Engineering Mechanics: Dynamics
Engineering Mechanics: Dynamics
8th Edition
ISBN: 9781118885840
Author: James L. Meriam, L. G. Kraige, J. N. Bolton
Publisher: WILEY
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Chapter 8.6, Problem 119RP
To determine

The natural frequency of the inverted pendulum and any other restriction in the solution.

Expert Solution & Answer
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Answer to Problem 119RP

The natural frequency of the inverted pendulum is 12π2kb2ml2gl.

The restriction imposed on the system is k>mgl2b2.

Explanation of Solution

Draw the schematic diagram.

Engineering Mechanics: Dynamics, Chapter 8.6, Problem 119RP

Write the expression for the kinetic energy of the system (T).

  T=12m(lθ˙)2        (I)

Here, the mass of the pendulum is m, the length of the pendulum is l.

Write the expression for the potential energy of the system (V).

  V=mgl(1cosθ)+12k(δ+bsinθ)2+12k(δbsinθ)2        (II)

Here, the acceleration due to gravity is g, the spring stiffness is k and the initial spring compression is δ.

Write the expression for the total energy of the system (E).

  E=T+V        (III)

Write the expression for the natural frequency of the system (fn).

  fn=ωn2π        (IV)

Here, the natural circular frequency is ωn.

Conclusion:

Substitute, Equation (I) and Equation (II) in Equation (III).

  E=12m(lθ˙)2mgl(1cosθ)+12k(δ+bsinθ)2+12k(δbsinθ)2        (V)

Differentiate Equation (V) and equate to 0.

  dEdt=0ddt[12m(lθ˙)2mgl(1cosθ)+12k(δ+bsinθ)2+12k(δbsinθ)2]=0ml2θ¨mglsinθ+k(δ+bsinθ)(bcosθ)+k(δbsinθ)(bcosθ)=0        (VI)

Rewrite Equation (VI) for small values of θ.

  ml2θ¨mglθ+k(δ+bθ)(b)+k(δbθ)(b)=0ml2θ¨mglθ+kb(δ+bθ+bθδ)=0ml2θ¨mglθ+2kb2θ=0θ¨mglml2θ+2kb2ml2θ=0θ¨+[2kb2ml2gl]θ=0        (VII)

Obtain the value of ωn from Equation (VII) as:

  ωn2=2kb2ml2glωn=2kb2ml2gl

Substitute, 2kb2ml2gl for ωn in Equation (IV) .

  fn=12π2kb2ml2gl        (VIII)

Thus, the natural frequency of the inverted pendulum is 12π2kb2ml2gl.

From Equation (VIII), we get;

  2kb2ml2gl>0

  2kb2ml2>glk>mgl2b2        (IX)

Thus, the restriction imposed on the system is k>mgl2b2.

