ON A PARAMETER CHARACTERIZING THE MAGNITUDE OF THE DYNAMIC FREQUENCY RANGE OF SEISMIC VIBRATION SENSORS

V. F. Gorbunov, G. S. Zhartovskii

(Presented by the scientific seminar of the Department of Mining Machines, Mine Transport and Mining Mechanics)

To create a fixed reference point in vibration sensors, seismic systems of very different designs are widely used. It is known [1] that, in the vibrometric mode, the frequency range is bounded above by the frequencies of the higher resonances of the springs, and below by the natural frequency of the fundamental tone \(f_0\). The ratio \(f_{01}: f_0\), where \(f_{01}\) is the frequency of the first overtone, determines the width of the dynamic frequency range of sensor suspensions; for different designs it is different. Let us try to establish certain quantitative relationships among various suspensions widely used in practice. The table gives the values of the natural frequencies \(f_0\) and the frequencies of the first overtones \(f_{01}\) for suspensions, which we shall need for the subsequent analysis.

Table

In the last row of the table, a variety of helical suspension with the number of turns \(i=1\) is considered. In the formulas, \(m\) denotes the masses of the springs. The approach to estimating the value of the dynamic frequency range of pendulum systems with elastic hinges differs somewhat

in connection with scaling by means of levers and will be considered separately.

Starting from the table, one can write the frequency of the first overtone for suspensions in the general form

\[ f_{01}=\kappa \sqrt{\frac{C}{m}}, \tag{1} \]

where \(\kappa\) is a coefficient characterizing the design of the suspension. The radical expression \(\frac{C}{m}\) attracts attention. Physically it denotes a certain specific quantity, namely: the stiffness of the suspension per unit mass of its springs. Obviously, the larger the ratio \(\frac{C}{m}\), the wider the dynamic frequency range. From the value of the ratio \(\frac{C}{m}\) one can predict the behavior of the inertial system in dynamics (in the sense of frequency properties). In our opinion, this ratio, or parameter, should form the basis for selecting types of suspensions for the practical purposes of wide-range measurements, and for designing new elastic systems for sensors and vibration-protective devices, since it reflects the perfection of the elastic structure.

The concentrated mass \(M\) practically does not enter expression (1), if one neglects its insignificant influence on the coefficient \(\kappa\). This means that the layout of the seismic system in the zone of the fundamental tone is arbitrary: by choosing the corresponding value of \(M\) one can obtain the required frequency \(f_0\) in the presence of an elastic structure with stiffness \(C\), while the elastic structure should be chosen with the maximum parameter \(\frac{C}{m}\).

It should be noted that such a formulation of the problem is not new. It has been reflected in the theory of beams of minimum weight, where optimal shapes of elastic elements are sought that possess maximum stiffness at their minimum weight [2].

Let

\[ \frac{C}{m}=\varepsilon . \]

Then, for a cantilever suspension,

\[ \varepsilon_{\mathrm{k}}= \frac{3EI_{\mathrm{k}}}{l_{\mathrm{k}}^{3}m_{\mathrm{k}}} = \frac{3Eb_{\mathrm{k}}h_{\mathrm{k}}^{3}g}{12\,b_{\mathrm{k}}h_{\mathrm{k}}l_{\mathrm{k}}^{4}\gamma} = \left(a\frac{h_{\mathrm{k}}}{2l_{\mathrm{k}}^{2}}\right)^2, \tag{2} \]

where \(a\) is the speed of sound in the spring material with specific weight \(\gamma\).

For a guided-clamped suspension,

\[ \varepsilon_{\mathrm{p}}= \frac{24EI_{\mathrm{p}}}{l_{\mathrm{p}}^{3}m_{\mathrm{p}}} = \frac{24Eb_{\mathrm{p}}h_{\mathrm{p}}^{3}g}{2\cdot 12\,b_{\mathrm{p}}h_{\mathrm{p}}l_{\mathrm{p}}^{4}\gamma} = \left(a\frac{h_{\mathrm{p}}}{l_{\mathrm{p}}^{2}}\right)^2. \tag{3} \]

As is seen from formulas (2) and (3), under otherwise equal conditions (\(l_{\mathrm{k}}=l_{\mathrm{p}}\); \(h_{\mathrm{k}}=h_{\mathrm{p}}\) and the same material), the guided-clamped suspension has a dynamic frequency range twice as large as that of the cantilever suspension, since their parameters \(\varepsilon\) differ from one another by a factor of four.

In real devices the ratio of the ranges is close to two. Thus, two vibrometer sensors with the same \(f_0=17\) Hz have an upper measurement limit, restricted by the frequencies of the first overtone, of \(\sim 150\) Hz for the cantilever suspension, and \(\sim 320\)--\(350\) Hz for the guided-clamped suspension. This ratio is maintained over wide limits of variation of \(f_0\): approximately from 5 to 80 Hz.

