BDA 31103 – VIBRATION

CHAPTER 2 –

SINGLE DEGREE

OF FREEDOM

SYSTEM

MR MOHD AMRAN BIN HJ. MADLAN

Faculty of Mechanical & Manufacturing Engineering

Universiti Tun Hussein Onn Malaysia

2

Free Vibration of Single Degree of

Freedom (SDOF)

� A system is said to undergo free vibration when it oscillates only under an initial disturbance with no external forces acting after initial disturbance

3

Introduction - SDOF

� One coordinate (x) is sufficient to specify the position of the mass at any time

� There is no external force applied to the mass

� Since there is no element that cause dissipation of energy during the motion of the mass, the amplitude of the motion remains constant with time, undamped system

4

Introduction - SDOF

� If the amplitude of the free vibration diminished gradually over time due to the resistance offered by the surrounding medium, the system are said to be damped

� Examples of free vibration: oscillations of the pendulum of grandfather clock, the vertical oscillatory motion felt by a bicyclist after hitting a road bump, and the swing of a child on a swing under an initial push.

5

6

7

Free Vibration of an Undamped

Translation System

� Equation of Motion using Newton’s Second Law

� Select a suitable coordinate to describe the position of the mass or rigid body

� Determine the static equilibrium configuration of the system and measure the displacement of the mass or rigid body

� Draw the free body diagram of the mass or rigid body when a positive displacement and velocity are given

� Apply Newton’s second law of motion

8

Free Vibration of an Undamped

Translation System

� Newton’s second law

� Applied to undamped SDOF system

A spring-mass system in horizontal position

xmtF &&rr

=)( θ&&rrJtM =)(

For rigid body undergoing

rotational motion

9

Free Vibration of an Undamped

Translation System

A spring-mass system in horizontal position

xmkxtF &&rr

=−=)(

0=+ kxxm &&

10

Free Vibration of an Undamped

Translation System

� Equation of Motion using other methods

� D’Alembert’s Principle

� Principle of Virtual Displacements

� Principle of Conservation of Energy

0=+ kxxm &&

� Spring-Mass System in Vertical Position

For static equilibrium

11

stkmgW δ==

Wxkxm st ++−= )( δ&&

0=+kxxm&&

� The solution can be found assuming,

substituting

characteristic equation eigenvalues12

stCetx =)(

0)()(2

2

=+ stst CekCedt

dm 0)( 2 =+ kmsC

02 =+ kmsni

m

ks ω±=

−±=2

12

1

=m

knω

� The general solution,

where C1 and C2 are constants

using

where A1 and A2 are new constants and can be determine

from the initial conditions13

titi nn eCeCtxωω −+= 21)(

tAtAtx nn ωω sincos)( 21 +=

tite ti ααω sincos ±=±

� The initial conditions at t = 0

14

02

01

)0(

)0(

xAtx

xAtx

n&& ===

===

ω

Hence, . Thus the solution

subject to the initial conditions is given by

nxAxA ω/ and 0201&==

tx

txtx n

n

n ωω

ω sincos)( 00

&+=

� Free vibration of an undamped: Harmonic

Motion

15

)sin()( 00 φω += tAtx n

where A0 and are new constants, amplitude

and phase angle respectively: 0φ

2/12

02

00

+==

n

xxAA

ω&

= −

0

01

0 tanx

x n

&

ωφ

amplitude

phase angle

The nature

of harmonic

oscillation

can be

represented

graphically

in the figure

16

Example 1:

Consider a small spring about 30 mm long,

welded to a stationary table (ground) so that

it is fixed at the point of contact, with 12 mm

bolt welded to the other end, which is free to

move. The mass of the system is 49.2 x 10^-

3 kg. The spring constant, k = 857.8 N/m.

Calculate the natural frequency and period

of system.

17

Example 1: Solution

18

srad

xm

kn 132

102.49

8.8573=== −ω

Natural frequency:

In hertz:

The period:

Hzf nn 21

2==

πω

sf

Tnn

0476.012===

ωπ

Example 2: Harmonic

Obtain the free response of

in the form

Initial condition are and

19

)(1282 tfxx =+&&

tAtAtx nn ωω sincos)( 21 +=

mx 05.0)0( =

smx /3.0)0( −=&

Example 2: Solution

20

tt

ttx

8sin0375.08cos05.0

8sin8

3.08cos05.0

−=

−+=