PRESENTATION OF NORMAL MODE

1. NORMAL MODE
PREPARED BY
ABHINAV KUMAR

2. CONTENT
INTRODUCTION
NORMAL COORDINATES AND NORMAL
MODES
SYMMETRIC AND ANTISYMMETRIC MODES
KINETIC AND POTENTIAL ENERGIES IN
NORMAL COORDINATES

3. INTRODUCTION
 A normal mode of an oscillating system is a pattern of motion in which all parts of the system move
sinusoidally with the same frequency and with a fixed phase relation. The free motion described by
the normal modes takes place at the fixed frequencies. These fixed frequencies of the normal modes
of a system are known as its natural frequencies or resonant frequencies. A physical object, such as a
building, bridge, or molecule, has a set of normal modes and their natural frequencies that depend
on its structure, materials and boundary conditions. When relating to music, normal modes of
vibrating instruments (strings, air pipes, drums, etc.) are called "harmonics" or "overtones".
 The most general motion of a system is a superposition of its normal modes. The modes are normal in
the sense that they can move independently, that is to say that an excitation of one mode will never
cause motion of a different mode. In mathematical terms, normal modes are orthogonal to each
other.
 In physics and engineering, for a dynamical system according to wave theory, a mode is a standing
wave state of excitation, in which all the components of the system will be affected sinusoidally
under a specified fixed frequency.
 Because no real system can perfectly fit under the standing wave framework, the mode concept is
taken as a general characterization of specific states of oscillation, thus treating the dynamic system
in a linear fashion, in which linear superposition of states can be performed.

4. NORMAL COORDINATES AND NORMAL MODES
A normal coordinate is a linear combination of Atomic cartesian displacement
coordinates that describe the coupled motion of all the items that comprise a
molecule. A normal mode is the coupled motion of all the Atoms described by
a normal coordinates
If A2 = 0
X1 = A1 cos (𝜔1 𝑡 + ∅1 )
X2 = A1 cos (𝜔1 𝑡 + ∅1)
If A1 = 0
X1 = A2 cos (𝜔2 𝑡 + ∅2 )
X2 = - A2 cos (𝜔2 𝑡 + ∅2)
If A2 = 0, the two masses oscillate together in phase with frequency 𝜔1 and If A1 = 0 the two masses oscillate with
frequency 𝜔2 opposite to each other i.e. out of phase by 𝜋 radian. The two such modes of oscillation involving a
single frequency are called normal mode of vibration of the system. Thus for a given normal mode all the
coordinates X1 and X2 oscillate with same frequency.

5. Equation of motion
mẍ1 + kx1 + k’(x1-x2) = 0
mẍ2 + kx2 - k’(x1-x2) = 0
Adding equation of motion
𝑚
𝑑2
𝑑𝑡2 𝑥1 + 𝑥2 + 𝑘 𝑥1 + 𝑥2 = 0
𝑑2
𝑑𝑡2
𝑥1 + 𝑥2 + 𝜔2
0 𝑥1 + 𝑥2 = 0
Subtracting equation of motion
𝑚
𝑑2
𝑑𝑡2 𝑥1 − 𝑥2 + (𝑘 + 2𝑘′
) 𝑥1 − 𝑥2 = 0
𝑑2
𝑑𝑡2 𝑥1 − 𝑥2 + (𝜔2
0 + 2𝜔𝑐
2) 𝑥1 − 𝑥2 = 0
Two new coordinates definition
𝑋1 =
𝑥1 + 𝑥2
2

