Sharanabasava C Pilli Principal, KLE Societys

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ME65 MECHANICAL VIBRATIONS
Sharanabasava C PilliPrincipal, KLE Societys
College of Engineering and Technology,
Udyambag, Belgaum-590008Email
scpilli_at_yahoo.co.in
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CHAPTER 7 CONTINUOUS VIBRATIONS

• Introduction
• Vibration of string
• Longitudinal vibrations of rods
• Torsional vibrations of rods
• Eulers equation for beams
• Simple problems
• transverse vibration of string
• longitudinal vibration of rods

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PROBLEM FORMULATION

• An independent spatial variable is chosen. This
  represents the displacement from a reference
  position when the system is in its equilibrium
  position.
• Free body diagrams of a representative
  differential element are drawn at an instant.
• The Newtons law is applied to the free body
  diagrams. Appropriate kinematic conditions and
  constitutive relations are applied to derive a
  partial differential equation.
• Appropriate boundary conditions, dependent on the
  end supports of the member are formulated.
• Appropriate initial conditions are formulated.

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STRING VIBRATION
Applying the Newtons law to an elemental length
the governing differential equation is
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STRING VIBRATION Contd
The free vibrations equation can be solved by the
Fourier method or separation of variables.
The solution is written as a product of
a function Y(x) which depends on length x and
a function T(t) which depends upon time t only.
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STRING VIBRATION Contd
The constants A and B are evaluated from the
boundary conditions and constants C and D are
evaluated from the initial conditions
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STRING VIBRATION Contd
The function yn(x) is called the nth normal mode,
or characteristic mode.
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STRING VIBRATION Contd
The particular vibration that occurs is uniquely
determined by the specified initial conditions.
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EXAMPLE 1
A string of length L fixed at its end has a large
initial tension T N / m. It is plucked at x L
/ 3 through a distance y0 and released.
Determine the subsequent motion.
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EXAMPLE 1 Contd.
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EXAMPLE 1 Contd.
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EXAMPLE 1 Contd.
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EXAMPLE 1 Contd.
initial conditions
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EXAMPLE 1 Contd.
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EXAMPLE 1 Contd.
Using Fourier Transformation
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EXAMPLE 1 Contd.
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EXAMPLE 1Contd.
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EXAMPLE 1Contd.
The constants are
and
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EXAMPLE 2
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EXAMPLE 2 Contd.
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EXAMPLE 2 Contd.
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EXAMPLE 2 Contd.
Using Fourier transformation
or
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EXAMPLE 2 Contd.
Integrating
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EXAMPLE 3
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EXAMPLE 3 Contd.
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EXAMPLE 3 Contd.
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EXAMPLE 3 Contd.
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EXAMPLE 3 Contd.
Evaluating the constants
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EXAMPLE 4
A bar is fixed at one end and is pulled at the
other end with a force P. The force is suddenly
released. Investigate the vibration of bar.
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EXAMPLE 4 Contd
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EXAMPLE 4 Contd
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EXAMPLE 4 Contd
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EXAMPLE 4 Contd
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EXAMPLE 4 Contd
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EXAMPLE 4 Contd
and
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EXAMPLE 5
Natural frequencies can be found from boundary
conditions
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EXAMPLE 5 Contd
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EXAMPLE 5 Contd
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RECAP

• The general solution for free transverse
  vibration
• of string by the method of separation
  of
• variables is reviewed.
• Problems of transverse vibrations of string and
  
• longitudinal vibration of rods are
  illustrated.
• The examples illustrated are with initial
  conditions
• the combination polynomials and
• harmonic functions
• initial displacement and zero velocity
• zero initial displacement and finite initial
  velocity.