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2. A mass m is connected to a light rigid rod of length & that is free to pivot about its bottom end (figure 1). The angular deflection of the rod from the vertical is 0. A torsion spring produces a torque -k(0 - 0o), where k is the spring stiffness and is the angle at which the spring provides zero torque, both of which are constant. The motion is opposed by a damping force –cld, where c is a constant and ê is time. The angular deflection is governed by the ODE de me2d²0 + cl² dt +k(0-00) - mgl sin 0 = 0, df² (1) where g is the acceleration due to gravity. mg l Figure 1: A mass connected to a rod that is free to pivot about its bottom end. (a) Suppose circumstances are such that the first term involving the second derivative in (1) is negligible in comparison with the other terms, so that it can simply be ignored. Show that the resulting first-order ODE can be written in nondimensional form de sin 0 - B(0-00), dt (2)
= Tt for where the nondimensional time t is related to the dimensional time î by î time scale T, and ß is a nondimensional parameter. Determine T and 6 as part of your solution. (b) Identify the formulae needed to (i) find the critical points of (2) and (ii) determine the stability of those points. Use these to complete the script that plots the bifurcation diagram for 0 = 0.1 on MATLAB GRADER (see Written Assignment 3 - Question 2(b) on MyUni). Your mark for this part will be determined by MATLAB GRADER. You do not need to include anything for this part in your written submission. (c) Figure 2 shows part of the bifurcation diagram derived from the simplified model (2) for such a device when 00 = 0.1. 10 stable - unstable 5 0 -5 -10 0 0.5 1 1.5 2 В Figure 2: Bifurcation diagram for (1) with 0 = 0.1. = i. Suppose the rod is rotated radians clockwise from 0, held steady for a moment, then released. What will happen to the rod for ß = 1? ii. Suppose the rod is doubled in length, keeping all other dimensional parameters equal, and the rod is set up as before. What will happen to the rod this time? 0*