On the Determination of Hysteresis Energy Losses in Shafts

V. K. Nechaev, A. T. Bolgov

In a number of power installations with piston internal-combustion engines (for example, in marine installations with a two-node form of vibration), the hysteresis of the shafting material is the main source of damping of torsional vibrations.

In calculations of resonant torsional vibrations of the shafts of such installations, hysteresis energy losses are usually evaluated on the basis of data from the literature. A considerable number of works have been devoted to the study of hysteresis; these present both theoretical investigations and the results of experiments on determining hysteresis losses in specimens made of steels of certain specific grades. However, for most steels currently used for the manufacture of shafts, such data are not available. Under these conditions, a reliable estimate of hysteresis losses in a shaft proves difficult.

Similar difficulties were encountered by the authors in connection with the need for a sufficiently accurate estimate of hysteresis losses in comparatively small smooth cylindrical shafts of an experimental motor installation. For the experimental determination of these hysteresis losses, a special apparatus was built in the internal-combustion-engine laboratory of the Tomsk Polytechnic Institute.^1

The purpose of the present article is to describe the apparatus and to set forth the method used for processing the experimental results.

The method adopted by the authors for determining the coefficients of hysteretic damping is based on an analysis of the curves of free damped torsional vibrations of the shaft under investigation.

The shaft under investigation (Fig. 1) was suspended in a vertical position. Its upper end was fastened to a rigid frame built into the wall. At the lower end of the shaft a heavy disk 1 (hereinafter called the flywheel) with anchors 3 was attached (Fig. 2). To excite torsional vibrations

Fig. 1

^1 Such an apparatus may also be used for fatigue tests.

of the shaft–flywheel system, two electromagnets were installed opposite the armatures, fixed rigidly on special brackets. The windings of the electromagnets were supplied with alternating current from a synchronous generator G, whose rotor was driven by a direct-current electric motor D (Fig. 2). As a result of the action of the electromagnets on the flywheel armatures, torsional oscillations were excited in the shaft–flywheel system, the frequency of which was equal to twice the frequency of the supply current. To obtain oscillation frequencies lower than the frequency corresponding to twice the minimum stable speed of the electric motor, a selenium rectifier was connected into the electromagnet circuit (Fig. 2). To obtain sufficiently strong torsional oscillations of the shaft, the frequency of the current supplying the electromagnets was selected (by a corresponding change in the speed of the motor-generator) so that resonant oscillations of the shaft–flywheel system were excited. The amplitudes of the resonant oscillations of the flywheel could be varied by changing the strength of the supply current, through regulation of the excitation of the synchronous generator.

Fig. 2

After the electromagnets were switched off by means of a knife switch, the established forced oscillations of the flywheel (with constant amplitude) passed into damped free oscillations, which could be recorded in the following manner. An induction pickup D of seismic type was fastened to the end of the shaft under the flywheel (Fig. 1). During torsional oscillations of the flywheel, an alternating emf was induced in the pickup, whose magnitude at any instant was proportional to the angular velocity of the flywheel oscillations. The output voltage of the pickup was fed to an integrating circuit (Fig. 3); the output voltage of the latter, proportional to the angular displacements of the flywheel (or to the twist of the shaft), was fed to the input of a low-frequency amplifier and then to the loop oscillograph MPO-2.

Fig. 3

As an example, Fig. 4 shows one of the oscillograms of the damped vibrations of the system in Fig. 1 obtained in this way. On the oscillogram, time marks are also applied by means of the second loop of the oscillograph; these are a recording of an alternating voltage with a frequency of 100 Hz from the audio-frequency generator ZG-2A (Fig. 3).

Fig. 4

To determine the scale of the recording of the vibration amplitudes (taking into account the gain of the entire electrical channel), an optical device was used, consisting of an illuminator 1, a mirror 2 fixed on the shaft 3 being tested (near the flywheel), and a stationary scale 4 (Fig. 5). When steady vibrations of the shaft–flywheel system had been established, an oscillogram was recorded as described above, and simultaneously the amplitude of the flywheel vibrations was measured by the optical method. By comparing the results of measuring the vibration amplitudes \(\Phi\) by the optical method with the vibration amplitudes on the oscillogram, the required scale of the vibration recording by the oscillograph was determined. To establish the dependence of the recording scale of the vibrations on the vibration amplitude, such experiments were also carried out at different amplitudes of the flywheel.

