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HELICOPTER PERFORMANCE STABILITY AND CONTROL COURSE CODE: AE4314
Name: DEEPAK PAUL TIRKEY
ASSIGNMENT NUMBER 04 Student number: 4590929
Question 1. Flapping blade angle variation for the advancing blade for one complete revolution Answer : My chosen helicopter is MBB Bo105
Locks inertia number ๐พ =๐๐๐๐ 4
๐ผ๐ฝ = 5.07
where ฯ = 1.225 ๐๐
๐3 [sea level condition] ; a = lift curve slope = 6.113 /rad ; c = chord = 0.27 m ;
R = radius = 4.91 m ; ๐ผ๐ฝ= 231.7 ๐๐
๐2 [ data source : Helicopter Flight Dynamics โPadfield]
Induced Velocity in hover ๐๐ = โ๐
2๐๐๐ 2 = 10.8 m/sec ; where W = 2200 x 9.81 = 21582 N
Induced inflow ratio ๐๐ = ๐๐
ฮฉ๐ =
10.8
218 = 0.05
Flapping solution in hover ฮฒ (t) = ๐ท๐๐๐(t) + ๐ท๐๐๐๐(t)
Particular solution ๐ฝ๐๐๐๐ก(t) = ๐พ
8( ฮธ -
4
3๐๐ ) =
5.07
8(8 -
4
30.05) = 5.03 degree; where ฮธ = 8 degree(given)
[๐ฝ๐๐๐๐ก(t) = 5.03 degree is constant coning angle ]
Homogenous solution ๐ฝโ๐๐(t) = ๐ฝ0๐โ๐พฮฉ๐ก
16 [cos (ฮฉโ1 โ (๐พ
16)
2
. ๐ก) +๐พ
16
โ1โ(๐พ
16)
2 sin (ฮฉโ1 โ (
๐พ
16)
2
. ๐ก)]
MS Excel was used to calculate and plot ฮฒ (t) = ๐ฝโ๐๐(t) + ๐ฝ๐๐๐๐ก(t) ,
were azimuth angle ฯ =ฮฉ t , ๐ฝ0 = ๐ฝ๐๐๐๐ก ๐ค๐๐ ๐โ๐๐ ๐๐
Fig. 01 Flapping blade angle ฮฒ (ฯ) = ฮฒhom(ฯ) + ฮฒpart(ฯ) variation for one revolution
0
2
4
6
8
10
12
0 5 10 15 20 25 30
be
ta (
de
gre
e)
PSI ( x 15 degree)
Beta_hom +Beta_parti
coning angle(Beta_parti)
HELICOPTER PERFORMANCE STABILITY AND CONTROL COURSE CODE: AE4314
Name: DEEPAK PAUL TIRKEY
ASSIGNMENT NUMBER 04 Student number: 4590929
Question 2. Angle of attack variation for the advancing blade for one complete revolution at different radii positions on the blade varying from 0 to R Answer : Angle of attack of blade element located at a distance r from the rotation axis
๐ถ๐ = ฮธ - ๐๐+๐ท
ฮฉ๐
Now to calculate from the relation ๐ฝ (๐ก ) = ๐ฝโ๐๐ (t) + ๐ฝ๐๐๐๐ก
(t) = ๐ฝโ๐๐ (t) [as ๐ฝ๐๐๐๐ก(t) is a constant term]
Writing ๐ฝ (๐ก ) = ๐ฝ0[A+B]
Were A = (โ๐พฮฉ
16) ๐
โ๐พฮฉ๐ก
16 [cos (ฮฉโ1 โ (๐พ
16)
2
. ๐ก) +๐พ
16
โ1โ(๐พ
16)
2sin (ฮฉโ1 โ (
๐พ
16)
2
. ๐ก)]
And B = ๐โ๐พฮฉ๐ก
16 [โ (ฮฉโ1 โ (๐พ
16)
2
)sin (ฮฉโ1 โ (๐พ
16)
2
. ๐ก) +๐พฮฉ
16cos (ฮฉโ1 โ (
๐พ
16)
2
. ๐ก)]
Simplifying [A+B] , cosine terms from both A and B cancel each other
[A+B] = (โ)๐โ๐พฮฉ๐ก
16 sin (ฮฉโ1 โ (๐พ
16)
2
. ๐ก) [(
๐พ
16)
2ฮฉ
โ1โ(๐พ
16)
2+ ฮฉโ1 โ (
๐พ
16)
2
]
Simplifying the [ ] term further
[A+B] = (โ)๐โ๐พฮฉ๐ก
16 sin (ฮฉโ1 โ (๐พ
16)
2
. ๐ก) [ฮฉ
โ1โ(๐พ
16)
2]
or ๐ (๐ญ ) = (โ)๐๐ฮฉ ๐
โ๐ฮฉ๐ญ๐๐
โ๐โ(๐
๐๐)
๐๐ฌ๐ข๐ง (ฮฉโ๐ โ (
๐
๐๐)
๐. ๐ญ)
using the ฮฒ (t ) value , ๐ผ๐ at each radial location was calculated using MS EXCEL and are plotted below.
