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 analysis of diesel crankshaft vibration, including amplitude, resonance, and shearing stresses.
| Identifier | ExFiles\Box 132\1\ scan0114 | |
| Date | 18th March 1939 | |
| 376 DIESEL CRANKSHAFT VIBRATION e₂ = (I₁ω² / M₁)e₁ = (I₁ω² / GJ)e₁ e₂ = e₁ - (Col. 8 / Col. 9)e₁ = 1 - (0.814 x 10⁶ / 13.7 x 10⁶) = 1 - 0.0595 = 0.9405 At cylinder No. 3, e₃ = e₂ - (Col. 8 / Col. 9) = 0.9405 - (1.579 x 10⁶ / 13.7 x 10⁶) = 0.9405 - 0.1150 = 0.8255 Amplitude of Torsional Vibration Forced torsional vibration of the crankshaft is induced by variations in the gas-pressure torque. The torque curve is very irregular, but it repeats itself every two revolutions (in a four-stroke engine), and the torque, therefore, can be represented by the well-known Fourier series with harmonic components of the ½, 1, 1½, 2, 2½, etc., orders. By equating the work done by the harmonic torque during one cycle to the work absorbed by damping (elastic hysteresis), we can determine the amplitude A of the vibration. In Fig. 7 line OB represents the work done by the force P at resonance; line OC, the work done by the damping forces per cycle. The two curves intersect at E, the point of energy equilibrium, which corresponds to an amplitude A = Oγ. Amplitude Oα shows that at that stage less energy is given up than is put into the system, while with an amplitude Oz the energy absorbed by the damping medium exceeds the energy put into the system by the vibration. Combining the work done by force P with a damping force which increases with the amplitude of vibration but always tends toward point E of energy equilibrium, has the effect of reducing the resonance peaks. We have what may be called free vibration of the system, and the work done by all cylinders per cycle may be expressed by the Equation U Σe = P B R Σe ...... (10) where U is the harmonic torque = P x B x R ................. (11) Σe, the vector sum of ordinates of the normal elastic curve for any particular firing order and for the critical speeds of the different orders is found from the vector diagram (Table V). As already mentioned, the ordinates of a normal elastic curve represent the amplitudes of vibration of the moving masses in the various cylinders. P, the harmonic coefficient in lb. per sq. in. per unit crank radius as given by Prof. Lewis (Figs. 9a and 9b). R, the crank radius, 2.25 in. B, the area of the cylinder bore, 11 sq. in. March 18, 1939 Figure 8 the magnitude of the torque T and can be expressed by the equation S = Tr{Capt. F. W. Turner - Finance} / J where T = Σ I ω²e is the total torque applied to each end of the shaft (Fig. 11, also Table I, col. 8), in lb.-in. per radian of deflection at mass No. 1. r = d/2, is the radius of the shaft in in., and J = (π d⁴)/g = (π d⁴)/32 = 8 (for a 3-in. shaft) is the polar moment of inertia of the shaft. As one radian is equal to 57.3 deg., the shear stress in any section of the shaft for 1 deg. deflection at mass No. 1 (see Table I, col. 12, and Fig. 12) is S = Tdg / (2πd⁴ x 57.3) = (ΣIω²e / 303.5) lb. per sq. in. per deg .. (13) If the natural frequency of vibration f = 18,440 cycles per min. (Table I) and the moment of inertia of the system Σ I e² = 0.7926 (Table I, col. 11), the amplitude of vibration of the crank farthest from the flywheel may be expressed by A = (5230 UΣe) / (Ie² f²) deg. or A = (5230 x 11 x 2.25 x PΣe) / (0.7926 x 340 x 10⁶) = 0.0005 PΣe . . . . . . . . . . (12) At resonance the maximum amplitude of vibration at the free end of the crankshaft is Aₘₐₓ. = Aγ deg. where γ is the magnification factor (Equation 8). Harmonic coefficients P (Equation 10) for four-stroke Diesel engines with a connecting-rod crank radius ratio of 4.25, for critical speeds of the ½ to 12th order and for i.m.e.ps. of 20 to 140 lb. per sq. in. have been calculated by Professor Lewis and are shown graphically in Figs. 9a and 9b. The values there given can be used also for other connecting rod/crank radius ratios. Harmonic analysis of the engine torque curve can be effected by means of a special instrument called a harmonic analyzer, as well as by the analytical method, which analysis is beyond the scope of this article. Shearing Stresses According to the investigations of St.{Capt. P. R. Strong} Venant, elastic deformation of a long shaft in torsion produces a shear stress which depends only on Equivalent Length and Moment of Inertia of Each Crank Unit The equivalent length of the crank unit (Fig. 8) is the length l of a shaft of the same diameter d₁ as the journal which, if subjected to the same torque, will show the same torsional deflection. It can be calculated by means of a well-known formula developed by B. C. Carter, according to which the equivalent length from A to B is l = (2b + 0.8h) + 0.75 (d₁⁴ / d₂⁴) a + 1.5r (d₁⁴ / hw³) = 7 in. Figure 9 FIG. 9A HARMONIC COEFFICIENT P ORDER 0.5 TO 5.5 FIG. 9B HARMONIC COEFFICIENT P ORDER 6 TO 12 Automotive Industries | ||