Forced Vibration and Resonance

Forced Vibration

•

Is the vibration in a system as a result of impulse received from another system

vibrating nearby

OR

•

Is type of vibration in which a force is repeatedly applied to a mechanical system

RESONANCE

•

Is the tendency of a system to oscillate at maximum amplitude at certain

frequencies from another system.

OR

•

Is a phenomenon that occurs when the frequency at which a force is periodically applied is

equal or near equal to one of the natural frequencies of the system on which it acts

OR

•

Is a large increase in amplitude of vibration in a body when it interacts with

another vibrating body

OR

•

Is the phenomena where by the response of the system that is set into forced vibration

when the driving frequency is equal to the natural frequency of the responding system.

NB:

A resonance is said to occur when a body or system is set into vibration or

oscillation at its own natural frequency as a result of impulses received from another

system which is vibrating at the same frequency.

OR

A Resonance is said to occur when the amplitude of an object’s oscillations are

increased by the matching vibrations of another object

Example of Resonance

1. A group of troupes was marching towards the bridge the bridge collapsed even

before it is approached.

2. If a very loud sound is produced near the mouth of the glass bottle, the

glass is likely to break.

3. The buildings are likely to collapse following the occurrences of the earth quake

4. Applied when turning the knob of a radio. This occurs when changing the natural

frequency of the receiver

,

it matches the transmission frequency of the radio station.

When the two frequencies match ,energy transfer occurs and we listen to the selected

channel

Resonance in a Closed Pipe

•

When a turning fork is sounded at the top of a tube with one end open and the

other closed, the air in the tube vibrate freely (resonates) at a certain length of

a tube. The resonance is observed as a loud sound produced in the tube when

the proper length obtained

Consider the figure below

+ 풄 = ^{흀.................................}(i)

풍_ퟏ

Considering the end correction,

•

ퟒ

For second harmonic or first overtone is produced when the length is

increased to 풍_ퟐ

+ 풄 = ^{ퟑ흀 ............................}(ii)

Considering the end correction, then 풍_ퟐ

•

ퟒ

Now, consider the two equations

흀

풍 + 풄 =

→ 풄 = − 풍

…

............. (iii)

ퟏ

ퟒ

ퟑ흀

→ 풄 = ^ퟑ흀− 풍

풍 + 풄 =

…..............(iv)

ퟐ

ퟒ

C

흀

ompare the two equations ((iii) and (iv))

= ^ퟑ흀− 풍

= ^ퟑ흀− ^흀

=

흀

− 풍

→ 풍 − 풍

ퟐ

→ 풍 − 풍

ퟐ ퟏ

ퟏ

ퟐ

ퟒ

ퟐ

(

)

∴ 풘풂풗풆풍풆풕풉, 흀 = ퟐ 풍_ퟐ− 풍_ퟏ

(

)

∴ 푽 = 흀풇 = ퟐ 풍_ퟐ− 풍_ퟏ풇

Whereby:

V

is the speed of sound in air column and

ƒ

is frequency of sound in air

Class Assignment

1.

A turning fork of frequency 512 Hz is sounded at the mouth of a tube closed at one

end with a movable piston. It is found that resonance occurs when the column of

air is 18cm long and again when the column is 51cm long. Find wave length and

velocity of sound in air

(ANS:

흀

= 0.66m and V_{A =}338m/s)

2. In a closed pipe, the first resonance is at 23cm and second at 73cm. determines the

wave length of the sound and the end correction of pipe (ANS: c = 0.002 m, 흀 = 1.0 m)

3. A resonance tube produces a loud sound for the first time when the length of the air

column is 17 cm and a loud sound at the second time when the length of the air column

is 51 cm .The turning fork frequency used is 500 Hz .Determine the speed of the air in

the tube (ANS: V = 340 m/s)

4. The first resonance in the tube of resonance occurs when the length of the air

column is 20 cm. What are the lengths of air column in the second resonance and

third resonance respectively (ANS: 60 cm and 100 cm respectively)

5. (a) Identify three characteristics of sound which distinguish one note from another.

