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1. IMPERIAL COLLEGE OF SCIENCE, TECHNOLOGY & MEDICINE
MECHANICAL ENGINEERING DEPARTMENT
2M SOLID MECHANICS
VIBRATION LECTURES
October 2005
Lecture 1 Introduction
Lecture 2 Free motion of undamped SDOF systems
Lecture 3 Free motion of damped SDOF systems
Lecture 3A Lagrange’s Equation of Motion
Lecture 4 Forced motion of SDOF systems
Lecture 5 Vibration Isolation
Lecture 6 Base excitation of SDOF systems
Lecture 7 Introduction to 2-DOF systems
Textbooks
B. H. Tongue 2002. Principles of Vibration, Oxford University Press
D. J. Inman 1994. Engineering vibration, Prentice-Hall International
MI - 2M SM Vibration Lecture 1 1

2. IMPERIAL COLLEGE OF SCIENCE, TECHNOLOGY & MEDICINE
MECHANICAL ENGINEERING DEPARTMENT
OVERVIEW
• Vibration Lectures - 10 lectures
• Dynamics Lectures - 6 lectures
• Further notes and tutorial questions in:
2M SM - Lecture notes on Vibration (Yellow booklet)
See the MED Web for all handouts.
MI - 2M SM Vibration Lecture 1 2

3. IMPERIAL COLLEGE OF SCIENCE, TECHNOLOGY & MEDICINE
MECHANICAL ENGINEERING DEPARTMENT
LECTURE 1 - INTRODUCTION
Objectives:
• To provide a road map for the Vibrations &
Dynamics Course
• To give an overview of vibration problems in
engineering
• To review available solution techniques
MI - 2M SM Vibration Lecture 1 3

4. IMPERIAL COLLEGE OF SCIENCE, TECHNOLOGY & MEDICINE
MECHANICAL ENGINEERING DEPARTMENT
ROAD MAP
SOLID
MECHANICS
STATICS DYNAMICS MATERIALS ……..
RIGID BODY VIBRATION 1-DOF
DYNAMICS ANALYSIS SYSTEMS
STEADY FORCES UNSTEADY FORCES N-DOF
RIGID BODIES FLEXIBLE BODIES SYSTEMS
CONTINUOUS
SYSTEMS
MI - 2M SM Vibration Lecture 1 4

5. IMPERIAL COLLEGE OF SCIENCE, TECHNOLOGY & MEDICINE
MECHANICAL ENGINEERING DEPARTMENT
VIBRATION MODES OF A PLATE
MI - 2M SM Vibration Lecture 1 5

6. IMPERIAL COLLEGE OF SCIENCE, TECHNOLOGY & MEDICINE
MECHANICAL ENGINEERING DEPARTMENT
TIME DOMAIN VS FREQUENCY DOMAIN - I
• For vibration analysis, we General arbitrary motion
need to describe the
motion of each point on a
structure.
• If the motion is arbitrary,
we need to describe the TIME
motion in the time
domain.
• If the motion is harmonic, Harmonic motion
we can reduce this
requirement by working in
the frequency domain.
TIME
Can be expressed as x(t)=Asinωt
MI - 2M SM Vibration Lecture 1 6

7. IMPERIAL COLLEGE OF SCIENCE, TECHNOLOGY & MEDICINE
MECHANICAL ENGINEERING DEPARTMENT
TIME DOMAIN VS FREQUENCY DOMAIN - II
AMPLITUDE
We observe
the motion of a FREQUENCY
given point at a
given
frequency
MI - 2M SM Vibration Lecture 1 7

8. IMPERIAL COLLEGE OF SCIENCE, TECHNOLOGY & MEDICINE
MECHANICAL ENGINEERING DEPARTMENT
EXAMPLE OF FREQUENCY DOMAIN ANALYSIS
Vibration shape at natural frequency Vibration shape at any other
=>Mode shape frequency =>Operating deflection
shape
Natural frequency
MI - 2M SM Vibration Lecture 1 8

