1

MECHANICAL VIBRATION

MME4425/MME9510

Prof. Paul Kurowski

2

TEXT BOOKS

REQUIRED RECOMMENDED

3

MME4425b web sitehttp://www.eng.uwo.ca/MME4425b/2012/

Design Center web sitehttp://www.eng.uwo.ca/designcentre/

4

Software used:

SolidWorksDesign 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 www.solidworks.com/SEK

Use SEK-ID = XSEK12

Select release 2012-2013

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

RIGID BODY MOTION

How many rigid body motions?

11

Discrete system

Mass, stiffness and damping are separated

Distributed system

Mass, stiffness and damping are NOT separated

DISCRETE SYSTEM VS. DISTRIBUTED SYSTEM

12

DISCRETE SYSTEM VS. DISTRIBUTED SYSTEM

1DOF.SLDASM 2DOF.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

Homogenous equation

SINGLE DEGREE OF FREEDOM SYSTEM, LINEAR VIBRATIONS

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

18Where A and Ф are found from initial conditions

FINDING GENERAL SOLUTION OF HOMOGENEOUS EQUATION

By guessing solution

ωn – natural angular frequency

found from system properties

19

FINDING GENERAL SOLUTION OF HOMOGENEOUS EQUATION

Using complex numbers method

20

and

Since is linear, then the sum of two solutions is also a solution

We have found two solutions to equation

Using Euler’s relations:

The equation can be re-written as:

Where A and Ф are found from initial conditions

FINDING GENERAL SOLUTION OF HOMOGENEOUS EQUATION

Using complex numbers method

21

FINDING GENERAL SOLUTION OF HOMOGENEOUS EQUATION

Using Laplace transformation

Taking Laplace transform of both sides

Using (5), (6)

22

2

[ ( )] ( ) (0)

[ ( )] ( ) (0) (0)

L x t sX s x

L x t s X s sx x

Laplace transformation

23Inman p 619

Laplace transformation

24

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.

QUANTITIES CHARACTERIZING VIBRATION

25

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

QUANTITIES CHARACTERIZING VIBRATION

26

QUANTITIES CHARACTERIZING VIBRATION

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 displacementL

ine

s o

f co

nst

an

t a

cce

lera

tion

Nomogram for specifying acceptable limits of sinusoidal vibration (Inman p 18)

27

LINEAR SDOF

28

Linear spring400000N/m

10kg mass

Base

SDOF.SLDASM

LINEAR SDOF

29Results of modal analysis

LINEAR SDOF

30

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.

34pendulum 02.SLDPRT

PENDULUM SDOF

35

2 sin 0

0

1

2

2

ml mgl

l g

g

l

gf

l

lT

g

Equations of motion method

PENDULUM SDOF

36

37

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

Energy method

2

2

2

.

( ) 0

1( )

2( cos )

1( ( ) ( cos )) 02

sin 0

sin 0

0

T U const

dT U

dt

T m l

U mg l l

dm l mg l l

dt

ml mgl

l g

l g

g

l

PENDULUM SDOF

40

TORSIONAL SDOF

41disk 01.SLDPRT

TORSIONAL SDOF

4

2

2(1 )

torsion

T JG

L

J r

EG

JGk

L

0torsionI k

polar moment of inertia of cross-section

42

TORSIONAL SDOF

21

2J mr

43Inman p 32

ROLER SDOF

roler.SLDASM

44

ROLER SDOF

45

2/

k

m J r

k = k1 + k2 = 2000N/m

m = 75.4kg

r = 0.1m

J = 0.3770kgm2

2

20004.2 /

75.4 0.38 / 0.10.66

rad s

f Hz

ROLER SDOF

Inman p 32

46

ROLER SDOF

47

rotation.SLDASM

MASS AT THE END OF BEAM

48

MASS AT THE END OF BEAM

cantilever.SLDPRT

mass 2.7kg

3

3EI

kl

49

RING

Ring.SLDASM

50

HOMEWORK 1

1. Derive equation of motion of SDOF using energy method

2. Find amplitude A and tanΦ for given x0, v0

3. Find natural frequency of cantilever, l=400mm, Φ=5mm, E=2e11Pa, m=2.7kg. Confirm with SW Simulation

4. Work with exercises in chapter 19 – blue book

51

Model file trifilar.sldasm

Configuration trifilar

Model type solid

Material as shown

Supports as shown

Objectives

Find the natural frequency of trilifar

TORSONAL SDOF TRIFILAR

1060 alloy

Custom material

E = 10MPaρ = 1kg/m3

very soft, very low density

1060 alloy

Fixed support

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

0.845 M kg mass of platform0.1 R m radius of platform

0.004225 J kgm^2 mass moment of inertia of platform

0.1 R m radius of attachment of wires0.5 L m length of wires

9.81 g m/s 2̂ gravitatonal acelleration

6.26 rad/s natural frequency1.00 Hz natural frequency

56Trifilar can be used to find moments of inertia of objects placed on rotating platform

2

2P

MASS P

M gRJ J

L

TORSONAL SDOF TRIFILAR

57

2

2

2

P

P

PP

M gR

J L

M gRJ

L

spur gear.SLDPRT