Hence state the physical factors which correspondingly define the mentioned characteristics

(b) A resonance tube whose one end is closed and other open, resonance to a

note of frequency 560Hz when the length of the air column is 15cm. determine the

wave length of this sound in air. What is the shortest length of the air column

which resonates in similar conditions to a note of frequency 1000 Hz (ANS: a.

frequency, Loudness (amplitude) and Quality of music note (Timbre) .b L₂= 0.0504m

6. A turning fork of frequency 250Hz is used to produce resonance in an opened pipe.

Given that the velocity of sound in air is 350m/s. find the length of tube which gives

)

(a) First resonance

(b) Third resonance (ANS: L = 1.4m)

7.

The length of a closed pipe is 160mm. calculate the wavelength and the frequency

of (i) The first overtone (ii) The third harmonic

(

λ = 0.213, f₂

≈1500,Hz, f₃= 2500Hz)

8. A pipe closed at one end has a length of 100 cm. If the velocity of sound in air of

the pipe is 340m/s. Calculate the frequency of;

(a) The fundamental (f₀= 85 Hz)

(b) The first overtone ( f₁= 255 Hz)

Resonance in a closed Pipe (without end correction, c)

•

푣

휆

From: 푣 = 휆 푓

→ 푓 =

But

푙 = ^휆

→ 휆 = 4푙

4

풗

=

∴ 풇풖풏풅풂풎풆풏풕풂풍 풇풓풆풒풖풆풏풄풚 , 풇_ퟎ

ퟒ풍

For the 1^stovertone (2^nd

harmonic)

•

푣

휆

From: 푣 = 휆푓

→ 푓 =

4푙

푙 = ^3휆

→ 휆 =

But

4

3

풗

ퟒ풍

ퟑ풗

풗

=

= ퟑ ×

, 풃풖풕 풇

풇_ퟏ=

=

ퟎ

ퟒ풍

ퟑ

ퟒ풍

∴ 풕풉풆 풇풊풓풔풕 풐풗풆풓풕풐풏풆, 풇_ퟏ= ퟑ풇_ퟎ

For the 2^ndovertone (3^rdharmonic)

•

풗

흀

From: 풗 = 흀풇

→ 풇 =

ퟓ흀

ퟒ풍

But

풍 =

→ 흀 =

ퟒ

ퟓ

풗

ퟒ풍

ퟓ풗

풗

=

= ퟓ ×

, 풃풖풕 풇

풇_ퟐ=

=

ퟎ

ퟒ풍

ퟓ

∴ 풕풉풆 풔풆풄풐풏풅 풐풗풆풓풕풐풏풆, 풇_ퟐ= ퟓ풇_ퟎ

Generally the 풏^풕풉풐풗풆풓풕풐풏풆 풊풏 풂 풄풍풐풔풆풅 풑풊풑풆 풊풔 품풊풗풆풏 풃풚 ;

(

)

풇_풏= ퟐ풏 + ퟏ 풇_ퟎ… … … … … 푤ℎ푒푟푒 푛 = 1,2,3,4 푒푡푐

Since the resonance tube (closed at one end) produces odd harmonics then the

equation of the length of tube (length of air column) is given by

퐧훌

퐋 =

→ 퐧 = ퟏ, ퟑ, ퟓ, ퟕ … ..

ퟒ

Resonance in Opened Pipe

Consider the diagram below

Fundamental note (First resonance)

•

흀

ퟐ

From: 풗 = 흀풇

→ 풇 = ^풗,

But

풍 =

→ 흀 = ퟐ풍

흀

풗

=

∴ 푭풖풏풅풂풎풆풏풕풂풍 풇풓풆풒풖풆풏풄풚, 풇_ퟎ

ퟐ풍

For the first overtone (second harmonic/second resonance)

풗

From: 풗 = 흀풇

→ 풇 =

흀

But

풍 = 흀

푽

풗

풇 =

푽

푳

ퟐ

풗

× ^풗= ퟐ ×

= ퟐ풇 , 풔풊풏풄풆 풇

=

ퟎ

ퟏ

풍

ퟐ풍

흀

ퟐ풍

∴ 푭풊풓풔풕 풐풗풆풓풕풐풏풆, 풇_ퟏ= ퟐ풇_ퟎ

For the 2^ndovertone (

T

hird harmonic/third resonance)

