Friction of the Piston for Damping Crankshaft Vibrations
Abstract Generated abstract
The paper examines piston friction as a possible contributor to the damping or excitation of resonant torsional vibrations in internal combustion engine crankshafts, addressing the inadequacy of energy balances based only on shaft hysteresis and hydrodynamic bearing losses. It derives expressions for crankshaft torque components produced by dry piston friction, velocity dependent piston friction, and friction caused by lateral piston pressure from inertia forces, using simplified kinematic relations and harmonic decomposition. The analysis shows that piston friction can generate harmonic torque components, especially of the first and second orders, that may either damp or reinforce forced shaft vibrations depending on phase, and therefore should be included in vibration energy balance calculations.
Full Text
FRICTION OF THE PISTON AS A FACTOR IN DAMPING CRANKSHAFT VIBRATIONS
V. K. NECHAEV
Attempts to draw up an energy balance of resonant torsional vibrations of the crankshaft of an internal-combustion engine, taking into account only the principal known sources of damping—hysteresis in the shaft metal (Lewis [1]), viscous friction and other hydrodynamic losses in the shaft supports (Widler [2], Shannon [3], Draminsky [4])—as a rule end in failure [5, 6].
The sum of the calculated quantities of energy loss per cycle of vibration due to hysteresis and hydrodynamic losses in the shaft supports often proves to be considerably smaller than the work performed per cycle of vibration by the resonant harmonic components of the torques on the shaft, produced by the engine cylinders.
This indicates the presence in the oscillatory system of other sources of damping of the vibrations of the shaft of the engine unit, apart from those indicated above (impacts in the clearances of the links of connecting-rod-and-crank mechanisms, so-called structural friction in the joints of the unit’s shafting, etc.).
I. Geiter [5], in considering the question of the sources of damping of torsional vibrations, included as one such source of damping the additional friction forces between the piston and the cylinder liner that arise here as a result of a change in the diagram of normal pressures \(N\) due to torsional vibrations of the shaft.
It is shown below that the source of damping here may be not only these additional friction forces, but also those friction forces between the liner and the piston that exist even in the complete absence of torsional vibrations in the shaft and with perfectly uniform rotation of the shaft (first of all, the friction of the piston rings), and even, in general, in the absence of side pressure of the piston on the cylinder liner.
Let us assume that these latter forces \(R\) (not depending on the magnitude of the side pressure \(N\)) obey the law
\[ R = R_0 + k v_n, \]
where
\(R_0\) is the constant component of the friction force (“dry” friction); \(k\) is the coefficient of the linear, “velocity,” component of the force; \(v_n\) is the instantaneous velocity of the piston.
The force \(R\) produces on the crankshaft a certain variable torque \(M_R\), whose magnitude is determined from the obvious relation
\[ M_R \cdot \eta_1 = R \cdot v_n, \]
where \(\eta_1\) is the instantaneous value of the angular speed of rotation of the crankshaft.
Hence
\[ M_R = R \cdot \frac{v_n}{\eta_1} = \left(R_0 + k v_n\right)\cdot \frac{v_n}{\eta_1}, \]
or
\[ M_R = M'_R + M''_R, \]
where
\[ M'_R = R_0 \cdot \frac{v_n}{\eta_1}, \tag{1} \]
\[ M''_R = \frac{k v_n^2}{\eta_1}. \tag{2} \]
Here \(M'_R\) and \(M''_R\) are, respectively, the components of the friction torque on the shaft produced by the constant and the “velocity” components of the force \(R\).
In the presence of torsional vibrations in the shaft (monoharmonic in the simplest case) with frequency \(h\eta_1\), the true speed of rotation of the crankshaft is
\[ \eta_1 = \eta + \Phi h\eta \sin (h\eta t + z), \]
where \(\Phi\) is the angular amplitude of vibration of this crankshaft. But for small \(\Phi\), \(\eta\), and low orders of vibration \(h\), approximately
\[ \eta_1 = \eta \]
and, as usual,
\[ v_n = r\eta \left(\sin z + \frac{1}{2}\mu \sin 2z\right), \tag{3} \]
where
\[ \mu = r/l. \]
Now
\[ M'_R = R_0 r\left(\sin z + \frac{1}{2}\mu \sin 2z\right), \tag{4} \]
or
\[ M'_R = R_0 r \sin \eta t + \frac{1}{2}\mu R_0 r \sin 2\eta t, \tag{5} \]
that is, the constant force \(R_0\) of piston friction produces on the shaft two (more precisely, more, if a more accurate expression is adopted for \(v_n\)) harmonic components of the friction torque.
It follows from this that, in the absence of other forces acting on the piston (for example, when the engine is being cranked with the cylinder heads removed at low speeds), the constant force of dry friction between the piston and the cylinder bore may be the cause of the occurrence and development in the shaft of the power unit of forced vibrations of low orders \((h = 1; 2)\).
In the presence of gas or inertial forces acting on the piston, each of the torques entering the right-hand side of expression (5) may manifest itself in strengthening or weakening (that is, in posi-
tive or negative damping) of forced oscillations of the shaft, depending on the phase angle between this oscillation and the component \(M'_R\) of the moment under discussion (at least of the first and second orders).
Thus, dry friction on the piston must enter, as a definite item, into the energy balance of the forced oscillatory process. Let us note that the force \(R\) does not give a constant component of moment on the shaft and, consequently, cannot be covered by the constant component of the engine torque.