Fig. 0.2 Angle of attack variation with radial and azimuth
5
5.5
6
6.5
7
7.5
8
0 10 20 30 40 50
An
gle
of
atta
ck (
de
gre
e)
PSI (x 15)
r 0.1 r 0.2
r 0.3 r 0.4
r 0.5 r 0.6
r 0.7 r 0.8
r 0.9 r 1
HELICOPTER PERFORMANCE STABILITY AND CONTROL COURSE CODE: AE4314
Name: DEEPAK PAUL TIRKEY
ASSIGNMENT NUMBER 04 Student number: 4590929
Question 3. Coning angle , longitudinal and lateral disc tilt angles. Answer : Coning angle a0 = ฮฒpart(t) = 5.03 degree
Longitudinal disc tilt angle a1 = 0 degree Lateral disc tilt angle b1 = 0 degree
Assume that the helicopter is fixed to the ground but has a body pitch rate q and roll rate p. Question 4. Advancing blade angular velocities
Blade angular velocities: i ฮฉ sinฮฒ + q sinฯ cosฮฒ โ p cosฯ cosฮฒ
j q cosฯ + p sinฯ - k ฮฉ cosฮฒ + p cosฯ sinฮฒ โ q sinฯ sinฮฒ
Question 5. Flapping equation when the helicopter is having body rates p and q and is fixed to the ground Kinetic Energy:
T = 1
2๐ผ [(๐ ๐๐๐ ๐ + ๐ ๐ ๐๐๐ โ )
2+ (ฮฉ ๐๐๐ ๐ฝ + ๐ ๐๐๐ ๐ ๐ ๐๐๐ฝ โ ๐ ๐ ๐๐๐ ๐ ๐๐๐ฝ)2] eq. 01
[Note: neglecting the i axis contribution of blade angular velocities towards Kinetic Energy calculation]
Writing the above eq. 01 for simplicity T = 1
2๐ผ[๐ด + ๐ต]
Lagrange equation of motion:
๐
๐๐ก(
๐๐
๐) โ
๐๐
๐๐ฝ+
๐๐
๐๐ฝ= ๐๐ฝ were ๐๐ฝ is the aerodynamic moment about the flapping hinge ๐๐(๐ก)
Evaluating the first term Lagrange EoM
Now ๐๐
๐ =
1
