On the Experimental Determination of the Moments of Inertia of Bodies of Revolution

V. K. Nechaev

The bifilar suspension method (or bifilar torsion pendulum) is one of the most convenient and accurate methods for the experimental determination of the moments of inertia of bodies of revolution about their central geometric axes. By this method, the moments of inertia of the masses of flywheels of automotive-tractor engines, rotors of small electrical machines, gear wheels, crankshafts, etc., can most conveniently and simply be determined.

Fig. 1

In practice, here (Fig. 1), instead of the ideal mathematical threads assumed by the theory of the bifilar pendulum, there is usually used…

steel wire of a diameter corresponding (in tensile strength to the weight of the part being investigated).

But sometimes it proves structurally difficult to provide a hinged fastening, with small friction, of the upper and lower ends of these wires. In a number of cases, finally, instead of hinges, rigid fastening of the wire ends in the corresponding devices—clamps—is used. But in this case the wire, obviously, definitively loses the properties of a mathematical thread, exhibiting during oscillations of the tested part its bending stiffness. Below a method is presented for estimating and allowing for the experimental error associated with this.

Each of the wires is a long rod, loaded at the lower end by a considerable longitudinal tensile force \(V\) and subjected to bending by a certain horizontal force \(H\) (Fig. 2). The equation of bending of this rod depends on the method of fastening its ends. With hinged (frictionless) fastening of the lower end of the rod, the bending equation has the simplest form:

\[ EIy''=-M_x=Vy-Hx, \]

where \(E\) is the modulus of the first kind, \(I\) is the equatorial moment of inertia of the cross section of the rod (wire). With the notation

\[ \frac{V}{EI}=v; \]

\[ \frac{H}{EI}=h; \]

\[ \alpha=\sqrt{v}=\sqrt{\frac{V}{EI}}, \tag{1} \]

the bending equation takes the form:

Fig. 2

\[ y''-vy=-hx \tag{2} \]

and has the general solution in hyperbolic functions:

\[ y=A\,\operatorname{sh}\alpha x+B\,\operatorname{ch}\alpha x+\frac{hx}{v}. \]

Initial conditions:

\[ \left. \begin{array}{ll} y=0, & \text{for } x=0 \quad \text{(hinge)},\\ y'=0, & \text{for } x=l \quad \text{(clamping)} \end{array} \right\} \]

determine here the constants of integration:

\[ A=-\frac{h}{\alpha v\,\operatorname{ch}\alpha l};\qquad B=0. \]

Now the equation of the elastic curve of the deformed rod (wire) can be written in final form:

\[ y=-\frac{h}{a v}\,\frac{\operatorname{sh}\alpha x}{\operatorname{ch}\alpha l}+\frac{h x}{v}. \]

For \(x=l\):

\[ y=y_{\max}=\frac{h l}{v}\left(1-\frac{\operatorname{th}\alpha l}{\alpha l}\right) \]

or

\[ y_{\max}=\frac{H}{V}\,l\left(1-\frac{\operatorname{th}\alpha l}{\alpha l}\right). \]

Consequently, when the lower end of the rod (wire) is deflected from the vertical by some amount \(y_{\max}\), at the end of this rod there arises a horizontal reactive (elastic) force:

\[ H=\frac{y_{\max}}{l}\,\frac{V}{\left(1-\dfrac{\operatorname{th}\alpha l}{\alpha l}\right)}. \]

In the case of an absolutely flexible thread this horizontal force would have the magnitude:

\[ H_0=\frac{y_{\max}}{l}\,V, \]

which can easily be established from elementary geometrical considerations.

