Wide-Range Pendulum-Type Vibration Sensor

G. S. Zhartovsky

(Submitted by the Department of Mining Machines, Mine Transport, and Mining Mechanics)

For measuring vibrations of mechanical objects, seismic vibration sensors have become widespread; in them the reference point is created artificially by means of seismic, or otherwise inertial, systems, an essential element of which is an elastic suspension. Usually the elastic suspension simultaneously serves as the motion guide for providing linear or torsional oscillations. The frequency properties of vibration sensors depend on how it is made. For a vibrometric regime, one always strives to make the natural frequency \(f_0\) as low as possible, since it limits the lower boundary of the measurement frequency range. This is sometimes associated with an inadmissibly large static deflection of the seismic mass, increasing the dimensions and worsening the operating properties of the sensors. The upper limit of measurements in frequency is restricted either by loss of stability of the elastic suspension under the action of unmeasured vibration components, or by the influence of the distributed masses of the springs, i.e., by high resonances \(f_i\). It is convenient to evaluate the dynamic properties of seismic systems by the magnitude of the dynamic frequency range, expressed through the ratio \(\dfrac{f_i}{f_0}\). For measurement purposes the most important is the first overtone \(f_1\). In serial domestic sensors intended for measurements in the infrasonic frequency range, the value \(\dfrac{f_1}{f_0}\) is 15–30, which does not always satisfy practical requirements. An analysis of the most widespread types of elastic suspensions shows that the problem of substantially extending the dynamic frequency range can be satisfactorily solved by means of pendulum suspensions using an elastic frictionless hinge—the “spring cross,” which provides a dynamic frequency range an order of magnitude higher than the values cited. The present work is devoted to the study of the properties of magnetoelectric vibration sensors with a pendulum suspension on an elastic hinge.

As is known, the natural frequency of a pendulum suspension (Fig. 1) is determined by the expression

\[ f_0=\frac{1}{2\pi}\sqrt{\frac{c}{I}},\ \text{Hz}, \]

where \(c\) is the stiffness of the springs with respect to torsional oscillations;

\(I = Ml_p^2\) is the moment of inertia of the seismic mass \(M\);

\(l_p\) is the distance from the center of mass to the point of rotation (under real conditions \(l_p\) takes into account the influence of the lever mass).

The stiffness with respect to torsional oscillations can be determined from the following considerations. Under the action of the weight \(P\), the mass \(M\) will be deflected by the amount of the static deflection \(\lambda_{\mathrm{st}}\), related to the natural frequency by the dependence

\[ \lambda_{\mathrm{st}} \simeq 2.48 \left(\frac{10}{f_0}\right)^2 \ \mathrm{mm}. \]

Each of the springs forming the elastic hinge will be loaded practically only by the bending moment

\[ M_{\mathrm{bend}}=\frac{m_0}{n}=\frac{P \cdot l_p}{n}, \]

where \(n\) is the number of springs in the hinge.

Fig. 1. Operating diagram of a pendulum suspension with an elastic hinge

The deformation of the springs due to the force \(P\) compressing the vertical spring is small; therefore its effect may be neglected. Thus, by solving the bending equations for springs treated as beams with rigidly fixed ends, loaded only by a bending moment, it is easy to obtain an expression for the stiffness of the elastic hinge. Let us consider this for the example of a horizontal spring.

\[ EI \frac{d^2 y}{d x^2}=M_{\mathrm{bend}}=\frac{m_0}{n}, \qquad \frac{dy}{dx}=\varphi(x)=\frac{m_0}{EI \cdot n}\cdot x. \]

But since \(c\varphi=\dfrac{m_0}{n}\), then

\[ c=\frac{EI}{l_{\mathrm{pr}}}\cdot n, \]

where \(l_{\mathrm{pr}}\) is the working length of the spring.

Knowing the upper limit of vibration-displacement measurement, one can readily calculate the stresses in the springs and estimate the service life of the sensor (the number of permissible cycles from the Wöhler curves for the spring material), since in the post-resonance region the deflections of the seismic mass are equal to the external excitation. If the spring length and the upper measurement limit are kept constant, the stresses in the springs increase with increasing natural frequency of the sensor. For springs of rectangular cross section this dependence can be calculated and represented as

\[ \frac{\sigma'}{\sigma''}=\left(\frac{f_0'}{f_0''}\right)^{2.3}, \]

and for a circular cross section,

\[ \frac{\sigma'}{\sigma''}=\left(\frac{f_0'}{f_0''}\right)^{1.2}. \]

Here \(\sigma'\) and \(f_0'\) are taken for one suspension, and \(\sigma''\) and \(f_0''\) for another.

