Perching Kinematics Analysis in MATLAB

• We carried out a simple analysis of the perching process in order to better understand what kind of reaction we'll get from the springs. The equation we found for the vertical displacement as s function of time after impact is: x(t)=A*e^(sqrt(k/m)*i*t)+B*e^(-sqrt(k/m)*i*t)-(mg)/k where A and B are constants determined by the initial velocity, m, g, and k. Further, we had to find the time at which the first peak of displacement (maximum recoil) occurs. The formula for this is: t = sqrt(m/k)*arctan((v0/g)*sqrt(k/m)). Some of the plots we made are shown below. The spring constant seen in the plots is the constant for the leg of the plane. The recoil distance refers to the amount the plane "bounces" back up after impact causes the springy legs to extend.

• Keeping Mass Constant, Changing Spring Constant:

• Keeping Spring Constant Same, Changing Mass:

• Studying the Effect of Different Initial Velocities on the Maximum Recoil versus Distance From Wall when Engine is Shut Off:

• Given an initial height, we wanted to find the maximum recoil for different spring constants. We made plots with various masses and found that there is a negligible difference within the range of reasonable masses (0.1kg-2kg):

• We also wanted to find what initial velocity is necessary to have a certain desired drop distance. In this simulation air resistance is not considered. For small distances this won't be a significant problem, but for larger distances it must be considered. From this plot it is evident that we need to have the peak of the plane's trajectory relatively close to the wall if we are to keep our initial x-velocity within reason.