Answer Happy

(f) Express the Equation of Motion in Equation (1) in terms of the displacement θ1​(t) about an equilibrium position θe​, i.e., θ(t)=θe​+θ1​(t). Simplify this nonlinear equation, if possible, but do not linearise it yet. Do not substitute values for θe​ yet. (g) Consider small motions of θ1​(t) about equilibrium θe​, i.e., ∣θ1​(t)∣≪1, and develop the linearised Equation of Motion for θ1​(t) about θe​. Do not substitute values for θe​ yet. Ignore terms with powers greater than one and/or products of θ1​(t) and θ˙1​(t). Note the approximations for small angles ∣θ1​(t)∣≪1 : (a) cos(θ1​)≈1−21​θ12​+241​θ14​+…; (b) sin(θ1​)≈ θ1​−61​θ13​+…; and (c) 1−cos(θ1​)≈21​θ12​−241​θ14​+… 5 h) For your stable equilibrium position θe​, show that the linearised Equation of Motion is θ¨1​+ωn2​θ1​=0. (i) From the above linearised Equation of Motion for free vibration, determine the expression for the natural frequency. (j) Consider the following numerical values: L=3[ m],M=100 [kg], m=200 [kg], and g= 9.81[ m/s2]. If possible, design a reasonable value of the rotational spring constant Kr​[ N− m/rad] such that 0<=θe​<=π/4 and the resulting θe​ equilibrium position is stable. What is the natural frequency of vibration ωn​[rad/sec] for the resulting linearised Equation of Motion about θe​ ? Hint: Use the relationships developed from Equation (2) and Equation (3).
J0​θ¨+Kr​θ−Lg(M+21​m)sinθ=0