2๐ผ [2-2(๐ ๐๐๐ ฮฉ๐ก + ๐ ๐ ๐๐ฮฉ๐ก)] and
๐
๐๐ก(
๐๐
๐) = ๐ผ[ + ๐ฮฉ ๐ ๐๐๐ โ ๐ฮฉ ๐๐๐ ๐] with ฯ = ฮฉt
Now expand the B term of eq. 01 B = [(ฮฉ ๐๐๐ ๐ฝ)2 + (๐ ๐๐๐ ๐ โ ๐ ๐ ๐๐๐ )2๐ ๐๐2๐ฝ + 2ฮฉ ๐๐๐ ๐ฝ๐ ๐๐๐ฝ(๐ ๐๐๐ ๐ โ ๐ ๐ ๐๐๐ )] = [(ฮฉ ๐๐๐ ๐ฝ)2 + (๐ ๐๐๐ ๐ โ ๐ ๐ ๐๐๐ )2๐ ๐๐2๐ฝ + ฮฉ ๐ ๐๐2๐ฝ(๐ ๐๐๐ ๐ โ ๐ ๐ ๐๐๐ )] = [(ฮฉ ๐๐๐ ๐ฝ)2 + 0 + ฮฉ ๐ ๐๐2๐ฝ(๐ ๐๐๐ ๐ โ ๐ ๐ ๐๐๐ )] Middle term became 0 as ๐ฝ is small and neglecting ๐2, ๐2, ๐. ๐ terms Now evaluating the second term of Lagrange EoM
๐๐
๐๐ฝ=
1
2๐ผ[โ2ฮฉ2 ๐๐๐ ๐ฝ๐ ๐๐๐ฝ + 2ฮฉ๐๐๐ 2๐ฝ(๐ ๐๐๐ ๐ โ ๐ ๐ ๐๐๐ ) ]
= ๐ผ[โฮฉ2๐ฝ + ฮฉ(๐ ๐๐๐ ๐ โ ๐ ๐ ๐๐๐ ) ] assuming ๐ฝ is small Third term of the Lagrange EoM is zero
HELICOPTER PERFORMANCE STABILITY AND CONTROL COURSE CODE: AE4314
Name: DEEPAK PAUL TIRKEY
ASSIGNMENT NUMBER 04 Student number: 4590929
Now evaluating the complete Lagrange EoM:
๐ผ + ๐ผ๐ฮฉ ๐ ๐๐๐ โ ๐ผ๐ฮฉ ๐๐๐ ๐ + ๐ผฮฉ2๐ฝ โ ๐ผฮฉ๐ ๐๐๐ ๐ + ๐ผฮฉ๐ ๐ ๐๐๐ = ๐๐(๐ก) Retaining the ๐ฝ terms on LHS and dividing throughout by ๐ผ
+ ฮฉ๐๐ท = โ๐ฮฉ๐ ๐๐๐๐ + ๐ฮฉ๐ ๐๐๐๐ + ๐ด๐(๐)
๐ฐ eq. 02
Question 6. Disc tilt angles a0 , a1 and b1 Angle of attack of a blade element
๐ผ = ๐ โ๐๐+๐ฝ ๐โ๐ ๐ ๐๐๐ ๐โ๐๐๐ ๐๐๐
ฮฉ๐
Evaluate aerodynamic moment about flapping hinge:
๐ด๐
๐ฐ=
1
๐ผโซ ๐ (๐ โ
๐๐+๐ฝ ๐โ๐ ๐ ๐๐๐ ๐โ๐๐๐ ๐๐๐
ฮฉ๐)
1
2๐(ฮฉ๐)2๐๐ ๐๐
๐
0
=๐๐๐
๐ผ.
ฮฉ2
2โซ (๐0 โ
๐๐
ฮฉ๐ ๐ฅโ
๐ฝ
ฮฉ+
๐ ๐๐๐ ๐
ฮฉ+
๐๐ ๐๐๐
ฮฉ) ๐ฅ3๐ 4 ๐๐ฅ
1
0 taking ๐ฅ =
๐
๐ ; ๐ = ๐ฅ๐ ; ๐๐ = ๐ ๐๐ฅ
=๐๐๐๐ 4
๐ผ.