For cylindrical helical springs the parameter \(\varepsilon\) depends substantially on the number of turns. Indeed,

\[ \varepsilon_{\mathrm{v}} = \frac{Gd^{4}}{8\,iD^{3}m_{\mathrm{v}}} = \frac{4\gamma d^{4}g}{8\gamma(i\pi dD^{2})^{2}} = \frac{Gd}{2\gamma} \left(\frac{d}{\pi iD^{2}}\right)^{2}. \tag{4} \]

Formula (4) is valid for \(i \ge 1\). To expand the frequency range one can also vary the quantities \(d\) and \(D\). The limit of such variation is determined by the maximum permissible stresses in the turn during deformation of the spring, since increasing the ratio \(f_{01}:f_0\) by changing \(d\), \(D\), or \(i\) is associated with an increase in the stressed state of the spring. Consequently, if there is a reserve with respect to the permissible stress, it is advisable to design a vibration system on helical springs, operating over a wide frequency spectrum, with one working turn. A single-turn suspension has better dynamic properties in another sense as well: a multi-turn spring is prone to loss of stability and has low natural frequencies of flexural vibrations in the transverse direction. A single-turn suspension was used to design small-size vibration sensors and damping devices. With its aid it was possible to extend the upper limit of the frequency range to \(700\text{--}800\) Hz at \(f_0 = 10\) Hz.

Elastic systems with a large value of the parameter \(\varepsilon\), at identical deformations, are more highly stressed. Evidently, this is where the possibility of obtaining large values of \(f_{01}:f_0\) in more heavily loaded systems is manifested. Analysis of the stressed state of suspensions is consistent with this conclusion; therefore let us consider their stressed state. Figure 1 shows the elastic lines of cantilever and movable-clamped suspensions. For comparison, let us specify certain initial conditions. Suppose that their natural frequencies \(f_0\) are equal. This will mean equality of stiffnesses (for the same mass \(M\)) and of the magnitudes of the static deflections. Let the spring thicknesses also be identical \((h_k = h_p)\), and let the suspensions be made with equal overall dimensions \((l_k = l_p)\). Under these assumptions it is necessary to correct the width of the springs. This can be done by equating the stiffnesses:

\[ C_k = C_p; \qquad \frac{3Eb_kh_k^{3}}{12l_k^{3}} = \frac{24Eb_ph_p^{3}}{12l_p^{3}}. \]

Therefore,

\[ b_k = 8b_p. \tag{5} \]

Fig. 1. Schemes of cantilever and movable-clamped suspensions of vibration sensors

Under the action of the weight \(P\), for the cantilever, at the point of maximum stresses the bending moment is

\[ M_k^{\max} = Pl_k, \]

and the stress is

\[ \sigma_k^{\max} = \frac{M_k^{\max}}{W_k} = \frac{6Pl_k}{b_kh_k^{2}}. \tag{6} \]

For the second suspension,

\[ M_p^{\max} = \frac{\pm 6EI_p}{l_p^{3}}\lambda_{\mathrm{st}}. \]

But since

\[ \lambda_{\mathrm{st}}=\frac{P}{C_{\mathrm{p}}}=\frac{Pl^{3}}{24\cdot EI_{\mathrm{p}}}, \]

then

\[ M_{\mathrm{p}}^{\max}=\pm\frac{Pl_{\mathrm{p}}}{4}. \]

Accordingly, the maximum stresses are

\[ \sigma_{\mathrm{p}}^{\max}=\frac{M_{\mathrm{p}}^{\max}}{W_{\mathrm{p}}} =\pm\frac{6\cdot Pl_{\mathrm{p}}}{4\cdot b_{\mathrm{p}}h_{\mathrm{p}}^{2}}. \tag{7} \]

From formulas (5), (6), and (7) we obtain the stress ratio:

\[ \frac{\sigma_{\mathrm{p}}^{\max}}{\sigma_{\mathrm{k}}^{\max}} = \frac{6\cdot Pl_{\mathrm{p}}b_{\mathrm{k}}h_{\mathrm{k}}^{2}} {6\cdot 4\cdot Pl_{\mathrm{k}}b_{\mathrm{p}}h_{\mathrm{p}}^{2}} =2. \]

Thus, under identical comparison conditions, the flexibly clamped suspension has a value of the parameter \(\varepsilon\) four times larger while being in a more highly stressed state.

