Lecture-5- 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 (5)
Mechanical Vibrations
ME 242
Dr. Mohamed S. Salem
ةعمجملا ةعماج
ةسدنهلا ةيلك
ةيعانصلا و ةيكيناكيملا ةسدنهلا مسق

2. HomeWork
Determine the equivalent spring constant for the system shown. The weight is uniformly
distributed and can be assumed as one force at the rod’s center of gravity.
Submission deadline: Sunday 14/09/2025 @ 12:00 am -No late
submission is accepted

3. HomeWork
Determine the equivalent spring constant for the system shown. The weight is uniformly
distributed and can be assumed as one force at the rod’s center of gravity.

4. HomeWork
Determine the equivalent spring constant for the system shown. The weight is uniformly
distributed and can be assumed as one force at the rod’s center of gravity.
•F
1= force in the left spring (??????
1=1×10
5
N/m).
•F
2 =force in the right spring (??????
2=2×10
5
N/m).
•The rod’s weight Wacts at its center of gravity (0.5L=0.3
m from the left spring, 0.1 m from the right end).
•From Force Balance in the Vertical direction:
W = F
1 + F
2
•By taking moments about the left spring = 0:
??????
2=
�∗0.3
0.4
=0.75 �??????
2 ∗ 0.4 −� ∗0.3=0
Then, ??????
1=0.25 �
�
1=
??????
1
??????
1
=
0.25 ??????
??????
1
,�
2=
??????
2
??????
2
=
0.75 ??????
??????
2
Deflection at load point (linear along the rigid rod):
??????�
????????????
�
2−�
1
=
0.3
0.4
�
????????????=
??????
??????
????????????
=�
1+
0.3
0.4
�
2−�
1=
0.25 ??????
??????
1
+0.75 ∗(
0.75 ??????
??????
2
−
0.25 ??????
??????
1
)
1
??????
????????????
=
0.25
??????
1
+0.75 ∗(
0.75
??????
2
−
0.25
??????
1
) ➔ ??????
????????????=2.91×10
5
N.m

5. Example 4
Shown is a simple single-degree-of-freedom model of a wheel mounted on a spring. The
friction in the system is such that the wheel rolls without slipping. Determine the mathematical
model and calculate the natural frequency of oscillation using the energy method. Assume that
no energy is lost during the contact.

6. Example 4

7. Example 5
Use the energy method to determine the equation of motion of the pendulum system
(the rod l is assumed massless) shown in the figure (Note: )

8. Example 5

9. Example 6
Derive the mathematical model and determine the natural frequency corresponding to the free
vibrations of the torsional dynamic vibration absorber sketched in this figure.
Assume the rod connecting the two masses is rigid and massless, and that the displacements
are small, such that the four springs’ directions remain unchanged. Also assume the motion
takes place in a horizontal plane such that gravitational effects can be neglected.
Known are m = 1kg, k = 100 N/m, l 1 = 0.25 m, and l 2 = 0.5 m.

10. Example 6
Knowns are m = 1kg, k = 100 N/m, l 1 = 0.25 m, and l 2 = 0.5 m.
The total energy of this system E

11. SolutionofDifferentialEquationofMotion
??????.??????
°°
+??????.??????=??????
1.Thedisplacement(x)
2.Thevelocity(x
o)
3.Theacceleration(x
oo)
•Functionintime(t)
•Howtosolvethedifferentialequationofmotiontodetermine(x)valuethatfitthegivenequation

12. SolutionofDifferentialEquationofMotion
�??????
°°
+ �??????=0→1
??????�=c�
��
→2Assumethegeneralsolutionofequation CandSareconstantstobedetermined
??????
°
�=cs�
��??????
°°
�=cs
2
�
��
then,
SubstitutioninEq.1,
and��
��
≠0
���
2
�
��
+�c�
��
=0
��
��
(��
2
+�)=0
(��
2
+�)=0→3 AuxiliaryorCharacteristicequation
�1,2=±
�
−=±????????????�
�
S
1,2aretherootsoreigenvaluesofthe
Characteristicequation
??????=−1 ??????�: NaturalFrequency

13. ??????�=c�
��
→2
SolutionofDifferentialEquationofMotion
SubstitutioninEq.2,�1,2=±????????????�
??????�=c1�
????????????��
+c2�
−????????????��
→4
SubstitutioninEq.4,
Euler’sFormula;�
±????????????��
=cos??????��±??????sin??????��
??????�=c1cos??????��+??????sin??????��+c2cos??????��−??????sin??????��
??????�=�1+�2cos??????��−c1+c2??????sin??????��
??????�=�1cos??????��+�2sin??????��→5A
1,2constantstobedeterminedforminitialconditions
att=0 ??????0=�1 ??????��??????
°
�0=�2??????�=??????
°
??????
°
�=−�1??????�sin??????��+�2??????�cos??????��
�1=??????��
2
??????
°
=
�
??????�
SubstitutioninEq.5,

14. SolutionofDifferentialEquationofMotion
??????�=�1cos??????��+�2sin??????��→5 �1=??????�,�
2
??????
°
=
�
??????�
????????????=??????
�cos??????
�??????+
??????�°
??????�
sin??????
�??????→6
�2=��cos∅�
��=�
1
2
+�
2
2
=
�
??????
°
??????�
�1=��sin∅�
2
=??????���??????����
IntroducingA
1,2inEq.5,
∅�=tan
−1
�1
�2
=tan
−1
??????�??????�
??????�
°
=??????ℎ??????�������
Where,
Wehave
??????�=��sin??????��+∅�→8
??????
�
2
+
??????�
??????�
=��sin∅�cos??????��+��cos∅�sin??????��
=��sin??????��cos∅�+��cos??????��sin∅�
sin(�+�)=sin�cos�+cos�sin�

