On the Experimental Determination of the Moments of Inertia of Moving Masses of Machines

S. I. Shubovich

When carrying out a number of dynamic calculations (for example, in studies of the process of acceleration of a machine, etc.), it becomes necessary to determine the actual moments of inertia of the moving masses of a machine, reduced to some axis of rotation. This problem, as is known, is solved by analytically or experimentally determining the moments of inertia of the individual parts and then reducing them to the specified axis of rotation. In the case when this generally accepted method is for some reason unacceptable (when there are no sufficiently detailed drawings of the parts and, at the same time, disassembly of the machine is undesirable), the reduced moment of inertia of all the moving masses of the machine can be determined by a method based on measurements of the run-down time of the machine; for brevity, this method will hereinafter be called the run-down method.

The possibility of determining the reduced moments of inertia of all the moving masses of a machine by the run-down method follows from analysis of the basic run-down equation:

\[ \Theta \frac{d\omega}{dt} + M = 0, \tag{1} \]

where \(\Theta\) is the total moment of inertia of all the moving masses of the machine, reduced to the axis of some shaft of the machine,

\(\omega\) is the angular velocity of rotation of this shaft,

\(M\) is the moment of resistance to rotation of the shaft during run-down (i.e., the moment causing the machine to stop), reduced to the axis of the same shaft.

During the run-down of a machine without external load, the resisting moment \(M\) will be the friction moment in the machine \(M_{tr}\), which in the general case may be represented as the sum of at least two terms: the moment of the forces of “viscous” friction and the moment of the forces of dry friction; therefore the equation of the friction moment is written in the form:

\[ M_{tr} = K \omega^n + M_c, \tag{2} \]

where \(K\) is a proportionality coefficient depending mainly on the load of the rubbing pairs and the viscosity of the oil, \(n\) is the exponent of the dependence of friction on the speed of motion, and \(M_c\) is the moment due to dry friction, i.e., friction forces not depending on the speed of motion.

After substituting expression (2) for the resisting moment into equation (1) and after some transformations, the differential equation of run-down can be represented in the following form:

\[ \frac{d\omega}{dt} + \frac{K}{\Theta}\omega^n + \frac{M_c}{\Theta} = 0. \tag{3} \]

The friction exponent \(n\) in the case of “viscous” friction (which will also be the most probable in machines with small ventilation losses) is equal to unity.

The solution of differential equation (3), for \(n=1\) and under the initial conditions \(t=0,\ \omega=\omega_0\), will have the following form:

\[ \omega=\left(\omega_0+\frac{M_c}{K}\right)e^{-\frac{k}{\Theta}t}-\frac{M_c}{K}, \tag{4} \]

where \(\omega_0\) is the initial run-down speed, \(t\) is the run-down time, i.e., the time during which the speed decreases from the value \(\omega_0\) to \(\omega\). Let us write expression (4) for the run-down of a machine with the reduced moment of inertia of the moving masses \(\Theta\), and for the same machine, under the same conditions, but with the artificially changed value of the moment of inertia of the masses to some value \(\Theta_1\).

\[ \left. \begin{aligned} \omega&=\left(\omega_0+\frac{M_c}{K}\right)e^{-\frac{k}{\Theta}t}-\frac{M_c}{K},\\[6pt] \omega&=\left(\omega_0+\frac{M_{c1}}{K_1}\right)e^{-\frac{k_1}{\Theta_1}t_1}-\frac{M_1}{K_1}. \end{aligned} \right\} \tag{5} \]

If the change in the moment of inertia of the moving masses of the machine from the value \(\Theta\) to \(\Theta_1\) does not cause qualitative or quantitative changes in the friction of the machine, then \(K_1=K\) and \(M_{c1}=M_c\), and the simultaneous solution of equations (5) gives us the following relation:

\[ \frac{t_1}{t}=\frac{\Theta_1}{\Theta}. \tag{6} \]

It is not difficult to notice that the presence in the machine of only viscous friction or only dry friction will not change the relation obtained. Moreover, if it is assumed that the friction in the machine is only velocity friction [1], obeying the law:

\[ M_{tr}=K_{\mathrm{e}}\omega^n, \]

where \(K_{\mathrm{e}}\) is the equivalent coefficient of friction, and \(n\) is the friction exponent, having the value \(0<n<1\), then in this case relation (6) will also be valid.

