MECHANICAL VIBRATIONS ME 65

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MECHANICAL VIBRATIONS(ME 65)

• Session 2
• By
• Dr. P.Dinesh
• Sambhram Institute of Technology
• Bangalore

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In Session 2

• Phenomenon of Beats
• Harmonic motion in complex form
• Addition of Harmonic motions
• Analytical
• Graphical

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• A simple harmonic motion can be expressed as
• xA sin?t
• A amplitude of vibration
• ? circular frquency, rad/sec
• Consider a vector of length equal to amplitude A
  and rotating at a speed of ? rad/sec. After time
  t it will be at an angle ?t wrt ref. axis (X
  axis)
• Projection of this vector on Y axis gives the
  diplacement vector at time t.

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Y
P
A
?t
X
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• x A sin?t , projection on Y axis disp vector
• x ?A(sin?t 90º) , gives velocity vector
• x ?2A (sin?t 180º) , gives accn. vector
• All vectors rotate in CCW direction

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Beats Phenomenon

• x1 aSin?1t and x2 b Sin?2 t moving in same
  direction ,?1 is not equal to ?2,resultant motion
  is not harmonic
• Phase difference between x1 and x2 keeps changing
• Resultant ampl. is (ab) when in phase and (a-b)
  when out of phase.
• Resultant amplitude continuously varies from
  (ab) a (a-b)
• This is Beats occuring with a frequency of ?2- ?1

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Harmonic motion in Complex form

• If r is vector magnitude, ? the angle it makes
  with ref.
• The diplacement vector xrei?t
• The velocity vector (dx/dt) i?x
• The acceleration vector (dx/dt)2 -?2x
• Vel. Leads disp by 90º and accn. Vector leads
  vel. Vector by 90º

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Addition of Harmonic motions

• Two vectors rep.SHM can be added
• Analytically
• x1 A1Sin?t , x2 A2Sin(?tf)
• x x1 x2 the resultant motion is SHM
• x A Sin(?t?), A res. Ampl., ?Ang. Of Resul.
  with x1

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• Add the following vectors analytically
• x1 4cos(?t 10º) and x2 6sin(?t 60º)
• x x1x2 A sin(?t ?)
• A sin(?t ?) 4cos(?t 10º) 6sin(?t 60º)
• Expanding both sides
• A(sin?tcos ?cos?tsin?)4cos?tcos10º -
  4sin?tsin10º 6sin?tcos60º 6cos?tsin60º

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• sin?t(A cos?) cos ?t(Asin?) sin?t(2.305)
  cos?t(9.135)
• Therefore, A cos? 2.305
• A sin? 9.135
• The resultant amplitude A2(Acos?) 2 (Asin? )2
• A 9.42
• tan ? (Asin?/Acos?)(9.135/2.305) 76º
• Or x 9.42sin(?t76º)

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• Graphical Method
• Putting two vectors in same form
• x24cos(?t10º) 4cos(?t10º90º)
• 4sin(?t100º)
• x1 6 sin(?t60º)
• Draw ox the reference axis.Draw oa of length 6
  units(60mm) wrt ox at 60º.Draw ob of 4 unit
  length (40 mm) at 100º to ox.

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• Complete the vector diagram by drawing lines
  parallel to oa and ob to get point c. measure oc
  which will be equal to A 9.42 units and ? the
  angle of oc wrt ox will be 76º

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Problems

• a) Add the following harmonics analytically and
  check the solution graphically.
• X1 3sin(?t30º), X2 4 cos(?t10º)
• b) Split the harmonic motion x 10
  sin(?t30º)into two harmonic motion, one having a
  phase angle of zero and 46º.

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• c) A body is subjected to two harmonic motions
  x1 15sin(?t 30º) and x2 8cos(?t60º). What
  extra harmonic should be given to bring it to
  static equilibrium
• d) A harmonic displacement is given by
• x(t) 6 sin(20t60º), find a) frequency and
  period of motion b) max. displacement, velocity
  and acceleration.

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Conclusion Session 2

• What is Beat phenomenon
• Methods of adding two harmonics analytically and
  graphically