Lecture-3- mechanical vibrations, Mohamed salem.pdf

Vibration course for mechanical engineering by Dr. Mohamed Sameh

1. Dr. Mohamed Sameh Salem
Contact Info:-
Email: [email protected]
Mobile: [phone]
1Lecture (3)
Mechanical Vibrations
ME 242
Dr. Mohamed S. Salem
ةعمجملا ةعماج
ةسدنهلا ةيلك
ةيعانصلا و ةيكيناكيملا ةسدنهلا مسق

3. DampingElement(c)
•Themechanismbywhichthevibrationalenergy isgradually convertedintoheat
orsound isknownasdamping.(Dissipatedenergy).
•Dampingforceexistsonlyifthereisrelativevelocitybetweenthetwoendsof
the damper.

4. DampingElement(c)
DampingTypes
•ViscousDamping.Whenmechanicalsystemsvibrateinafluidmediumsuchasair,gas,
water,oroil,theresistanceofferedbythefluidtothemovingbodycausesenergytobe
dissipated.
•Dry-FrictionDamping.Herethedampingforceisconstantinmagnitudebutoppositein
directiontothatofthemotionofthevibratingbody.
•MaterialorSolidorHystereticDamping:(Nonrigid)Whenamaterialisdeformed,energy
isabsorbedanddissipatedbythematerial.

5. DampingElement(c)
a)TranslationalDamper(N.s/m)
•Force
•DissipativeEnergy(DE)
�=????????????
°
1
��=
2
????????????
°
2
b)TorsionalDamper(N.s.m/rad)
•Torque
•DissipativeEnergy(DE)
??????=??????????????????
°
1
��=
2
??????????????????
°
2
C:DampingCoefficient(N.s/m)

6. CombinationofDampers(C
eq)
•ParallelDampers(samedeflection)
•Dampersinseries(differentdeflection)
�????????????=�1+�2+⋯+�??????
�????????????
1 11
�1�2
=++⋯+
1
�??????
DampingElement(c)

7. EquivalentSpringandDampingConstants(TranslationalSystem)

8. EquivalentSpringandDampingConstants(RotationalSystem)

9. EquivalentSpringandDampingConstants (Translational-RotationalSystem)

10. Equivalent stiffness of multiple springs
Springs could be connected in:-

11. Proof of combination of springs
In parallel connected springs, the
springs are deflected together by the
same amount of deflection.

12. Proof of combination of springs (cont.)Deflection of Beams and Plates

13. k
1
k
2
Series or Parallel?

14. Problems slide: Parallel or Series?
m
K
beam
K
Spring
(a)
m
K
Spring
K
beam
(b)

16. Example 1.5.5 (continued)

17. Problem
Determine the equivalent spring constant of the system shown

18. Problem
Determine the equivalent spring constant of the system shown

19. Combination of torsional springs
Rao, 6
th
edition
Parallel or series?

20. Combination of torsional springs
Rao, 6
th
edition

22. Example
Determine the equivalent stiffness of the system. Assume that the beam and the
wire rope has the same modulus of elasticity E
Simplify Equivalent
Solution
The bending stiffness of the beam element is
The tension stiffness of the wire rope is
The equivalent stiffness of the system is

23. Static deflection of springs
Vertical or inclined springs deflects due to the weight of other elements
in the system.
The deflected position of the system represents the equilibrium
position of the systems vibrations
m
m
x
(t)
Δ
s
Equilibrium
position
Unloaded
spring has no
deflection
Once a mass is added, the
spring deflects statically byΔ
s
In vibrations, x is measured from
equ. Pos.

24. Static deflection of springs
m
m
x
(t)
Δ
s
Equilibrium
position
Unloaded spring has
no deflection
Once a mass is added, the spring
deflects statically
In vibrations
k
k k
kΔ
s
m
mg
m
K(x+Δ
s)
mg
Static equilibrium, ∑F=zero, i.e.
kΔ
s=mg Dynamic equilibrium, ∑F=ma, i.e. -kx=ma

25. Horizontal springs and static deflection
Horizontal spring can have static deflection if the moment of its force balances the moment of
the weight.
Example: Determine the static deflection of the spring k for the following system.
O
m
1 g
kΔ
s
r
2
r
1
∑M
O=zero
m
1 g r
2 – k Δ
s r
1 = 0.0
Then,
Δ
s=(m
1g r
2) / (k r
1)

26. Problems slide

27. Problems slide

28. Problems slide

29. Problems slide

30. Problems slide

31. Problems slide

32. Problems slide
Model the following system as a disk attached to a torsional
spring of an equivalent stiffness.

33. Problems slide

34. Problems slide
Determine the static deflection of the spring in the following figure. Neglect the mass of the
beam.

35. Problems slide
Determine the static deflection of the spring in the following figure. Neglect the mass of the
beam.
Problems slide
Determine the static deflection of the spring in the following figure. Neglect the mass of the
beam.

36. Home Work slide
Determine the static deflection of the spring such that the uniform rod is horizontal. The free
length of two springs are not equal.
Submission deadline: Sunday 14/09/2025 @ 12:00 am -No late
submission is accepted

37. Dr. Mohamed Sameh Salem
Associate professor, Mechanical Power Engineering
[email protected]
Tel: [phone]