Indeed, the solution of equation (3) for \(M_c=0\) and \(0<n<1\) will be an expression of the following form

\[ \omega=\left[\omega_0^{\,1-n}-\frac{K_{\mathrm{e}}}{\Theta}(1-n)t\right]^{\frac{1}{1-n}}. \]

Therefore a system of equations analogous to system (5) is written as:

\[ \left. \begin{aligned} \omega&=\left[\omega_0^{\,1-n}-\frac{K_{\mathrm{e}}}{\Theta}(1-n)t\right]^{\frac{1}{1-n}},\\[6pt] \omega&=\left[\omega_0^{\,1-n_1}-\frac{K_{\mathrm{e}1}}{\Theta_1}(1-n_1)t_1\right]^{\frac{1}{1-n_1}}. \end{aligned} \right\} \]

Hence, for \(n = n_1\) and \(K_a = K_{a1}\), we obtain the previous relation (6), which shows that, with unchanged limits of the run-down speeds and unchanged friction conditions in the machine, the run-down time is always proportional to the reduced moment of inertia of the moving masses.

Under practical conditions, a change in the moment of inertia of the moving masses of a machine can be accomplished by fitting an additional mass onto the machine shaft, for example by fitting a flywheel or by attaching some other rotating parts.

In this connection, it is more convenient to write expression (6) in the following form

\[ \frac{t_1}{t}=\frac{\Theta+\Theta_g}{\Theta}, \]

where \(\Theta_g\) is the moment of inertia of the additional mass.

Hence:

\[ \Theta=\Theta_g \frac{1}{\dfrac{t_1}{t}-1} \tag{7} \]

Thus, in order to determine the reduced moment of inertia of a machine, it is necessary to put the machine into operation, bring the rotational speed of its shaft, to which the moment of inertia is reduced, up to some speed \(\omega_0\), disconnect the machine from the energy source, and measure the run-down time \(t\) to some speed \(\omega\)¹). Then the reduced moment of inertia of the moving masses should be changed and the experiment repeated to measure the run-down time over the same speed limits—\(t_1\).

From the experimentally obtained quantities \(t\), \(t_1\), and the known moment of inertia \(\Theta_g\)²), the moment of inertia \(\Theta\) is found from equation (7).

To illustrate what has been said, let us give an example. In determining the moment of inertia of the rotor of an electric motor by the method described above, the author obtained the following results:

a) the average run-down time of the rotor from the initial speed \(n_0 = 1480\) rpm to the final \(n_k = 0\), \(t = 8.9\) sec,

b) the average run-down time of the rotor with a pulley fitted on it (over the same speed limits) \(t_1 = 40.5\) sec,

c) the moment of inertia of the pulley (found by the method of oscillation on a bifilar suspension) \(\Theta_g = 0.45\ \text{kg}\cdot\text{cm}\cdot\text{sec}^2\).

Now the required moment of inertia of the rotor is determined from formula (7)

\[ \Theta=0.45 \frac{1}{\dfrac{40.5}{8.9}-1} =0.127\ \text{kg}\cdot\text{cm}\cdot\text{sec}^2. \]

In order to ascertain the reliability of the result obtained, the moment of inertia of the rotor was determined a second time by the method of oscillation on a bifilar suspension, and in this case its value proved to be \(0.126\ \text{kg}\cdot\text{cm}\cdot\text{sec}^2\). If this latter value of the moment of inertia is taken as the true value, then the error in determining the moment of inertia by the run-down method lies within 1%.

¹) In order to increase the accuracy of time measurements, it is advisable to set as large a speed range as possible.

²) If the moment of inertia of the additional mass is unknown, it is not difficult to determine it by the generally accepted method [2].

However, the run-down method cannot always provide such high accuracy. This can be seen from an analysis of the relative root-mean-square errors of the function

\[ \Theta = f(\Theta_g,\, t,\, t_1) \]

as a function of the relative errors of the arguments \(t\) and \(t_1\).

The root-mean-square error of a function of several variables is found, as is known [3], from the following relation:

\[ \sigma_y = \pm \sqrt{ \left(\frac{dy}{dx}\right)^2 \sigma_x^2 + \left(\frac{dy}{dz}\right)^2 \sigma_z^2 + \ldots + \left(\frac{dy}{dt}\right)^2 \sigma_t^2 }; \]

if

\[ y = f(x, z, \ldots t), \]

where \(\sigma_y, \sigma_x, \sigma_z, \ldots \sigma_t\) are the root-mean-square errors of the quantities \(y, x, z, \ldots t\). In our case this relation is written in the following form:

\[ \sigma_\Theta = \pm \sqrt{ \left(\frac{d\Theta}{d\Theta_g}\right)^2 \sigma_{\Theta g}^{\,2} + \left(\frac{d\Theta}{dt}\right)^2 \sigma_t^2 + \left(\frac{d\Theta}{dt_1}\right)^2 \sigma_{t_1}^{\,2} }. \]

