Mechanical Vibrations - UNESP · Harmonic Response of a water tank The column of the water tank...
Transcript of Mechanical Vibrations - UNESP · Harmonic Response of a water tank The column of the water tank...
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Mechanical Vibrations
Prof. Paulo J. Paupitz Gonçalves
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Equations of Motion
Newton's Method
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Undamped Vibration (sdof)
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Displacement – Velocity Plane
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Displacement – Velocity Plane
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Example 2.1Harmonic Response of a water tank
The column of the water tank shown in the figure is 300 ft high and is made of reinforced concrete with a tubular cross section of inner diameter 8 ft and outer diameter 10 ft. The tank weighs 6 x 105 lb when filled with water. By neglecting the mass of the column and assuming the Young s modulus of reinforced concrete as 4 x 106 psi, determine the following:
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Example 2.1
Harmonic Response of a water tank
Question a) Find the natural frequency and the natural time period of transverse vibration of the water tank.
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Example 2.1
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Example 2.1
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Example 2.1
Harmonic Response of a water tank
Question b) Find the the vibration response of the water tank due to an initial transverse displacement of 10 in.
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Example 2.1
Harmonic Response of a water tank
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Example 2.1
Harmonic Response of a water tank
Question c) Find the maximum values of the velocity and acceleration experienced by the water tank.
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Example 2.1
Harmonic Response of a water tank
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Response due to Impact – Ex. 2.2
A cantilever beam carries a mass M at the free end as shown. A mass m falls from a height h onto the mass M and adheres to it without rebounding. Determine the resulting transverse vibration of the beam.
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Response due to Impact – Solution
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Response due to Impact – Solution
Conservation of Momentum
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Response due to Impact – Solution
Since free vibration of the beam with the new mass (M + m) occurs about its own static equilibriumposition, the initial conditions of the problem can be stated as
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Problem 2.6
The maximum velocity attained by the mass of a simple harmonic oscillator is 10 cm/s, and the period of oscillation is 2 s. If the mass is released with an initial displacement of 2 cm, find
(a) the amplitude(b) the initial velocity(c) the maximum acceleration(d) the phase angle
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Problem 2.6
(a)
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Problem 2.6
(d) the phase angle
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Problem 2.6
(b) the initial velocity
(c) the maximum acceleration
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Compound Pendulum
Small angles:
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Compound Pendulum
Small angles:
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Centre of Percussion
Comparing this natural frequency with the natural frequency of a simple pendulum
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Centre of Percussion
we can find the length of the equivalent simple pendulum:
Replacing by , where is the radius of gyration of the body about O
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Centre of Percussion
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Centre of Percussion
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Centre of Percussion
This equation shows that, no matter whether the body is pivoted from O or A, its natural frequency is the same. The point A is called the centre of percussion.
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Centre of Percussion