1. MECHANICAL VIBRATION
MME4425/MME9510
Prof. Paul Kurowski

2. TEXT BOOKS
REQUIRED
RECOMMENDED

3. MME4425b web site
Design Center web site

4. Software used:
SolidWorks
Design and assembly of mechanisms and structures
SolidWorks Simulation (add-in to SolidWorks)
Structural analysis
Motion Analysis (add-in to SolidWorks)
Kinematic and dynamic analysis of mechanisms
Excel

5. SolidWorks 2012 installation and activation instructions:
Go to
Use SEK-ID = XSEK12
Select release
When prompted enter serial number for activation

6. WHAT IS THE DIFFERENCE BETWEEN
DYNAMIC ANALYSIS AND VIBRATION ANALYSIS?

7. DIFFERENCE BETWEEN A MECHANISM AND A STRUCTURE
Structure is firmly supported, mechanism is not.
Structure can only move by deforming under load. It may be one time deformation when the load is applied or a structure can vibrate about its neutral position (point of equilibrium).
Generally a structure is designed to stand still.
Mechanism moves without deforming it components. Mechanism components move as rigid bodies.
Generally, a mechanism is designed to move.

8. DIFFERENCE BETWEEN A MECHANISM AND A STRUCTURE
STRUCTURES
MECHANISMS

9. RIGID BODY MOTION

10. How many rigid body motions?

11. DISCRETE SYSTEM VS. DISTRIBUTED SYSTEM
Mass, stiffness and damping are separated
Distributed system
Mass, stiffness and damping are NOT separated

12. DISCRETE SYSTEM VS. DISTRIBUTED SYSTEM
1DOF.SLDASM DOF.SLDASM
Discrete system
Mass, stiffness and damping are separated
Distributed system
Mass, stiffness and damping are NOT separated

13. swing arm 01.SLDASM
swing arm 02.SLDASM
Discrete system
Mass, stiffness and damping are separated
Distributed system
Mass, stiffness and damping are NOT separated

14. Discrete system
Vibration of discrete systems can be analyzed by Motion Analysis tools such as Solid Works Motion or by Structural Analysis such as SolidWorks Simulation based on the Finite Element Analysis
Distributed system
Vibration of distributed systems must be analyzed by structural analysis tools such as SolidWorks Simulation based on the Finite Element Analysis.

15. SINGLE DEGREE OF FREEDOM SYSTEM
LINEAR VIBRATIONS

16. SINGLE DEGREE OF FREEDOM SYSTEM, LINEAR VIBRATIONS
Homogenous equation

17. FINDING GENERAL SOLUTION OF HOMOGENEOUS EQUATION
By guessing solution
How to solve this?
We guess solution based on experience that the solution will be in the form:
A – magnitude of amplitude
Ф – initial value of sine function
ωn – angular frequency

18. FINDING GENERAL SOLUTION OF HOMOGENEOUS EQUATION
By guessing solution
ωn – natural angular frequency found from system properties
Where A and Ф are found from initial conditions

19. FINDING GENERAL SOLUTION OF HOMOGENEOUS EQUATION
Using complex numbers method

20. FINDING GENERAL SOLUTION OF HOMOGENEOUS EQUATION
Using complex numbers method
We have found two solutions to equation
and
Since
is linear, then the sum of two solutions is also a solution
Using Euler’s relations:
The equation can be re-written as:
Where A and Ф are found from initial conditions

21. FINDING GENERAL SOLUTION OF HOMOGENEOUS EQUATION
Using Laplace transformation
Taking Laplace transform of both sides
Using (5), (6)

22. Laplace transformation

23. Laplace transformation
Inman p 619

24. QUANTITIES CHARACTERIZING VIBRATION
Average value of amplitude is
But average value of
is zero.
Therefore, average value of amplitude is not an informative way to characterize vibration.
for this reason we use mean-square value (variance) of displacement:
Square root of mean square value is root mean square (RMS).
RMS values of are commonly used to characterize vibration quantities such as displacement, velocity and acceleration amplitudes.

25. QUANTITIES CHARACTERIZING VIBRATION
Displacement
Velocity
Acceleration
These quantities differ by the order of magnitude or more, hence it is convenient to use logarithmic scales.
The decibel is used to quantify how far the measured signal x1 is above the reference signal x0

26. QUANTITIES CHARACTERIZING VIBRATION
Lines of constant displacement
For a device experiencing vibration in the frequency range 2-8Hz:
The maximum acceleration is 10000mm/s^2
The maximum velocity is 400mm/s
Therefore the maximum displacement is 30mm
Lines of constant acceleration
Nomogram for specifying acceptable limits of sinusoidal vibration (Inman p 18)

27. LINEAR SDOF

28. LINEAR SDOF
10kg mass
Linear spring
400000N/m
Base
SDOF.SLDASM

29. Results of modal analysis
LINEAR SDOF
Results of modal analysis

31. Trigonometric relationship between the phase, natural frequency, and initial conditions.
Note that the initial conditions determine the proper quadrant for the phase.

32. PENDULUM SDOF

33. PENDULUM SDOF Galileo Galilei lived from 1564 to 1642.
Galileo entered the University of Pisa in 1581 to study medicine. According to legend,
he observed a lamp swinging back and forth in the Pisa cathedral. He noticed that the
period of time required for one oscillation was the same, regardless of the distance of
travel. This distance is called amplitude.
Later, Galileo performed experiments to verify his observation. He also suggested that
this principle could be applied to the regulation of clocks.

34. PENDULUM SDOF
pendulum 02.SLDPRT

35. PENDULUM SDOF
Equations of motion method

38. The energy method is suitable for reasonably simple systems.
The energy method may be inappropriate for complex systems, however. The reason is
that the distribution of the vibration amplitude is required before the kinetic energy
equation can be derived. Prior knowledge of the “mode shapes” is thus required.

39. PENDULUM SDOF
Energy method

40. TORSIONAL SDOF

41. TORSIONAL SDOF
polar moment of inertia of cross-section
disk 01.SLDPRT

42. TORSIONAL SDOF

43. ROLER SDOF
roler.SLDASM
Inman p 32

44. ROLER SDOF

45. ROLER SDOF Inman p 32 k = k1 + k2 = 2000N/m m = 75.4kg r = 0.1m
J = kgm2

46. ROLER SDOF

47. MASS AT THE END OF BEAM
rotation.SLDASM

48. MASS AT THE END OF BEAM
mass 2.7kg
cantilever.SLDPRT

49. RING
Ring.SLDASM

50. HOMEWORK 1 Derive equation of motion of SDOF using energy method
Find amplitude A and tanΦ for given x0, v0
Find natural frequency of cantilever, l=400mm, Φ=5mm, E=2e11Pa, m=2.7kg. Confirm with SW Simulation
Work with exercises in chapter 19 – blue book

51. TORSONAL SDOF TRIFILAR
1060 alloy
Model file trifilar.sldasm
Configuration trifilar
Model type solid
Material as shown
Supports as shown
Objectives
Find the natural frequency of trilifar
Fixed support
Custom material
E = 10MPa
ρ = 1kg/m3
very soft, very low density
1060 alloy
Restraint in radial direction to force torsional mode
trifilar.SLDASM

52. TORSIONAL SDOF BIFILAR

53. TORSIONAL SDOF TRIFILAR

54. TORSIONAL SDOF TRIFILAR
Using energy method:

55. TORSIONAL SDOF TRIFILAR

56. TORSONAL SDOF TRIFILAR
Trifilar can be used to find moments of inertia of objects placed on rotating platform

57. spur gear.SLDPRT