Lecture-4- 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 (4)
Mechanical Vibrations
ME 242
Dr. Mohamed S. Salem
ةعمجملا ةعماج
ةسدنهلا ةيلك
ةيعانصلا و ةيكيناكيملا ةسدنهلا مسق

2. TypesofVibratingSystem
Force
ForcedVib.Sys.
spring
spring
Damper
Mass
Outof Scope
FreeVib.Sys.
FreeDampedVib.Sys.
1.ForcedundampedVib. Sys.
2.ForceddampedVib. Sys.

3. FreeVibrationSystem
Freevibration:
meansthatthemassissetintomotionduetoinitialdisturbancewithno
externallyappliedforceotherthanthespringforce,damperforce,or
gravitationalforce.

4. FreeVibrationofanUndampedSystem
•Selectasuitablecoordinatetodescribethepositionofthemassorrigidbodyinthesystem.
•Determinethestaticequilibriumconfigurationofthesystemandmeasurethe
displacementofthemassorrigidbodyfromitsstaticequilibriumposition.
•Drawthefree-bodydiagramofthemassorrigidbodywhenapositivedisplacementand
velocityaregiventoit.Indicatealltheactiveandreactiveforcesactingonthemassorrigid
body.
•ApplyNewton’ssecondlawofmotiontothemassorrigidbodyshownbytheFreebody
diagram.

5. FreeVibrationofanUndampedSystem
•Newton’sSecondLawofMotion
•PrincipleofConservationofEnergy
•D’AlembertsPrinciple
•PrincipleofVirtualDisplacements.

6. •Tostudythesystemdynamics,atanytime(t),
weneedtoknowthefollowing:
1.Thedisplacement(x)
2.Thevelocity(x
o)
3.Theacceleration(x
oo)
??????
∘
=
�??????
��
, ??????
∘∘
=
�??????
∘
��
Didweneeddatafromthestaticstateofthesystem?
EquationofMotionofaSpring-MassSystem in
VerticalPosition(Newton’s2
nd
Law)

7. Equation of Motion
Equation of Motion
Newton’s 2
nd
Law Energy Method
Natural frequency, ω
n
Damping Ratio, ξ

9. •AtBalance:(Static)
Σ????????????����=??????
•FreebodydiagramAtBalance:
+??????.??????−????????????=??????
??????.??????=????????????
??????=
??????.??????
??????
??????.??????
EquationofMotionofaSpring-MassSystem in
VerticalPosition(Newton’s2
nd
Law)

10. •Atdisplacement(x):(dynamicinitiation)
Σ????????????����=????????????
°°
•FreebodydiagramAtmotionstart: ??????(??????+??????)
+??????.??????− ????????????+??????=??????.??????
°°
??????=
??????.??????
??????
+??????.??????−??????
??????.??????
??????
− ??????.??????=??????.??????
°°
−??????.??????=??????.??????
°°
??????.??????
°°
+ ??????.??????=??????
Fromstaticstate
differentialequationofMotion
EquationofMotionofaSpring-MassSystem in
VerticalPosition(Newton’s2
nd
Law)

11. •Formostoffreevibrationsystems,wecanneglectbothofwightforceandstaticdeflection
force.
Σ????????????����=????????????
°°
−??????.??????=??????.??????
°°
??????.??????
°°
+??????.??????=??????
differentialequationofMotion
EquationofMotionofaSpring-MassSystem in
VerticalPosition(Newton’s2
nd
Law)

12. Energy Method
Conditions
•Total energy is conservative (E = constant)
•Zero energy addition
•Zero energy loss

13. EquationofMotionofaSpring-MassSystem in
Vertical Position (Conservation of Energy)
1
2
�=�??????
2
1
2
�=�??????
°
2
??????
??????�
(�+�)=0
PrincipleofConservationofEnergy
�+�=??????����??????��
KineticEnergy
potentialEnergy
°
21
2
1
�??????+�??????=??????����??????��
2 2
°°
�.??????+�.??????=0

14. Notethefollowingaspectofthespring-masssystem:
�??????=
�
�
=
Whenthemassvibratesinaverticaldirection,wecancomputethenaturalfrequency
andtheperiodofvibrationbysimplymeasuringthestaticdeflection.Wedon’tneedto
knowthespringstiffnesskandthemassm.
??????
??????��
��
??????�??????
�= =
????????????
��

15. Example 1
Derive the equation of motion for the system shown below and find the
natural frequency.
෍??????=�ሷ??????=−
13
5
�??????
�ሷ??????+
13
5
�??????=0
ሷ??????+
13
5
�
�
??????=0 ??????
�=
13
5
�
�
Solution

16. Example 2
↺෍??????
�=??????
�
ሷ??????
−�??????��??????�??????=��
2ሷ??????
�ሷ??????+?????? �??????�??????=0.0
ሷ??????+
??????
�
??????=0.0
The +ve direction is in the direction of the angle
θ. Assume that sinθ=θ, then the linear form of
the EoM is shown. Note that ሷ?????? and ?????? are
functions of time (i.e., ሷ?????? (�) and ??????�)
Derive the equation of motion for the system shown below and find the
natural frequency if m=10 kg and k=1000 N/m. the rod is massless
When??????isverysmall,thensin??????=??????
ForLumpedMass: ?????? = ��
2

17. Example 3
Use the energy method to determine the equation of motion of the simple pendulum
(the rod l is assumed massless) shown in the figure

18. Example 3

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