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 idiosyncrasies and harmonic analysis of valve mechanisms.
| Identifier | ExFiles\Box 56\2\ Scan084 | |
| Date | 15th January 1929 guessed | |
| 10 VALVE-MECHANISM IDIOSYNCRASIES tions per minute in valve-closed position. The record indicates a considerable increase in frequency during valve lift and the resultant damping effect on the vibration. This is particularly evident in the lower record, which is that of the tenth harmonic. It can be seen at the left of the diagram that the spring vibrations have virtually died out before the valve begins its next lift. Summary.—The rational steps in the selection of a combination of spring and cam to result in minimum vibration are: (1) Harmonic analysis of the valve-lift curve (2) Determination of the minimum frequency of the spring to keep it out of resonance with harmonics of high amplitude, up to the maximum engine-speed (3) Selection of the spring with reference to equation (2) and to fit load requirements and stress and frequency limitations APPENDIX Harmonic Analysis.—A complete discussion of the subject of harmonic analysis is beyond the scope of this paper. For this the reader is referred to Engineering Mathematics, by C. P. Steinmetz¹, and an article on Wave Form Analysis by P. M.{Mr Moon / Mr Moore} Lincoln². A modification of the Fischer-Hinnen method can be used to advantage in the analysis of the valve-lift curve. A simplification results if the analysis starts with zero degrees at the center of valve lift, as shown in Fig. 21. This brings the maximum value of all harmonics at zero degrees, and the equation can be written y = a₀ + a₁ cos θ + a₂ cos 2θ + ... + aₙ cos nθ (3) in which y is the valve lift, a₀ is the valve lift averaged over 360 deg. of the camshaft, aₙ is the maximum value of the nth harmonic, θ is the cam angle from the top of the valve lift, and n is the order number of the harmonic, a whole number. The analysis of harmonics consists, first, of finding the value of a₀, which is the area under the valve-lift curve divided by the length of one cycle, or 360 deg., as shown in Fig. 21. This area can be found either by a planimeter or by plotting the curve on cross-section paper and counting the squares enclosed. The coefficients a₁, a₂, ... aₙ are then found. Starting with the highest, say n = 30, ordinates of the lift curve are measured 360/n, or 12 deg. apart, beginning at θ = zero degrees. In this way a column of n ordinates is obtained, which is added together and divided by n, as shown in the sample calculation in Table 2. To obtain the maximum value of the nth harmonic, a₀ must be subtracted from the above average. These higher harmonics can be neglected on account of their small magnitude. When n = 15 is reached, however, a₀ must be subtracted to obtain a₁₅. Likewise, to obtain a₁₄ it is necessary to subtract a₃₀ and a₁₅. So, for any lower harmonic, all its multiples previously obtained must be deducted. Table 2 shows a sample analysis for five harmonics of cam No. 4 in Table 1. VALVE-MECHANISM IDIOSYNCRASIES 11 TABLE 2—SAMPLE ANALYSIS OF HARMONICS 18th Harmonic Cam Angle, Deg. | Lift, In. 0 | 0.438 20 | 0.362 40 | 0.145 60 | 0 80 | 0 100 | 0 120 | 0 140 | 0 160 | 0 180 | 0 200 | 0 220 | 0 240 | 0 260 | 0 280 | 0 300 | 0 320 | 0.145 340 | 0.362 Total | 1.452 Average | 0.0807 Subtracting a₀ | –0.0796 a₁₈ = +0.0011 17th Harmonic Cam Angle, Deg. | Lift, In. 0 | 0.438 21.2 | 0.353 42.3 | 0.113 63.6 | 0 84.8 | 0 106.0 | 0 127.2 | 0 148.5 | 0 169.7 | 0 190.9 | 0 211.1 | 0 233.0 | 0 254.2 | 0 275.4 | 0 296.3 | 0 317.3 | 0.113 339.0 | 0.353 Total | 1.370 Average | 0.0805 Subtracting a₀ | –0.0796 a₁₇ = +0.0009 16th Harmonic Cam Angle, Deg. | Lift, In. 0 | 0.438 22.5 | 0.342 45.0 | 0.074 67.5 | 0 90.0 | 0 115.5 | 0 135.0 | 0 157.5 | 0 180.0 | 0 205.5 | 0 225.0 | 0 247.5 | 0 270.0 | 0 295.5 | 0 315.0 | 0.074 337.5 | 0.324 Total | 1.270 Average | 0.0794 Subtracting a₀ | –0.0796 a₃₂ = –0.0002 15th Harmonic Cam Angle, Deg. | Lift, In. 0 | 0.438 24 | 0.329 48 | 0.038 72 | 0 96 | 0 120 | 0 144 | 0 168 | 0 192 | 0 216 | 0 240 | 0 264 | 0 288 | 0 312 | 0.038 336 | 0.329 Total | 1.172 Average | 0.0782 Subtracting a₃₀ | –0.0001 a₁₅ + a₄₅ = –0.0014 Subtracting a₄₅ | –0.0001 a₁₅ = –0.0015 14th Harmonic Cam Angle, Deg. | Lift, In. 0 | 0.438 25.7 | 0.312 51.4 | 0.012 77.2 | 0 102.8 | 0 128.6 | 0 154.3 | 0 180.0 | 0 205.7 | 0 231.4 | 0 257.0 | 0 283.0 | 0 308.5 | 0.012 334.3 | 0.312 Total | 1.086 Average | 0.0776 Subtracting a₀ | –0.0796 a₂₈ = –0.0020 Subtracting a₄₂ | –0.0001 a₁₄ = –0.0019 Acceleration Harmonics.