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).
Page from a publication discussing forced vibration and elastic hysteresis in diesel engine crankshafts.
| Identifier | ExFiles\Box 132\1\ scan0112 | |
| Date | 18th March 1939 | |
| 374 DIESEL CRANKSHAFT VIBRATION Forced Vibration If in addition to the weight of the masses (Figs. 1 and 2) representing the flywheel, pistons, connecting rods, crankpins and crank arms, we consider the gas pressure acting on the pistons, as represented by the indicator diagram of the Diesel engine, we find that the latter induces torsional vibration of the shaft, and “forced vibration” of the system results. The frequency f of free vibration of the shaft and the frequency f₁ of forced vibration are independent of each other; but when, at a certain engine speed, the frequency of natural vibration of the shaft coincides with the forced frequency, the phenomenon of “resonance” appears, and if there are no damping forces in the system the amplitude of vibration can increase indefinitely. Resonance occurs whenever the ratio of natural frequency f in cycles per min. to the shaft speed in r.p.m. is one of the following series of numbers (or harmonic orders): k = 1/2, 1, 1 1/2, 2, 2 1/2, 3, 3 1/2, 4, 4 1/2, 5, 5 1/2, 6, 6 1/2, etc. (for a four-stroke engine). The speed N at which resonance occurs is called the critical speed. N = f / k For the six-cylinder engine of our example (see Table I), for which f = 18,440 cycles per min., the various important critical speeds are as follows: N₅.₅ = 18,440/5.5 = 3350 r.p.m. for the 5.5. harmonic; N₆. = 18,440/6. = 3000 r.p.m. for the 6. harmonic; N₇. = 18,440/7. = 2640 r.p.m. for the 7. harmonic; N₈. = 18,440/8. = 2300 r.p.m. for the 8. harmonic, and so on. When there is resonance, or when the engine is running at a critical speed, the stresses in most cases may become large enough to cause shaft breakage, and to prevent such breakage the elastic properties and the dynamic force of the system must be suitably proportioned and the operating-speed range of the engine must be so limited as not to include any of the more important critical speeds. Magnification Factor The ratio of the amplitude of forced vibration at N r.p.m. to the amplitude of free vibration is called the magnification factor γ = h₁ / h = 1 / (1 - (p/n)²) ... (6) where p is the frequency of free vibration, and n the frequency of forced vibration. From Equation (6) we see that γ is positive when p is less than n, negative when p is greater than n (Curve C) and infinite (without damping) when p is equal to n. Fig 4 shows the magnification factor with and without damping for different frequency ratios. We notice that without damping, the amplitude increases indefinitely when the frequency of natural vibration of the shaft is equal to the frequency of forced vibration (p/n = 1), and with the aid of Equation (6) a resonance curve may be constructed that is similar to Curve A of Fig. 4. If the ratio p/n is small, or the frequency of forced vibration is below the frequency of natural vibration of the shaft, the magnification factor is also small and will be close to unity. As a condition of resonance is approached, the magnification factor grows rapidly; but in dealing with the actual crankshaft, where damping forces are present, Curve A will become a Curve B, the vibration amplitude will never reach an infinite value, and the resonance curve will be represented by a curve similar to B. Damping Elastic Hysteresis Damping forces may be either external or internal. External damping forces include such items as friction in bearings or between piston and cylinder wall, while the internal damping force consists of the molecular friction of the material of the vibrating crankshaft. Engine friction is comparatively small and can be neglected, and in the following we will consider only the internal molecular friction of the crankshaft material, which results in the generation of heat. Flexing of the shaft in torsional vibration at below its elastic limit, under continuously changing torque, requires the expenditure of energy. The greater part of this energy is stored in the shaft; however, a certain amount of the energy is absorbed by the molecular friction of the material, this being called “elastic hysteresis.” Tests show that when a shaft is subjected to torsional moments, the stress-strain curve obtained does not exactly follow Hooke’s law, but forms a closed loop (Fig. 5). The area MNPQ, which is called the hysteresis loop, represents the energy dissipated during each cycle, and is a measure of the damping properties of the crankshaft material. Rowett, Canfield and Dorey have made important contributions to our knowledge on the subject of the damping characteristics of materials. Doctor Dorey gives the value of the hysteresis constant C as 2100 for nickel-chromium steel, with a critical stress of 120,000 lb. per sq. in., when stressed below the critical range. It is an accepted rule of practice that for reverse torsion in the abovementioned steel, the permissible safe working stress is 10 per cent of the ultimate tensile strength, or 10,000 lb. per sq. in. On the basis of Dr. Dorey’s experiments, Dr. W. Wilson gives the following expression for the maximum stress at resonance: S = C √Q lb. per sq. in., ... (7) and for the magnification factor at resonance. γ = C / √Q , . . . . . . (8) where S is the maximum vibration stress at resonance in lb. per sq. in., and Q the free-vibration stress in lb. per sq. in. In our case Q = sA, [Figure 4: A graph showing Magnification Factor vs{J. Vickers} Ratio p/n, with curves for damping and without damping.] Figure 4 [Figure 5: A graph showing a Stress-Elongation hysteresis loop.] Figure 5 HYSTERESIS LOOP March 18, 1939 Automotive Industries | ||