formulae to find out radius of a given curve or curve section

8/4/2019 Formulae to Find Out Radius of a Given Curve or Curve Section

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Formulae to find out radius of

a given curve or curve section

By

Jayesh [email protected]


2/10

Formulae to find out radius of a given curve or curvesection

Formulae to find out radius of a given curve or

curve section

By

Jayesh Jaiswal

[email protected] Page 2


3/10


Figure 1

Here, r = radius of curve

c = chord passing through curve ( 0 < c 2r )p = perpendicular distance between curve and chord c ( 0 < p r )

k = angle inscribed inside the curve ( 90 k 180 )

[A] k in terms of c and r

From graphical observations:

For 0 c r or c (0, r]

k = 180 - [(c/r)30] -------------------------------------------------(1)

For r c 7r/4 or c [r, 7r/4]

k = 180 - [(c/r) (30 + 10)] + 10

k = 190 - [(c/r)40] -------------------------------------------------(2)

For 7r/4 c 2r or c [7r/4, 2r]

k = 180 - [(c/r) (30 + 90)] + 150k = 330 - [(c/r)120] -----------------------------------------------(3)

Considering figure (2)

k1 = 180 - ()

Therefore, k1 = 180 - k2 --------------------------------------------------(4)

Hence,

for c (0, r]

() = (c/r)30 ------------------------------------------------------(5)

for c [r, 7r/4]



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() = [(c/r)40] - 10 ---------------------------------------------(6)

for c [7r/4, 2r]

() = [(c/r)120] - 150 ------------------------------------------(7)

If,

For 0 c r or c (0, r]k2 = [(c/r)30] or [c(30) r(0)]/r ------------------------------(8)

For r c 7r/4 or c [r, 7r/4]k2 = [c(40) r(10)]/r --------------------------------------------(9)

For 7r/4 c 2r or c [7r/4, 2r]k2 = [c(120) r(150)]/r ----------------------------------------(10)

Deviation degree [of (2), (3) from (1)]

For 0 c r or c (0, r]

= 0 ----------------------------------------------------------------(11)

For r c 7r/4 or c [r, 7r/4]

= 180 - [(c/r)30] - 190 + [(c/r)40]

therefore, = 10[(c/r) 1] --------------------------------------(12)

For 7r/4 c 2r or c [7r/4, 2r]

= 180 - [(c/r)30] - 330 + [(c/r)120]therefore, = [(c/r)90] 150therefore, = 30[(3c/r) 5] ------------------------------------(13)

Deviation degree [of (3) from (2)]

' = 190 - [(c/r)40] - 330 + [(c/r)120]

therefore, = [(c/r)80] 140therefore, = 20[(4c/r) 7] --------------------------------------------(14)

[B] k in terms of p and r

for p (0, r/8]

k = 180 - [(p/r)240] ---------------------------------------------(15)

for p [r/8, r/2]

k = 160 - [(p/r)80] ----------------------------------------------(16)

for p [r/2, r]



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k = 150 - [(p/r)60] ----------------------------------------------(17)

Deviation degree [of (16), (17) from (15)]

For 0 p r/8 or p (0, r/8] = 0 ---------------------------------------------------------------(18)

For r/8 p r/2 or p [r/8, r/2]

= 160 - [(p/r)80] - 180 + [(p/r)240]

therefore, = [(p/r)160] 20therefore, = 20[(8p/r) 1] -----------------------------------(19)

For r/2 p r or p [r/2, r]

= 150 - [(p/r)60] - 180 + [(p/r)240]therefore, = [(p/r)180] 30

therefore, = 30[(6p/r) 1] -----------------------------------(20)

Deviation degree [of (17) from (16)]

' = 150 - [(p/r)60] - 160 + [(p/r)80]therefore, = [(p/r)20] 10

therefore, = 10[(2p/r) 1] -------------------------------------------(21)

Correlating c and p:

For 0 c r or c (0, r] & for p (0, r/8]180 - [(p/r)240] = 180 - [(c/r)30]

therefore [(p/r)240] [(c/r)30] = 0

therefore (8p/r) (c/r) = 0

c/p = 8 -------------------------------------------------------------(22)

For r c 7r/4 or c [r, 7r/4] & for p [r/8, r/2]

