Influence of External Disturbances on the Operation of the Elastic Element of a Pressure Gauge
Abstract Generated abstract
This paper examines how external mechanical vibrations affect the elastic tubular spring and readings of industrial pressure gauges. The author models the tubular spring as an approximate single degree of freedom system, derives an expression for its natural frequency using Lagrange’s equations, and compares calculated frequencies with experiments on a vibration stand for gauges with different upper measurement limits and loading conditions. The experiments indicate that natural frequency and the acceleration required to initiate resonance increase with the measurement limit, while forced vibrations can produce pointer swings far exceeding permissible errors. The study concludes that ordinary pressure gauges are unsuitable for vibrating installations without design improvements.
Full Text
EFFECT OF EXTERNAL DISTURBANCES ON THE OPERATION OF THE ELASTIC ELEMENT OF A PRESSURE GAUGE
V. T. TIKHONOV
(Presented by the scientific seminar of the Department of Theoretical Mechanics)
Pressure gauges produced by the industry of our country are precise measuring instruments.
However, in a number of cases, when they are installed on machines that vibrate during operation, they give false readings that do not reflect the actual pressure values. As a result, the possibility of accidents occurring in production cannot be ruled out.
This makes understandable the concern of enterprises that use pressure gauges, which have notified us of this and have asked us to determine the causes producing false readings of pressure gauges and to outline ways of eliminating these causes.
The task of the present investigation was to study the influence of certain external mechanical actions on the readings of pressure gauges—in particular, the influence of harmonic oscillations.
As is known, the principal working element of a pressure gauge is a tubular spring; therefore, before the investigation, certain questions concerning the oscillation of the tubular spring were first considered. To determine completely the deformations arising in a tubular spring during oscillations, it was necessary to establish the dependence of the displacements of all points of the system on time. A tubular spring is a system with an infinite number of degrees of freedom. The study of the oscillations of such systems becomes possible only with considerable simplifications. Having studied the character of the oscillations of the pressure-gauge spring, we came to the conclusion that, to a certain approximation, the oscillations of the tubular spring can be related to the change in the central node. In this case the scheme of action of the forces of the spring is reduced to a system with one degree of freedom (Fig. 1).
Fig. 1. Computational scheme of the operation of the spring
The derivation of the equation for the natural frequency of oscillations of the elastic element of the pressure gauge was carried out by means of Lagrange’s equations of the second kind in generalized coordinates.
Dependence of the natural frequency of oscillations of a tubular spring on its parameters (generally accepted notation).
\[ \frac{d}{dt}\left(\frac{\partial T}{\partial \dot q}\right)-\frac{\partial T}{\partial q}=Q. \tag{1} \]
Expressing \(Q\) through the potential energy of the oscillating spring, we have
\[ \frac{d}{dt}\left(\frac{\partial T}{\partial \dot q}\right)-\frac{\partial T}{\partial q}=-\frac{\partial u}{\partial q}. \tag{2} \]
Taking into account the independence, with respect to time, of the constraints imposed on the system and the small magnitude of the oscillations of the system, we have
\[ T=\frac{a\dot q^{2}}{2}, \quad \text{where } a=\mathrm{const}. \tag{3} \]
For stationary constraints, the potential energy \(u\) is only a function of the coordinate \(q\). If the equilibrium position of the system is taken as the origin for measuring the potential energy, then \(u=0\), and therefore \(Q=0\), i.e. \(\dfrac{du}{dq}=0\).
Expanding \(|q|\) in powers of \(q\), beginning with terms no lower than the second degree with respect to \(q\), we have
\[ u(q)=\frac{1}{2}cq^{2}. \tag{4} \]
Substituting expressions (3) and (4) into equation (1–2), we obtain
\[ a\ddot q+cq=0. \tag{5} \]
Taking \(\rho=\dfrac{r+R}{2}\) and considering the mass of the spring \(M\) to be uniformly distributed along the arc of a circle of radius \(\rho\), we have the kinetic and potential energy
\[ T=\frac{1}{2}M\rho^{2}\dot\varphi^{2} \quad \text{and} \quad u=\frac{1}{2}c\varphi^{2} \tag{6} \]
Using equations (1), (2), (6), (7), we obtain
\[ M\rho^{2}\ddot\varphi+c\varphi=0. \tag{7} \]
The solution of this equation will be
\[ \varphi=A\sin(\omega t+\alpha), \]
\[ \omega=\sqrt{\frac{c}{M\rho^{2}}}. \tag{9} \]
Substituting into equation (9) of the circular frequency the value of \(c\), and expressing the mass through the weight, as well as reducing the circular frequency \(\omega\) to the ordinary frequency \(\nu\), equal to the number of complete oscillations per unit time, we shall have
\[ \nu=\frac{\sqrt{\dfrac{Fq}{\rho\Theta P}}}{2\pi}. \tag{10} \]
Having determined experimentally the dependence of the angle of twisting of the spring on the magnitude of the disturbing force according to the loading scheme shown in Fig. 1, we construct curves of the natural frequency of the pressure gauge versus the upper limit of measurement (Fig. 2). The points are marked with crosses.
Experimental investigations of the natural frequency of pressure gauges were carried out on the vibration stand \(DK\).
We assume that the natural frequency of a single-coil tubular spring is at the same time also the natural frequency of the pressure gauge, since the spring is the principal working element of the latter.
