Question

5. Consider the following cart with solid wheels which is attached to a wall by a spring of stiffness k. There are several assumptions that can make this problem more straightforward to solve (it is up to you to decide if these assumptions are reasonable for real life situations): In i. The wheels do not slip while rolling. ii. The center of mass of the entire cart is at point G, at the center of the wheel k-10K ii. The bungee cord is attached to the cart such that the spring force is acting through G m, 1500 kg iv. There is no friction between the wheels and the cart R, = 0.5m x 0.5 m ˇ If the cart was pulled away from the wall by xo meters and released from rest, use the Laplace transform method to determine the resulting motion of the cart. Hint: use energy methods to find the equivalent mass, then use the equivalent mass with a standard simple harmonic motion model. a. b. Plot the response of the cart for 10 seconds. Does this behavior seem sensible? Which assumption from above should be eliminated to develop a more realistic dynamic model of the behavior?

Answer 1

Given that, k = 10K N/m m_c = 1500kg I_w = 3kg m^2 R_w = 0.5m x_0 = 0.5 m for rolling the wheel rotate about instantaneous centre. Now, by using energy method, K middot E = 1/2 m_c x^2 + 1/2 R theta^2 where, theta = x/r and P middot E = 1/2 kx^2 Theta = x/r \r\n Energy method is, d/dt(K middot E + P middot E) = 0 = > d/dt(1/2 m_c x^2 + 1/2R(x/r)^2 + 1/2 kx^2) m_c d^2 x/dt^2 + I/x^2 d^2x /dt^2 + kx = 0 = > d^2 x/dt^2 (m_c + R/x^2) + kx = 0 where, (m_c + R/r^2) = Equivalent mass Substitute m_c = 1500 R = 3 x = 0.5 = > 1500 + 3/0.5^2 = 1512 kg = > (1512) d^2 x/dt^2 + 10000 x = 0 where, x = A1512s^2 e^st + 10000 Ae^s^2 = 0 = > 1512s^2 + 10000 = 0 \r\n Therefore s = plusminus i (2.57) Now, x(t) = A_1 e^s_1 t + A_2 e^s_2 t