﻿ Newtontime.com - Gyroscopic Torque Explained

### The Gyroscopic Effect

#### by N. O. Williams

```10  CLS : PRINT: REM Numerical integration program to calculate gyroscopictorque for a rotating ring which is simultaneously tilted.
15  REM              Filename:-  GyroTorque.txt
20  INPUT "INPUT NO. OF INTEGRATION ELEMENTS PER RING QUADRANT - USE A HIGHER NUMBER FOR GREATER ACCURACY"; Q
25  I = 4 * Q
30  INPUT "INPUT TOTAL MASS OF RING IN KILOGRAMS"; M#
35  INPUT "RADIUS OF RING IN METRES"; R#
40  INPUT "INPUT SPIN SPEED OF RING IN R.P.M"; NS#
45  INPUT "INPUT TILT SPEED OF RING IN R.P.M"; NT#: PRINT
50  REM convert angular speeds to radians per second
55  K# = 3.1415926536#: WS# = 2 * K# * NS# / 60: WT# = 2 * K# * NT# / 60
60  REM ratio of tilt speed/spin speed
65  C# = NT# / NS#
70  REM **********************************************************************
75  PRINT "PHI-DIRECTION, ACCELERATIONS NORMAL TO SPIN PLANE"
80  REM set initial values for near end of first element
85  MO# = 0: T1# = 0: P1# = 0: D1# = 0: V1# = 0: A1# = WS# ^ 2 * 2 * C# * R#
90  FOR J = 1 TO I / 2
95  REM Calculate angles (Radians) Theta & Phi, at far end of element
100 T2# = J * 2 * K# / I: P2# = C# * T2#
105 REM calculate displacement at far end of element
110 D2# = R# * SIN(T2#) * SIN(P2#)
115 REM calculate velocity at far end of element
120 V2# = WS# * R# * (COS(T2#) * SIN(P2#) + C# * SIN(T2#) * COS(P2#))
125 REM calculate acceleration at far end of element
130 A2# = WS# ^ 2 * R# * (-SIN(T2#) * SIN(P2#) + 2 * C# * COS(T2#) * COS(P2#) - C# ^ 2 * SIN(T2#) * SIN(P2#))
135 REM calculate force at middle of element (= mass x acceleration)
140 DF# = M# * (A1# + A2#) / (2 * I)
145 REM calculate gyro-moment at middle of element (= force x offset)
150 DM# = DF# * R# * COS((T1# + T2#) / 2)
155 REM summation of moments and set initial values for next element
160 MO# = MO# + DM#: T1# = T2#: P1# = P2#: D1# = D2#: V1# = V2#: A1# = A2#
165 NEXT J
170 REM torque = mass x rad^2 x w(spin) x w(tilt)
175 TT# = M# * 4 * R# ^ 2 * K# ^ 2 * NS# * NT# / 3600: CM# = 2 * MO#
180 PRINT "CALCULATED TORQUE DUE TO FORCES NORMAL TO SPIN PLANE = "; CM#; "Nm":PRINT
185 PRINT "RATIO OF CALCULATED NORMAL FORCES ONLY TORQUE/FORMULA TOTAL TORQUE  = "; CM# / TT#: PRINT
190 REM **********************************************************************
195 PRINT "THETA-DIRECTION, ACCELERATIONS PARALLEL TO SPIN PLANE"
200 REM set initial values for near end of first element
205 NO# = 0: T1# = 0: P1# = 0: E1# = R#: U1# = 0: B1# = WS# ^ 2 * -R#
210 FOR J = 1 TO I / 2
215 REM Calculate angles (Radians) Theta & Phi, at far end of element
220 T2# = J * 2 * K# / I: P2# = C# * T2#
225 REM calculate displacement at far end of element
230 E2# = R# * (1 - COS(T2#))
235 REM calculate velocity at far end of element
240 U2# = WS# * -R# * (SIN(T2#))
245 REM calculate acceleration at far end of element
250 B2# = WS# ^ 2 * -R# * (COS(T2#))
255 REM calculate force at middle of element (= mass x acceleration)
260 EF# = M# * (B1# + B2#) / (2 * I)
265 REM calculate gyro-moment at middle of element (= force x offset)
270 EM# = EF# * R# * SIN((T1# + T2#) / 2) * SIN((P1# + P2#) / 2)
275 REM summation of moments and set initial values for next element
280 NO# = NO# + EM#: T1# = T2#: P1# = P2#: E1# = E2#: U1# = U2#: B1# = B2#
285 NEXT J
290 REM torque = mass x rad^2 x w(spin) x w(tilt)
295 TT# = M# * 4 * R# ^ 2 * K# ^ 2 * NS# * NT# / 3600: CN# = -2 * NO#
300 PRINT "CALCULATED TORQUE DUE TO FORCES PARALLEL TO SPIN PLANE = "; CN#; "Nm":PRINT
305 PRINT "RATIO OF CALCULATED PARALLEL FORCES ONLY TORQUE/FORMULA TOTAL TORQUE = "; CN# / TT#: PRINT:PRINT
310 REM ***********************************************************************
315 PRINT "SUMMING TORQUES DUE TO FORCES IN NORMAL & PARALLEL DIRECTIONS":PRINT
320 PRINT "SUM OF CALCULATED TORQUES = "; CM# + CN#; "Nm" :PRINT:PRINT
325 PRINT "FORMULA-BASED TOTAL GYROSCOPIC TORQUE = "; TT#; "Nm":PRINT:PRINT
330 PRINT "RATIO OF TOTAL CALCULATED/FORMULA TORQUES = "; (CM# + CN#) / TT#
335 END
```