A skydiver of mass m = 70 kg makes a high-altitude jump from a plane and wants to determine his maximum velocity. The differential equation of motion of the skydiver during the free fall can be derived by applying Newton’s second law which yields
[url removed, login to view] = mg-R (1.1)
Where v denotes the velocity, R is the aerodynamic drag force and g = [url removed, login to view] m/s^2 is the acceleration of gravity. The aerodynamic drag is proportional to the square of the velocity
R= av^2 (1.2)
Where a= [url removed, login to view] Ns^2/m^2.
Develop a MATLAB program to integrate numerically equation (1.1) over the time interval [0, tf] to find the maximum velocity of the sky-diver (you may need to experiment with the value of the final time tf to find the terminal velocity).
Determine also the acceleration of the sky-diver over the simulation time interval[0, tf]. Illustrate the results of your calculations by plotting the velocity and acceleration of the skydiver as a function of time. Discuss and critically appraise the results of your calculations.
A single degree of freedom model of a vehicle traveling at speed v over a rough road surface is shown in Figure 2 (a). The vehicle has a mass of m = 1500 kg, the suspension has a spring-damper system of a spring constant k = 400 kN/m and of a coefficient of viscous damping c = 2000 kg/s. The road surface varies harmonically (sinusoidally) with an amplitude Y = [url removed, login to view] m and a wavelength (the length of one cycle) of l =6 m.
The behaviour of the vehicle can be investigated by using a base vibration mode (see Figure 2 (b)). The vertical response (vibrations) x(t) of the vehicle are then governed by the second order differential equation of motion given as
mx .. (t) + cx sin cos Ω. (t) + kx (t) = kY sin Ωt + cYΩ cos Ωt (2.1)
Where Ω is the frequency of excitation due to the road surface given as
Ω = [1/ (3.6)] ([2 π *v]/l) (2.2)
Where the speed v is expressed in km/h.
Develop a MATLAB model (using an ode m-function and a relevant ode solver)) and a Simulink model to solve the equation of motion (2.1) and simulate the response the vehicle over the time interval [0, 20]s when it travels at two different speeds, 100 km/h and 55 km/h, respectively. Verify the correctness of your solution by comparing the results obtained from both models. Illustrate the results by plotting the vertical displacements the vehicle as a function of time. Discuss and critically appraise the simulation results, explaining why the vehicle is subject to different vibration levels at the two speeds.