited into relatively high amplitudes.

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(a) (b)

Figure 6.9. Measured speed of two cars: (a) car 1, (b) car 2, with linear fittings

Figure 6.10. Displacements of the plate at measured points by simulation (blue curve)

compared with experimental results (red curve) for two separate moving cars with long time gap

C ar 1 s pe ed ( m /s ) C ar 2 s pe ed ( m /s ) D is p la ce m en ts ( m m )

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Figure 6.11. Spectrum of free vibration of the plate structure at 1st measured point at

initial car speeds of 1.11 m/s and 1.79 m/s with long time gap