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16.338 Lab Report #2: Kapitsa's Stable Inverted Pendulum - LMPT

16.338 Lab Report #2: Kapitsa's Stable Inverted Pendulum - LMPT

1 Introduction The 1978

1 Introduction The 1978 Nobel Laureate in Physics, Pyotr Leonidovich Kapitsa, discovered during the 1940’s that an inverted pendulum can be stabilized when forced with high frequency vertical oscillations. The present laboratory assignment investigates this phenomenon with a jigsaw-mounted inverted pendulum, as depicted in Figure 1. Pendulum θ L λ cos ωt Jigsaw Figure 1: Experimental Setup 2 Dynamic Model We develop the governing equations from a moment balance about the pivot point. ∑ Mpivot : I ¨θ = 1 2 Lmλω2 cos ωt sin θ + 1 mgL sin θ (1) 2 Where the center of mass was taken at 1/2L. The right-hand-side terms account for forcing and gravity, respectively. The proper moment of inertia, I = 1/3mL 2 , and the small angle approximation sin θ ≈ θ are introduced and the expression is simplified. ¨θ − 3 2 ( g L + λ L ω2 cos ωt ) θ = 0 (2) We have neglected rotational friction for the present derivation, but the damping effects will be discussed in a later section. 1

3 Analytical Investigation We investigate the behavior of the dynamic model with two analytical treatments. First, with an approximate stability boundary derived from the broader theory of the Mathieu equation. Second, from an approximate solution form that is particular to this problem. 3.1 The Mathieu Equation Equation 2 is a special form of Hill’s equation known as the Mathieu equation [1], for which there are no general closed-form solutions. The Mathieu equation can be written d 2 y + (a − 2q cos 2z) y = 0 (3) dz2 We can put Equation 2 into this form by making the arguments of the cosine functions identical. d 2 ( ) y 2 2 ( dz + − 3g 2 ω 2L − 3λ ) 2L cos 2z y = 0 (4) The coefficients of Equation 3 gives a = − 6g ω 2 L q = − 3λ L (5) (6) Approximate stability boundaries have been determined in the previously given reference by using the perturbation method. Figure 2 depicts the boundaries of interest, where the present system has values for a = f(ω) around -0.01 and for q = const. at -0.015. The low frequency ]mathieu Figure 2: Approximate Stability boundaries for the Mathieu equation stability boundary for the Mathieu equation with parameters in this range has been found to be approximately a = − q2 2 which predicts the critical stabilization frequency to be (7) ω crit = 2 λ √ gL 3 (8) 3.2 Approximate Solution Towards the end of finding an approximate solution for the slow oscillations, we conjecture that the solution is composed of a slow, large amplitude motion superimposed on a fast, small amplitude oscillation θ = θ 1 + θ 2 cos ωt (9) 2

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