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unit6_viewnote2 - MIT 8.03 Fall 2005 Analysis of the Driven...

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± MIT 8.03 Fall 2005 Analysis of the Driven Triple Pendulum We wish to study the behavior of three pendulums, each of mass m and length L , in the conﬁguration shown below under the inﬂuence of a variable displacement η = η 0 cos( ωt ) at the top end. ﬁrst ﬁnd the normal mode frequencies by considering the case in which the system is un-driven, i.e. η ( t ) = 0. then proceed to ﬁnd the amplitudes of oscillations of the pendulums as a function of the driving frequency ω . begin by setting a convenient coordinate system and labeling the relevant variables. η θ 2 θ 3 θ 1 L L L x x x 3 2 1 The ﬁgure below shows the forces acting on the individual pendula. 0 1 T 1 0 1 T 2 T 3 Τ 2 mg mg 0 1 mg T 3 For small angles, T 1 3 mg , T 2 2 mg and T 3 mg . If all three angles are zero and the system is at rest then this follows immediately. The acceleration in the y-direction is negligible for small angles. Note that sin θ 1 =( x 1 η ) /L ,s in θ 2 x 2 x 1 ) /L and sin θ 3 x 3 x 2 ) /L . Then, the equations of motion for the pendulums (for small oscillations) are m ¨ x 1 = T 2 sin θ 2 T 1 sin θ 1 2 mg ( x 2 x 1 ) /L 3 mg ( x 1 η ) /L m ¨ x 2 = T 3 sin θ 3 T 2 sin θ 2 mg ( x 3 x 2 ) /L 2 mg ( x 2 x 1 ) /L m ¨ x 3 = T 3 sin θ 3 ≈− mg ( x 3 x 2 ) /L, where we have made the small angle

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This note was uploaded on 02/18/2011 for the course PHYS 320 taught by Professor Martinconner during the Spring '10 term at Open Uni..

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unit6_viewnote2 - MIT 8.03 Fall 2005 Analysis of the Driven...

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