EFFECT OF DISPLACEMENT OF A BODY’S CENTER OF GRAVITY ON THE ACCURACY OF DETERMINING THE MOMENT OF INERTIA BY THE BIFILAR SUSPENSION METHOD

V. V. TURKIN

(Presented by the scientific seminar of the Department of Applied Mechanics)

The determination of the moments of inertia of the mass of bodies of complex shape is usually carried out by experimental methods, one of the most widespread of which is the method of oscillations on a bifilar suspension.

As is known, the moment of inertia with respect to an axis passing through the center of gravity of a body is calculated by the formula

\[ J_z=\frac{Ga^2}{l}\,\frac{T^2}{4\pi^2}, \tag{1} \]

where \(G\) is the weight of the body;
\(2a\) is the distance between the threads;
\(l\) is the length of the threads;
\(T\) is the period of oscillations of the body on the bifilar suspension.

Formula (1) is valid in the case where the central axis of the body, with respect to which the moment of inertia is determined, coincides with the axis of symmetry of the bifilar suspension. Usually these axes do not coincide. As a result, an error arises, the greater the larger the displacement of the axes. With a considerable displacement, formula (1) becomes entirely unsuitable.

In the present article, the results are given of an analysis of the accuracy of the method as a function of the relative displacement of the above-mentioned axes for three possible types of displacement: linear, angular, and mixed.

Linear displacement

In this case the central axis of the body is displaced in the plane of the bifilar suspension parallel to its axis by a distance \(e_x\) (Fig. 1).

If a body suspended in this way is turned about the central axis through an angle \(\varphi\) and released without initial velocity, then the center of gravity of the body will oscillate along this axis, and its position will be determined by the coordinates \(x_c=e_x,\ y_c=0\), and

\[ z_c=l(1-\cos\psi)-(a+e_x)\sin\Theta. \tag{2} \]

As can be seen from Fig. 1, the angles \(\psi_1\), \(\psi_2\), and \(\Theta\) are related to the angle of twist of the bifilar suspension \(\varphi\) by the equations:

\[ \sin \psi_1 = \frac{a+e_x}{l}\sin \varphi, \tag{3} \]

\[ \sin \psi_2 = \frac{a-e_x}{l}\sin \varphi, \tag{4} \]

\[ \sin \Theta = \frac{l}{2a}\left(\cos\psi_2-\cos\psi_1\right). \tag{5} \]

It is expedient to simplify this system of equations. To this end one can always choose values of the quantities \(\dfrac{2a}{l}\) and \(\varphi_0\), convenient for practice, for which the angles \(\psi_1\), \(\psi_2\), and \(\Theta\) will be so small that it will be possible, with sufficient accuracy, to take \(\sin\psi_1 \simeq \psi_1\), \(\sin\psi_2 \simeq \psi_2\), \(\sin\Theta \simeq \Theta\), \(\cos \Theta \simeq 1\) for any displacement \(e_x<a\).

Fig. 1.

In this case, instead of the complicated system of exact equations (2), (3), (4), (5), we obtain the simple approximate formulas:

\[ \psi_1 = \frac{a+e_x}{l}\varphi, \tag{6} \]

\[ \psi_2 = \frac{a-e_x}{l}\varphi, \tag{7} \]

\[ \Theta = \frac{l}{e_x}\varphi^2, \tag{8} \]

\[ z_c = \frac{a^2-e_x^2}{2l}\varphi^2, \tag{9} \]

whose accuracy, for \(\varphi<10^\circ\) and \(\dfrac{2a}{l}<1\), is practically not lower than that of the original equations. For such values of \(\varphi\), \(\dfrac{2a}{l}\), and with an oscillation frequency of about \(1\) cps, convenient for visual reading, the displacement, velocity, and acceleration of the body’s center of gravity will be so small that the vertical inertial forces may be neglected. Then the differential equations of motion of the center of gravity in projections on the axes \(x\), \(y\), \(z\) coincide with the equilibrium equations, from which, under the stated conditions, we obtain with sufficient accuracy the values of the thread tensions

\[ T_1=\frac{G}{2a}\left(a-e_x\right), \tag{10} \]

\[ T_2=\frac{G}{2a}\left(a+e_x\right). \tag{11} \]

