- An Inverted Pendulum Bar Of Mass M And Length L Also Has An End Mass M A Rotational Spring K Is At The Pivot At The Ba 1 (169.61 KiB) Viewed 34 times
An inverted pendulum bar of mass m and length L also has an end mass M. A rotational spring K, is at the pivot at the base of the pendulum. When the pendulum is vertical, the generalised coordinate e angle, measured clockwise from the vertical, is zero. When 0 = 0, the rotational spring does not provide any torsional resistance. The acceleration of gravity is denoted by g. Answer the following questions: (a) Develop an expression for the total mass moment of inertia Jo about the pivot. (b) Develop an expression for the kinetic energy T(0,0). (c) Develop an expression for the potential energy V(0) due to gravity and the spring. (d) Using Lagrange's Equation for free vibrations, verify that the governing nonlinear Equation of Motion is as shown below: 1 Joë+K,0-Lg (M+m) sin0 = 0. (e) The equilibrium positions for a SDOF system with generalised coordinate 0 may be deter- mined using the potential energy function V(0) as follows: dV de 10₂ d²v d0² le, = 0. For a SDOF system, the stability of a specific equilibrium position 0, is assured if the follow- ing condition is satisfied: > 0. (1) i. Develop an expression for the equilibrium configurations 0₂. ii. For each of the physically possible equilibrium solutions: A. Determine a numerical value of 0, or the transcendental equation containing 0₂. (2) Now, using the above equilibrium and stability requirements, examine the stability for the present problem and determine the stability of each physically possible equilibrium position 0e. Address the following items: 2 (3) B. Examine the stability of the configuration for the each value of 0. C. Determine restrictions on the physical parameters m, L, M, and K, that are necessary to ensure stability. (f) Express the Equation of Motion in Equation (1) in terms of the displacement 0₁ (1) about an equilibrium position 0e, i.e., 0(t) = 0 + 0₁ (1). Simplify this nonlinear equation, if possible, but do not linearise it yet. Do not substitute values for 0, yet. (h) For your stable equilibrium position 0e, show that the linearised Equation of Motion is 0₁ +020₁ = 0. (g) Consider small motions of 0₁ (1) about equilibrium 0e, i.e., 01 (1)| < 1, and develop the lin- earised Equation of Motion for 0₁ (1) about 0e. Do not substitute values for 0, yet. Ignore terms with powers greater than one and/or products of 01 (t) and 0₁ (t). Note the approx- imations for small angles 0₁ (1)| << 1: (a) cos(0₁) 1-10+0+...; (b) sin(0₁) 0₁-0+...; and (c) 1-cos(01) 10-2401 +.... (4) (i) From the above linearised Equation of Motion for free vibration, determine the expression for the natural frequency. (1) 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 K, [N- m/rad] such that 0 <= 0₂ <= π/4 and the resulting 0e equilibrium position is stable. What is the natural frequency of vibration on [rad/sec] for the resulting linearised Equation of Motion about 0₂? Hint: Use the relationships developed from Equation (2) and Equation (3).