m 1 b k 3. A tuning fork is an example of a "resonant system", that is, one that has a low damping ratio. A model of a (half) tuning fork is a rod of mass m with air drag modeled as a damper connected approximately at its middle, and a torsional spring at the base of the cantilever. Assume the cantilever rotates back and forth about its pivot by a small angle o. Because other forces are much larger, you can reasonably neglect the effect of gravity in this model. The moment of inertia of a cantilevered beam pinned at one end is equal to ml2, where I is its length. (a) What is the natural frequency of this system if m= 0.1 kg and k = 2548 Nm/rad, and 1 = 0.1 m. Please give the value in rad/s and Hz. (b) Find an upper bound on the damping coefficient b that insures that the damping ratio of the tuning fork is no greater than = 0.01. (c) For this damping ratio, what is the damped natural frequency of the system? (d) For this damping ratio, what is the decay rate (the size of the exponent multiplying by t) of the envelope function describing the diminishing amplitude of the oscillations?

m 1 b k 3. A tuning fork is an example of a "resonant system", that is, one that has a low damping ratio. A model of a (half) tuning fork is a rod of mass m with air drag modeled as a damper connected approximately at its middle, and a torsional spring at the base of the cantilever. Assume the cantilever rotates back and forth about its pivot by a small angle o. Because other forces are much larger, you can reasonably neglect the effect of gravity in this model. The moment of inertia of a cantilevered beam pinned at one end is equal to ml2, where I is its length. (a) What is the natural frequency of this system if m= 0.1 kg and k = 2548 Nm/rad, and 1 = 0.1 m. Please give the value in rad/s and Hz. (b) Find an upper bound on the damping coefficient b that insures that the damping ratio of the tuning fork is no greater than = 0.01. (c) For this damping ratio, what is the damped natural frequency of the system? (d) For this damping ratio, what is the decay rate (the size of the exponent multiplying by t) of the envelope function describing the diminishing amplitude of the oscillations?

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

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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

+
b
3.
A tuning fork is an example of a "resonant system", that is, one that has a low damping ratio.
A model of a (half) tuning fork is rod of mass m with air drag modeled as a damper connected
approximately at its middle, and a torsional spring at the base of the cantilever. Assume the cantilever
rotates back and forth about its pivot by a small angle o. Because other forces are much larger, you
can reasonably neglect the effect of gravity in this model. The moment of inertia of a cantilevered beam
pinned at one end is equal to ml2, where I is its length.
(a) What is the natural frequency of this system if m= 0.1 kg and k = 2548 Nm/rad, and 1 = 0.1
m. Please give the value in rad/s and Hz.
(b) Find an upper bound on the damping coefficient b that insures that the damping ratio of the
tuning fork is no greater than = 0.01.
(c) For this damping ratio, what is the damped natural frequency of the system?
(d) For this damping ratio, what is the decay rate (the size of the exponent multiplying by t) of the
envelope function describing the diminishing amplitude of the oscillations?

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