DYNAMICS PROBLEMS

A commercial aircraft has a total mass M. It has a landing gear with a total of 16 wheels. Each wheel has a moment of inertia I has a radius r. The total mass of all the wheels is m. μ_s and μ_k are the coefficients of static and kinetic friction between the aircraft wheels and the runway. Assume that coefficient of rolling friction is zero on the runway.

Figure 1 Figure of an aircraft landing on the runway.

1. The aircraft is towed from the parking area by a truck that pulls the aircraft with a linear force F. What is the initial linear acceleration of the aircraft?
2. Typical landing speed of the commercial aircraft is V_l =  250 km/h. Assume that all 16 wheels of the aircraft touch down at the same time. Also consider that the aircraft undergoes following phases after it lands.
   1. The slipping phase after the touchdown and before pure rolling starts.
   2. The rolling phase when the wheels are undergoing pure rolling and no brakes have been applied
   3. Braking phase when the pilot engages the brakes on the wheels. Braking is characterized by a torque τ_b that acts in a direction opposite to direction of rotation of the wheels. Ignore all the aerodynamic forces after touchdown.
   For each of the above cases draw the free body diagram of one of the wheels and clearly show the direction of the frictional force acting between the wheel and the runway.
3. Find the magnitude of the frictional force on each wheel and deceleration of the aircraft in phases (a) and (b).
4. In the slipping phase, how long from the point of touchdown will it take for the wheel to start rolling?
5. The following table shows typical values of the coefficient of friction

   Kinetic friction when dry	0.4
   Static friction when dry	0.9
   Kinetic friction in ice	0.15
   Static friction on ice	0.4

   While landing on an icy runway, would it take more time for the wheels to start rolling or less?
6. In the braking phase, describe how the frictional force between the tire and the runway will vary if braking torque on the wheels is gradually increased. In order to be able to land on short runways, it is necessary that the braking torque be optimum. Calculate this optimum braking torque for the wheel. Assume that all the wheels have brakes on them. (Hint: Is there a possibility of the tire slipping again as the brake force is increased? Will slamming the brakes hard always result in a better braking action?)
7. It is said that one needs to operate the brakes more gently in case the runway (or the road) is icy. How do you justify this statement considering the values given in 5?

Figure 2 Forces acting on the airplane when it is being towed by the truck.