Fig. 5

As is known, the hysteresis work \(dA_z\) in an elementary volume \(dV\) of the shaft material during one vibration cycle can be represented by the expression

\[ dA_z = s \tau^q dV, \tag{1} \]

where

\[ \begin{aligned} \tau&\text{ — the maximum shear stress arising in this elementary volume during a vibration cycle,}\\ s&\text{ — the true hysteresis constant,}\\ q&\text{ — the hysteresis exponent.} \end{aligned} \]

The quantities \(s\) and \(q\) depend chiefly on the material of the shaft and its heat treatment, and to a lesser degree on the preceding conditions under which the shaft operated.

By integrating (1) it is easy to obtain an expression for the energy loss \(A_2\) due to hysteresis in a straight cylindrical shaft. This expression for \(A_2\), as applied to the case of oscillations of the shaft in the experimental apparatus of Fig. 1, with a certain amplitude \(\Phi\) of oscillation of the lower end of the shaft (at the flywheel), may be written in the form:

\[ A_2 = S\left(\frac{c}{W_p}\right)^q V \Phi^q \tag{2} \]

or, in a somewhat different form,

\[ A_2 = a\left(\frac{c}{W_p}\right)^q \Phi^q d^2 l, \tag{3} \]

where

\[ S=\frac{2s}{2+q}, \]

\[ c=\frac{GI_p}{l}, \]

\[ W=\frac{\pi d^3}{16}. \]

Here

\(S\) and \(a\) are hysteresis constants, related to one another by the evident relation

\[ S=\frac{4}{\pi}a, \]

\(G\) is the modulus of the second kind for the shaft material,

\(c\) is the stiffness of the shaft being tested,

\(I_p\) and \(W_p\) are, respectively, the polar moment of inertia and the section modulus of the shaft cross section;

\(d\) and \(l\) are the diameter and length of the shaft (in deriving expressions (2) and (3), the influence of the shaft’s own mass on the form of the elastic curve of the shaft during its torsional oscillations was not taken into account).

The hysteresis characteristics \(S\), \(a\), and \(q\) entering into (2) and (3) can be determined experimentally in the following way.

The energy dissipated during one \(n\)-th cycle of the damped oscillations of the flywheel in the apparatus of Fig. 1 is determined as the difference of the potential energies of the system at the beginning and at the end of the cycle:

\[ \Delta \Pi_n=\Pi_n-\Pi_{n+1}=\frac{c}{2}\left(\Phi_n^2-\Phi_{n+1}^2\right). \]

Neglecting all other energy losses in the system (apart from hysteresis losses), one may write¹

\[ A_2 \simeq \Delta \Pi_n=\frac{c}{2}\left(\Phi_n^2-\Phi_{n+1}^2\right). \]

The successive decreasing values \(\Phi_0, \Phi_1, \Phi_2 \ldots\) of the amplitudes of the free damped oscillations of the flywheel are easily measured directly from the recorded oscillograms (Fig. 6), taking into account the previously determined scale of the recording.

Now one can calculate the values \(\Delta \Pi\) for a number of subsequent cycles

\[ \Delta \Pi_0=\frac{c}{2}\left(\Phi_0^2-\Phi_1^2\right), \]

\[ \Delta \Pi_1=\frac{c}{2}\left(\Phi_1^2-\Phi_2^2\right) \]

\[ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot \]

\[ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot \]

and present these results on a graph (Fig. 7) in the coordinates \(\Delta \Pi\) (or \(A_2\)) and \(\Phi_n\), or else on a graph (Fig. 8) in the coordinates \(\lg A_2\) and \(\lg B_2\). Here, in application to (2), the notation adopted is

\[ B_2=\frac{c}{W_p}\Phi_n. \]

Fig. 6

Fig. 7

If, for the material of the shaft being tested, the basic expressions (2) and (3) are valid, then all the points on the graph (Fig. 8) will lie on a single inclined straight line described by the equation

\[ \lg A_2 = \lg(SV) + q \lg B_r, \]

which is readily obtained from (2) or (3). In our experiments (Fig. 8), these expressions were fully confirmed.

Fig. 8.

