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Torsional Vibration

Torsional Vibration 76 AIRCRAFT ENGINEERIN G March, 1938 A Formula Giving Close Approximations for Calculating Fundamental Frequencies By Li Teng-Ko, B.SC H E usual cases that occur in torsional vibration calculations are of the type shown in Fig. 1. where p = Value of equa l mass. c = Value of equa l stiffness. sp = Value of the first mass, s being generally large. rc = Value of the stiffness of the first portion. n = Number of equal masses. For the lower frequencies the general solution for this case is given by where β is determined from the equation, Or neglecting , Equatio n (2) becomes By Formula (5) = 0·093. f = 82·1 v.p.s. which give n values of β corresponding to n By Formula (4) = 0·09295. frequencies. f = 82·0 v.p.s. TABL E I. R. & M. 1303 gives the exact solution, f = 82·0 v.p.s. Comparison of Formula (6) with exact solutions for liquation (3). Willie Value Example 2. Liberty Engine. n — 6. of r of n c = 4·5 X 10°, rc = 2·5 x 10°, r = 0·556 (Formul a (Exact ) (Formul a (Exact ) Remark s (therefore using Equation (7)), p = 0·59, 6) 0 0 0 0 0 — — sp = 49·90, s = 84·6. I n Exact . — — By Formula (6) = 0·1076. 2 Exact . — — f = 94·6 v.p.s. 1·5 — — — 20° 46 ' 20° 40' 0·3545 0·3529 By Formula (5) =0·1114. '' 2 — 3 14° 15' 14° 12 ' 0·2456 0·2453 Almos t exact . f = 97·8 v.p.s. 4 10° 48' 10° 48 ' 0·1874 0·1874 Exac t when n = 4 and I t is interesting to note tha t Formula (6) will By Formula (4) =0·1111. over . give exact solutions for Equation (3) when 0·5 n — — — — f = 97·6 v.p.s. r = o, 1 an d 2 for all values of n, an d for inter­ 2 0·2227 0·2337 15° 52' 13° 31 ' mediate values of r, Formula (6) will give very R. & M. 1304 gives the exact solution, '' 3 10° 0' 10° 15 ' 0·1736 0·1779 '' 4 8° 11 ' 8° 18' 0·1423 0144 4 close results and almost exact values for n f = 97·5 v.p.s. 6 6° 0 ' 6° 2 ' 0·1051 '' 0·1045 '' — greater than 3 or 4, as shown in th e following I t will be noted that Formula (4) gives almost table. the exact results and the simpler Formula (5) The solution of Equation (2) or (3) can only is sufficiently accurate for practical purpose. From the table given above it will be noted be done by trial and error method. If only The derivation of these formulæ is rational fundamental frequencies are required, the and is given below : following formulae will give very close and in tha t for r < 1, i t is more close t o use instead Equatio n (2) can b e writte n in th e form, some cases, exact results. of sin and therefore Equation (1) becomes Actual calculations of fundamental fre­ To show the closeness of results given by or less exactly quencies show tha t the values of and of — nβ Formulas (4), (5) and (6) in actual engine calculations, the following two examples are are generally small enough to allow the use of taken from R. & M. 1303 an d 1304. sin = , cos = 1 an d ta n = Example 1. Tornado Mk. III Engine. n = 8. or neglecting c = 53·3 X 10°, rc = 4·6 x 106, r = 0·868 — nβ. Substituting these approximate values, (therefore using Equation (7)), p = we obtain the following quadratic equation 6·925, sp = 987·5, s = 142·5. By Formula (6) = 0·09075. th e solution of which gives Formula (4) an d * Se e "Strengths of Shafts in Vibration," by J . Morris [Crosby f = 80·1 v.p.s. Formulæ (5) an d (6) follow accordingly. Lockwood). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Aircraft Engineering and Aerospace Technology Emerald Publishing

