Three cylindrical rollers of equal length and material are arranged as shown in end viewbelow. Cylinders 2A and 2B have radiusr and cylinder 1 has radius 4r. Their axes are parallelto each other and to that of the fixed concave cylindrical surface, of radius 6r, that supportsthem. They roll without slip. Assuming no damping and following the steps below,determine the frequency of small free rolling oscillations.(a)For rotation 6, of the cylinder 1, determine the resulting rotation of the lowercylinders. [Hint: Does it look like an epicyclic gear train?]When the upper roller rotates with velocity 6, determine the translational androtational velocities of the centre of mass of the lower rollers. [Hint: In an epicyclicgear train the centres of these would be supported on the planet carrier.](b)(c)Write an expression for the kinetic energy of the whole system atsome instant during the oscillation when the rotational velocity of cylinder 1 is , .(d)Write an expression for the gravitational potential energy of the system when thecylinder 1 has rotated through angle 0, from its equilibrium position. Use theequilibrium position as the potential energy datum.(e)Assuming no damping, apply the principle of conservation of energy to the systemto derive the governing differential equation of motion. [Energy]=0(f)By assuming small oscillation amplitude, linearise the differential equationobtained in part (e). Then determine the natural frequency of oscillation of theoscillating motion.

a)Let angular velocity of 1 =ω1that of each of other two cylinder be ω2. Hence for bigger cyl,…

Three cylindrical rollers of equal length and material are arranged as shown in end view below. Cylinders 2a and 2B have radius r and cylinder 1 has radius 4r. Their axes are parallel to each other and to that of the fixed concave cylindrical surface, of radius 6r, that supports them. They roll without slip. Assuming no damping and following the steps below, determine the frequency of small free rolling oscillations. (a) For rotation 6, of the cylinder 1, determine the resulting rotation of the lower Mlin

Three cylindrical rollers of equal length and material are arranged as shown in end view below. Cylinders 2a and 2B have radius r and cylinder 1 has radius 4r. Their axes are parallel to each other and to that of the fixed concave cylindrical surface, of radius 6r, that supports them. They roll without slip. Assuming no damping and following the steps below, determine the frequency of small free rolling oscillations. (a) For rotation 6, of the cylinder 1, determine the resulting rotation of the lower Mlin

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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Concept explainers

Free undamped vibrations

Whenever an object/system displaces in either side of its equilibrium position with the help of an external/internal means, then the object/system moves regularly or randomly on both sides of its equilibrium position. This motion is called vibration. Generally, there are three types of vibration: longitudinal vibration, transverse vibration, and torsional vibration.

Question

Three cylindrical rollers of equal length and material are arranged as shown in end view
below. Cylinders 2A and 2B have radiusr and cylinder 1 has radius 4r. Their axes are parallel
to each other and to that of the fixed concave cylindrical surface, of radius 6r, that supports
them. They roll without slip. Assuming no damping and following the steps below,
determine the frequency of small free rolling oscillations.
(a)
For rotation 6, of the cylinder 1, determine the resulting rotation of the lower
cylinders. [Hint: Does it look like an epicyclic gear train?]
When the upper roller rotates with velocity 6, determine the translational and
rotational velocities of the centre of mass of the lower rollers. [Hint: In an epicyclic
gear train the centres of these would be supported on the planet carrier.]
(b)
(c)
Write an expression for the kinetic energy of the whole system at
some instant during the oscillation when the rotational velocity of cylinder 1 is , .
(d)
Write an expression for the gravitational potential energy of the system when the
cylinder 1 has rotated through angle 0, from its equilibrium position. Use the
equilibrium position as the potential energy datum.
(e)
Assuming no damping, apply the principle of conservation of energy to the system
to derive the governing differential equation of motion. [Energy]=0
(f)
By assuming small oscillation amplitude, linearise the differential equation
obtained in part (e). Then determine the natural frequency of oscillation of the
oscillating motion.

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