ANALYSIS OF DAMPING FORCES IN A MOTOR INSTALLATION BY THE INTERCHANGEABLE FLYWHEEL METHOD

V. K. NECHAEV

In the author's article [1] it was shown that, in order to improve the accuracy and reliability of modern factory calculations of resonant torsional vibrations of shafts in motor installations with piston engines, it is necessary to accumulate experimental data on the damping forces in these installations.

But it was also noted there that these experimental data can be useful in calculations only if they separately characterize at least the principal types of damping in the installation.

Among such principal types of damping in installations of the multicylinder engine–flywheel type are viscous (linear) damping in the engine mechanism and hysteretic (nonlinear) damping in the shaft of the entire installation, including the engine shaft. Viscous damping in the mechanism of each engine cylinder is characterized by the coefficient \(k\), and hysteretic damping—in the material of the installation shaft—by the hysteresis constant \(a\) and the hysteresis exponent \(q\)¹).

However, the experimental determination, separately, of the coefficients \(k\), \(a\), and \(q\) in any given specific motor installation is a very difficult experimental problem, requiring the performance of many complex tests with the installation as a whole and with its individual units and elements. The processing of the experimental results presents no less difficulty.

Below, a comparatively simple method is proposed for the separate experimental determination of the coefficients \(k\), \(a\), and \(q\) for any specific motor installation, based on processing the results of three torsiograph recordings of this installation at critical speeds of the same order, but each time with a different (interchangeable) flywheel. These flywheels must have different moments of inertia, but, if possible (in order to keep the loads on the bearings unchanged), the same weight.

Obviously, when the flywheel is changed, the frequency \(\omega\) and the form of the free vibrations of the shaft will change, and the critical engine speeds will also shift; in connection with the shift of the latter, the amplitudes \(M_h\) of the resonant harmonic exciting moments on the shaft may also change.

Consequently, the three different values of the moment of inertia \(\Theta'\), \(\Theta''\), \(\Theta'''\) of the interchangeable flywheel correspond to their own values of the frequency of free vibrations of the shaft \(\omega'\), \(\omega''\), \(\omega'''\), the absolute amplitude of vibrations of the first mass \(\Phi'_{1abc}\), \(\Phi''_{1abc}\), \(\Phi'''_{abc}\), the amplitudes of the resonant moments \(M'_h\), \(M''_h\), \(M'''_h\), as well as their own values of the coefficients \(\lambda'\), \(\lambda''\), \(\lambda'''\), \(\mu'\), \(\mu''\), \(\mu'''\) and \(\nu'\), \(\nu''\), \(\nu'''\). The coefficients \(k\), \(a\), and \(q\), however, determined by the characteristics of the engine and the material of the shaft of the motor installation, will remain unchanged.

¹) We retain here in full the notation adopted in article [1]. The necessary additional explanations will be given below.

Now, for each of these three cases one can also write the corresponding equation of the energy balance under resonant oscillations of the motor installation under consideration:

\[ \left. \begin{aligned} \lambda' M_h' \Phi'_{1\mathrm{abs}} &= \mu' k\omega' {\Phi'_{1\mathrm{abs}}}^{2}+\nu' a{\Phi'_{1\mathrm{abs}}}^{q},\\ \lambda'' M_h'' \Phi''_{1\mathrm{abs}} &= \mu'' k\omega'' {\Phi''_{1\mathrm{abs}}}^{2}+\nu'' a{\Phi''_{1\mathrm{abs}}}^{q},\\ \lambda''' M_h''' \Phi'''_{1\mathrm{abs}} &= \mu''' k\omega''' {\Phi'''_{1\mathrm{abs}}}^{2}+\nu''' a{\Phi'''_{1\mathrm{abs}}}^{q}. \end{aligned} \right\} \tag{1} \]

The simultaneous solution of equations (1) should give us the required values of the coefficients \(k\), \(a\), and \(q\) in the given installation.

From (1)

\[ \left. \begin{aligned} B'&=k+\frac{\nu'a}{\mu'\omega'}\,{\Phi'_{1\mathrm{abs}}}^{q-2},\\[4pt] B''&=k+\frac{\nu''a}{\mu''\omega''}\,{\Phi''_{1\mathrm{abs}}}^{q-2},\\[4pt] B'''&=k+\frac{\nu'''a}{\mu'''\omega'''}\,{\Phi'''_{1\mathrm{abs}}}^{q-2}, \end{aligned} \right\} \tag{2} \]

where it is denoted that

\[ \left. \begin{aligned} B'&=\frac{\lambda' M_h'}{\mu'\omega'\Phi'_{1\mathrm{abs}}},\\[4pt] B''&=\frac{\lambda'' M_h''}{\mu''\omega''\Phi''_{1\mathrm{abs}}},\\[4pt] B'''&=\frac{\lambda''' M_h'''}{\mu'''\omega'''\Phi'''_{1\mathrm{abs}}}. \end{aligned} \right\} \tag{3} \]