•

풗

흀

From: 풗 = 흀풇

→ 풇 =

ퟑ흀

ퟐ풍

But

풍 =

→ 흀 =

ퟐ

ퟑ

풗

= ퟑ ×

풕풉풆풏, 풇_ퟐ

=

ퟐ풍

ퟑ

ퟐ풍

풗

=

∴ 풕풉풆 푺풆풄풐풏풅 풐풗풆풓풕풐풏풆, 풇_ퟐ= ퟑ풇_ퟎ, 풔풊풏풄풆 풇_ퟎ

ퟐ풍

푡ℎ

(

)

Generally the 푛 표푣푒푟푡표푛푒 표푓푎푛 표푝푒푛 푝푖푝푒 푖푠 푒푥푝푟푒푠푠푒푑 푎푠 풇_풏= 풏 + ퟏ 풇_ퟎ

Whereby (n = 1,2,3,4.............

)

•

Since the resonance tube(opened at both ends) produces integral multiples of

harmonics, then the equation of the length of tube (length of air column) at 풏^풕풉

harmonic is given by

풏흀

푳 =

→ 풏 = ퟏ, ퟐ, ퟑ, ퟒ, … ..

ퟐ

Example

1. A turning fork of frequency 256 Hz is sounded at the mouth of a tube closed at

one end with a movable. It is found that resonance occurs when the column of

air is 15 cm long and again when the column is 80 cm long. Determine the

velocity of sound in air.

Soln:

Given: L₁= 15 cm, L₂= 80 cm, f = 256Hz, V = ?

(

)

From: 푽 = 흀풇 = ퟐ 풍_ퟐ− 풍_ퟏ풇

(

)

(

)

∴ 푽 = 흀풇 = ퟐ 풍_ퟐ− 풍_ퟏ풇 = ퟐퟓퟔ 풙ퟐ ퟎ. ퟖퟎ − ퟎ. ퟏퟓ = ퟓퟏퟐ 풙 ퟎ. ퟔퟓ = ퟑퟑퟐ. ퟖ 풎/풔

Class Activity

1. The speed of sound waves in air is found to be 340m/s. Find;

(a) The fundamental frequency

(b) The frequency of the 3^rdharmonic

(c) The frequency of 9^thharmonic (d) The frequency of 51^stharmonic

Given that the sound waves are probating in a closed pipe of length 700m.

ANS: f₀

=

121.5Hz, f₃= 850.5 Hz, f₉= 2308.5 Hz, f₅₁= 12514.5 Hz

2. In a closed pipe, the first resonance is at 23 cm and second at 73 cm. Determine the

wavelength of the sound and the end correction of the pipe. (ANS: 흀 = ퟏ 풎, 풄 = ퟐ 풄풎)

3. A pipe closed at one end has a length of 10cm. If the velocity of sound in the

air of the pipe is 340m/s.Calculate the frequency of;

(a) The fundamental

(b) 1^stovertone

(ANS: f₀= 850 Hz, f₁= 2550 Hz)

4.

A pipe closed at one and has a length of 2.46m. Find the frequency of the

fundamental and the first two overtones. Take 343m/s as the speed of sound in

air.

(

ANS: f₀= 34.85Hz, f1 = 104.55 Hz, f2 =174.25 Hz)

5. When a tuning fork of 512Hz is sounded at the top of the measuring cylinder

which contains water. The first resonances are observed when the length of

the air column (the distance from the mouth to the level of the water is 50 cm)

and the second resonance is observed when the length of the air column (the

distance from the mouth to the level of water) is 80 cm; using these

observations. Calculate the velocity of water in air.(ANS: v = 307 .2m/s)

Beats

•

A beat is a rise or fall in loudness of sound when two sources of sound of

nearly equal frequencies produce sound together.

•

The Beat frequency (number of beats):

Is the difference between the two frequencies of sound

That is B_f= f₁– f₂or f₂– f₁

Example

1. A 256Hz turning fork produces sound at the same time with a 249Hz turning

fork. What is the beat frequency? (ANS: Bf = 7Hz)

What is the beat frequency when a 262 Hz and 266 Hz turning forks are

sounded together? (ANS: B_F= 4 Hz)

2.