9. IMPERIAL COLLEGE OF SCIENCE, TECHNOLOGY & MEDICINE
MECHANICAL ENGINEERING DEPARTMENT
WAYS OF REPRESENTING A STRUCTURE
• SPATIAL DESCRIPTION
• Finite element model
• Mass-spring model
ω=82 Hz
• MODAL DESCRIPTION
• Use natural frequencies and mode shapes
• RESPONSE DESCRIPTION
• Use the actual behaviour of the structure
MI - 2M SM Vibration Lecture 1 9

10. IMPERIAL COLLEGE OF SCIENCE, TECHNOLOGY & MEDICINE
MECHANICAL ENGINEERING DEPARTMENT
VIBRATION MEASUREMENTS
Shaker + “sand” method Shaker + Laser method
MI - 2M SM Vibration Lecture 1 10

11. IMPERIAL COLLEGE OF SCIENCE, TECHNOLOGY & MEDICINE
MECHANICAL ENGINEERING DEPARTMENT
CONSEQUENCES OF VIBRATION
• Noise
• Discomfort
• Malfunction
• Shutdown
• Structural failure
MI - 2M SM Vibration Lecture 1 11

12. IMPERIAL COLLEGE OF SCIENCE, TECHNOLOGY & MEDICINE
MECHANICAL ENGINEERING DEPARTMENT
TOTAL STRUCTURAL FAILURE - I
MI - 2M SM Vibration Lecture 1 12

13. IMPERIAL COLLEGE OF SCIENCE, TECHNOLOGY & MEDICINE
MECHANICAL ENGINEERING DEPARTMENT
TOTAL STRUCTURAL FAILURE - II
Engine casing Turbine disk
MI - 2M SM Vibration Lecture 1 13

14. IMPERIAL COLLEGE OF SCIENCE, TECHNOLOGY & MEDICINE
MECHANICAL ENGINEERING DEPARTMENT
NOISE & TRANSMISSION OF VIBRATION
SHAKER
(FORCE INPUT)
CAR ENGINE
MOUNTS
CHASSIS POINT
(RESPONSE OUTPUT)
MI - 2M SM Vibration Lecture 1 14

15. IMPERIAL COLLEGE OF SCIENCE, TECHNOLOGY & MEDICINE
MECHANICAL ENGINEERING DEPARTMENT
FREQUENCY RESPONSE OF A HELICOPTER
THESE MODES OF VIBRATION MAY BE EXCITED BY
THE FORCES DUE TO ROTATION
MI - 2M SM Vibration Lecture 1 15

16. IMPERIAL COLLEGE OF SCIENCE, TECHNOLOGY & MEDICINE
MECHANICAL ENGINEERING DEPARTMENT
NOISE & VIBRATION TRANSMISSION FROM
HEAVY MACHINERY
MI - 2M SM Vibration Lecture 1 16

17. IMPERIAL COLLEGE OF SCIENCE, TECHNOLOGY & MEDICINE
MECHANICAL ENGINEERING DEPARTMENT
RAIL CORRUGATION
MI - 2M SM Vibration Lecture 1 17

18. IMPERIAL COLLEGE OF SCIENCE, TECHNOLOGY & MEDICINE
MECHANICAL ENGINEERING DEPARTMENT
BRIDGE FLUTTER
The unsteady
aerodynamic
forces caused by
the wind excited a
torsional mode of
the bridge,
causing total
structural failure.
MI - 2M SM Vibration Lecture 1 18

19. IMPERIAL COLLEGE OF SCIENCE, TECHNOLOGY & MEDICINE
MECHANICAL ENGINEERING DEPARTMENT
GAS TURBINE BLADE VIBRATION
MI - 2M SM Vibration Lecture 1 19

20. IMPERIAL COLLEGE OF SCIENCE, TECHNOLOGY & MEDICINE
MECHANICAL ENGINEERING DEPARTMENT
WHAT IS THE OBJECTIVE ?
• Find the critical frequencies
• Determine the excitation
• …………………….
• MINIMISE THE RESPONSE
Higher the response, more likely is noise,
malfunction, failure due to high cycle fatigue.
Σ FORCES
RESPONSE = [SYSTEM DESIGN]
∑ DAMPING
• Modify the design
• Change operating conditions
• Increase damping
MI - 2M SM Vibration Lecture 1 20