—The acceleration harmonics can be derived from equation (3), for valve harmonics, by two differentiations A = (d²y) / (dt²) = ω²n² × (a₁ cos θ + a₂ cos 2θ + ... + aₙ cos nθ) (4) in which A is valve acceleration; ω is the angular velocity of the camshaft, in radians per sec.; and the other symbols are the same as in previous equations. For any harmonic, say the nth, the value of its acceleration equals –(ωn)² × aₙcos nθ, and the maximum value is bₙ = (ωn)² × aₙ (5) in which bₙ is the maximum acceleration and is 1 – (ωn)² times the valve-lift harmonic. It can be seen that, for any specific cam-and-spring mechanism, the frequency of the spring is proportional to ωn at resonant speeds, so that under these conditions the actuating force on the spring due to harmonics is proportional to the valve-lift harmonic. This can be expressed bₙ = KF²aₙ (6) in which F is the frequency of the spring. Derivation of the General Equation of Spring-Vibration Amplitude.—The equation for the amplitude of forced vibration of a damped elastic system having one degree of freedom under the action of a periodic disturbing force³ is C = a / √[(1 – T²/Tₛ²)² + (T²Y²/Tₛ⁴)] in which C is the amplitude of forced vibration, a is the deflection that would be produced by the maximum disturbing force if applied statically, T is the period of the natural vibration of the system, Tₛ is the period of the disturbing force, and Y is a quantity depending on the magnitude of the damping forces. When the disturbing force is in resonance with the natural period of the system, T = Tₛ, and C = a / Y; but a = P / K, where P is the maximum value of the disturbing force and R is the rate of the spring; and Y = Δ√(MR), where Δ is a damping constant and M is the active mass of the spring. Therefore, C = P / Δ√(MR). Since, according to Ricardo⁴, F = K, √(R/M), then C = KP/(Δ × F) (7) The disturbing force P, however, depends on the value of the acceleration harmonic of the valve motion which is in resonance at any given camshaft speed and also on certain spring characteristics. In other words, a given acceleration harmonic at a certain camshaft speed produces different disturbing forces on springs of different dimensions even though they have the same frequency. P being the disturbing force; m, a mass dependent on the linear-unit weight of the spring wire and the mean diameter of spring coil; bₙ, the value of the acceleration harmonic in resonance; d, the wire diameter, in inches; and D, the mean diameter of the coil, in inches, this disturbing force can be expressed: P = mbₙ, also m = K₁d²D, and therefore P = K₁d²D × bₙ. Substituting for P in equation (7), C = K₂(d² × D × bₙ)/(Δ × F) A further simplification results if we substitute for bₙ the value given in equation (6), as follows: C = K₃(d² × D × F² × aₙ)/Δ Substituting equation (1) for F, we have C = K₄(d²aₙ)/(D × N × Δ) which is equation (2) in the text of this paper. [Image of a graph with y-axis 'Valve Lift' and x-axis 'Camshaft Angle, deg' from 0 to 360. A symmetric S-shaped curve is shown, with a horizontal line indicating 'a₀'.] FIG. 21—VALVE-LIFT CURVE FOR HARMONIC ANALYSIS ¹McGraw-Hill Publishing Co., 1911. ²See Electrical Journal, July, 1908, p. 386. ³See Applied Elasticity, by S. Timoshenko and J.{Mr Johnson W.M.} M.{Mr Moon / Mr Moore} Lessells, p. 330; Westinghouse Technical Night School Press, 1925. ⁴See The Internal Combustion Engine, by H.{Arthur M. Hanbury - Head Complaints} R.{Sir Henry Royce} Ricardo, vol. 2, p. 210; D.{John DeLooze - Company Secretary} Van Nostrand Co., 1923. | ||