Since the magnitude of the moment of inertia of the added mass \(\Theta_g\) can always be determined with negligibly small error, \(\sigma_{\Theta g}=0\), and then

\[ \sigma_\Theta = \pm \sqrt{ \left(\frac{d\Theta}{dt}\right)^2 \sigma_t^2 + \left(\frac{d\Theta}{dt_1}\right)^2 \sigma_{t_1}^{\,2} }. \]

Random errors in measuring time (with a properly operating stopwatch) do not depend on the magnitude of the measured time interval, since the error usually arises only because the observer is unable to record the required instant absolutely precisely by starting or stopping the stopwatch. In view of this, one may take \(\sigma_t=\sigma_{t_1}\). Then the equation for the root-mean-square error of the required moment is written in the following form:

\[ \sigma_\Theta = \pm \sqrt{ \left(\frac{d\Theta}{dt}\right)^2 + \left(\frac{d\Theta}{dt_1}\right)^2 }\cdot \sigma_t . \]

Hence, after substituting the values of the partial derivatives and carrying out certain transformations, we obtain the final expression for the root-mean-square error of the experimental determination of the moment of inertia \(\Theta\):

\[ \sigma_\Theta = \pm \Theta_g \frac{ \sqrt{\left(\frac{t_1}{t}\right)^2 - 1} }{ \left(\frac{t_1}{t} - 1\right)^2 } \cdot \frac{\sigma_t}{t}, \tag{8} \]

where \(\dfrac{\sigma_t}{t}\) is the relative root-mean-square error of the time measurements of the run-down without the added mass.

Dividing equation (8) by (7) gives the expression for the relative root-mean-square error of the required moment of inertia

\[ \frac{\sigma_\Theta}{\Theta} = \pm \frac{ \sqrt{\left(\frac{t_1}{t}\right)^2 - 1} }{ \frac{t_1}{t} - 1 } \cdot \frac{\sigma_t}{t}; \]

which can be written in the more convenient form:

\[ \frac{\delta_\Theta}{\Theta}=\pm m\frac{\delta_t}{t}, \tag{9} \]

where \(m\) is the coefficient of increase of the relative quadratic error, which, as follows from expression (9), is a function of the ratio \(\frac{t_1}{t}\) and is equal to:

\[ m=\frac{\sqrt{\left(\frac{t_1}{t}\right)^2-1}}{\frac{t_1}{t}-1}. \tag{10} \]

The dependence \(m=f\left(\frac{t_1}{t}\right)\) is presented graphically in Fig. 1.

From analysis of expression (10) it follows that, for values of \(\frac{t_1}{t}\) approaching unity, \(m\) tends to infinitely large values.

Fig. 1

Consequently, however small the relative errors in the time measurements may be, the final result for the value of \(\Theta\) may prove to be completely unusable in accuracy if only the ratio \(\frac{t_1}{t}\) does not differ sufficiently from unity.

Thus, when determining the moment of inertia of masses by the run-out method, the additional mass must be chosen so that, with the existing error in the measurements of the time \(t\), the required accuracy in determining the moment of inertia \(\Theta\) is ensured.

Thus, for example, suppose it is required to determine the moment of inertia of a mass by the rundown method with a relative quadratic error of \(\pm 2\%\), while the relative error in the measurements of the rundown time \(\frac{\varepsilon_t}{t}\) is \(\pm 1\%\). To ensure the specified accuracy, it is evidently necessary to satisfy the condition \(m \leqslant 2\). According to the graph in Fig. 1, this condition is ensured only when

\[ \frac{t_1}{t} \geqslant 2.25, \]

which corresponds to \(\Theta_g \geqslant 1.25\Theta\).

With a further increase in the moment of inertia of the added mass \(\Theta_g\), the error in determining the sought moment of inertia will continue to decrease, but the effect of reducing the error will be the weaker, the larger \(\frac{t_1}{t}\) is; therefore there is no point in resorting to excessively large added masses \(\Theta_g\), especially since in this case new errors may arise, associated with an increase in the load in the bearings of the shaft on which the added mass is mounted.

REFERENCES

1. K. T. Shatalov. Problems of experimental investigations of torsional vibrations of engine shafts. In the collection of articles Dynamics and Strength of Crankshafts, Publishing House of the Academy of Sciences of the USSR, 1948.

2. S. V. Serensen et al. Dynamic Strength in Mechanical Engineering, Mashgiz, 1945.

3. Mechanical Engineering, encyclopedic handbook, vol. 1, book one, Mashgiz, 1947.