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10.7 The normal modes of a vibrating system

An eigenvector decomposition is not only useful for computing the inverse of a matrix or

other functions of a matrix, it also provides a way for analyzing characteristics of dynamical

systems. As an example, a simple model for the oscillations of a vibrating molecule is

shown here. This system is the prototype of a vibrating system that has di erent modes

of vibration. The natural modes of vibration are usually called the normal modes of that

system. Consider the mechanical system shown in gure (10.7). Three particles with

mass m are coupled by two springs with spring constants k. It is assumed that the three

masses are constrained to move along a line. The displacement of the masses from their

equilibrium positions are denoted with x1 , x2 and x3 respectively. This mechanical model

can considered to be a grossly oversimpli ed model of a tri-atomic molecule such as CO2

or H2 O.

m1 m2

m m

m

x1 x2 k3

x3

k1 k2

Figure 10.7: De nition of variables for a simple vibrating system.

Each of the masses can experience an external force Fi , where the subscript i denotes

10.7. THE NORMAL MODES OF A VIBRATING SYSTEM 121

the mass under consideration. The equations of motion for the three masses is given by:

mx1 = k(x2 x1 ) + F1

;

mx2 = 2 x1 ) + k(x3 x2 ) + F2 (10.76)

;k(x ; ;

mx3 = 3 x2 ) + F3

;k(x ;

For the moment we will consider harmonic oscillations, i.e. we assume that the both the

driving forces Fi and the displacements xi vary with time as exp The displacements

;i!t.

x1 , x2 and x3 can be used to form a vector x, and summarily a vector F can be formed

from the three forces F1 , F2 and F3 that act on the three masses.

Problem a: Show that for an harmonic motion with frequency ! the equations of motion

can be written in vector form as:

!

m!2 I x = 1 F

A; k (10.77)

k

with the matrix A given by

0 1

B1 0

C:

;1

A=@ A

2 (10.78)

;1 ;1

0 1

;1

The normal modes of the system are given by the patterns of oscillations of the system

when there is no driving force. For this reason, we set the driving force F in the right hand

side of (10.77) momentarily to zero. Equation (10.77) then reduces to a homogeneous sys-

tem of linear equations, such a system of equations can only have nonzero solutions when

the determinant of the matrix vanishes. Since the matrix A has only three eigenvalues,

the system can only oscillate freely at three discrete eigenfrequencies. The system can

only oscillate at other frequencies when it is driven by the force F at such a frequency.

Problem b: Show that the eigenfrequencies !i of the vibrating system are given by

s

!i = kmi (10.79)

where i are the eigenvalues of the matrix A.

Problem c: Show that the eigenfrequencies of the system are given by:

s s

k !3 = 3k :

!1 = 0 !2 = m (10.80)

m

Problem d: The frequencies do not give the vibrations of each of the three particles

respectively. Instead these frequencies give the eigenfrequencies of the three modes

of oscillation of the system. The eigenvector that corresponds to each eigenvalue

CHAPTER 10. LINEAR ALGEBRA

122

gives the displacement of each particle for that mode of oscillation. Show that these

eigenvectors are given by:

01 01 01

1 1 1

(1) = 1 B 1 C (2) = 1 B 0 C (3) = 1 B@C

v

^ v

^ v

^

@A @A A (10.81)

p p p

3 2 6

;2

1 1

;1

Remember that the eigenvectors can be multiplied with an arbitrary constant, this

constant is chosen in such a way that each eigenvector has length 1.

Problem e: Show that these eigenvectors satisfy the requirement (10.54).

Problem f: Sketch the motion of the three masses of each normal mode. Explain phys-

ically why the third mode with frequency !3 has a higher eigenfrequency than the

second mode !2 .

Problem g: Explain physically why the second mode has an eigenfrequency !2 = pk=m

that is identical to the frequency of a single mass m that is suspended by a spring

with spring constant k.

Problem h: What type of motion does the rst mode with eigenfrequency !1 describe?

Explain physically why this frequency is independent of the spring constant k and

the mass m.

Now we know the normal modes of the system, we consider the case where the system is

driven by a force F that varies in time as exp For simplicity it is assumed that the

;i!t.

frequency ! of the driving force di ers from the eigenfrequencies of the system: ! = !i .

6

The eigenvectors v(n) de ned in (10.81) form a complete orthonormal set, hence both the

^

driving force F and the displacement x can be expanded in this set. Using (10.59) the

driving force can be expanded as

X

3

F= v(n) (^(n) F) :

^v (10.82)

n=1

Problem i: Write theP

displacement vector as a superposition of the normal mode dis-

placements: x = 3 cn v(n) , use the expansion (10.82) for the driving force and

n=1 ^

insert these equations in the equation of motion (10.77) to solve for the unknown co-

e cients cn . Eliminate the eigenvalues with (10.79) and show that the displacement

is given by:

1 X v(n) (^(n) F) :

3^ v

x =m (10.83)

(!2 !2 )

n;

n=1

This expression has a nice physical interpretation. Expression (10.83) states that the total

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