ฮฉ2
2โซ (๐0๐ฅ3 โ ๐๐๐ฅ
2 โ๐ฝ
ฮฉ๐ฅ3 +
๐ ๐๐๐ ๐
ฮฉ๐ฅ3 +
๐๐ ๐๐๐
ฮฉ๐ฅ3) ๐๐ฅ
1
0
= ๐พ.ฮฉ2
2[
๐0
4โ
๐๐
3โ
๐ฝ
4ฮฉ+
๐ ๐๐๐ ๐
4ฮฉ+
๐๐ ๐๐๐
4ฮฉ]
๐ด๐
๐ฐ=
๐พฮฉ2
8(๐0 โ
4๐๐
3) โ
๐พ๐ฝ ฮฉ
8+
๐พ ๐ฮฉ ๐๐๐ ๐
8+
๐พ ๐ฮฉ ๐ ๐๐๐
8 eq. 03
Now rewriting eq. 02 using eq. 03
+ ฮฉ2๐ฝ = โ2ฮฉ๐ ๐ ๐๐๐ + 2ฮฉ๐ ๐๐๐ ๐ +๐พฮฉ2
8(๐0 โ
4๐๐
3) โ
๐พ๐ฝ ฮฉ
8+
๐พ ๐ฮฉ ๐๐๐ ๐
8+
๐พ ๐ฮฉ ๐ ๐๐๐
8
+๐พ๐ฝ ฮฉ
8+ ฮฉ2๐ฝ =
๐พฮฉ2
8(๐0 โ
4๐๐
3) +
๐พ ๐ฮฉ ๐๐๐ ๐
8+
๐พ ๐ฮฉ ๐ ๐๐๐
8 โ 2ฮฉ๐ ๐ ๐๐๐ + 2ฮฉ๐ ๐๐๐ ๐
+๐ธฮฉ
๐ + ฮฉ๐๐ท =
๐ธฮฉ๐
๐(๐ฝ๐ โ
๐๐๐
๐) + ฮฉ (
๐ธ ๐
๐+ ๐๐) ๐๐๐ฮฉ๐ + ฮฉ (
๐ธ ๐
๐ โ ๐๐ ) ๐๐๐ฮฉ๐ eq. 04 (using ๐ = ฮฉ๐ก)
Assuming solution for the above eq. 04 of the form ๐ฝ = ๐0 โ ๐1 ๐๐๐ ฮฉ๐ก โ ๐1 ๐ ๐๐ฮฉ๐ก
= ๐1ฮฉ ๐ ๐๐ฮฉ๐ก โ ๐1ฮฉ ๐๐๐ ฮฉ๐ก
= ๐1ฮฉ2 ๐๐๐ ฮฉ๐ก + ๐1ฮฉ2 ๐ ๐๐ฮฉ๐ก
Now substituting the values of ๐ฝ, , on the LHS of eq. 04 LHS of eq.04
โก ๐1ฮฉ2 ๐๐๐ ฮฉ๐ก + ๐1ฮฉ2 ๐ ๐๐ฮฉ๐ก +๐ธ
๐๐1ฮฉ2 ๐ ๐๐ฮฉ๐ก โ
๐ธ
๐๐1ฮฉ2 ๐๐๐ ฮฉ๐ก + ฮฉ2๐0 โ ๐1ฮฉ2 ๐๐๐ ฮฉ๐ก โ ๐1ฮฉ2 ๐ ๐๐ฮฉ๐ก
โก ฮฉ2๐0 + (๐1ฮฉ2 โ๐ธ
๐๐1ฮฉ2 โ ๐1ฮฉ2) ๐๐๐ ฮฉ๐ก + (๐1ฮฉ2+
๐ธ
๐๐1ฮฉ2 โ ๐1ฮฉ2) ๐ ๐๐ฮฉ๐ก
โก ฮฉ2๐0 โ๐ธ
๐๐1ฮฉ2 ๐๐๐ ฮฉ๐ก +
๐ธ
๐๐1ฮฉ2 ๐ ๐๐ฮฉ๐ก
Equating the above expression with the RHS of eq. 04 and comparing constant, sine and cosine terms Constant term
ฮฉ2๐0 =๐พฮฉ2
