From the Rolls-Royce experimental archive: a quarter of a million communications from Rolls-Royce, 1906 to 1960's. Documents from the Sir Henry Royce Memorial Foundation (SHRMF).
Technical paper on the torsional vibration of diesel engines, including frequency calculations.
| Identifier | ExFiles\Box 132\1\ scan0117 | |
| Date | 25th March 1939 | |
| 402 1116 Torsional Vibration of Diesel By O.{Mr Oldham} MALYCHEVITCH, A.{Mr Adams} E.{Mr Elliott - Chief Engineer} THE three-mass equivalent system is shown in Fig. 13 and the normal deflection curve of the six-cylinder crankshaft for second-mode vibration in Fig. 14. According to Equation (3) the torsional rigidities of the two shaft lengths in Fig. 13 are Ma = GJ / L1 = (12 × 10⁶ × 8) / 21.5 = 4.47 × 10⁶ lb-in. per radian Mb = GJ / L2 = (12 × 10⁶ × 8) / 16 = 6 × 10⁶ lb-in. per radian (16) The phase velocity for the three-mass system can be found from the following quadratic equation: aω⁴ - bω² + c = 0 or ω² = (-b ± √(b² - 4ac)) / 2a After substituting the values for I, the moment of inertia, and M, the torsional rigidity, from Fig. 14 and Equation (16) we get a = (IaIbI₇) / (MaMb) = 162 × 10⁻¹⁵ b = (IaIb)/Ma + (IaI₇)/Ma + (IaI₇)/Mb + (IbI₇)/Mb = 377 × 10⁻⁸ c = Ia + Ib + I₇ = 11.32 Substituting these values in the above equations we have: 162 × 10⁻¹⁵ ω⁴ - 377 × 10⁻⁸ ω² + 11.32 = 0 ............ (17) ω₁² = 3,300,000 rad²-sec.⁻² or ω₁ = 1800 radians per sec. f₁ = 17,200 cycles per min. ω₂² = 19,734,000 rad²-sec.⁻² or ω₂ = 4440 radians per sec. f₂ = 42,400 cycles per min. The two roots of Equation (17), ω₁ and ω₂, give us the approximate values of the frequencies for the first and second modes of vibration. The frequency of the first mode, ω₁ = 1800 radians per sec., was found in the foregoing in solving for the frequency of the equivalent two-mass system. The approximate values of ω₁ can be made use of in the calculation of Table I for the frequency of a seven-mass system representing the six-cylinder engine including the flywheel (Fig. 11). We will try first the natural frequency ω₁² = 3,300,000 rad² sec.⁻² and multiply this value by the inertia I (col. 5), which gives us the torque per unit of angular deflection of each mass in lb.-in. per radian. Column 7 gives the inertia torque of each mass in lb.-in. for an amplitude of one radian at mass No. 1. Column 8 gives the total torque in lb.-in., that is, the sum of the value in column 7 plus the previous value in column 8. The value of the inertia torque of mass No. 6 in column 8 represents the sum of the torques of all discs to the left of the node, and must be equal to the inertia torque of the flywheel located to the right of the node. Then the last entry in column 8 will equal zero if the value assumed for the phase velocity ω₁² corresponds to the natural frequency. If ω₁² = 3,300,000 rad²-sec.⁻² is too low, the value of the total torque in column 8 will be positive (as it is in our case). Suppose that in the next trial ω₁² is taken too high; then by interpolation we find that ω₁² = 3,700,000 rad²-sec.⁻² will make the last torque equal to zero (col. 8). The corresponding frequency is given by Equation (2) Four-Cylinder Engine f = 26148 cycles per min. Frequency Calculation ONE-NODE VIBRATION ω² = 7,500,000 rad²-sec.⁻² TABLE IV Figs. 17-18-19. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |---|---|---|---|---|---|---|---|---|---|---|---| | Cyl. Mass No. | d | l | Eq. 14 | Iω² | ε Eq. 9 | Iω²ε | ΣIω²ε | M = GJ/l Eq. 3 | ΣIω²ε/M | Iε² | S Eq. 13 | | 1 | 3 | 7 | 0.22 | 1.65 × 10⁶ | 1.0000 | 1.65 × 10⁶ | 1.65 × 10⁶ | 13.7 × 10⁶ | 0.1200 | 0.2200 | ± 5430 | | 2 | 3 | 7½ | 0.22 | 1.65 × 10⁶ | 0.8800 | 1.45 × 10⁶ | 3.10 × 10⁶ | 12.8 × 10⁶ | 0.2400 | 0.1703 | 10250 | | 3 | 3 | 7 | 0.22 | 1.65 × 10⁶ | 0.6400 | 1.066 × 10⁶ | 4.17 × 10⁶ | 13.7 × 10⁶ | 0.3050 | 0.0917 | 13700 | | 4 | 3 | 8 | 0.22 | 1.65 × 10⁶ | 0.3400 | 0.56 × 10⁶ | 4.7 × 10⁶ | 12 × 10⁶ | 0.3950 | 0.0254 | 15300 | | FLY | ... | ... | 11.5 | 86.2 × 10⁶ | -0.055 | -4.7 × 10⁶ | 0 | ... | ... | 0.0345 | ... | | | | | | | | | | ΣIε² = 0.5419 | | | | Automotive Industries | ||