Torsional Vibration

TengKo, Li
Aircraft Engineering and Aerospace Technology , Volume 10 (3): 1 – Mar 1, 1938
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Publisher
Emerald Publishing
Copyright
Copyright © Emerald Group Publishing Limited
ISSN
0002-2667
DOI
10.1108/eb030284
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Abstract

76 AIRCRAFT ENGINEERIN G March, 1938 A Formula Giving Close Approximations for Calculating Fundamental Frequencies By Li Teng-Ko, B.SC H E usual cases that occur in torsional vibration calculations are of the type shown in Fig. 1. where p = Value of equa l mass. c = Value of equa l stiffness. sp = Value of the first mass, s being generally large. rc = Value of the stiffness of the first portion. n = Number of equal masses. For the lower frequencies the general solution for this case is given by where β is determined from the equation, Or neglecting , Equatio n (2) becomes By Formula (5) = 0·093. f = 82·1 v.p.s. which give n values of β corresponding to n By Formula (4) = 0·09295. frequencies. f = 82·0 v.p.s. TABL E I. R. & M. 1303 gives the exact solution, f = 82·0 v.p.s. Comparison of Formula (6) with exact solutions for liquation (3). Willie Value Example 2. Liberty Engine. n — 6. of r of n c = 4·5 X 10°, rc = 2·5 x 10°, r = 0·556 (Formul a (Exact ) (Formul a (Exact ) Remark s (therefore using Equation (7)), p = 0·59, 6) 0 0 0 0 0 — — sp = 49·90, s = 84·6. I n Exact . — — By Formula (6) = 0·1076. 2 Exact . — — f = 94·6 v.p.s. 1·5 — — — 20° 46 ' 20° 40' 0·3545 0·3529 By Formula (5) =0·1114. '' 2 — 3 14° 15' 14° 12 ' 0·2456 0·2453 Almos t exact . f = 97·8 v.p.s. 4 10° 48' 10° 48 ' 0·1874 0·1874 Exac t when n = 4 and I t is interesting to note tha t Formula (6) will By Formula (4) =0·1111. over . give exact solutions for Equation (3) when 0·5 n — — — — f = 97·6 v.p.s. r = o, 1 an d 2 for all values of n, an d for inter­ 2 0·2227 0·2337 15° 52' 13° 31 ' mediate values of r, Formula (6) will give very R. & M. 1304 gives the exact solution, '' 3 10° 0' 10° 15 ' 0·1736 0·1779 '' 4 8° 11 ' 8° 18' 0·1423 0144 4 close results and almost exact values for n f = 97·5 v.p.s. 6 6° 0 ' 6° 2 ' 0·1051 '' 0·1045 '' — greater than 3 or 4, as shown in th e following I t will be noted that Formula (4) gives almost table. the exact results and the simpler Formula (5) The solution of Equation (2) or (3) can only is sufficiently accurate for practical purpose. From the table given above it will be noted be done by trial and error method. If only The derivation of these formulæ is rational fundamental frequencies are required, the and is given below : following formulae will give very close and in tha t for r < 1, i t is more close t o use instead Equatio n (2) can b e writte n in th e form, some cases, exact results. of sin and therefore Equation (1) becomes Actual calculations of fundamental fre­ To show the closeness of results given by or less exactly quencies show tha t the values of and of — nβ Formulas (4), (5) and (6) in actual engine calculations, the following two examples are are generally small enough to allow the use of taken from R. & M. 1303 an d 1304. sin = , cos = 1 an d ta n = Example 1. Tornado Mk. III Engine. n = 8. or neglecting c = 53·3 X 10°, rc = 4·6 x 106, r = 0·868 — nβ. Substituting these approximate values, (therefore using Equation (7)), p = we obtain the following quadratic equation 6·925, sp = 987·5, s = 142·5. By Formula (6) = 0·09075. th e solution of which gives Formula (4) an d * Se e "Strengths of Shafts in Vibration," by J . Morris [Crosby f = 80·1 v.p.s. Formulæ (5) an d (6) follow accordingly. Lockwood).

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Aircraft Engineering and Aerospace Technology – Emerald Publishing

Published: Mar 1, 1938

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