15. Graphical representation of the motion of a harmonic
oscillator

16. Example 7:Spring-MassSystem
�??????
°°
+�??????=0
Given:m=45kg
k=500N/m
att=0X= 5mm,X
o=2mm/s
??????=��sin(??????��+∅�)
�??????=
�
�
=
45
500
=3.33�??????�/���
??????
°
=
�??????
=�
�??????�cos??????��+∅
�
0.005=��sin3.33 ×0+∅� 0.005=��sin∅�
��
att=0,X=5mm
att=0,X
o=2mm/s 0.002=3.33 ×��cos3.33 ×0 + ∅�0.0006=��cos∅�
1
2
�
??????
°°
+
�
??????=0

17. 1
2
0.005=��sin??????
0.0006=��cos??????
��sin??????=0.005
??????�.1
??????�.2 =0.0006
tan∅=0.833 ∅=�??????�
−1
0.833=83.15
°
��cos??????
0.005
��=
sin??????
0.005
=
�??????�(83.15°)
=0.005�=5��
��=??????=5��
??????
°°
=
��
=−��??????
2
sin??????��+∅�
att=0 X=5mm
??????=��sin??????��+∅�
��
�??????
°
??????
°
=
�??????
=�??????cos??????�+∅
����
Example 7:Spring-MassSystem

18. Example 8:HarmonicResponseofa
WaterTank
•Thecolumnofthewatertankshowninthefigureis91.5
mhighandismadeofreinforcedconcretewithatubular
crosssectionofinnerdiameter3.048mandouterdiameter
2.438m.Thetankweighs272155kgwhenfilledwithwater.
Byneglectingthemassofthecolumnandassumingthe
Young’smodulusofreinforcedconcreteas2.7x10
10
Pa,
determinethefollowing:
a.Thenaturalfrequencyandthenaturaltimeperiodof
transversevibrationofthewatertank.
b.Thevibrationresponseofthewatertankduetoaninitial
transversedisplacementof0.254mandX
o
=0.
c.Themaximumvaluesofthevelocityandacceleration
experiencedbythewatertank.
??????=��sin(??????��+∅�)

19. �=
3??????�
�
3
=
3×2.7 ×10
10
×2.745
91.5
3
=290244.55�/�
�=91.5�,??????=2.7×10
10
�
2
�
,
��=3.048�,
??????�=2.438�,�=272155��Given:
64
�??????
??????
�= �
4
−�
4
64
??????
= 3.048
4
−2.438
4
=2.745�
4
??????�=
�
=
�
290244.55
272155
=1.033�??????�/���
�
�
2??????2×??????
??????= = =6.082���
??????1.033
a.Thenaturalfrequencyandthenaturaltimeperiod
Example 8:HarmonicResponseofa
WaterTank

20. Example 8:HarmonicResponseofa
WaterTank
b.Thevibrationresponseofthewatertankduetoaninitialtransversedisplacementof0.254m
??????�=��sin??????��+∅�
0.254=��sin1.033×0+∅�
0.254=��sin∅�att=0,X=0.254 m
att=0,X
o=0m/s 0=1.033 ×��cos(1.033 ×0 + ∅�) 0=��cos∅�
1
2
??????�.1
??????�.2 =0
tan∅=
1
0
∅=�??????�
−1
1
0
=90
°
��cos??????
0.254
��=
sin??????
0.254
=
�??????�(90°)
=0.254�
�� sin ∅� = 0.254

21. c.Themaximumvaluesofthevelocityandaccelerationexperiencedbythewatertank.
Example 8:HarmonicResponseofa
WaterTank
??????�=0.254sin1.033�+
??????
2
�????????????
??????
°°
���=�??????
2
=0.254×1.033
2
=0.271�/���
2
�????????????
??????
°
�=��??????�=0.254 ×1.033=0.262�/���

22. FreeVibrationofanUndampedTorsional
System
�
�
??????J�
�= ??????
•Fromthetheoryoftorsionofcircularshafts
M
t:TorsionalMoment
G:ModulusofRigidity(N/m
2)
J
o:PolarMomentofInertia(m
4)
L:Lengthoftheshaft(m)
??????:AngleofTwist(rad)
k
t:TorsionalSpringConstant(N.m/deg)
J�=
??????
32
�
4
��
��=
??????
=
??????J�??????
??????�
=
????????????�
4
32�

23. FreeVibrationofanUndampedTorsional
System
•DifferentialEquationofMotion
��??????
°°
+��??????=0
�??????=
��
���
�
??????
°°
+
��
??????=0
J
o:PolarMassMomentofInertia(kg.m
2)
�??????=
2??????
??????
�
�
??????�
�=
2??????

24. •SolutionofDifferentialEquationofMotion
FreeVibrationofanUndampedTorsional
System
A
1,2constantstobedeterminedforminitialconditions
att=0 ??????0=�1 ??????��??????
°
�0=�2??????�=??????
°
�1=??????��
2
??????
°
=
�
??????�
??????
°
??????�=�1cos??????��+�2sin??????��→9
�=−�1??????�sin??????��+�2??????�cos??????��

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