8(๐0 โ
4๐๐
3) or ๐๐ =
๐ธ
๐(๐ฝ๐ โ
๐๐๐
๐)
HELICOPTER PERFORMANCE STABILITY AND CONTROL COURSE CODE: AE4314
Name: DEEPAK PAUL TIRKEY
ASSIGNMENT NUMBER 04 Student number: 4590929
Sine term
๐พ
8๐1ฮฉ2 = ฮฉ (
๐พ ๐
8 โ 2๐ ) or ๐๐ = (
๐
ฮฉ โ
๐๐๐
๐ธฮฉ )
Cosine term
โ๐พ
8๐1ฮฉ2 = ฮฉ (
๐พ ๐
8+ 2๐) or ๐๐ = (
โ ๐
ฮฉโ
๐๐๐
๐ธฮฉ)
Assume that the helicopter is in forward flight and has body rate q Question 7. Perform analytically the integral of blade aerodynamic moment and compare with given expression Aerodynamic moment is given as:
๐๐ = โซ ๐ถ๐๐ผ ๐ผ1
2๐(ฮฉ๐ + ๐๐๐๐ ๐ผ๐ ๐ ๐๐๐)2๐
0๐ ๐๐. ๐ Where ๐ผ = ๐ โ
๐ ๐ ๐๐๐ผ๐+๐๐+๐โ๐๐ ๐๐๐ ๐+๐ฝ๐ ๐๐๐ ๐ผ๐ ๐๐๐ ๐
ฮฉ๐+๐ ๐๐๐ ๐ผ๐ ๐ ๐๐๐
Performing integration with the ๐ฝ part
๐๐1 = ๐๐ถ๐๐ผ ๐1
2 โซ ๐(ฮฉ๐ + ๐๐๐๐ ๐ผ๐ ๐ ๐๐๐)2๐
0 ๐๐. ๐
= ๐๐ถ๐๐ผ ๐1
2 ๐ โซ (ฮฉ2๐3 + ๐2๐ cos ๐ผ๐
2 ๐ ๐๐๐2 + 2ฮฉ๐ ๐2๐๐๐ ๐ผ๐ ๐ ๐๐๐)๐
0 ๐๐
= ๐๐ถ๐๐ผ ๐1
2 ๐ [
ฮฉ2๐ 4
4+
ฮฉ2๐ 4
2 (
๐๐๐๐ ๐ผ๐
ฮฉ๐ )
2
๐ ๐๐๐2 +2 ฮฉ2๐ 4
3(
๐๐๐๐ ๐ผ๐
ฮฉ๐ ) ๐ ๐๐๐] Where
๐๐๐๐ ๐ผ๐
ฮฉ๐ = ๐
= ๐๐ช๐๐ถ ๐๐น๐ฮฉ๐ ๐
๐ ๐ฝ [
๐
๐+
๐
๐๐๐ ๐๐๐๐๐ +
๐
๐๐ ๐๐๐๐]
Performing integration with ๐ฝ ๐๐๐๐ถ๐+๐๐+๐โ๐๐ ๐๐๐๐+๐ท๐ฝ ๐๐๐๐ถ๐ ๐๐๐๐
ฮฉ๐+๐ฝ ๐๐๐๐ถ๐ ๐๐๐๐ part
๐๐2 = ๐๐ถ๐๐ผ ๐1
2 โซ (๐ ๐ ๐๐๐ผ๐ + ๐๐ + ๐ โ ๐๐ ๐๐๐ ๐ + ๐ฝ๐ ๐๐๐ ๐ผ๐ ๐๐๐ ๐)(ฮฉ๐ + ๐๐๐๐ ๐ผ๐ ๐ ๐๐๐)
