ON THE DEPENDENCE OF THE CHARACTERISTICS OF DAMPING FORCES IN MOTOR INSTALLATIONS ON THE MOMENT OF INERTIA OF THE FLYWHEEL

A. T. BOLGOV

(Presented by Prof. V. K. Nechaev, Doctor of Technical Sciences)

The forces damping the torsional vibrations of the shafts of motor installations with reciprocating internal-combustion engines are evaluated by means of various damping coefficients.

In the existing literature on torsional vibrations, as a rule, the numerical values of damping coefficients are made dependent only on the type of installation (automobile, marine, etc.) and on the general (principal) form of the curve of resonant vibrations (single-node, two-node vibrations, etc.).

V. K. Zhitomirsky [1] indicates that the values of damping coefficients according to Wydler also depend on the order of the resonant vibrations.

V. K. Nechaev [2] showed that the numerical values of the generalized (total) damping coefficients according to the theories of Wydler and Lewis depend on many factors, in particular on the magnitudes of the moments of inertia of the masses of the installation (the design parameters of the system) and on the order of the resonant vibrations.

To investigate some of these dependences, experiments were conducted by the author, under the supervision of Professor-Doctor V. K. Nechaev, on an experimental installation created in the internal-combustion-engine laboratory of the Tomsk Polytechnic Institute.

The purpose of the present article is to set forth some results of these experiments.

The torsional diagram of the experimental installation is shown in Fig. 1.

Fig. 1

The installation described consisted of: a direct-current electric motor (with shunt excitation), a single-cylinder engine of type L-3, and a replaceable flywheel. The replaceable flywheel was connected to the engine by means of a long shaft (section 3—4 in Fig. 1). The long shaft was used in order to reduce the natural frequency of vibrations of the installation shaft to values corresponding to the natural frequencies of the shafts of marine installations with multi-cylinder engines.

Torsional vibrations in the installation described were excited by the inertia forces of the reciprocating masses of the engine crank mechanism. With such “inertial” excitation of vibrations, there is no need to manufacture a special exciter, and it becomes possible to determine analytically with precision the harmonic components of the exciting torque. The principal shortcoming of inertial excitation is that with it one can obtain sufficiently large harmonics of the exciting torque only of the first three or four orders. In our case we confined ourselves to analyzing vibrations of the second \((h=2)\) and third \((h=3)\) orders.

The cylinder head of the engine was removed. The shaft of the installation was driven in rotation by an electric motor. The speed of the electric motor could be regulated over a wide range by means of rheostats connected into the armature and field circuits of the electric motor.

Three interchangeable flywheels of different sizes were manufactured. The flywheels were assigned conventional indices: the smaller—II, the medium—III, and the larger—IV. Any one of these flywheels could be installed and securely fastened at the right-hand end of the long shaft 3—4 (Fig. 1).

Thus it became possible to carry out an experimental study of vibrations in three test installations that differed from one another only in the magnitude of the moment of inertia \(\Theta_4\) of the right-hand mass (the interchangeable flywheel) in Fig. 1. Below these installations are denoted by the indices II, III, IV, corresponding to the indices of the flywheels installed in them.

In all the test installations the following parameters remained constant (Fig. 1):

The values of the moments of inertia of the interchangeable flywheels \(\Theta_4\) and of the stiffnesses \(C_{3-4}\) of section 3—4 of the shaft are given in Table 1. Also given there are the values of the angular frequencies of the free vibrations of the test systems \(\omega\) and the corresponding calculated resonant speeds \(n_{\text{res}}\) (for single-node resonant vibrations of the 2nd and 3rd orders). The frequencies of the free vibrations \(\omega\) were calculated in the usual way by the Tolle method.

Each of the installations II, III, IV was subjected to torsiographing over a whole series of engine-shaft rotational speeds. The engine speeds (i.e., the electric-motor speeds) were selected in a sufficiently wide interval so as to make it possible to record the critical (resonant) speeds of the installation shaft corresponding to resonances with the harmonic components of the second and third orders of the inertial exciting torque on the engine shaft.

When torsiographing the installation, an electric torsiograph with a seismic-type induction pickup¹ was used, with recording of vibrations by means of cathode-ray and loop oscillographs, as well as a Geiger mechanical torsiograph with belt drive.

The pickup of the electric torsiograph was installed at the end of shaft 3—4 near the interchangeable flywheel. Thus, the torsiographed section of the installation shaft was the section at mass 4.

Table 1

From the results of processing the torsiograms (for each installation separately), resonance curves were constructed. From the maxima of these curves, the resonant amplitudes of oscillations of the 4th mass, $\Phi_{4\text{abs}}$, and the corresponding critical rotational speeds of the engine shaft were determined.

Table 2 gives the experimental values of the resonant speeds $n_{\text{res}}$ and amplitudes $\Phi_{4\text{abs}}$, as well as the values of the resonant oscillation amplitudes of the first, $\Phi_{1\text{abs}}$, and third (cylinder), $\Phi_{3\text{abs}}$, masses of the installation, calculated in the usual way from the known values of $\Phi_{4\text{abs}}$ and the mode-shape curve of the free oscillations of the shaft.

Table 2

From Tables 1 and 2 it is seen that the discrepancy between the experimental and calculated values of the resonant speeds does not exceed 3%. Therefore, in what follows, the calculated values of the resonant speeds were used.

The data in Tables 1 and 2 make it possible, by means of the corresponding calculations, to determine the experimental values of the damping coefficients in the experimental installations described above. The results of these calculations are given in Table 3.

The calculation procedure naturally depends on the adopted method for estimating and taking into account damping forces in motor installations. Below are the relations used by the author, following from the methods presently known for calculating the amplitudes of resonant torsional vibrations.