The principle of operation of the stand is as follows: the field of a magnet, supplied with direct current, vibrates a coil through which alternating current passes, the amplitude and frequency of which can be regulated. The vibrating table is rigidly connected to the coil and reproduces mechanical vibrations according to the law \(x = a \sin \omega t\).
By changing the magnitude of the acceleration of the stand, the magnitude of the disturbing force is changed.
Fig. 2. Curve of the natural frequency of a pressure gauge versus the upper limit of measurement
Dependence of the Natural Frequency of Pressure Gauges on the Upper Limit of Measurement
Determination of the natural frequency of pressure gauges was carried out for a loaded and an unloaded pressure gauge. Loading was performed at \(1/4\), \(2/4\), and \(3/4\) of the upper limit of measurement.
Comparing the experimental and calculated results for pressure gauges of diameter 150 mm, we see that these results agree relatively well. The course of the experimental frequency curve is analogous to the calculated one and confirms the latter (Fig. 2). The points of the experimental curve are marked with triangles.
Dependence of the Swing of the Pressure-Gauge Pointer at Resonance on the Upper Limit of Measurement
It is practically important to know the magnitude of the pointer swing in the case of resonance of the pressure-gauge spring. This swing depends not only on the frequency and amplitude of the external force, but also on the parameters of the system. Both in free vibrations and in forced vibrations, the entire process is characterized by two parameters of the system: the natural frequency and the damping decrement.
In the pressure-gauge industry, instead of the damping decrement, the term “quality factor” of the system is often introduced, which is equal to
\[ D = \frac{\sqrt{mc}}{n}, \]
where \(m\) is the oscillating mass, \(c\) is the stiffness, and \(n\) is the coefficient of the force
...of braking, which will consist of the frictional force of the pressure-gauge mechanisms and the forces of internal friction of the elastic element.
If the dependence of the displacement amplitude on the ratio of the natural frequency to the frequency of the forced vibrations is represented graphically, the well-known graph of curves is obtained (Fig. 3).
Analysis of the graph shows that, as the quality factor increases, the displacement amplitude also increases; at small values of the quality factor the amplitude reaches a maximum in the subresonant region.
As is evident from the expression for the quality factor, the latter depends mainly on the stiffness of the system \(c\) and on the moment of inertia of the mass. On this basis one might have expected that pressure gauges whose springs have a higher quality factor should have a larger pointer swing at resonance.
The experiments carried out by us clearly confirm this proposition, especially for 150 mm pressure gauges (Fig. 2).
Fig. 3. Graphical representation of the dependence of the vibration amplitude on the ratio of natural frequencies at various quality factors of the pressure gauge
Magnitude of the disturbing force of acceleration necessary to bring the pressure gauge into resonance
Oscillations in a system under the action of a disturbing force are determined not only by the parameters of the system, but also by the magnitude of the disturbing force. In order for a mechanical system to begin resonating, coincidence of the system frequency with the frequency of the forced vibrations alone is insufficient. It is obvious that the higher the quality factor of the system, the greater the disturbing force required for the given system to begin resonating.
Our experiments established the dependence of the onset of resonance of a pressure gauge on the change in acceleration on the test stand.
It turned out that the higher the measurement limit, the greater the acceleration required for the pressure gauge to begin resonating. This is also readily explained by the fact that the quality factor of the pressure gauge increases with increasing measurement limit, i.e., with an increase in the stiffness and mass of the spring.
Influence of the frequency of forced vibrations at specified accelerations on the magnitude of the pressure-gauge pointer swing
We carried out a study of the influence of the frequency of forced vibrations, at a constant magnitude of acceleration, on the readings of pressure gauges. For this purpose, pressure gauges with upper measurement limits of 10, 25, 40, and 100 kg/cm² were taken and subjected to vibration at accelerations 1, 2, and 3 times greater than the acceleration of free fall of a body.
The frequency was varied within the range from 20 to 350 Hz.
Experimental investigations showed that, under vibration, pressure gauges have several separate resonance points at a frequency of the forced vibrations below 150 Hz. Moreover, the first resonance points coincide with those obtained earlier.
On the basis of the experimental investigations carried out, we consider it possible to draw the following conclusions:
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The natural frequency of pressure gauges increases with an increase in the upper limit of measurement.
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As the upper limit of measurement increases, a greater acceleration is required to bring the pressure gauge into resonance.
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The experimentally found magnitude of the swing of the pointer of pressure gauges at a forced-oscillation frequency in the range from 18 to 350 Hz in most cases exceeded the permissible errors of the pressure gauges by up to sixteen times.
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Ordinary pressure gauges conforming to GOST 8625-95 cannot operate under vibration conditions, since a considerable swing of the pointer occurs under the direct action of vibration, reaching up to 20% of the upper limit of measurement.
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It is necessary to improve the design of pressure gauges for their operation on vibrating bases.
REFERENCES
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S. A. Bernstein. A New Method for Determining the Frequencies of Oscillation of Elastic Systems, 1939.
S. A. Bernstein. A new method for calculating oscillation frequencies and its application to stability problems. Proceedings of the Military Academy of Mechanization and Motorization, 1940.
S. P. Strelkov. Introduction to the Theory of Oscillations. Mashgiz, 1950.
V. I. Fedoseev. Elastic Elements of Precision Instrument Making. Mashgiz, 1958.