๐
0๐๐. ๐
Now integrating each term:
๐๐3 = ๐๐ถ๐๐ผ ๐1
2 โซ (๐ ๐ ๐๐๐ผ๐)(ฮฉ๐ + ๐๐๐๐ ๐ผ๐ ๐ ๐๐๐)
๐
0๐๐. ๐ = ๐๐ช๐๐ถ ๐๐น๐ฮฉ๐ ๐
๐ [
๐๐
๐+
๐๐ ๐
๐๐๐๐๐]
๐๐4 = ๐๐ถ๐๐ผ ๐1
2 โซ (๐๐)(ฮฉ๐ + ๐๐๐๐ ๐ผ๐ ๐ ๐๐๐)
๐
0๐๐. ๐ = ๐๐ช๐๐ถ ๐๐น๐ฮฉ๐ ๐
๐ [
๐๐
๐+
๐๐ ๐
๐๐๐๐๐]
๐๐5 = ๐๐ถ๐๐ผ ๐1
2 โซ (๐ฝ)(ฮฉ๐ + ๐๐๐๐ ๐ผ๐ ๐ ๐๐๐)
๐
0๐๐. ๐= ๐๐ช๐๐ถ ๐๐น๐ฮฉ๐ ๐
๐ [
๐ฮฉ+
๐
๐ฮฉ๐๐๐๐]
๐๐6 = ๐๐ถ๐๐ผ ๐1
2 โซ (โ๐๐ ๐๐๐ ๐)(ฮฉ๐ + ๐๐๐๐ ๐ผ๐ ๐ ๐๐๐)
๐
0๐๐. ๐= ๐๐ช๐๐ถ ๐๐น๐ฮฉ๐ ๐
๐ [
โ๐
๐ฮฉ๐๐๐ ๐ +
โ๐ ๐
๐ฮฉ๐๐๐๐๐๐๐ ๐]
๐๐7 = ๐๐ถ๐๐ผ ๐1
2 โซ (๐ฝ๐ ๐๐๐ ๐ผ๐ ๐๐๐ ๐)(ฮฉ๐ + ๐๐๐๐ ๐ผ๐ ๐ ๐๐๐)
๐
0๐๐. ๐= ๐๐ช๐๐ถ ๐๐น๐ฮฉ๐ ๐
๐ [
๐ท๐
๐๐๐๐ ๐ +
๐ท๐๐
๐๐๐๐๐๐๐๐ ๐]
Now ๐๐/๐ผ๐๐=๐๐1- (๐๐3 + ๐๐4 + ๐๐5 + ๐๐6 + ๐๐7) /๐ผ๐๐
= ๐พฮฉ2 1
2 ๐ [
1
4+
1
2๐2 ๐ ๐๐๐2 +
2
3๐ ๐ ๐๐๐] โ๐พฮฉ2 1
2 [
๐๐
3+
๐๐ ๐
2๐ ๐๐๐] โ ๐พฮฉ2 1
2 [
๐๐
3+
๐๐ ๐
2๐ ๐๐๐]
โ๐พฮฉ2 1
2 [
4ฮฉ+
๐
3ฮฉ๐ ๐๐๐] +๐พฮฉ2 1
2 [
๐
4ฮฉ๐๐๐ ๐ +
๐ ๐
3ฮฉ๐ ๐๐๐๐๐๐ ๐] โ๐พฮฉ2 1
2 [
๐ฝ๐
3๐๐๐ ๐ +
๐ฝ๐2
2๐ ๐๐๐๐๐๐ ๐]
Now flapping equation is given as: + ฮฉ2๐ฝ = โ2๐ฮฉ ๐ ๐๐๐ + ๐๐/๐ผ๐๐
+ ๐ธ
๐ฮฉ [๐ +
๐๐
๐๐๐๐๐]+ฮฉ๐๐ท [๐ +
๐ธ ๐
๐๐๐๐๐ +
๐ธ ๐๐
๐๐๐๐๐๐] = ๐ธฮฉ๐ ๐
๐ ๐ฝ(๐ + ๐๐) โ ๐ธฮฉ๐ ๐
๐(๐๐ + ๐๐)
+๐ธฮฉ๐ ๐
๐ ๐๐๐๐ (
๐
๐๐ฝ๐ โ ๐๐ (๐๐ + ๐๐))
+๐ธฮฉ๐
๐๐ ๐๐๐๐+๐ธฮฉ
๐
๐๐๐ ๐ ๐๐๐๐๐
โ๐ธฮฉ๐ ๐
