Abstract

An expression for the period of a conical Jolly balance spring is deduced theoretically and the result is verified by experimental data. By some assumptions which simplify the equation of motion it is reduced to that for simple harmonic motion from which the period is obtained. The period is:

T=2π(M+m03k)hmg
Mo is the mass of the load, m0 is the mass of the spring, h is the extension of the spring for a load m and C = (r2/r1 − 1)
K=5[(c+1)2+1]2(c+2)3(160+480c+696c2+620c3+360c4+135c5+30c6+3c7)
r2 is the radius of the large end coil, r1 is the radius of the small end coil. Very approximately, K = 1+C/3 so that
m03k=m02+r2/r1

© 1928 Optical Society of America

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References

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  1. Kelvin and Tait. “Natural Philosophy,”  2, arts, pp. 604–607.
  2. Timoshenko and Lessells. “Applied Elasticity,” p. 32.
  3. G. F. C. Searle’s “Experimental Harmonic Motion,” Art. 30.
  4. Rayleigh’s “Theory of Sound,” 1, Art. 88. Second edition.
  5. A. Sommerfeld, coupled“Oscillations of a Helical Spring,” J.O.S.A. & R.S.I.,  7, p. 5291923.
    [Crossref]

1923 (1)

A. Sommerfeld, coupled“Oscillations of a Helical Spring,” J.O.S.A. & R.S.I.,  7, p. 5291923.
[Crossref]

Kelvin,

Kelvin and Tait. “Natural Philosophy,”  2, arts, pp. 604–607.

Lessells,

Timoshenko and Lessells. “Applied Elasticity,” p. 32.

Rayleigh’s,

Rayleigh’s “Theory of Sound,” 1, Art. 88. Second edition.

Searle’s, G. F. C.

G. F. C. Searle’s “Experimental Harmonic Motion,” Art. 30.

Sommerfeld, A.

A. Sommerfeld, coupled“Oscillations of a Helical Spring,” J.O.S.A. & R.S.I.,  7, p. 5291923.
[Crossref]

Tait,

Kelvin and Tait. “Natural Philosophy,”  2, arts, pp. 604–607.

Timoshenko,

Timoshenko and Lessells. “Applied Elasticity,” p. 32.

J.O.S.A. & R.S.I. (1)

A. Sommerfeld, coupled“Oscillations of a Helical Spring,” J.O.S.A. & R.S.I.,  7, p. 5291923.
[Crossref]

Natural Philosophy (1)

Kelvin and Tait. “Natural Philosophy,”  2, arts, pp. 604–607.

Other (3)

Timoshenko and Lessells. “Applied Elasticity,” p. 32.

G. F. C. Searle’s “Experimental Harmonic Motion,” Art. 30.

Rayleigh’s “Theory of Sound,” 1, Art. 88. Second edition.

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Tables (2)

Equations (15)

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T = 2 π ( M + m 0 3 k ) h m g
K = 5 [ ( c + 1 ) 2 + 1 ] 2 ( c + 2 ) 3 ( 160 + 480 c + 696 c 2 + 620 c 3 + 360 c 4 + 135 c 5 + 30 c 6 + 3 c 7 )
m 0 3 k = m 0 2 + r 2 / r 1
X = n P G ρ 4 ( r 1 2 + r 2 2 ) ( r 1 + r 2 )
U 2 π n ( r 1 2 + r 2 2 ) ( r 1 + r 2 ) = V θ ( r 2 + r 1 2 ) ( r + r 1 )
E 0 = 0 2 π n 1 2 ( σ r d θ ) V 2
E = 1 2 m 0 3 K U 2 K = 5 [ ( c + 1 ) 2 + 1 ] 2 ( c + 2 ) 3 ( 160 + 480 c + 696 c 2 + 620 c 3 + 360 c 4 + 135 c 5 + 30 c 6 + 3 c 7 ) C = ( r 2 r 1 1 )
E = 1 2 ( M + m 0 3 K ) U 2
W = 1 2 ( m g h ) X 2
d E d t + d W d t = 0
( M + m 0 3 K ) d 2 x d t 2 + ( m g h ) X = 0 d 2 x d t 2 = [ ( m g h ) ( M + m 0 K 3 ) ] X
T = 2 π [ m g / h ( M + m 0 / 3 K ) ] 1 / 2 = 2 π ( M + m 0 3 K ) h m g
C = 0 , K = 1 and T = 2 π ( M + m 0 3 ) h m g
C = r 2 r 1 1 , K = 2 r 1 + r 2 3 r 1
m 0 3 K = m 0 2 + r 2 / r 1