Question 12.1: The main landing gear sketched in Fig. 12.2 is fitted with t......
The main landing gear sketched in Fig. 12.2 is fitted with two wheels, on a live axle, and has two sets of disk brakes. These were sometimes found to lock when the brakes were applied firmly at very low speeds, resulting in a skid, during which severe foreand-aft vibration of the gear occurred if the runway was wet.
The two accelerometers shown, fitted at the top and bottom of one brake back-plate, were used to monitor the vibration. From the readings, it was found that the mode shape could be represented by rotation of the lower parts of the gear, essentially the wheels, tires and brakes, about a point 0.70 m above the ground line. The mass properties were represented by a mass M = 80.3 kg, at the axle, and a moment of inertia about the axle I = 2.0 kgm². The equivalent mass, m, referred to the ground line was then given by:
M (0.4)^{2} + I = m(0.7)^{2} (A)From Eq. (A), the equivalent mass, m = 30.3 kg.
A vibration test on the unit had previously shown the non-dimensional damping coefficient to be about 0.05 of critical under static conditions.
In one measured trial, at a ground speed of 1.5 m/s, the oscillation, at 29 Hz, was found to grow at a rate corresponding to a damping coefficient of – 0.04 of critical, and to limit at an amplitude of about ± 6 mm, as measured at the lower accelerometer. The vertical static load on the landing gear was 45 000 N.
(a) Explain why the skidding oscillations only occurred when the runway was wet, and why they grew at a rate corresponding to about – 0.04 of critical damping.
(b) Explain why the oscillation was found to limit at about ± 6 mm, at the lower accelerometer, at a ground speed of 1.5 m/s.
Part (a):
From Eq. (12.10), the effective non-dimensional damping coefficient due to the friction force only is
where all the quantities are referred to the ground line. We now need the slope of the coefficient of friction versus speed curve. This was not measured, but Fig. 12.3 shows averaged published data for tires on both wet and dry concrete. At speeds below about 10 m/s, (d \mu / d \nu ) is seen to be equal to about – 0.02 per m/s for wet concrete, but close to zero for dry concrete.
In Eq. (B), for the system operating on wet concrete at low speed, numerical values are
P = 45 000 N, the static vertical load on the gear;
m = 30.3 kg, the equivalent mass of the system referred to the ground line;
\omega_{n} = 2\pi f_{n} = 2\pi \times 29 = 182 rad/s, the natural frequency;
(d \mu / d \nu ) = – 0.02, the slope of the \mu \mathrm{versus} \nu curve at low speed.
Substituting these values into Eq. (B) gives \gamma _{F} = – 0.082. However, the damping coefficient measured under static conditions, \gamma _{0}, was +0.05, so the predicted net damping coefficient is – 0.032, close to the measured value, – 0.04.
In dry conditions at low speed, (d \mu / d \nu ) ≈ 0, and the net damping coefficient remains at about the original value of 0.05, explaining why divergent oscillations did not occur.
Part (b):
From Eq. (12.12) the amplitude would be expected to limit when:
With \nu_{1} = 1.5 \mathrm{m/s, and} ω_{n} = 182 rad/s, Eq. (C) gives x_{max} = 0.0082 m, or 8.2 mm. This is the vibration displacement defined at the ground. From the dimensions in Fig. 12.2, it can be seen that this corresponds to:
\frac{0.55 \times 8.2}{0.70} = 6.4 mm
at the lower accelerometer, which is close to the measured value of 6 mm.