Thus, due to the flexural rigidity of the wire (caused by the rigid fixing of its upper end), the torsional rigidity\(^1\) of the entire bifilar-pendulum system increases from the value

\[ c_0=H_0\frac{2a}{\varphi} \tag{3} \]

to the value

\[ c=H\frac{2a}{\varphi}, \tag{3'} \]

that is, it increases in the ratio

\[ \beta=\frac{H}{H_0}=\frac{1}{1-\dfrac{\operatorname{th}\alpha l}{\alpha l}}. \tag{4} \]

(Here \(2a\) is the distance between the axes of the two wires, and \(\varphi\) is the instantaneous angle of deviation of the tested part from its equilibrium position.) Therefore the torsional rigidity of the bifilar pendulum, which enters into the differential equation of its motion,

\[ \Theta\varphi''+c\varphi=0, \tag{5} \]

must be determined by the expression:

\[ c=\beta H_0\frac{2a}{\varphi}. \]

\(^1\) The torsional rigidity of a bifilar pendulum gives the magnitude of the restoring moment arising when the suspended mass is rotated about its vertical axis through an angle \(\varphi=1\) radian and acting about this axis.

But

\[ H_0 = V \operatorname{tg}\varepsilon, \]

that is,

\[ c=\beta V \frac{2a}{\varphi}\operatorname{tg}\varepsilon, \]

or, for small amplitudes of oscillation,

\[ c=\beta \varepsilon V \frac{2a}{\varphi}. \]

Here

\[ \varepsilon=\frac{a\varphi}{l}. \]

Now

\[ c=\beta V \frac{2a^2}{l}, \]

or, taking into account that

\[ V=\frac{G}{2}, \]

where \(G\) is the weight of the tested part,

\[ c=\beta \frac{Ga^2}{l}. \]

Therefore, instead of (5) one may write:

\[ \Theta \varphi''+\beta \frac{Ga^2}{\Theta l}=0. \]

Hence the angular frequency of free oscillations of the bifilar pendulum under consideration is

\[ \Theta \omega=\sqrt{\beta\frac{Ga^2}{\Theta l}}\ \mathrm{sec}^{-1} \]

and the period of free oscillations is

\[ T=2\pi\sqrt{\frac{\Theta l}{\beta Ga^2}}\ \mathrm{sec}. \]

Now the error of the experimental determination of the moment of inertia \(\Theta\) of the part will be determined by the quantity

\[ \delta_1=\sqrt{\beta}-1, \tag{7} \]

that is, the true value of the moment of inertia of the part is

\[ \Theta=\sqrt{\beta}\Theta_0=(1+\delta_1)\Theta_0. \tag{8} \]

The quantity \(\Theta_0\) entering here is computed on the basis of the usual relation for the period of free oscillations of a bifilar pendulum:

\[ T=2\pi\sqrt{\frac{\Theta_0 l}{Ga^2}} \]

from the experimentally measured values of the quantities \(G,l,a\), and \(T\):

\[ \Theta_0=\frac{T^2}{4\pi^2}\cdot\frac{Ga^2}{l}. \tag{9} \]

According to the preceding,

\[ \delta_1=\frac{1}{\sqrt{1-\dfrac{\operatorname{th}\alpha l}{\alpha l}}}-1 \tag{10} \]

where \(\alpha l\) is a parameter determined by the expression:

\[ \alpha l=l\sqrt{\frac{V}{EI}}=l\sqrt{\frac{G}{2EI}} \tag{11} \]

and characterizing the relative stiffness of the wires.

The error \(\delta_1\) approaches zero as the parameter \(\alpha l\) increases (Fig. 3).

Fig. 3

Under ordinary practical conditions (comparatively large values of \(G\) and \(l\), with sufficiently small \(I\)), the hyperbolic tangent \(\operatorname{th}\alpha l\) is very close to unity. In this case the practically exact expression will be:

\[ \delta_1=\frac{1}{\sqrt{1-\dfrac{1}{\alpha l}}}-1 \]

or, after expanding its right-hand side in a series:

\[ \delta_1=\frac{1}{2\alpha l}+\frac{3}{8\alpha^2l^2}+\cdots . \]

Consequently, for sufficiently large values of the parameter \(\alpha l\), one may take:

\[ \delta_1\simeq\frac{1}{2\alpha l} \tag{12} \]

or, according to (10):

\[ \delta_1=\frac{1}{2l}\sqrt{\frac{2EI}{G}}=\frac{1}{16}\sqrt{\frac{2\pi E}{G}\,\frac{d^2}{l}} \tag{13} \]