Now from (2)

\[ \left. \begin{aligned} B'-B''&=\frac{\nu'a}{\mu'\omega'}\,{\Phi'_{1\mathrm{abs}}}^{q-2} -\frac{\nu''a}{\mu''\omega''}\,{\Phi''_{1\mathrm{abs}}}^{q-2},\\[4pt] B'-B'''&=\frac{\nu'a}{\mu'\omega'}\,{\Phi'_{1\mathrm{abs}}}^{q-2} -\frac{\nu'''a}{\mu'''\omega'''}\,{\Phi'''_{1\mathrm{abs}}}^{q-2}. \end{aligned} \right\} \tag{4} \]

We introduce a new coefficient

\[ C=\frac{B'-B''}{B'-B'''}, \tag{5} \]

which, according to (3), is determined by the expression

\[ C= \frac{ 1-\left(\dfrac{\Phi''_{1\mathrm{abs}}}{\Phi'_{1\mathrm{abs}}}\right)^{q-2} \cdot\dfrac{\nu''}{\nu'}\cdot\dfrac{\mu'\omega'}{\mu''\omega''} }{ 1-\left(\dfrac{\Phi'''_{1\mathrm{abs}}}{\Phi'_{1\mathrm{abs}}}\right)^{q-2} \cdot\dfrac{\nu'''}{\nu'}\cdot\dfrac{\mu'\omega'}{\mu'''\omega'''} } \]

or

\[ C= \frac{ 1-\beta_{12}\delta_{21}^{q-2}\cdot\dfrac{\nu''}{\nu'} }{ 1-\beta_{13}\delta_{31}^{q-2}\cdot\dfrac{\nu'''}{\nu'} }. \tag{6} \]

Here the following notations have been adopted:

\[ \left. \begin{aligned} \delta_{21}&=\frac{\Phi''_{1abc}}{\Phi'_{1abc}};\quad \delta_{31}=\frac{\Phi'''_{1abc}}{\Phi'_{1abc}},\\[4pt] \beta_{12}&=\frac{\mu'\omega'}{\mu''\omega''};\quad \beta_{13}=\frac{\mu'\omega'}{\mu'''\omega'''} \end{aligned} \right\}. \tag{7} \]

Let us rewrite expression (6) in the form

\[ C\left(1-\beta_{13}\delta_{31}^{q-2}\cdot\frac{\nu'''}{\nu'}\right) = 1-\beta_{12}\delta_{21}^{q-2}\frac{\nu''}{\nu'} \tag{8} \]

or

\[ C\gamma_{13}=\gamma_{12}, \]

where

\[ \left. \begin{aligned} \gamma_{12}&=1-\beta_{12}\delta_{21}^{q-2}\cdot\frac{\nu''}{\nu'},\\[4pt] \gamma_{13}&=1-\beta_{13}\delta_{31}^{q-2}\cdot\frac{\nu'''}{\nu'} \end{aligned} \right\}. \tag{9} \]

Expressions (8), (9) are the basis for calculating the desired damping characteristics \(k\), \(a\), and \(q\) in the motor installation from torsiographing data for the installation shaft with three different interchangeable flywheels.

According to (5), the coefficient \(C\) depends on the coefficients \(B'\), \(B''\), and \(B'''\). But, as expressions (3) show, these coefficients may be regarded as known. Indeed, the absolute amplitudes of the resonant oscillations of the first mass of the installation, \(\Phi'_{1abc}\), \(\Phi''_{1abc}\), and \(\Phi'''_{1abc}\), which enter into (3), are readily determined from the recorded torsiograms; the amplitudes of the resonant exciting harmonic moments \(M_h\) can be found by known methods of harmonic analysis of the torque curve of one engine cylinder, constructed from the results of indicator diagrams. The quantities \(\omega'\), \(\omega''\), and \(\omega'''\) are found by ordinary calculations and, if necessary, are checked by torsiographing. Finally, the coefficients \(\lambda\) and \(\mu\) entering into (3) are readily determined from the formulas

\[ \lambda=\pi\sum \varphi_i, \]

\[ \mu=\pi Fr^2\sum \varphi_i^2, \]

where \(\sum \varphi_i\) is the geometric sum of the relative amplitudes of the free vibrations of the reduced masses of the engine cylinders, determined with the aid of a phase diagram of the exciting moments,

\[ \sum \varphi_i^2 \]

is the algebraic sum of the squares of the relative amplitudes of these same masses,

and \(F\) and \(r\) are the piston area and the crank radius.

Thus the coefficients \(B'\), \(B''\), and \(B'''\) will be known from (3) in their numerical value; correspondingly, from (5) the numerical value of the coefficient \(C\) in equation (8) will also be determined. The coefficients \(\beta_{12}\), \(\beta_{13}\), \(\delta_{21}\), \(\delta_{31}\) are likewise calculated without particular difficulty on the basis of formulas (7).

