§4.3 composition of shm if a point mass moves in several shm, its state of motion should be...

§4.3 Composition of SHM

• If a point mass moves in several SHM, its state of motion should be described by the addition of SHM.

• The displacement of the net vibration ( 总振动 ) can be obtained by the summation of every component ( 分量 ) of the vibrational reference vectors ( 振动参考矢量 ).

Vibrational reference vectors ( 振动参考矢量 )

t+

M

M0

AA

P P0

Fig. 4.2 the circle of reference of SHM.

x

y

4.3.1 The addition of the two vibrations with same direction and same frequency

1. The equation of compositive vibrations

• Now we are considering that a point mass moves in two SHM on a line and these two SHMs have the same vibratory direction and identical frequency.

• As the two vibrations can have different magnitudes and different initial phase, so they have individual reference circle of their own.

Suppose that at some moment, the displacements of the two vibrations respectively are

)cos(

)cos(

222

111

tAx

tAx

Since the x1 and x2 are on the same line (called x-axis), the total displacement should be the addition of x1 and x2, e. g.

(4.16)

t

x1x2

t

)2/cos(2

)2/cos(2

2

1

tx

tx

)sinsin(sin

)coscos(cos

)sinsincos(cos

)sinsincos(cos

)cos()cos(

2211

2211

222

111

2211

21

AAt

AAt

ttA

ttA

tAtA

xxx

As the two vibrations have the same frequency, the reference vectors A1 and A2 will rotate at the same angular velocity.

A1 and A2 will have the same angle between them (see Fig. 4.5) all the way through their rotating. The vector A will rotate at the same angular velocity. The x-component of A is

2211

2010

0

coscos

cos

AA

xx

Ax

A1A2

Fig. 4.5 Parallelogram rule ( 平行四 边形法则）

1

2

y10

y20

ox20

x10

2211

20100

sinsin

sin

AA

yyAy

A1

A2

A A1

Therefore,

)cos(

)sinsincos(cos

sinsincoscos

)sinsin(sin

)coscos(cos

2211

2211

tA

ttA

AtAt

AAt

AAtx

The above result shows that the compositive vibration of the two SHM is also a simple harmonic motion. It has the same angular frequency and a new initial phase factor and a new amplitude. The phase factor and new amplitude can be calculated as follows:

(4.17)

2. The amplitude and the initial phase factor of the compositive vibration

Using Pythagorean theorem （勾股定理） , the magnitude of the new vibration is given by

2121

22

21

22211

22211

20

20

cos2

sinsincoscos

AAAA

AAAA

yxA

(4.18)

The initial phase factor can be found as

2211

2211

0

0

coscos

sinsintan

AA

AA

x

y

These two parameters can be determined by the initial conditions only.

(4.19)

3. Discussions

• The compositive ( 合成的 ) vibration is not only a SHM but also its frequency is still the same as those of the component vibrations.

• The amplitude and the phase of the resultant vibration depend on the amplitudes and initial phases of the two vibrations.

• Two special cases

(1) 1 - 2 = 2k (k = 0, 1, 2, …)

In this case

21

2122

21

212122

21

2

cos2

AA

AAAA

AAAAA

This means that when the initial two vibrations are in phase, their reference vectors of SHM are at a line and in the same direction and the compositive amplitude is in its maximum status.

This means that when the two initial SHM are out of phase, the sum amplitude is minimum.

||

2

cos2

21

2122

21

212122

21

AA

AAAA

AAAAA

(2) 1 - 2 = (2k+1) (k = 0, 1, 2, …)

4.3.2 The composition of the two vibrations with the same direction and different frequency

• If the component vibrations have different frequency, the two rotating reference vectors A1 and A2 will have different angular velocity. So the angle between the two reference vectors will be a function of time, not only depending on the amplitudes and initial phases of A1 and A2 but also, the amplitude of the compositive vibration will change with time.