In the equation of motion of a body suspended on a bifilar, with respect to the vertical axis \(z_1\) passing through the center of gravity,

\[ J_{z_1}\frac{d^2\varphi}{dt^2}=M \]

the restoring moment will have the form

\[ M=-(T_1\sin\psi_1 h_1+T_2\sin\psi_2 h_2). \tag{12} \]

On the basis of the foregoing, the perpendiculars dropped from the point \(C_0\) onto the direction of the horizontal projections of the forces \(T_1\) and \(T_2\), with the accepted degree of accuracy, will be equal to

\[ h_1=a+e_x,\qquad h_2=a-e_x. \]

After substituting in (12) the values of \(h_1, h_2, T_1, T_2\) and replacing the sines of the angles by the angles, we obtain the expression for the restoring moment, with the accuracy usually adopted, in the form

\[ M=-\frac{G}{l}\left(a^2-e_x^2\right)\varphi . \tag{13} \]

Thus, the differential equation of small oscillations of a body on a bifilar in the presence of a displacement of the center of gravity has the form

\[ J_{z_1}\frac{d^2\varphi}{dt^2}+\frac{G}{l}\left(a^2-e_x^2\right)=0 \]

and the period of oscillation

\[ T=2\pi\sqrt{\frac{J_{z_1}l}{G(a^2-e_x^2)}}. \tag{14} \]

Therefore the moment of inertia of the body with respect to the central axis should be determined from the formula

\[ J_{z_1}=\frac{G(a^2-e_x^2)}{l}\,\frac{T^2}{4\pi^2}, \tag{15} \]

when the displacement \(e_x\) is known.

Usually \(e_x\) is unknown and the moment of inertia is determined by formula (1), as a result of which an error is introduced, whose expression will be

\[ \delta_T=\frac{J_{z_1}-J_z}{J_{z_1}}=-\frac{e_x^2}{a^2-e_x^2}=-\frac{\varepsilon_x^2}{1-\varepsilon_x^2}, \tag{16} \]

where

\[ \varepsilon_x=\frac{e_x}{a}. \]

The minus sign in (16) indicates that \(J_z>J_{z_1}\).

To verify the results obtained, experiments were carried out on a bifilar with parameters: \(2a=100\) mm, \(l=400\) mm, and with a body having

in the form of a circular cylinder with height \(h = 220\) mm, diameter \(d = 2r = 5\) mm, weight \(G = 33.6\) g, and moment of inertia with respect to the central axis perpendicular to the cylinder axis

\[ J_{an} = \frac{m}{12}\left[3r^2 + h^2\right] = 1.38\ \text{g}\cdot\text{cm}\cdot\text{sec}^2 \tag{17} \]

The cylinder, provided with graduations, was suspended on the bifilar threads horizontally at various displacements of the center of gravity; at each displacement the period of oscillation about the vertical axis was measured. By formula (16) the theoretical error was determined, and by the formula

\[ \delta_e = \frac{J_{an} - J_e}{J_{an}} \tag{18} \]

the experimental error, where

\[ J_e = J_z = \frac{Ga^2}{l}\,\frac{T^2}{4\pi^2}. \]

The results of the experiments and calculations, presented by the graph in Fig. 2, confirm the correctness of formulas (15) and (16), and also indi-

Fig. 2.

cate the possibility of the appearance of a significant error in determining the moment of inertia of a mass when using formula (1).