We arrive at an analogous conclusion when considering helical springs. Restricting ourselves to taking into account only shear stresses, that is, neglecting bending of the coils [3], we obtain

\[ \tau^{\max}=\frac{8\cdot PD}{\pi d^{3}}. \]

The initial data for the comparison are analogous to the preceding case. Let \(D_{1}=D_{2}\) (under the subscript \(1\) we shall mean the multi-turn spring, and under the subscript \(2\)—the single-turn spring). From the condition of equality of stiffnesses we correct the wire diameters. Obviously,

\[ d_{1}=\sqrt[4]{i}\,d_{2}, \]

and since

\[ \frac{\tau_{1}^{\max}}{\tau_{2}^{\max}}=\left(\frac{d_{2}}{d_{1}}\right)^{3}, \]

then

\[ \tau_{2}^{\max}=i^{3/4}\cdot\tau_{1}^{\max}. \]

The ratio of the parameters is

\[ \frac{\varepsilon_{1}}{\varepsilon_{2}} = \frac{d_{1}^{2}}{i^{2}D_{1}^{4}}\cdot\frac{D_{2}^{4}}{d_{2}^{2}} = \left(\frac{d_{1}}{i d_{2}}\right)^{2} = i^{3/4}. \]

Comparing the ratio of the parameters \(\varepsilon\) and the values of \(\tau^{\max}\), one may say that the single-turn suspension, in the dynamic sense, differs by better properties, while its stressed state is considerably higher than that of the multi-turn one.

We have limited ourselves to the case in which the suspensions are acted upon only by static forces caused by the weight of the mass. While observing identical calculation conditions, other forces could also have been taken. The choice of forces is arbitrary, since in the present case it is not the absolute values of the stresses that are of interest, but their ratios.

In conclusion, let us consider why pendulum guides with an elastic hinge of the “spring cross” type possess a very large dynamic frequency range. The torsional stiffness of an elastic hinge consisting of several flat springs or of circular cross section is determined by expression [4]:

\[ C_{\mathrm{k.p}}=\frac{nEI_{\mathrm{k.p}}}{l_{\mathrm{k.p}}}, \]

where \(n\) is the number of springs in the elastic hinge (usually 3–4);
\(l_{\mathrm{k.p}}\) is the working length of the springs.

The torsional stiffness can be reduced to a linear dimension through the length of the pendulum \(l_p\), namely:

\[ C'_{\text{k.p}}=\frac{C_{\text{k.p}}}{l_p^2} =\frac{n E I_{\text{k.p}}}{l_{\text{k.p}} l_p^2}. \]

Otherwise we would not be able to compare an elastic hinge with a flexure-clamped suspension.

For example, in order to ensure that the pivot point of the pendulum lever is stable, without “drift” under deformation, the working length of the springs is made short (\(1\text{--}2\) mm). The thickness and width of the springs are of the same order as in a flexure-clamped suspension. Therefore the value of the hinge stiffness, referred to the mass of its springs, becomes two orders of magnitude higher than in the suspensions considered:

\[ \varepsilon_{\text{k.p}}= \frac{C'_{\text{k.p}}}{m_{\text{k.p}}} = \frac{n E b_{\text{k.p}} h_{\text{k.p}}^3}{l_p^2 \cdot 12 \cdot b_{\text{k.p}} l_{\text{k.p}}^2 h_{\text{k.p}}\gamma} = \frac{1}{12\,l_p^2} \left( \frac{a h_{\text{k.p}}}{l_{\text{k.p}}} \right)^2 . \tag{8} \]

Let us take, for comparison, real values of the spring dimensions of an elastic hinge and a flexure-clamped suspension. Let \(h_{\text{p}}=h_{\text{k.p}}=0.1\) mm; \(l_{\text{k.p}}=1\) mm; \(l_p=20\) mm; \(l_{\text{p}}=30\) mm and the spring material be the same. These data are taken for magnetoelectric sensors with a natural frequency \(f_0=.17\) Hz for instruments with electrical correction. The sensors are approximately identical in dimensions and weight. They differ only in their suspensions and, naturally, in the upper frequency of the measurement range. From expression (3)

\[ \varepsilon_{\text{p}}=\frac{a^2\cdot 10^{-2}}{30^4}, \]

and from (8)

\[ \varepsilon_{\text{k.p}}=\frac{a^2\cdot 10^{-2}}{12\cdot 20^2}. \]

Therefore

\[ \frac{\varepsilon_{\text{k.p}}}{\varepsilon_{\text{p}}} = \frac{30^4}{12\cdot 20^2} \simeq 170. \]

This ratio provides the pendulum sensor with a frequency range at least an order of magnitude wider than that of the one being compared. The elastic hinge is, in this case, sufficiently highly stressed. To ensure a large number of operating cycles under alternating deformations of the hinge springs, the material is selected from high-quality alloy steel.

REFERENCES

1. Yu. I. Iorish. Vibrometry, 1963.
2. K. A. Abgaryan. On the theory of beams of minimum weight. In: Strength Calculations. Issue 8, Mashgiz, 1962.
3. L. E. Andreeva. Elastic Elements of Instruments, Mashgiz, 1962.
4. G. S. Zhartovsky. Wide-range pendulum-type vibration transducer. Izvestiya TPI. Collection 146. 1966.