Above we considered the case of clamping of the rods (wires) in their upper section, with a hinged (frictionless) connection of their lower ends to the part under investigation. But in practice cases are possible in which, owing to the design features of the fastening or to considerable friction, the upper and lower ends of the suspensions (wires) are clamped simultaneously. In this case, in the lower section of the suspension (at the point where it is fastened to the part under investigation) there arises a certain clamping moment \(M\) (Fig. 4).

Now, in the section \(x\) of the rod (wire), the bending moment acting will be:

\[ M_x=-Vy-M+Hx \tag{14} \]

Consequently, here:

\[ EIy''=-M_x=Vy-Hx+M \]

or

\[ y''=vy-hx+m, \tag{15} \]

where \(v\) and \(h\) are determined by formulas (1), and

\[ m=\frac{M}{EI}. \tag{16} \]

The differential equation (15) has the general solution:

\[ y=A\,\operatorname{sh}\alpha x+B\,\operatorname{ch}\alpha x+\frac{hx}{v}-\frac{m}{v} \tag{17} \]

where, as before,

\[ \alpha=\sqrt{\frac{V}{EI}}. \]

Fig. 4

Initial conditions:

\[ \begin{aligned} y&=0 &&\text{for } x=0,\\ y'&=0 &&\text{for } x=0 \quad \text{(clamping)},\\ y'&=0 &&\text{for } x=l \quad \text{”} \end{aligned} \]

give here:

\[ A=-\frac{h}{\alpha v};\qquad B=\frac{h}{\alpha v}\,\frac{\operatorname{ch}\alpha l-1}{\operatorname{sh}\alpha l} \]

\[ m=\frac{h}{\alpha}\,\frac{\operatorname{ch}\alpha l-1}{\operatorname{sh}\alpha l}. \tag{18} \]

Now, instead of (15), one may write:

\[ y=-\frac{h}{\alpha v}\sinh \alpha x+\frac{h}{\alpha v}\, \frac{(\cosh \alpha l-1)(\cosh \alpha x-1)}{\sinh \alpha l} +\frac{hx}{v}. \]

Hence, for \(x=l\),

\[ y=y_{\max}=\frac{hl}{v}\left[1-\frac{2(\cosh \alpha l-1)}{\alpha l\sinh \alpha l}\right] \tag{19} \]

or

\[ y_{\max}=\frac{hl}{v}\left[1-\frac{2}{\alpha l\tanh \dfrac{\alpha l}{2}}\right]. \]

Finally, according to (1):

\[ y_{\max}=\frac{H}{V}\,l\left[1-\frac{2}{\alpha l\tanh \dfrac{\alpha l}{2}}\right]. \]

Now, analogously to (3), the coefficient of increase in the stiffness of the bifilar pendulum is

\[ \beta=\frac{1}{1-\dfrac{2}{\alpha l\tanh \dfrac{\alpha l}{2}}}. \]

Therefore here the relative error \(\delta_2\), entering into a formula similar to (8) for the true value of the moment of inertia \(\Theta\) of the part under investigation, is determined by the expression

\[ \delta_2=\sqrt{\beta}-1= \frac{1}{\sqrt{1-\dfrac{2}{\alpha l\tanh \dfrac{\alpha l}{2}}}}-1 \tag{20} \]

For \(\alpha l>10\)

\[ \tanh \frac{\alpha l}{2}\simeq 1 \]

and

\[ \delta_2=\frac{1}{\sqrt{1-\dfrac{2}{\alpha l}}}-1 \]

or, approximately (for sufficiently large \(\alpha l\)):

\[ \delta_2\simeq \frac{1}{\alpha l} \tag{21} \]

Comparison of expressions (12) and (21) shows that replacing the lower hinged (frictionless) connection of the suspensions (wires) with the part under investigation by complete clamping practically doubles (Fig. 3) the magnitude of the error \(\delta\), as compared with the case in which only the upper ends of the suspensions are clamped.