๐ ๐ฝ๐๐๐๐๐๐๐ โ ๐๐ฮฉ ๐๐๐๐ eq. 05
Question 8. Deduce disc tilt angles Assuming solution for the above eq. 05 of the form ๐ฝ = ๐0 โ ๐1 ๐๐๐ ๐ โ ๐1 ๐ ๐๐๐
= ๐1ฮฉ ๐ ๐๐๐ โ ๐1ฮฉ ๐๐๐ ๐ by using ๐ = ฮฉ๐ก
= ๐1ฮฉ2 ๐๐๐ ๐ + ๐1ฮฉ2 ๐ ๐๐๐
HELICOPTER PERFORMANCE STABILITY AND CONTROL COURSE CODE: AE4314
Name: DEEPAK PAUL TIRKEY
ASSIGNMENT NUMBER 04 Student number: 4590929
Now substituting the values of ๐ฝ, , on the LHS of eq. 05 LHS of eq.05
โก ๐1ฮฉ2 ๐๐๐ ๐ + ๐1ฮฉ2 ๐ ๐๐๐ +๐พ
8ฮฉ (๐1ฮฉ ๐ ๐๐๐ โ ๐1ฮฉ ๐๐๐ ๐ +
4๐
3๐1ฮฉ ๐ ๐๐2๐ โ
4๐
3๐1ฮฉ๐ ๐๐๐๐๐๐ ๐)
+ฮฉ2 (๐0 โ ๐1 ๐๐๐ ๐ โ ๐1 ๐ ๐๐๐ +๐พ ๐
6๐0 ๐๐๐ ๐ โ
๐พ ๐
6๐1 ๐๐๐ 2๐ โ
๐พ ๐
6๐1๐ ๐๐๐๐๐๐ ๐ +
๐พ ๐2
8๐0 ๐ ๐๐2๐ โ
๐ธ ๐๐
๐๐๐๐๐๐ ๐๐๐๐๐๐ โ
๐ธ ๐๐
๐๐๐๐๐๐๐ ๐๐๐๐๐) eq. 06
Simplifying the last two terms of eq. 06 :
โ ๐ธ ๐๐
๐๐๐๐๐๐ ๐๐๐๐๐๐ โก โ
๐พ ๐2
4๐1๐๐๐ ๐ ๐ ๐๐๐ ๐๐๐ ๐ โก โ
๐พ ๐2
4๐1 ๐ ๐๐๐ ๐๐๐ 2๐
โก โ ๐พ ๐2
4๐1 (
1
4๐ ๐๐๐ +
1
4๐ ๐๐3๐ )
โก โ ๐พ ๐2
16๐1๐ ๐๐๐ โ
๐พ ๐2
16๐1๐ ๐๐3๐
โ๐ธ ๐๐
๐๐๐๐๐๐๐ ๐๐๐๐๐ โก โ
๐พ ๐2
4๐1๐ ๐๐๐ ๐ ๐๐๐ ๐๐๐ ๐ โก โ
๐พ ๐2
4๐1๐ ๐๐2๐ ๐๐๐ ๐
โก โ ๐พ ๐2
4๐1 (
1
4๐๐๐ ๐ โ
1
4๐๐๐ 3๐ )
โก โ ๐พ ๐2
16๐1๐๐๐ ๐ +
๐พ ๐2
16๐1๐๐๐ 3๐
Rewriting eq. 06
โก ๐1ฮฉ2 ๐๐๐ ๐ + ๐1ฮฉ2 ๐ ๐๐๐ +๐พ
8ฮฉ2๐1 ๐ ๐๐๐ โ
๐พ
8ฮฉ2๐1 ๐๐๐ ๐ +
๐พ
8ฮฉ2 2๐
3๐1 (1 โ ๐๐๐ 2๐) โ