The damping coefficients \(k_0\) according to Wydler’s method were calculated from the expression \([3,6]^{1)}\):

\[ k_0=\frac{M_h}{F r^2 \Phi_{1abc}\omega \varphi_3}\ \frac{\text{kg sec}}{\text{cm}^3}, \]

where \(M_h\) is the amplitude of the harmonic exciting moment of the \(h\)-th order,

\(\varphi_3\) is the relative vibration amplitude of the 3rd mass,

\(F\) is the piston area (in our case \(F=28.3\ \text{cm}^2\));

\(r\) is the crank radius (in our case \(r=4.5\ \text{cm}\)).

The amplification coefficients \(\beta\) according to the Kerr Wilson method were calculated from the expression \([4]\)

\[ \beta=\frac{\Phi_{1abc}}{\Phi_{01}}, \]

where

\[ \Phi_{01}=\frac{M_h\cdot \varphi_3}{\omega^2\cdot \displaystyle\sum_{i=1}^{4}\Theta_i\varphi_i^2} \]

is the equilibrium amplitude of the first mass of the system.

The damping coefficients \(D\), according to V. K. Zhitomirskii, were determined by the formula \([1]\)

\[ D=\frac{1}{2\beta}. \]

To determine the amplification coefficients \(\beta_h\), according to I. Sh. Neiman, the formula \([5]\) was used

\[ \beta_h=\frac{\Phi_{1abc}}{\Phi_{1st}}, \]

where

\[ \Phi_{1st}=\frac{M_h}{c_{03}} \]

is the “static” amplitude of the first mass of the system;

\(c_{03}\) is the torsional stiffness of the shaft segment between the node and the 3rd mass of the system.

In calculating the hysteretic-damping coefficients \(a\) according to Lewis, the expression \([3]\) was used (with the conventionally adopted value of the exponent \(q=3\)):

\[ a_0=\frac{\pi M_h\varphi_3}{\varphi_{1abc}^{q-1}\cdot \nu}, \]

where

\[ \nu=\frac{4}{\pi}\sum_{i=1}^{3}V_i\left(\frac{16\cdot \Delta\varphi_i\cdot c_{i,i+1}}{\pi d^3}\right)^q \]

is a constant of the system (calculated

\(^{1)}\) These and the expressions given below are written for our case: a single-cylinder engine.

when calculating the frequencies of free vibrations and the elastic modes).
Here: \(V_i\) is the volume of the shaft section between masses \(i\) and \(i+1\),

\[ \Delta \varphi_i=\varphi_{i+1}-\varphi_i \]

is the relative twist of this shaft section,

\(c_{i,i+1}\) is the stiffness of the shaft section between masses \(i\) and \(i+1\),
\(d\) is the diameter of the shaft section,
\(q\) is the hysteresis exponent.

The amplitudes of the inertial harmonic exciting moments at the foundation [6] were determined from the expression:

\[ M_h=b\cdot \eta^2\ \text{kg cm}, \]

where

\[ \eta=\frac{\omega}{h} \]

is the mean angular velocity of the shaft corresponding to the resonant speed,

\(b\) is a coefficient taking into account the masses of the connecting-rod–crank mechanism of the cylinder and depending on the order of the moment; in our case, for the second-order harmonic \(b=54.69\cdot 10^{-4}\ \text{kg cm sec}^2\), and for the third-order harmonic \(b=20.5\cdot 10^{-4}\ \text{kg cm sec}^2\).

To characterize the degree of change in the moment of inertia of mass 4, Table 3 gives the ratios of the moment of inertia of mass 4 of the given installation to the moment of inertia of the same mass of installation II.

Table 3

From Table 3 it is seen that the damping coefficients depend on the magnitude of the moment of inertia of mass 4 of the installation and, in our experiments, proved to be different for vibrations of the second and third orders.

When the mass moment of inertia of the flywheel was increased by a factor of 3.8, the values of the damping coefficient according to Widder \(k_0\) decreased by a factor of 13.5, the values of the damping coefficient according to Lewis \(a\) decreased by a factor of 7.5; the values of the coefficients \(\beta\) and \(\beta_h\) increased by a factor of 5, etc.

The damping coefficients according to Widder \(k_0\) in resonant vibrations of the second order proved to be approximately 2.4 times greater than in vibrations of the third order; the amplification coefficients according to Ker Wilson \(\beta\) in resonant vibrations of the second order were approximately 2.4 times smaller than in vibrations of the third order, etc.

From these experimental data it follows that damping in motor installations of the same type, differing from one another only in the magnitude of the moment of inertia of one of the masses, can no longer be evaluated by a damping coefficient of one and the same magnitude.

The numerical value of the damping coefficient, even for a given specific installation, does not remain constant. It depends, for example, on the order of the resonant vibrations in the system.

References

1. Zhitomirskii V. K. Torsional vibrations of the shafts of aircraft piston engines. Oborongiz. Moscow, 1952.

2. 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.

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

4. Aircraft piston engines. Kinematics, dynamics, and strength calculation. Edited by I. Sh. Neiman and T. M. Melkumov. Oborongiz. Moscow, 1950.

5. Neiman I. Sh. Dynamics of aircraft engines. Oborongiz. Moscow–Leningrad, 1940.

6. Nechaev V. K. Theoretical torsion diagrams for the shaft of an internal-combustion engine. Proceedings of the Tomsk Industrial Institute, vol. 58, issue II, 1937.

7. Bolgov A. T. On certain characteristics of damping of torsional vibrations of installation shafts with piston internal-combustion engines. Candidate dissertation. TPI. Tomsk, 1956.

1. This torsiograph was developed and manufactured in the Internal Combustion Engines Laboratory of the Tomsk Polytechnic Institute. ↩