160 - [(p/r)80] = 190 - [(c/r)40]therefore, [(c/r)40] [(p/r)80] = 30

therefore, (c/r) (2p/r) =

therefore, c-2p = 3r/4

therefore, c = (3r+8p)/4 -----------------------------------------(23)

For 7r/4 c 2r or c [7r/4, 2r] & for p [r/2, r]150 - [(p/r)60] = 330 - [(c/r)120]

therefore, [(c/r)120] [(p/r)60] = 180

therefore, (2c/r) (p/r) = 3

therefore, 2c = 3r + ptherefore, c = (3r+p)/2 ------------------------------------------(24)



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[D] Clarifying (22) and Finding r:

For 7r/4 c 2r or c [7r/4, 2r] & for p [r/2, r]r = (2c p)/3 -----------------------------------------------------(25)

For r c 7r/4 or c [r, 7r/4] & for p [r/8, r/2]r = (4c 8p)/3 ---------------------------------------------------(26)

For 0 c r or c (0, r] & for p (0, r/8]Assuming, r = (xc yp)/3 -------------------------------------(27)

Putting c = r and p = r/8

y 8x + 24 = 0 --------------------------------------(27a)

Putting c = 3r/4 and p = r/16

y 12x + 48 = 0 -------------------------------------(27b)Multiplying (27b) by (-1)

-y + 12x 48 = 0 ------------------------------------(27c)Now, adding (27b) and (27c)

4x 24 = 0

therefore, x = 6 ---------------------------------------(27d)Putting (27c) in (27a)

y = 24 -------------------------------------------------(27e)

Putting (27d) and (27e) in (27)r = (6c 24p)/3

therefore, r = 2(c-4p), for c (0, r] and p (0, r/8]-------(28)

For c = 0 and p = 0, value of r is indefinite.

Here, c/p [2, )

Hence one can always select c and p for a given curve or curve section such thatc/p 2

Some Interesting Observations

Suppose,

r = radius of curve,

c = chord passing through curvep = perpendicular distance between curve and curve section

k = angle inscribed inside the curved = r p

-r d r0 k 180

0 p 2r



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0 c 2r

d = 0 p = r k = 90

d = r p = 0 k = 180

d = -r p = 2r k = 0

If p increases by r k decreases by 90If p decreases by r k increases by 90

If d increases by r k increases by 90If d decreases by r k decreases by 90

If p increases by r/p k decreases by 90/p

If p decreases by r/p k increases by 90/p

p = r k = 90p = (r + r) k = 0p = (r r) k = 180

c = 2p k = 90c = (0)p k = 180

c = 3p k = 120

c = 6p k = 150

c = (0)r k = 180

c = r/2 k = 165

c = r k = 150

k = 180- (c/r)30, [r c 2r]

Deviation ,


c

r 0

5r/4 2.5

3r/2 5

7r/4 7.5


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Double Circle Concept

Figure 2

Some Interesting Graphs

C K P K



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C deviation P C

Some Interesting Data

K1 C P

0 0 2r 360

30 r 15r/8 300

60 7r/4 3r/2 240

90 2r r 180

120 7r/4 r/2 120

150 r r/8 60

180 0 0 0

210 r r/8

240 7r/4 r/2

270 2r r

300 7r/4 3r/2


Figure 3


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330 r 15r/8

360 0 2r

Figure 4

C 0 r 7r/4 2r

K P Q P Q P Q P Q

0 2r 0

30 37r/20 2r 59r/20 7r/2 3r 4r

60 15r/20 9r/10 41r/40 63r/40 4r/5 9r/5

90 7r/20 r/2 13r/40 7r/8 0 r

12

0

3r/20 3r/10 r/40 21r/40 2r/5 3r/5

15

0

0 3r/20 23r/80 21r/80 7r/10 3r/10

18

0

0 0 3r/20 0 11r/20 0 r 0

21

0

3r/10 3r/20 13r/16 21r/80 13r/10 3r/10

24

0

9r/20 3r/10 43r/40 21r/40 8r/5 3r/5

27

0

13r/20 r/2 57r/40 7r/8 2r r

30

0

21r/20 9r/10 17r/8 63r/40 14r/5 9r/5

33

0

43r/20 2r 81r/20 7r/2 5r 4r

360

2r 0


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