We proceed to determine the “velocity” component of the moment on the shaft. From (2) and (3)
\[ M''_R = kr^2\eta\left(\sin\alpha+\frac{1}{2}\mu\sin2\alpha\right)^2, \tag{6} \]
or, after simple transformations,
\[ M''_R = kr^2\eta\left[ \frac{1}{2}+\frac{1}{8}\mu^2+\frac{1}{2}\mu\cos\alpha -\frac{1}{2}\cos2\alpha-\frac{1}{2}\mu\cos3\alpha -\frac{1}{8}\mu^2\cos4\alpha \right]. \]
Thus the “velocity” friction on the piston gives on the shaft a considerable constant moment
\[ \left(\frac{1}{2}+\frac{1}{8}\mu^2\right)kr^2\eta, \]
which does not damp and does not cause oscillations and is covered by a share of the constant component of the engine torque. The most significant harmonic component of the moment \(M''_R\) is the harmonic of the second order, with amplitude
\[ \frac{1}{2}kr^2\eta. \]
The work of the damping (or exciting) harmonic moments included in \(M'_R\) and \(M''_R\) can easily be calculated by the usual methods, with proper allowance for the phase angles between these moments and the shaft oscillations. But it is easy to see that, in practice, accounting for the effect of these moments can be of importance only for \(h=1\) and \(h=2\).
The variable lateral pressure on the piston, caused by gas pressure and by the forces of inertia of the connecting-rod–crank mechanism, even for \(\eta=\mathrm{const}\), gives a new friction force on the piston
\[ R_N=fN, \]
where the coefficient of friction \(f\) may have a complex (mixed) character.
This force \(N\) gives a new friction moment on the shaft
\[ M_N=R_N\cdot\frac{v_n}{\eta}, \]
where here \(M_N\) is already a function of three variables \(\eta\), \(\alpha\), and \(v_n\).
A harmonic analysis of this dependence \(M_N\) on the crank angle \(\alpha\) (and, for large amplitudes of torsional oscillations, also on time \(t\)) gives the harmonic components of the moment \(M_N\), which can also be taken into account in calculating the damping in the oscillatory system.
An analytical investigation of the moment \(M_N\) in general form is difficult because of the complex character of the variation of the lateral pres-
of \(N\) over the piston stroke. Here we shall confine ourselves to solving this problem only with respect to the force \(N_{\text{in}}\), produced by the inertial forces of the connecting-rod–crank mechanism of the cylinder,
\[ P_{\text{in}} = mr\eta_1^2 \cdot (\cos\alpha + \mu \cos 2\alpha). \]
This force gives the lateral pressure on the piston
\[ N_{\text{in}} = mr\eta_1^2(\cos\alpha + \mu \cos 2\alpha)\cdot \tan\beta \]
and causes the corresponding friction force
\[ R_N = fmr\eta_1^2(\cos\alpha + \mu \cos 2\alpha)\cdot \tan\beta. \]
We take, approximately,
\[ \tan\beta \simeq \sin\beta \]
and, consequently,
\[ \tan\beta \simeq \mu \sin\alpha, \]
where \(\beta\) is the angle of inclination of the connecting rod to the cylinder axis.
Then
\[ N = mr\eta_1^2(\cos\alpha + \mu \cos 2\alpha)\cdot \mu \sin\alpha, \]
or, neglecting the terms with \(\mu^2\) as small,
\[ N = A \sin 2\alpha, \]
where
\[ A = \frac{1}{2}mr\eta_1^2\mu. \]
We expand this function in a Fourier series, considering all negative half-waves of the sinusoid \(A\sin 2\alpha\) to be positive. Then we obtain
\[ N = \frac{2A}{\pi} - \frac{4}{3}\frac{A}{\pi}\cos 4\alpha - \frac{4A}{15\pi}\cos 8\alpha - \ldots \]
We shall restrict ourselves to the first two terms of the expansion
\[ N = \frac{2}{\pi}A - \frac{4}{3\pi}A\cos 4\alpha. \]
Now
\[ R_N = \frac{2}{\pi}fA - \frac{4}{3\pi}fA\cos 4\alpha \]
and
\[ M_N = \frac{R_N \cdot V_n}{\eta_1} = \left(\frac{2}{\pi}Af - \frac{4}{3\pi}fA\cos 4\alpha\right)\cdot r \cdot \left(\sin\alpha + \frac{1}{2}\mu\sin 2\alpha\right) \]
or
\[ M_N = \frac{2}{\pi}fAr\sin\alpha + \frac{A}{\pi}\mu fr\sin 2\alpha - \]
\[ -\frac{4}{3\pi}frA\sin\alpha\cos 4\alpha -\frac{2}{3\pi}\mu frA\sin 2\alpha\cos 4\alpha. \]
Thus, here again harmonic moments of the first, second, and higher orders appear. The damping effect of these moments can be taken into account analogously to the preceding case.
References
- F. Lewis. Torsional Vibration in the Diesel Engine, Trans. Soc. Nav. Arch., vol. 33, 1925.
- H. Wydler. Drehschringungen in Kolbenmaschinenanlagen, 1922.
- I. Shannon. Damping influences in torsional oscillation, Proc. IME, vol. 131, 1935.
- P. Draminsky. Craankshaft damping, Proc. IME, vol. 159, No. 46, 1948.
- I. Geiger. Dämpfung bei Drehschwingungen vor Motoren, ZdVDJ, vol. 78, 1934.
- A. T. Bolgov. On energy balances in torsional systems of motor installations with D.V.S. Proceedings of the Altai Institute of Agricultural Machine Building, Barnaul, 1957.