Experimental Determination of the Moment of Inertia
Abstract Generated abstract
The paper addresses experimental determination of the moment of inertia of an electric machine rotor or of a motor and working mechanism reduced to the motor shaft. Starting from the coast-down equation of motion, it discusses graphical differentiation of speed-time curves and a more accurate tachogenerator-capacitor method for obtaining deceleration. Because the friction torque varies with speed and is difficult to determine directly, the authors present two comparative procedures: adding a known flywheel when friction is appreciable, and applying an additional Prony-brake torque when friction is small. The resulting formulas allow the moment of inertia to be found from measured decelerations or capacitor currents at the same speed, and can subsequently be used to determine the unknown friction torque.
Full Text
EXPERIMENTAL DETERMINATION OF THE MOMENT OF INERTIA
A. I. ZAITSEV, M. P. TABINSKY
The moment of inertia of the rotor of a large machine, or the reduced moment of inertia of the motor–working mechanism system, is usually determined by the coast-down method [1; 2]. For this purpose the motor is accelerated to rated speed and disconnected from the mains, after which it is braked by coasting. The kinetic energy accumulated in the moving parts of the mechanical system motor–working mechanism is expended in overcoming friction forces, as a result of which the motor speed gradually decreases. The equation of motion during coast-down braking has the form
\[ - M_{\mathrm{fr}} = I_{\mathrm{red}} \frac{d\omega}{dt} = \frac{\pi}{30} I_{\mathrm{red}} \frac{dn}{dt}, \ \mathrm{kgm}, \tag{1} \]
where \(I_{\mathrm{red}}\) is the moment of inertia of the motor rotor and the moment of inertia of the working mechanism reduced to the motor shaft, \(\mathrm{kgm \, sec^2}\);
\(M_{\mathrm{fr}}\) is the friction torque, \(\mathrm{kgm}\);
\(\omega\) is the angular velocity, \(1/\mathrm{sec}\);
\(n\) is the rotational speed, \(\mathrm{rev}/\mathrm{min}\).
It follows from formula (1) that the required moment of inertia is equal to
\[ I_{\mathrm{red}} = - M_{\mathrm{fr}} \frac{30}{\pi} \frac{1}{\dfrac{dn}{dt}}, \ \mathrm{kgm \, sec^2}. \tag{2} \]
The derivative of speed with respect to time, \(\dfrac{dn}{dt}\), is determined by graphical differentiation of the coast-down curve \(n=f(t)\) (Fig. 1). Drawing a tangent at the point \(n=b\) to the curve \(n=f(t)\), one obtains
\[ \frac{dn}{dt} = \tan \alpha = - \frac{\overline{ab}}{\overline{ac}} \cdot \frac{\mu_n}{\mu_t}, \]
where \(\overline{ab}\), \(\overline{ac}\) are the lengths of the segments, cm;
\(\mu_n,\ \dfrac{\mathrm{rev}}{\mathrm{min}\cdot \mathrm{cm}};\ \mu_t,\ \dfrac{\mathrm{sec}}{\mathrm{cm}}\) are the scales of speed and time.
Consequently, the reduced moment of inertia will be equal to
\[ I_{\mathrm{red}}=M_{\mathrm{red}}\cdot \frac{30}{\pi}\,\frac{\overline{ac}\cdot\psi_t}{ab\cdot\psi_n},\ \mathrm{kg\,m\,s^2}. \]
The derivative \(\dfrac{dn}{dt}\) can be determined more accurately by the Itterberg method [2], by differentiating the voltage proportional to the motor speed, using a capacitor (Fig. 2). The tachogenerator TG, rigidly coupled to the motor shaft, gives a voltage proportional to the speed,
Fig. 1.
\[ U_{\mathrm{tg}}=\kappa\cdot n_{\mathrm{mot}}, \tag{4} \]
where \(\kappa\) is the speed coefficient of the tachogenerator, \(\dfrac{\mathrm{V}\cdot\mathrm{min}}{\mathrm{rev}}\).
A capacitor \(C\) and a milliammeter \(A\) are connected in series in the armature circuit of the tachogenerator. When the voltage \(U_{\mathrm{tg}}\) changes, a charging current flows through the capacitor, which is determined by the equation
\[ i_C=C\,\frac{dU_C}{dt}. \]
Neglecting the voltage drop across the milliammeter, one may assume that \(U_C\simeq U_{\mathrm{tg}}=\kappa\cdot n_{\mathrm{mot}}\); consequently, \(i_C=C\cdot\kappa\cdot\dfrac{dn_{\mathrm{mot}}}{dt}\), whence
\[ \frac{dn_{\mathrm{mot}}}{dt}=\frac{i_C}{C\cdot k},\quad \frac{\mathrm{rev}}{\mathrm{min}\cdot \mathrm{s}} \tag{5} \]
where \(i_C\) is the capacitor charging current, A;
\(C\) is the capacitance of the capacitor, F.
Substituting (5) into (3), we obtain
\[ I_{\mathrm{eq}}=-M_{\mathrm{tr}}\frac{30}{\pi}\frac{C\cdot k}{i_C}, \quad \mathrm{kg}\,\mathrm{m}\,\mathrm{s}^2 . \tag{6} \]
Considerable difficulty is caused by determining the friction moment, which is also some function of the speed, \(M_{\mathrm{tr}}=\varphi(n)\), chiefly because of the fan. Below, two methods are recommended by which this difficulty can be resolved.