Now in (8) there remains one unknown quantity \(q\), which enters it both explicitly (in the exponent of \(\partial\)) and implicitly through the quantities \(\nu'\), \(\nu''\), \(\nu'''\). The latter are determined from expressions of the form¹ [1]

\[ \nu=\sum \left[(\varphi_{i+1}-\varphi_i)\frac{c_{i,i+1}}{w_p}\right]^q d^2 l, \tag{10} \]

where \(c_{i,i+1}\) is the stiffness of the shaft in the section between the adjacent (\(i\)-th and \(i+1\)-st) masses of the installation,

\[ w_p \]

is the polar section modulus of the shaft of the installation in the given section,

and \(d\) and \(l\) are the diameter and length of each shaft section.

The root of equation (8), that is, the desired value of the hysteresis exponent \(q\) for the shaft of the given motor installation, can be found, for example, graphically, as the abscissa of the point of intersection of two curves \(\gamma_{12}\) and \(C\gamma_{13}\) as functions of \(q\) (Fig. 1). These curves are plotted from points computed by expressions (9) for a number of arbitrarily adopted values of the exponent \(q\). For the same values of \(q\), the quantities \(\nu'\), \(\nu''\), and \(\nu'''\) entering (9) are calculated by formulas (10).

Fig. 1

After finding the exponent \(q\), one can determine another characteristic of hysteresis losses—the constant \(a\). For this it is sufficient to use one of expressions (4). Thus, from the first of them,

\[ a= \frac{B'-B''} {\dfrac{\nu'}{\mu'\omega'}\,\Phi_{1abc}'^{\,q-2} - \dfrac{\nu''}{\mu''\omega''}\,\Phi_{1abc}''^{\,q-2}} . \]

All the quantities entering the right-hand side of this expression are already known.

Finally, from any of expressions (2) the coefficient \(k\) is also determined. For example,

\[ k=B'-\frac{\nu'a}{\mu'\omega'}\,\Phi_{1abc}'^{\,q-2}. \]

But here it is necessary to note that the accuracy of the results obtained for determining the coefficients \(k\), \(a\), and \(q\) by the method set forth above will depend to a considerable extent both on the accuracy of measurement of the resonance amplitudes and the accuracy of the computations, and on the degree to which the basic assumptions adopted above are valid for the given installation. These basic assumptions consisted in assuming that, in the motor installation, there is only viscous (linear) damping in the engine mechanism and hysteresis losses in the shaft of the installation.

The invalidity of such an assumption for the particular motor installation in question will be indicated, for example, by a considerable difference in the numerical values of the coefficient \(k\) found from each of formulas (2), greater than the natural difference caused by the inevitable errors in measuring the resonance amplitudes, calculating the frequencies \(\omega\), determining the amplitudes of the moments \(M_h\), etc. If such a large difference is found, a special investigation will be necessary in order to clarify the magnitudes and the character of the energy losses not taken into account in equations (1).

In some cases the invalidity of the assumption noted above is already obvious in advance; for example, in the case of a marine motor installation operating on a propeller. Here the propeller may be a source of considerable additional damping of shaft vibrations. In this case, to the right-hand side of each of equations (1) it is necessary to add the work absorbed by the damping of the propeller, determined by known expressions of the form

\[ A_{\text{propeller}}=\pi k_v \omega \varphi_v^2 \Phi_{1\text{abs}}^2, \]

where \(k_v\) is the damping coefficient of the propeller,

\[ \varphi_v \]

is the relative amplitude of the propeller vibrations (for \(\varphi_1=1\)).

But if the value of \(k_v\) is known in advance (for example, from other tests or on the basis of published experimental relationships), these additional terms in the right-hand sides of equations (1) will be completely known in their numerical values. The method of simultaneous solution of equations (1) will now differ little from that described above.

In conclusion, we note that the proposed method of separate determination of the coefficients \(k\), \(a\), and \(q\) from torsiographing data can be noticeably simplified if, in the given motor installation (with one and the same flywheel), it proves possible to observe and torsiograph three sufficiently strong resonance peaks. Then

\[ \nu'=\nu''=\nu''', \]

provided only that all these three peaks correspond to one and the same mode shape of free shaft vibrations (for example, with one node). Here equation (8) is simplified and takes the form

\[ C\left(1-\beta_{13}\nu_{31}^{q-2}\right)=1-\beta_{12}\nu_{21}^{q-2}. \]

The graphical solution of this equation no longer presents any substantial difficulties and gives the required value of the exponent \(q\). The two other coefficients \(a\) and \(k\) are determined as before. Thus, the need for interchangeable flywheels is eliminated here.

Finally, the possibility is not excluded of determining the coefficients $k$, $a$, and $q$ from the results of threefold torsiographing at one and the same critical speed, with one and the same flywheel, but under three different engine loads. Here the computational procedure is simplified still further.

REFERENCES

1. Nechaev V. K. On the question of the characteristics of damping forces in motor installations with piston internal-combustion engines. Proceedings of the Tomsk Polytechnic Institute, vol. 85, 1957.

1. We assume here that the entire shaft of the motor installation is made of one steel, with values of the coefficients \(a\) and \(q\) constant along the entire length of the shaft. ↩