Now, with a rectilinear character of the dependence of \(\lg A_2\) on \(\lg B_r\), the hysteresis exponent \(q\) is determined as the tangent of the angle of inclination of the straight line \(\lg A_2 = f(\lg B_r)\) to the abscissa axis, while the hysteresis constant \(S\) (or \(a\)) is calculated from measurement of the intercept cut off by this straight line on the ordinate axis and on the scale giving the quantity \(\lg(SV)\).

On the setup described above, tests were carried out, in particular, on a smooth cylindrical shaft with dimensions

\[ d = 3.0\ \text{cm}, \qquad l = 89.5\ \text{cm}, \]

made of mild normalized steel of composition \(0.07\%\) C; \(0.13\%\) Mn; \(0.15\%\) Si.

The stiffness of the shaft was \(c = 0.712 \cdot 10^{5}\ \text{kgcm}\). The frequency of the free torsional oscillations of the shaft–flywheel system (Fig. 1) was \(36.8\ \text{Hz}\), or \(2208\) cycles/min.

In the experiments described, the initial amplitudes of the flywheel oscillations were set in the interval from 10 to 75 minutes, which corresponded to amplitudes of the shear stresses in the shaft ranging from 40 to \(300\ \text{kg}/\text{cm}^{2}\). The oscillograms were measured with the aid of a toolmaker’s microscope.

To estimate the magnitude of the air resistance during oscillations of the flywheel, the authors used a procedure previously applied for an analogous purpose in the work of E. Luntz [2]. The armatures of the flywheel (Fig. 2) were fitted with wings made of thin tinplate. These wings had the same shape and were located at the same distance from the axis of the flywheel as the armatures themselves. Therefore the wings experienced air resistance of the same character as the armatures. Three pairs of wings were made, of equal length (in the radial direction) and of different height. When these wings were attached, the frontal area of the armatures was increased, respectively, by one and a half, two, and two and a half times. Four oscillograms of the damped oscillations of the flywheel were taken with the same initial amplitude \(\Phi_0\): without wings on the armatures and with each of the pairs of wings. As a result of processing the oscillograms by the method described above, four graphs analogous to Fig. 7 were obtained. These graphs proved to coincide completely with one another (within the accuracy of the experiments), which indicates the negligibly small role of air resistance in the damping of the free oscillations in the experimental setup.

By processing the graph in Fig. 8, the following values of the hysteresis characteristics were obtained for the material of the shaft under investigation:

\[ q = 2.5, \]

\[ S = 17.4 \cdot 10^{-10}\ \frac{\text{kgcm}}{\text{cm}^{3}}, \qquad a = 13.7 \cdot 10^{-10}\ \frac{\text{kgcm}}{\text{cm}^{3}} . \]

In the literature known to us there are no data on the coefficients of hysteresis damping for the steel investigated by us. Comparison of the results of our experiments with the data of other investigators is therefore difficult, especially since a strong dependence of the hysteresis indices on the composition and heat treatment of steel has long been established [3]. For a rough comparison we can use only the data of Lewis, who, for steel of composition \(0.17\%\) C; \(0.24\%\) Mn; \(0.02\%\) Si, gives, on the basis of Rowett’s experiments, the following figures [4]:

\[ q = 2.3, \]

\[ a = 15.85 \cdot 10^{-10} \quad \text{or} \quad S = 20.2 \cdot 10^{-10}. \]

It is evident that, for a reliable estimate of the hysteresis losses in the shaft of a designed installation, it is necessary to determine experimentally the coefficients \(S\), \(a\), and \(q\) for the steel from which the shaft of this installation is made, for example, by the method set forth in the present article.

References

1. K. T. Shatalov. Questions of experimental investigations of torsional vibrations of engine shafts. In: Dynamics and Strength of Crankshafts. Academy of Sciences of the USSR, Moscow–Leningrad, 1948.

2. E. B. Lunts. On damping in torsional vibrations. PMM, vol. I, no. 3, 1938.

3. S. Dorey. Elastic Hysteresis in Crankshaft Steels. Proc. Inst. Mech. Eng., Vol. 123, 1932.

4. F. Lewis. Torsional Vibration in the Diesel Engine. Trans. Soc. Nav. Arch. and Mar. Eng., Vol. 33, 1925.

1. To obtain sufficiently reliable values of the quantities \(S\), \(a\), and \(q\), it is necessary, in particular, to ensure as rigid a fastening as possible of the upper end of the shaft (Fig. 1). ↩