Pendulum suspensions are sometimes made on hinges with dry rolling or sliding friction. Bearings increase the sensitivity threshold of the sensor and make it impossible to measure vibrations, co-

measured, with “play” in the joints. In other words, constancy of the pendulum lever’s point of rotation is not ensured. Elastic hinges of the “spring cross” type are free of these shortcomings. The property of such a hinge to preserve, with high accuracy, the constancy of the point of rotation within 10–15° of lever rotation deserves special attention. An experimental check of this property was carried out on four crosswise arranged springs 150 mm long. Within 6–8°, the displacement of the point of rotation was 0.1–0.15 mm. Thus, springs with an effective length of 1.5–2 mm can ensure a deviation of the point of rotation of no more than 1–1.5 μm, and in the case of vibration measurements of several microns the above-mentioned deviations amount to hundredths of a micron. That is, such a seismic system can be recommended for precision measurements. This property is also confirmed by the nature of the purely moment loading of the springs, as assumed in our calculation of stiffness, since only with an identical radius of curvature of the springs, which is characteristic of purely moment loading, is a stable position of the axis of rotation possible.

In order for the seismic mass to accurately follow the motion of the object being measured, it is necessary to keep the stiffness value constant within the prescribed limits of deformation of the elastic suspension. It is customary to assume [1] that the springs of sensors operate in the linear zone if the magnitude of the maximum deflection of the springs is no more than 10% of the working length of the spring. This requirement must be taken into account when selecting the length of the lever. Thus, for example, for \(l_{\mathrm{pr}} = 2\) mm and a maximum measurement limit of \(\pm 2\) mm, the length of the lever, or more precisely \(l_1\), must be at least 20 mm.

Under difficult experimental conditions, any spatial orientation of the sensor and, consequently, of the seismic system may be required. This requirement may be incompatible with the chosen natural frequency, since a large static deflection \(\lambda_{\mathrm{st}}\) uses up the entire free travel of the sensitive element. On the other hand, there is a limit to the stable state of a pendulum suspension positioned vertically with the mass upward.

The condition of static stability can be determined as follows. Let the pendulum lever deviate from the vertical under the action of the weight of the mass \(P\) by an amount \(\varphi\). This deviation will be opposed by the restoring moment of the elastic hinge, \(c\varphi\). The equilibrium equation of the rod in the deflected state takes the form

\[ P \cdot l_p \cdot \varphi - c\varphi = 0. \]

From the condition of the deflected state \((\varphi \ne 0)\) we find

\[ c_{\mathrm{cr}} = P \cdot l_p. \]

For free oscillations of the pendulum, an inertial term is added to the equilibrium equation:

\[ P l_p \varphi - c\varphi = I \frac{d^2 \varphi}{dt^2}, \]

or

\[ \frac{d^2 \varphi}{dt^2} + \frac{c - P \cdot l_p}{I}\cdot \varphi = 0. \]

Obviously, when \(c_{\mathrm{cr}} = P l_p\), the natural frequency of the pendulum becomes zero, i.e., the system loses stability. Knowledge of the limiting value of the stiffness is especially necessary when designing sensors with natural frequencies of several hertz.

The choice of the minimum length of the pendulum arm from the condition that the springs operate in the linear region may require an additional correction if the sensor is to operate under the action of unmeasured lateral components. An analysis of the dynamic stability of suspensions under the action of parametric loads shows that, from unmeasured vibrations, inertial forces may arise which are applied to the seismic mass. When resolved, these alternating forces will act both along the pendulum arm, stretching or compressing it, and in the direction of the measurement axis, additionally swinging the suspension. Without touching on the special question of the occurrence of parametric resonances, it should be noted that the shorter the arm, i.e., the greater the angle \(\varphi\), the greater the axial component of the inertial forces arising from lateral vibrations. Obviously, this increases the sensitivity of the sensor to unmeasured vibrations. Having specified in advance the value of the lateral sensitivity \(\gamma\%\) and the possible \(\varphi_{\max}\) during operation, one can correct the value of \(l_p\). The lateral sensitivity is

\[ \gamma = \operatorname{tg}\varphi_{\max}\cdot 100\%. \]

The angle \(\varphi_{\max}\), as is seen from Fig. 1, is determined from the maximum measurement limit \(\xi_{0}^{\max}\) and the length of the arm

\[ \operatorname{tg}\varphi_{\max} \simeq \frac{\xi_{0}^{\max}}{l_p}, \]

therefore,

\[ l_p = \frac{\xi_{0}^{\max}}{\gamma}\cdot 100. \]