)cos(

)cos(

2222

1111

tAx

tAx

• Though the composite vibration is not SHM, it can still be periodic vibration as long as the ratio of the two frequencies is an integer or the inverse of the ratio is an integral fraction. This means that there exists a common basic frequency ( 基频 ) between the two frequencies. Any of them divided by the fundamental frequency will give a pure integer.

tt 3sin2.0sin

)cos(

)cos(

22

11

tAx

tAx

ttA

tAtA

xxx

2cos

2cos2

)cos()cos(

1212

21

21

Beat frequency:

)cos(

)cos(

22

11

tAx

tAx

9.02

1

4.3.3 Vibrational spectrum ( 振动谱 )

Opposite to the composition of the SHM, any complicated, periodic vibration can be expanded to a series of SHM. In other word, any complicated periodic function can be expressed by Fourier Series as:

tBBtAtAAts 2sintsin2coscos)( 21210

00

0 sincosn

nn

n tnBtnAA (4.20)

Where An and Bn are Fourier Constants. These constants can be determined mathematically. This procedure is called Spectral analysis ( 频谱分析 ).

Read the Chinese text book to get some general concept about the applications of spectral analysis.

Example: See Chinese text book on page 66. Squre wave can be expanded by Fourier series which is that

)5sin(

5

1)3sin(

3

1sin

4)( ttt

Utu

Plot it

First 10 terms

4.3.4 The composition ( 合成 ) of two vibrations with the same frequency but orthogonal ( 互相垂直的 ) directions.

Assume that an object moves in two SHM in a mutually perpendicular direction and these two SHMs have the same frequency.

Suppose that the two vibrations are along x-axis and y-axis respectively and then the vibrational equations can be written as

)cos(

)cos(

22

11

tAy

tAx(4.21)

In order to find the real path of the object in x-y plane, we have to delete t from the above equations of the simple harmonic motion and then the orbital equation of the object can be obtained.

)(sin)cos(2

122

1221

22

2

21

2

AA

xy

A

y

A

x(4.22)

Generally the above is an elliptic ( 椭圆的 ) equation. Let’s have a look at some special cases:

(1). If 2 - 1 = 2 k (k = 0, 1, 2, …), then we have

02

2122

2

21

2

AA

xy

A

y

A

x

This equation can also be written as

xA

Ay

A

y

A

x

A

y

A

x

1

2

21

2

21

00

This is a typical line equation which goes through the point (0,0) with slope of A2/A1.

(2). If 2 - 1 = (2 k+1) (k = 0, 1, 2, …), the equation (4.22) becomes

xA

Ay

A

y

A

x

1

2

2

21

0

This is also a line equation with slope of (-A2/A1).

(3). If 2 - 1 = (2 k+1) /2 (k = 0, 2, 4, …), the equation (4.22) becomes

122

2

21

2

A

y

A

x

When A1 = A2, it is a circle equation. In this case, from the equations of SHM, we know that the object moves in clockwise ( 顺时针 ) direction; back

If 2 - 1 = (2 k+1) /2 (k = 1, 3, 5, …), the equation (4.22) is the same as above and the end of vibrating vector will rotate anti-clockwise ( 逆时针 ).

Do you know how we can prove whether the rotation is clockwise or anti-clockwise? Here is an example.

)2

cos()cos(

cos)cos(

222

111

tAtAy

tAtAx

For simplicity, if we suppose = 1 radian s-1, the x and y can be found

Example: Suppose that the initial phase on x-axis is zero and the initial phase of y-axis vibration is /2. This is the case of

2 - 1 = /2 see backConsidering the simultaneous equations

as follows

x

y

t = 0 t = /3 t = /2

1

0

x = cos(t)y = cos(t+ /2)

1/2 0

-1 1

23

y

x＋ 1

＋ 2＋ 3

Summary to the lecture

222220

0

4

m

FA

220

0220

2,2

m

FAresonanceresonance

• forced vibration and resonance, characteristics of damped vibration (period, mechanical energy, frequency, amplitude)

22

0

2arctan

•Composition ( 合成 ) of SHM

(1) same frequency in the same direction,

(2) different frequencies in the same direction,

(3) same frequency in perpendicular direction