Angular displacement

If the central axis \(z_1\) of the body, with respect to which the moment of inertia is determined, intersects the bifilar axis \(z\) and forms with it an angle \(\varepsilon \ne 0\), then in calculating the moment of inertia by formula (1) an error will be allowed that is the greater, the larger \(\varepsilon\) is.

To determine the magnitude of the error due to angular displacement, we use the well-known expression for the moment of inertia with respect to some axis \(z\), passing through the origin of coordinates and making, with the axes \(x_1, y_1, z_1\), which are the principal axes of the body, the angles \(\alpha, \beta\) and \(\gamma=\varepsilon\):

\[ J_z = J_{x_1}\cos^2\alpha + J_{y_1}\cos^2\beta + J_{z_1}\cos^2\varepsilon \]

or

\[ J_z = J_{z_1}\left(k_1\cos^2\alpha + k_2\cos^2\beta + \cos^2\varepsilon\right), \]

whence the sought moment of inertia is expressed by the formula

\[ J_{z_1} = \frac{J_z}{k_1\cos^2\alpha + k_2\cos^2\beta + \cos^2\varepsilon}, \tag{19} \]

where \(J_z\) is still determined by formula (1), and

\[ k_1=\frac{J_{x_1}}{J_{z_1}} \quad \text{and} \quad k_2=\frac{J_{y_1}}{J_{z_1}} \]

are quite definite positive numbers characterizing the shape of the body and capable of being either less than or considerably greater than unity.

Thus, depending on the shape of the body and on the error in setting it on the bifilar suspension, the actual value of the sought moment of inertia \(J_{z_1}\) will be greater or less than \(J_z\), determined experimentally by formula (1). Therefore the relative error

\[ \delta = \frac{J_{z_1}-J_z}{J_{z_1}} = 1-\left(k_1\cos^2\alpha + k_2\cos^2\beta + \cos^2\varepsilon\right) \tag{20} \]

may be either positive or negative.

Fig. 3.

Consider a special case: \(\alpha=90^\circ\), then \(\beta=90^\circ+\varepsilon\) (Fig. 3)

\[ J_{z_1} = \frac{J_z}{k_2\sin^2\varepsilon + \cos^2\varepsilon}, \tag{21} \]

\[ \delta = (1-k_2)\sin^2\varepsilon. \tag{22} \]

The relative error, calculated by formula (22) for a number of values of \(k_2\) and \(\varepsilon\), is represented by the graph in Fig. 4.

It is seen from the graph that, for \(k_2\) considerably exceeding unity, the error even at small angular displacements will be very substantial.

Thus, for example, for a cylinder with the ratio of height to diameter

\[ \frac{h}{d}=12.2 \]

and with the axis \(z_1\) displaced relative to the bifilar axis by an angle \(\varepsilon=1^\circ\), we shall have

\[ k_2 \approx 100 \quad \text{and} \quad \delta=(1-k_2)\sin^2\varepsilon \approx -3\%. \]

General case

In the presence of linear and angular displacements, the moment of inertia of a body must be determined by the formula

\[ J_{z_1} = \frac{\dfrac{G}{l}\left(a^2-\varepsilon_x^2\right)\dfrac{T^2}{4\pi^2}} {k_1\cos^2\alpha+k_2\cos^2\beta+\cos^2\varepsilon}. \tag{23} \]

The error in determining moments of inertia by the usual formula (1) in this case is expressed as

\[ \delta = \frac{J_{z_1}-J_z}{J_{z_1}} = 1- \frac{k_1\cos^2\alpha+k_2\cos^2\beta+\cos^2\varepsilon} {1-\varepsilon_x^2}. \tag{24} \]

From formulas (23) and (24), the formulas given above for particular types of displacement are easily obtained.

Experimental verification of formulas (23) and (24) also confirmed their correctness.

Thus, when determining the moment of inertia of the mass of a body by the bifilar suspension method, the error due to displacement of the center of gravity in many practically important cases may reach considerable values and, apparently, is the most significant error of this method.

Fig. 4.