It also follows from (20) that, for a small value of the parameter \(\alpha l\), the error in determining the desired moment of inertia of the part under investigation by

ordinary elementary formula (9) may be quite substantial. Analogously to (13), here approximately:

\[ \varepsilon_{\delta_2}=\frac{1}{8}\sqrt{\frac{2\pi E}{G}}\frac{d^2}{l} \tag{22} \]

Formulas (13) and (22) should be taken into account when choosing the parameters (\(l\) and \(d\)) of the experimental setup that ensure determination of the moment of inertia of the part by the bifilar-suspension method with a sufficient degree of accuracy.

It was assumed above that the clamping of the upper (or upper and lower) ends of the suspensions was due to the design features of their fastening. But, obviously, clamping of these ends may also occur with hinged fastening if the friction in the hinges is sufficiently large. It is not difficult to show that, under the conditions of an ordinary experimental setup, such frictional clamping is practically not excluded.

Such clamping will occur in the case when the friction moment in the hinge, even at the maximum angular deviation \(\varphi_{\max}\) of the part under investigation, exceeds the value of the maximum clamping moment \(M_{\max}\), entering, for example, into expression (14) and determined by formula (18). From (18) and (16), for \(\varphi=\varphi_{\max}\):

\[ M_{\max}=\frac{H_{\max}}{\alpha}\,\frac{\operatorname{ch}\alpha l-1}{\operatorname{sh}\alpha l} \]

or

\[ M_{\max}=H_{\max}l\cdot \frac{1}{\alpha l\,\operatorname{th}\dfrac{\alpha l}{2}} \tag{23} \]

But according to (3) and (6)

\[ H_{\max}=\frac{c\varphi_{\max}}{2a}=\beta\,\frac{Ga}{2l}\varphi_{\max}, \]

therefore, instead of (23),

\[ M_{\max}=\beta\,\frac{Ga}{2}\,\varphi_{\max}\cdot \frac{1}{\alpha l\,\operatorname{th}\dfrac{\alpha l}{2}}. \]

The friction moment in the hinge is:

\[ M_{\mathrm{fr}}=\mu\,\frac{D}{2}\,\frac{G}{2}, \]

where \(D\) is the diameter of the hinge pin, \(\mu\) is the coefficient of friction, and \(G\), as before, is the weight of the part under investigation.

Now the condition for complete clamping of the end of the wire by the forces of dry friction in an ordinary hinge can be written in the form:

\[ M_{\mathrm{fr}}>M_{\max} \]

or

\[ \mu\,\frac{GD}{4}> \frac{Ga}{2}\,\varphi_{\max}\cdot \frac{1}{\alpha l\,\operatorname{th}\dfrac{\alpha l}{2}}. \]

For sufficiently large values of the parameter \(\alpha l\), when approximately

\[ \beta\simeq 1,\qquad \operatorname{th}\frac{\alpha l}{2}\simeq 1, \]

the condition for complete frictional clamping takes the following form:

\[ \mu > 2\varphi_{\max}\frac{a}{D}\cdot\frac{1}{\alpha l}. \]

Thus, for example, for values possible in practice:

\[ \varphi_{\max}=0.05; \qquad \alpha l=30; \qquad \frac{a}{D}=50 \]

we obtain

\[ \mu > 0.17, \]

which, under conditions of dry friction (and, moreover, static friction), obviously cannot be considered unlikely.

For

\[ \mu < 2\varphi_{\max}\frac{a}{D}\cdot\frac{1}{\alpha l} \tag{24} \]

the clamping will be partial, varying with the angle \(\varphi\).

To reduce the influence of this clamping on the accuracy of determining the moment of inertia of a part by the bifilar-pendulum method (Fig. 1), it is necessary to ensure sufficiently large values of the parameter \(\alpha l\) and of the ratio \(a/D\), with the friction coefficient \(\mu\) as small as possible.