๐พ
8ฮฉ2 2๐
3๐1ฮฉ๐ ๐๐2๐
+ฮฉ2๐0 โ ฮฉ2๐1 ๐๐๐ ๐ โ ฮฉ2๐1 ๐ ๐๐๐ + ฮฉ2 ๐พ ๐
6๐0 ๐๐๐ ๐ โ ฮฉ2 ๐พ ๐
12๐1 (1 + ๐๐๐ 2๐) โ ฮฉ2 ๐พ ๐
12๐1๐ ๐๐2๐ +
ฮฉ2 ๐พ ๐2
8๐0 ๐ ๐๐2๐ โ ฮฉ2
๐พ ๐2
16๐1๐ ๐๐๐ โ ฮฉ2 ๐พ ๐2
16๐1๐ ๐๐3๐ โ ฮฉ2
๐พ ๐2
16๐1๐๐๐ ๐ + ฮฉ2 ๐พ ๐2
16๐1๐๐๐ 3๐
Rearranging the above expression in terms of free, sine, cosine terms etc..
โก (๐พ
8ฮฉ2 2๐
3๐1 + ฮฉ2๐0 โ ฮฉ2 ๐พ ๐
12๐1) + (๐1ฮฉ2 +
๐พ
8ฮฉ2๐1 โ ฮฉ2๐1 โ ฮฉ2
๐พ ๐2
16๐1) ๐ ๐๐๐
+ (๐1ฮฉ2 โ๐พ
8ฮฉ2๐1 โ ฮฉ2๐1 + ฮฉ2 ๐พ ๐
6๐0 โ ฮฉ2
๐พ ๐2
16๐1) ๐๐๐ ๐ + โฏ ๐ก๐๐๐๐ ๐๐๐๐ก๐๐๐๐๐๐ โ๐๐โ๐๐ โ๐๐๐๐๐๐๐๐
Comparing free terms with the RHS free term of eq.05
(๐พ
8ฮฉ2 2๐
3๐1 + ฮฉ2๐0 โ ฮฉ2 ๐พ ๐
12๐1) = ๐ธฮฉ๐ ๐
๐ ๐ฝ(๐ + ๐๐) โ ๐ธฮฉ๐ ๐
๐(๐๐ + ๐๐)
Or ๐๐ = ๐ธ๐
๐ [๐ฝ(๐ + ๐๐) โ
๐
๐(๐๐ + ๐๐)]
Comparing sine terms with the RHS sine term of eq.05
(๐1ฮฉ2 +๐พ
8ฮฉ2๐1 โ ฮฉ2๐1 โ ฮฉ2
๐พ ๐2
16๐1) = ๐พฮฉ2 1
8 (
8
3๐๐ โ 2๐ (๐๐ + ๐๐)) โ 2๐ฮฉ
๐พ
8ฮฉ2๐1 (1 โ
๐2
2) = ๐พฮฉ2 1
8 (
8
3๐๐ โ 2๐ (๐๐ + ๐๐)) โ 2๐ฮฉ
Or ๐๐ = (
๐
๐๐ฝ๐โ ๐๐ (๐๐+๐๐))โ
๐๐๐
๐ธฮฉ
(๐โ ๐๐
๐)
Comparing cosine terms with the RHS cosine term of eq.05
(๐1ฮฉ2 โ๐พ
8ฮฉ2๐1 โ ฮฉ2๐1 + ฮฉ2 ๐พ ๐
6๐0 โ ฮฉ2
๐พ ๐2
16๐1) = ๐พฮฉ
1
8๐
(1
8๐1 +
๐2
16๐1) = โ
๐
ฮฉ
1
8+
๐
6๐0
Or ๐๐ =โ
๐
ฮฉ +
๐ ๐
๐๐๐
(๐+ ๐๐
๐)