If the influence of distributed masses on the forced oscillations of the seismic system under consideration can be substantially reduced by using a “spring cross,” thereby extending the upper limit of frequency measurements, then the use of a magnetoelectric transducer for converting the oscillations of the seismic mass into an electrical signal makes it possible to vary the value of the natural frequency \(f_0\) over wide limits. As noted above, the static deflection \(\lambda_{\text{st}}\) is associated with this frequency. For example, for a sensor with a natural frequency of 5 Hz, intended for measuring vibrations within the range \(\pm 3\) mm, it is necessary to provide a free travel of at least 16 mm. It is not always possible to construct such a sensor; therefore the static deflection is sometimes compensated by introducing lifting springs of relatively low stiffness and large travel in sensors measuring vertical vibrations [2]. When measuring horizontal vibrations, lifting springs are not installed, i.e., the sensors cannot operate in any spatial orientation. The introduction of lifting springs impairs the frequency characteristics of sensors at high frequencies owing to the manifestation of higher resonances of the compensating spring. A promising method is the electrical compensation of static deflection in sensors with a magnetoelectric transducer, which consists in the following. A direct current is passed through the transducer coil; interacting with the field of the permanent magnet, this current holds the seismic mass in one position or another depending on its magnitude and direction. The compensating force can easily be determined:

\[ F = B l I_k, \]

where \(B\) is the induction in the air gap of the transducer,

\(l\) — active length of the coil conductor.

\(I_k\) — compensation current.

Since the component of the weight of the mass that produces the static deflection varies according to a sine law depending on the spatial orientation of the sensor, it is easy to provide a device for regulating \(I_k\) in order to ensure normal operation of the sensor in any position.

This method was used [3] in the development of low-frequency magnetoelectric vibration sensors with a natural frequency of 4–6 cps and an uncompensated static deflection of 7–16 mm. With an induction in the gap of 2000 tesla, a number of turns of 700 \((R_{\text{load}} = 500\ \Omega)\), the compensation current was 10 mA. The sensors operated stably in any orientation.

Use of the frequency range up to the resonance frequency is possible only when the nonuniformity of the amplitude-frequency characteristic in the resonance zone does not exceed the permissible measurement error. For these purposes, one usually seeks to introduce effective damping of the oscillatory system. In magnetoelectric sensors, damping is carried out electromagnetically: by the Foucault currents of the coil frame and by the generated current of the coil. The damping effect can be made greater the higher the induction in the air gap of the magnetic system, since the degree of damping is proportional to the square of the induction. When designing a closed system (Fig. 2), it makes sense to make full use of the dimensions of the magnetic circuit (i.e., \(D_1\) should be equal to \(D_2\)) in order to obtain maximum sensitivity of the sensor, since the induction \(B\) in the gap does not depend on the mean gap \(D_{\text{av}}\), while the sensitivity is proportional to the induction and to the active length of the coil turns. Indeed, from Ohm’s law for a magnetic circuit it follows that

\[ \Phi = \frac{Iw}{R_k + R_m + R_v}, \]

where \(\Phi\) is the magnetic-induction flux;

\(Iw\) — magnetomotive force of the magnet;

\(R_k\) — magnetic resistance of the permanent magnet;

\(R_m\) — magnetic resistance of the magnetic circuit;

\[ R_v = \frac{\delta}{S_v} \]

— magnetic resistance of the air gap;

\(S_v\) — cross-sectional area of the air gap.

Fig. 2. Magnetic system of the vibration sensor

Under the realistic assumption that \(R_k + R_m \ll R_v\), we obtain

\[ \Phi \simeq \frac{Iw}{\delta}\cdot S_v. \]

Consequently,

\[ B = \frac{\Phi}{S_v} = \frac{Iw}{\delta}. \]

Thus, while maintaining a constant value of the working gap \(\delta\), the gain in sensitivity is obtained by a factor of

\[ \frac{D_2}{D_1} \]

and the induction in the gap does not change. A change in the value of \(D_{\text{av}}\), strictly speaking, may have a positive effect on the damping effect, but it is insufficient; therefore, in order to flatten the amplitude-frequency characteristic in the resonance zone—

Fig. 3. Effect of shunting the vibration sensor on the amplitude–frequency characteristic.