Fig. 2.
a) Method of the added flywheel
Usually, in a motor--working mechanism system, the friction moment is significant. To determine the moment of inertia referred to the motor shaft, in addition to the coast-down curve of the system, another coast-down curve is taken for the system with a flywheel mounted on the motor shaft, the moment of inertia of which is known. The increase in the friction moment due to the weight of the flywheel may be neglected. Denoting the unknown moment of inertia by \(I_x\), and the known moment of inertia of the flywheel by \(I_M\), the following equations of motion may be written:
\[ \left. \begin{aligned} I_x\frac{d\omega_1}{dt}&=M_{\mathrm{tr}} \\[6pt] \text{for } \omega_1=\omega_2&=N, \\[6pt] \left(I_x+I_M\right)\frac{d\omega_2}{dt}&=M_{\mathrm{tr}} . \end{aligned} \right\} \tag{7} \tag{8} \]
Since at the speed \(\omega_1=\omega_2=N\) the friction moment is one and the same, the left-hand sides of equations (7) and (8) may be equated:
\[ I_x\frac{d\omega_1}{dt} = \left(I_x+I_M\right)\frac{d\omega_2}{dt}, \]
whence the required value of the moment of inertia will be equal to
\[ I_c = I_M \frac{\dfrac{d\omega_2}{dt}}{\dfrac{d\omega_1}{dt} - \dfrac{d\omega_2}{dt}} = I_M \frac{\dfrac{dn_2}{dt}}{\dfrac{dn_1}{dt} - \dfrac{dn_2}{dt}} = I_M \frac{i_{c2}}{i_{c2} - i_{c1}} . \tag{9} \]
It is obvious that \(\dfrac{dn_2}{dt}\) and \(\dfrac{dn_1}{dt}\) (or \(i_{c2}\) and \(i_{c1}\)) are determined at one value of the speed.
b) Method of additional braking
In the case of a small friction moment \(M_{\mathrm{fr}}\), its increase from the weight of the flywheel cannot be neglected. Therefore, in addition to the run-down curve of the motor or of the motor–working mechanism system, a run-down curve is also taken with an additional friction moment, which is produced by a Prony brake (Fig. 3). The Prony brake consists of two bars (1),
Fig. 3.
to which the brake shoes (2) are attached. By means of two springs (3), the brake shoes are pressed against the pulley or against the motor shaft. The magnitude of the braking moment depends on the material of the brake shoes, on the force with which they are pressed against the pulley, and on the presence of lubrication.
The additional friction moment is determined with the aid of the dynamometer (4)
\[ M_{\mathrm{add}} = P \cdot l,\ \text{kgm}, \tag{10} \]
where \(P\) is the dynamometer reading, kg;
\(l\) is the shortest distance from the axis of the motor to the axial line of the dynamometer, m.
The magnitude of the additional braking torque is selected experimentally so that the deceleration of the motor (and the discharge current of the differentiating capacitor proportional to it) increases by a factor of 2–3.
At large values of the additional braking torque, the brake shoes must be lubricated and cooled with water. Before the experiment, the motor on which the Prony brake is mounted must operate for some time so that thermal equilibrium is established between the brake shoes and the pulley.
This method also gives two equations of motion
\[ I_x \frac{d\omega_1}{dt} = M_x, \tag{11} \]
\[ I_x \frac{d\omega_2}{dt} = M_x + M_{\text{add}}. \tag{12} \]
Substituting (11) into (12), we obtain
\[ I_x = \frac{M_{\text{add}}}{\dfrac{d\omega_2}{dt} - \dfrac{d\omega_1}{dt}} = \frac{M_{\text{add}}\cdot 30}{\pi\left(\dfrac{dn_2}{dt} - \dfrac{dn_1}{dt}\right)} = \frac{30\cdot P\cdot l\cdot C\cdot k}{\pi\left(i_{C2}-i_{C1}\right)}, \quad \text{kgm sec}^2 . \tag{13} \]
The quantities entering equation (13), \(\dfrac{dn_1}{dt}\) and \(\dfrac{dn_2}{dt}\) (or \(i_{C2}\) and \(i_{C1}\)), are determined at one value of the speed.
In the case of rapidly occurring braking, when it is difficult to register the value of the deceleration with a milliammeter, it is necessary to record the curves \(n=f(t)\), \(i_{C1}=\varphi_1(t)\), and \(i_{C2}=\varphi_2(t)\) with a loop oscillograph; for this purpose, instead of the milliammeter, the loop of a vibrator is connected into the circuit of the differentiating capacitor.
Conclusions
Using the methods described above, it is possible to determine with great accuracy the moment of inertia of the rotor of an electric machine or the moment of inertia, reduced to the motor shaft, of a motor–working-mechanism system.
The additional-brake method may be recommended for cases in which the friction torque of the machines is small and the run-down braking process is prolonged.
The additional-flywheel method is expedient to apply when the friction torque of the machine is considerable. In this case, the increase in the friction torque due to the weight of the additional flywheel may be neglected.
After determining the moment of inertia, the unknown friction torque of the machine or of the motor–working-mechanism system can be found by means of formula (6).
REFERENCES
- V. P. Andreev and Yu. A. Sabinin. Fundamentals of Electric Drive, GEI, 1956.
- A. Utterberg. Eine neue Methode zur Bestimmung der Leerlaufsverluste einer Maschine. ETZ, H. 45, p. 1158, 1912.
Received by the editors
in May 1962.