INFLUENCE OF AN INCREASE IN MEASURING FORCE ON THE CHARACTERISTIC OF AN ELASTIC DYNAMOMETER

G. L. KUFAREV

(Presented by Prof. A. M. Rozenberg, Doctor of Technical Sciences)

By an elastic dynamometer we shall mean a dynamometer in which the force is measured by the magnitude of the displacement of some elastic element under the action of the force being measured. Thus, in the schematic diagram of an elastic dynamometer (Fig. 1), the magnitude of the force \(P\) is evaluated from the displacement of the elastic bending beam 1, which is recorded by the recorder 2.

Fig. 1

If the displacement of the elastic element is measured in such a way that the increase in measuring force, i.e., the force required to displace the recorder, is negligibly small, then the characteristic of the dynamometer will coincide with the characteristic of the deformable elastic element (by the characteristic of the dynamometer is meant the relation between the measured force \(P\) and the reading of the recorder \(h\)). If, however, the increase in the measuring force of the recorder \(Q_0\) when measuring the force up to its maximum value \(P_{\max}\) proves to be substantial in comparison with the measured force, then the characteristic of the dynamometer will be determined as the sum of the characteristics of the elastic element and of the recorder and will be steeper (Fig. 2). In this case, the loading of the elastic element by the force \(P_{\max}\) will correspond to a recorder reading \(h_1\), smaller than \(h_0\), which would have occurred in the absence of an increase in measuring force.

Fig. 2

It is not difficult to calculate what fraction the segment \(h_1\) will constitute of the value \(h_0\). Let the force \(Q_0\) be \(x\) percent of \(P_{max}\), i.e.

\[ Q_0=\frac{x}{100}\cdot P_{max}. \]

Then

\[ k=\frac{h_1}{h_0}=\frac{P_{max}}{P_{max}+Q_0}=\frac{1}{1+\dfrac{x}{100}}. \]

Graphically, this dependence is shown in Fig. 3.

Fig. 3

If the transmission of the elastic displacement to the recorder is carried out through an equal-arm lever (Fig. 4), then the characteristic of the dynamometer will remain the same, since in this case as well the increase in measuring force at the maximum load \(P_{max}\) will be equal to \(Q_0\).

In a number of cases, when it is desirable to increase the sensitivity of the dynamometer, the elastic displacement is transmitted to the recorder through an amplifying mechanical lever. In this connection it should be borne in mind that the inclusion of an amplifying mechanical lever in the arrangement of an elastic dynamometer does not always lead to an increase in its sensitivity.

The rationality of using a mechanical lever for this purpose can be discussed only as applied to a specific case of force measurement by a specific recorder.

Let (Fig. 5) the maximum measured force be equal to \(P_{max} k2\).

Under the action of the force \(P_{max}\), the elastic element of the dynamometer bends by the amount \(h_0\). Measurement of the displacement \(h_0\) by the given recorder requires an increase in the measuring force by an amount \(Q_0 k2\), which is \(x\) percent of the maximum measured force \(P_{max}\).

Fig. 4

The straight lines \(1'\), \(2'\), \(3'\) are the characteristics of the dynamometer for a gear ratio of the mechanical lever \(n=1, 2, 3\) in the case where measuring the resulting displacements \(h_0, 2h_0, 3h_0\) did not require an increase in the measuring force.

Fig. 5

The straight lines 1, 2, 3 are the actual characteristics of the dynamometer with a recorder requiring an increase of the measuring force by an amount \(Q\) kg when measuring the displacement \(h_0\).

Straight line 4 is the characteristic of the recording instrument.

Characteristics 2 and 3 are obtained as follows: the doubled or tripled force \(Q_0\), corresponding to the doubled or tripled value of \(h_0\) by means of the mechanical lever, is once again doubled or tripled when transmitted to the elastic element through a lever with an arm ratio of \(2/1\) or \(3/1\). Thus, from Fig. 5 it follows that, when measuring a constant force \(P_{\max}\), in the case where the increment of measuring force proves significant, the reading of the recording instrument \((h_1, h_2, h_3)\) increases more slowly than the transmission ratio of the lever \((n=1, 2, 3)\). One may reason in the same way in the general case, when the gear ratio of the mechanical lever is equal to \(n\).

In this case the quantity \(k\) is determined as follows:

\[ k=\frac{h}{h_0}=\frac{nP_{\max}}{P_{\max}+n^2Q_0} =\frac{n}{1+n^2\frac{x}{100}} . \tag{1} \]

Graphically, this dependence for \(x=0, 1, 2, 5, 10\%\) is presented in Fig. 6.

It follows from the graph that, for each given value of \(x\), increasing \(n\) is meaningful only up to a certain limit \((n_{k\max})\), to which the maximum possible increase \(k_{\max}\) corresponds.

Examining dependence (1) for a maximum, we obtain

\[ n_{k_{\max}}=\frac{10}{\sqrt{x}}. \]

Substituting this value of \(n\) into dependence (1), we find

\[ k_{\max}=\frac{10}{2\sqrt{x}}=\frac{n_{k_{\max}}}{2}. \]

Fig. 6

Fig. 7 makes it possible, for each value of \(x\), to say what increase can be obtained by means of a mechanical lever and what transmission ratio should be chosen for it.

Fig. 7

In conclusion, it should be noted that all the arguments presented are valid only in the case of a rectilinear characteristic of the displacement recorder and do not take into account possible losses in the dynamometer system. If, however, there are losses in the system (friction, bending of the levers, etc.), then the transmission ratio \(k\) will be lower than the values given above.