Vertical axis: Dynamic coefficient

Horizontal axis: Vibration frequency, Hz

Curve labels:

• \(R_{\mathrm{sh}} = 100\,\Omega\)
• \(R_{\mathrm{sh}} = 560\,\Omega\)
• \(R_{\mathrm{sh}} = 1\,\mathrm{k}\Omega\)
• \(R_{\mathrm{sh}} = 1.6\,\mathrm{k}\Omega\)
• \(R_{\mathrm{sh}} = \infty\)

Parameters indicated in the figure:

\[ W = 1000 \text{ turns } (d = 0.06 \text{ mm}) \]

\[ B = 2000 \text{ gauss} \]

\[ \delta_y = 10 \text{ mm/Hz} \]

\[ R_{\mathrm{damp}} = 400\,\Omega \]

\[ f_0 = 8 \text{ Hz} \]

resonance, the sensor is shunted, knowingly accepting a loss of sensitivity and a certain increase in its natural frequency \(f_0\). Figure 3 shows an experimentally obtained family of amplitude–frequency characteristics of a magnetoelectric sensor with various shunting. For the given sensor parameters, the optimum shunt resistance is \(R_{\mathrm{sh}} = 560\ \Omega\).

It should be noted that the stability of the amplitude–frequency characteristics in the resonance region is associated with a change in the damping factor \(\beta\) of the sensor. For magnetoelectric sensors, a decrease in the damping factor is caused by an increase in the active resistance of the aluminum or copper coil frame and of the turns with temperature. This is shown in Fig. 4; for convenience, however, the values of the dynamic coefficient \(K_{\mathrm{d}}\), which is related to the damping factor at the resonance frequency by the dependence

\[ K_{\mathrm{d}} = \frac{1}{2\beta}. \]

are plotted along the abscissa axis.

As can be seen from the graph, the change in the damping factor with temperature is approximately \(2.5\%\) per \(10^\circ\mathrm{C}\).

Fig. 4. Thermal stability of the dynamic coefficient

All the considerations set forth above formed the basis for the design of sensors used in a number of vibration-measuring instruments; moreover, the sensors differed structurally from one another only in the stiffness of the springs [4, 5]. The design of a sensor with natural frequency \(f_0 = 17\) cps is shown in Fig. 5, and its photograph—in Fig. 6.

The magnetic system of the vibration sensor consists of a permanent cone-shaped magnet 1, with a pole piece 2 pressed onto it, and a cylindrical magnetic core 3. The permanent magnet is made of the alloy “Magnico,” and parts 2 and 3 of magnetically soft iron. The conical part of the magnet is inserted into the magnetic core and clamped by washer 4. In the radial gap of the magnetic system there is located the coil 5 of the magnetoelectric transducer, which is wound on a duralumin frame with lever 6. Parts 5 and 6 form the seismic mass of the elastic system of the vibration sensor, which, by means of two crosswise-arranged flat springs 7, is suspended from the boss of the magnetic core, rigidly connected to the duralumin housing 8 by screws. The housing is screwed shut with cover 9 and is provided with two terminal buttons—current leads 10—and a threaded boss for fastening to the object. Springs 7

Fig. 5. Sensor design

made of high-quality spring steel 0.06 mm thick. The working length of the springs is 1.5 mm. Thus, the flat springs form a frictionless hinge and ensure rotational motion of the seismic mass. To limit the travel of the transducer coil, a stop 12 with a slot, in which the limiting pin 13 moves, is attached to the magnetic circuit. The magnetoelectric transducer has the following parameters: induction in the air gap, 2500 tesla; number of coil turns, 1000 (PEL-II wire, 0.08 mm); coil resistance, 200 ohms.

Fig. 6. Sensor with the housing removed

Technical characteristics of the sensor:
velocity sensitivity — 10 mV/mm sec;
frequency range 15–1200 Hz (with nonuniformity of the amplitude-frequency characteristic in the resonance zone of ±12%);
upper limit of measured vibration-displacement amplitudes — 2 mm;
upper limit of measured accelerations (continuous mode) — 15 g;
upper limit of measured accelerations (short-time mode) — 30 g;
sensitivity to lateral vibrations, not more than — 10%;
temperature range ±60°C;
spatial orientation arbitrary;
sensor weight 180 g.

REFERENCES

1. L. D. Gik. Frequency characteristics of real seismic vibration sensors and their modeling. Proceedings of the conference “Automatic Control and Electrical Measurements.” Issue II. Novosibirsk, 1960.

2. Yu. P. Iorish. Vibrometry. 1963.

3. L. D. Gik, A. I. Pinchuk, F. D. Laptev, I. G. Mityukhin, V. I. Pechurkin, G. S. Zhartovskii. VVA 1 vibration meter. Registration certificate No. 48292, 1964.

4. L. D. Gik, G. S. Zhartovskii, I. G. Mityukhin, V. I. Pechurkin. Development of a vibration sensor. Registration certificate No. 48288, 1964.

5. K. B. Karandeev, L. D. Gik, I. G. Mityukhin, G. S. Zhartovskii, V. I. Pechurkin, F. D. Laptev, L. A. Tomilina, Ya. T. Zagorskii. Development of a vibration-measuring instrument for operation with an induction sensor. Registration certificate No. 48290, 1964.