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Chapter 8 Solutions

Engineering Mechanics: Dynamics

Ch. 8.2 - Prob. 11PCh. 8.2 - Prob. 12PCh. 8.2 - Prob. 13PCh. 8.2 - Prob. 14PCh. 8.2 - Prob. 15PCh. 8.2 - Calculate the natural frequency fn of vibration if...Ch. 8.2 - Prob. 17PCh. 8.2 - Prob. 18PCh. 8.2 - Prob. 19PCh. 8.2 - Prob. 20PCh. 8.2 - Prob. 21PCh. 8.2 - Prob. 22PCh. 8.2 - Prob. 23PCh. 8.2 - Prob. 24PCh. 8.2 - Prob. 25PCh. 8.2 - Prob. 26PCh. 8.2 - Prob. 27PCh. 8.2 - Prob. 28PCh. 8.2 - Prob. 29PCh. 8.2 - Prob. 30PCh. 8.2 - Prob. 31PCh. 8.2 - Prob. 32PCh. 8.2 - Prob. 33PCh. 8.2 - Prob. 34PCh. 8.2 - Derive the differential equation of motion for the...Ch. 8.2 - Prob. 36PCh. 8.2 - Determine the equation of motion for the system in...Ch. 8.2 - Prob. 38PCh. 8.2 - Prob. 39PCh. 8.2 - Prob. 40PCh. 8.2 - Prob. 41PCh. 8.2 - Prob. 42PCh. 8.2 - Prob. 43PCh. 8.2 - Prob. 44PCh. 8.3 - Prob. 45PCh. 8.3 - Prob. 46PCh. 8.3 - Prob. 47PCh. 8.3 - Prob. 48PCh. 8.3 - Prob. 49PCh. 8.3 - Prob. 50PCh. 8.3 - Prob. 51PCh. 8.3 - Prob. 52PCh. 8.3 - Prob. 53PCh. 8.3 - The 4-lb body is attached to two springs, each of...Ch. 8.3 - Prob. 55PCh. 8.3 - The motion of the outer frame B is given by xB = b...Ch. 8.3 - Prob. 57PCh. 8.3 - Prob. 58PCh. 8.3 - When the person stands in the center of the floor...Ch. 8.3 - Prob. 60PCh. 8.3 - Derive the equation of motion for the inertial...Ch. 8.3 - Prob. 62PCh. 8.3 - Prob. 63PCh. 8.3 - Prob. 64PCh. 8.3 - Prob. 65PCh. 8.3 - Prob. 66PCh. 8.3 - Derive and solve the equation of motion for the...Ch. 8.3 - Prob. 68PCh. 8.3 - Prob. 69PCh. 8.3 - Prob. 70PCh. 8.4 - The light rod and attached small spheres of mass m...Ch. 8.4 - Prob. 72PCh. 8.4 - The thin square plate is suspended from a socket...Ch. 8.4 - Prob. 74PCh. 8.4 - The 20-lb spoked wheel has a centroidal radius of...Ch. 8.4 - Prob. 76PCh. 8.4 - The uniform sector has mass m and is freely hinged...Ch. 8.4 - Prob. 78PCh. 8.4 - Prob. 79PCh. 8.4 - Prob. 80PCh. 8.4 - Prob. 81PCh. 8.4 - Prob. 82PCh. 8.4 - Prob. 83PCh. 8.4 - Prob. 84PCh. 8.4 - Prob. 85PCh. 8.4 - Prob. 86PCh. 8.4 - Prob. 87PCh. 8.4 - Prob. 88PCh. 8.4 - Prob. 89PCh. 8.4 - Prob. 90PCh. 8.4 - Prob. 91PCh. 8.4 - Prob. 92PCh. 8.4 - Prob. 93PCh. 8.4 - Prob. 94PCh. 8.4 - Prob. 95PCh. 8.4 - Prob. 96PCh. 8.5 - The 1.5-kg bar OA is suspended vertically from the...Ch. 8.5 - The light rod and attached sphere of mass m are at...Ch. 8.5 - A uniform rod of mass m and length l is welded at...Ch. 8.5 - The spoked wheel of radius r, mass m, and...Ch. 8.5 - Prob. 101PCh. 8.5 - The length of the spring is adjusted so that the...Ch. 8.5 - The body consists of two slender uniform rods...Ch. 8.5 - By the method of this article, determine the...Ch. 8.5 - Prob. 105PCh. 8.5 - Prob. 106PCh. 8.5 - Prob. 107PCh. 8.5 - Prob. 108PCh. 8.5 - Prob. 109PCh. 8.5 - Prob. 110PCh. 8.5 - Prob. 111PCh. 8.5 - Prob. 112PCh. 8.5 - Prob. 113PCh. 8.5 - Prob. 114PCh. 8.5 - Prob. 115PCh. 8.5 - Prob. 116PCh. 8.5 - Prob. 117PCh. 8.5 - The quarter-circular sector of mass m and radius r...Ch. 8.6 - Prob. 119RPCh. 8.6 - Prob. 120RPCh. 8.6 - Prob. 121RPCh. 8.6 - Prob. 122RPCh. 8.6 - Prob. 123RPCh. 8.6 - Prob. 124RPCh. 8.6 - Prob. 125RPCh. 8.6 - Prob. 126RPCh. 8.6 - Prob. 127RPCh. 8.6 - Prob. 128RPCh. 8.6 - Prob. 129RPCh. 8.6 - Prob. 130RPCh. 8.6 - Prob. 131RPCh. 8.6 - Prob. 132RPCh. 8.6 - Prob. 133RPCh. 8.6 - Prob. 137RPCh. 8.6 - Prob. 138RPCh. 8.6 - Prob. 139RPCh. 8.6 - Prob. 140RP
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