MECHANICAL VIBRATIONS ME 65

1
MECHANICAL VIBRATIONS(ME 65)

• Session 7
• Dr. P. Dinesh
• Sambhram Institute of Technology
• Bangalore

2
In this session 7

• Problems on undamped free vibrations contd.
• Review of session 4,5,6,7

3

• 1) Determine the natural frquency of the system,
  neglecting the mass of the rod.

a
k
l
m
4

• The forces on the pendulum are

a cos?
?
Xl?
kasin?
x
mx
mgsin?
mg
5

• According to Newtons second law of motion
• Sum of moments Restoring torque
• Restoring torque I? ml2 ?
• Taking moments about fixed point
• (mgsin?)l and (kasin?)(acos?)
• ml2 ? - (mgsin?)l - (kasin?)(acos?)
• For small values of ?, sin? ? and cos? 1

6

• ml2?mgl?ka2 ? 0
• Or ?((mglka2 )/ ml2 )? 0
• Or ?n v(mglka2/ ml2 ), rad/sec.

7

• 2) A cylinder of dia. D and mass M floats
  vertically in a liquid of mass density ?. It is
  depressed slightly and released, find the period
  of its oscillation. What will be the frequency if
  salty liquid of specific gravity of 1.2 is used.

x
8

• x is the displacement of cylinder
• A is the c/s area of cylinder
• When cylinder is depressed by x
• Inertia force Mx
• Vol. of liquid displaced Ax
• Mass of displ. Liquid ?Ax Restoring force
• From Newtons law
• Inertia force Restoring force 0

9

• Mx ?Ax 0
• x (?A/M) x 0
• Time period T 2p/?
• ? v(?A/M) rad/sec
• T (2pv(M/?A) , sec.
• for ? 1.2?, ? v(1.2?A/M) rad/sec.

10

• 3) A cylinder of mass M and radius r rolls
  without slipping on a cylindrical surface of
  radius R . Determine the natural frequency of the
  oscillation when the cylinder is displaced
  slightly from its equilibrium position.

11

• Cylinder dipl. by ?

?
R
(R-r)cos?
3
5
F
x
4
2
(R-r)(1-cos?)
r
1
12

• R? rF, F (R/r)? , F (R/r)?
• Linear dipl. of centre x (R-r)?
• x (R-r)?
• Rotational vel.of cylinder F ?
• KE of system Translational Rotational
• 1/2(m)(x)2 1/m I (F
  ?)2
• I Mass MI ½ mr2

13

• KE of system
• 1/2m(R-r?)21/2(½ mr2) ?2(R/r -1)2
• PE of system mgh mg(R-r)(1-cos?)
• d(KE PE)/dt 0
• Differentiating and noting for small ?,sin? ?
• ? (2g/(3(R-r))? 0
• ?n v(2g/3(R-r)) , rad/sec.

14

• 4) A homogenous sphere pf radius r and mass m is
  free to roll without slipping on a spherical
  surface of radius R. If the motion of sphere is
  restricted to a vertical plane, determine the
  frequency.
• Soln. Except the MI of sphere everything is same
  in the analysis of problem

15

• KE of system is
• 1/2m((R-r)?)2 1/2 ((2/5)mr2)((R/r)-1)2?2
• PE of system mgh mg(R-r)(1-cos?)
• d(KE PE)/dt 0
• ? (5g/7(R-r))? 0
• ?n v (5g/7(R-r)), rad/sec.

16

• 5)A torsion pendulum has to have a natural
  frequency of 5 Hz.What length of steel wire of
  diameter 2 mm. should be used for this pendulum.
  The inertia of the mass fixed at the free end is
  0.0098 kg-m2. Take C 0.85 x 1011 N/m2.
• Soln For torsional vibration ?n vkt/I

L
17

• And fn ?n / 2p
• Therefore, ?n 5 x 2p 10p rad/sec.
• We know kt T/? GJ/L , G (p/32)d4
• kt 3.336 x 104 / L ( d is in m)
• Hence, ?n vkt/I ,
• 10 p v(3.336 x 104 / LI)
• 10 p v(3.336 x 104 / L x 0.0098)
• L 1.35 cm.

18

• 6) Determine the natural frequency of the system.

k
k
a
r
m
19

• xr?, xr?,x r?
• x12(ra)?

x1
Spring Force
x
?
20

• KE of systemDue to rotation due to tranl.
• ½(I?)2 1/2mx2 , subs.for I1/2mr2 and x
• KE 3/4mr2 ?2
• PE PE due to two springs
• 2 x ½ x k x (ra)2?2 k (ra)2?2
• d(KE PE)/ dt 0
• ? (4k (ra)2)/(3mr2) ? 0

21

• This is in std. form and hence,
• ?n v (4k (ra)2)/(3mr2) , rad/sec
• Or
• Fn 1/2p v (4k (ra)2)/(3mr2) , Hz.

22
Review of Sessions 4,5,6,7

• Undamped free vibrations
• Single DOF system
• Newtons Method Accn.force restoring force
• Energy Method d(KE PE)/dt 0

23

• Rayleighs Method- Max.KE Max PE
• Effect of mass of spring on Natural frequency-
  1/3 mass of spring is added to the mass
• Problems

24

• End of Session 7