6. 𝑋2 =
𝑥1 − 𝑥2
2
𝑑2
𝑋1
𝑑𝑡2 + 𝜔1
2
𝑋1 = 0
𝑑2
𝑋2
𝑑𝑡2
+ 𝜔2
2
𝑋2 = 0
We see that the motion of coupled system is now described by two uncoupled differential equation each of which
describe a simple harmonic motion of single frequency 𝜔1 and 𝜔2 in term of single coordinate ( X1 X2 )
X1 = A1 cos (𝜔1 𝑡 + ∅1 )
X2 = A2 cos (𝜔2 𝑡 + ∅2)
Type equation here.These two simple hormonic motion obtained after decoupling the coupled equation are
called normal mode or simple modes. Each modes of vibration has its normal frequency 𝜔1 and 𝜔2 and is
described by coordinate ( X1 X2 ) known as normal coordinate. The amplitude and phase constant mode 1 are A1
and ∅1 and mode 2, A2 ∅2 respectively
The general motion of oscillating system is expressed by the coordinate ( X1 X2 )
𝑥1=𝑋1+𝑋2 𝑎𝑛𝑑 𝑥2=𝑋1−𝑋2

7. 𝑥1 = A1 cos (𝜔1 𝑡 + ∅1 ) + A2 cos (𝜔2 𝑡 + ∅2)
𝑥2 = A1 cos (𝜔1 𝑡 + ∅1 ) − A2 cos (𝜔2 𝑡 + ∅2)
We find that the displacement of any mass is a linear combination or superposition of two mode ( X1 and X2 )
oscillating simultaneously.
When the system is oscillating in one mode or other with 𝐴1 = 0 𝑜𝑟 𝐴2 = 0. the displacement of each mass ( X1 X2 )
depend on both mode frequency 𝜔1 𝑎𝑛𝑑 𝜔2 hence the motion is no longer simple hormonic.
SYMMETRIC AND ANTISYMMETRIC MODES
If one mode is absent then only the other mode described motion. For X2 = 0, X1 coordinate is responsible for the
motion
𝑋2 =
𝑥1−𝑥2
2
= 0 or 𝑥1 = 𝑥2
𝑋1 =
𝑥1+𝑥2
2
= 𝑥1= 𝑥2= = A1 cos (𝜔1 𝑡 + ∅1 )

9. Thus in mode 1 both masses have equal displacement have the same frequency 𝜔1and keep in phase this is
called symmetric mode
For X1 = 0, X2 coordinate described the motion
𝑋1 =
𝑥1+𝑥2
2
= 0 or 𝑥1= - 𝑥2
𝑋2 =
𝑥1−𝑥2
2
= 𝑥1= - 𝑥2= A2 cos (𝜔2 𝑡 + ∅2)
In mode two both mass have equal and opposite displacement but oscillate with the same frequency 𝜔2 this is
called antisymmetric mode.
We observed that in symmetric mode, the two oscillate vibrate as if there were no coupling between them. In
antisymmetric mode the coupling is working and the oscillator are vibrating out of phase with a frequency higher
then the frequency of a single spring mass system
KINETIC AND POTENTIAL ENERGIES IN NORMAL
COORDINATES
Kinetic energy T =
1
2
𝑚𝑥1
2
+
1
2
𝑚𝑥2
2

10. =
1
2
𝑚 ẋ1 + ẋ2
2 +
1
2
𝑚 ẋ1 − ẋ2
2
=
1
2
𝑚ẋ1
2
+
1
2
𝑚ẋ2
2
Potential energy V =
1
2
k𝑥1
2
+
1
2
k 𝑥2
2
+
1
2
k’(𝑥1 − 𝑥2)2
=
1
2
𝑘 (𝑋1+𝑋2)² + (𝑋1−𝑋2)² +
1
2
k’ 2𝑋2 ²
= 𝑘 𝑋1
2
+ 𝑋2
2
+ 2𝑘′𝑥2
2
Lagrangian
L =
1
2
𝑚 ẋ1
2
+ ẋ2
2
− 𝑘 𝑋1
2
+ 𝑋2
2
− 2𝑘′𝑋2
2
When the kinetic and potential energies are expressed in term of normal coordinates no cross term of normal
coordinates are present i.e. both T and v are homogenous quadratic function.