Tiltmeters and Angle Measuring Equipment

Randall D. Peters, copyright 2000

Department of Physics

1400 Coleman Ave.

Mercer University

Macon, Georgia 31207


The tiltmeter is an instrument whose output is determined by the mass distribution of the earth, since it responds to the local acceleration of gravity, g. Also sometimes called an inclinometer, particularly if its range is large; the response of the instrument is determined by the direction of g relative to its orientation. The principles of operation can be illustrated with common tools of carpentry. Consider a static plumb-bob, which would constitute a spherical pendulum if the system were dynamic. The plumb-bob orients itself along the direction of g, and thus defines the local vertical. Alternatively, a fluid bubble, contained by a tube, will determine one of the locus of directions, orthogonal to g, which constitute the local ``level".

There are two ways in which the output from a tiltmeter can be altered. The first and commonly understood case, is one which occurs when the platform holding the instrument experiences an acceleration, such as by rotation. It can result from either the instrument housing being moved with respect to the ground, or from the ground itself moving. The latter is the basis for earth studies using the horizontal seismometer; which, as a tiltmeter, is sometimes referred to as a ``garden-gate" pendulum. In effect, it is a ``variable g" pendulum in which geff = g sinb. The details of its oscillation, due to a disturbance, are determined by the convolution of the external forcing function with the instrument's response to an impulse (i.e., its Green's function). b is especially important because the period is given by:

T = 2 pi sqr({[I/( mL g sinb)]})

where I, m, and L are, for the pendulous element, the moment of inertia, mass, and perpendicular distance from axis to center of gravity (cg) respectively. Unlike the plumb-bob, practical instruments are constrained to one degree of freedom, responding only to the component of acceleration which is perpendicular to the axis-cg plane. Thus a complete description of the disturbance requires two tiltmeters at right angles to each other.

The second type of tiltmeter disturbance requires an instrument of very high resolution. In this case, the acceleration is virturally zero, but the direction of g is altered due to a distortion of the earth in which there is a global redistribution of its mass. For example, if one is properly positioned with a sufficiently sensitive instrument, it should be possible to observe changes in the direction of g due to free vibrations of the earth. These eigenmodes of the earth were first observed with strainmeters after large earthquakes (1). Additionally, however, an early generation of the tiltmeter to be described in detail later (illustrated in Figure 1) has produced evidence for 15 spheroidal and torsional free-earth modes excited by tidal forces acting on the rotating anelastic earth (2). The longest measured period was 54 min, consistent with the lowest order spheroidal eigenmode. This mode is one in which there are prolate to oblate deviations from the geoid, described by the Legendre P2 function. For the low order modes, excitation by tidal forces is expected to occur because of the global character of the forcing function, which stands in stark contrast to the ``delta-function" character of earthquakes. Thus the efficiency for generating long period oscillations is much greater. It is postulated that the process is one borne of rapid stress relaxation, a kind of internal ``fracture" while the earth is subjected to the 12 hr periodic stress of the tides.


In the past, the most common plane angle measurements have involved visual comparisons against a circularly divided scale. For example, the stator full circle of a student spectrometer would be typically scribed with 720 equally spaced fiducial marks; from which the unknown position of the rotating shaft could be directly read to within 1 min of arc (290 mrad) using an adjacent scale of 30 marks. By viewing through a microscope and interpolating, the practical limiting accuracy of instruments of this type is about 1 sec of arc (4.8 mrad).

Another common instrument, of optical type, has been the precision auto-collimator (3). It will typically measure to 0.5 mrad over a range of 1000 mrad. It is essentially a telescope which superimposes a movable graduated scale on the image. It requires that an optically flat front surface mirror be attached to the rotating object.

One of the simplest, and most common of all angle measuring techniques is that of the optical lever. It has been employed in many physical instruments, such as the Cavendish balance for measuring the Newtonian gravitational constant, G. In this technique, a collimated beam of visible light, such as from a He-Ne laser, is reflected from a front surface mirror attached to the rotating object. The angle change of the reflected beam (twice the amount of mechanical rotation) is determined using the definition of angle in radians; i.e., the ratio of subtended circular arc to the corresponding radius. An overly idealized analysis would suggest that there should be no limit to the resolving power of this instrument, if the path length following reflection could be extended without limit. In actuality, it is very difficult to make measurements to better than 0.1 mrad because of (i) beam divergence, and (ii) increased susceptibility to environmental vibrations at larger path length. Moreover, it is not trivial to generate good permanent records with an optical lever.

For measurement of very small changes in angle, the classic method has been one involving interferometric techniques. Using methods pioneered by A. A. Michelson, measurements in the neighborhood of 0.25 mrad have become routine. Using modern refinements (4), an instrument was built with a resolution better than 1 ×10-10 rad and a dynamic range in excess of 200 dB.

This impressive level of performance would have been impossible without modern electronics. In their original form, all of the aforementioned angle measuring instruments have required that the data be read and recorded by an individual. The collection of data sets of sufficient size for good statistical confidence was a gruelling task in which the possibility for human errors was significant. This has naturally led to the development of instruments which are suitable for automation. Among early versions, the unknown angle was converted to an appropriate electrical signal by means of (i) resistance change using a calibrated slidewire, (ii) phase change between balanced ac inductive circuits such as the selsyn (synchro/resolver) system, and (iii) change in capacitance or inductance which can alter a bridge circuit or the frequency of an oscillator.

For small angles, where the arc length can be reasonably approximated by the associated chord length, one of the most popular sensors has been the linear variable differential transformer (LVDT). The LVDT has even been modified to measure rotation by means of a cardioid shaped rotor, but the performance of the resulting angle sensor is poor compared to the latest capacitive types. All of these inherently analog instruments have been largely replaced in recent years by digital devices, of which the optical encoder is the most common. The increasing presence of digital systems for information processing and display, particularly the advent of the personal computer, has made these digital sensors very attractive. Nevertheless, for some applications they have serious shortcomings that should not be overlooked (5).


As mechanical displacement sensors, capacitive devices have a number of advantages: (i) insignificant loading with no direct mechanical contact, (ii) high stability and reproducibility, and (iii) ease of manufacture resulting in low costs. As compared to inductive sensors, they have been less employed, but that is expected to change due to recently introduced technologies. Their greatest assets had to await electronic developments that could deal effectively with problems that derive from high output reactance and sensitivity to stray capacitance. Prior to amplifiers such as integrated circuit types which use the field effect transistor, it was difficult to exploit their advantages. For an idealized system, one solution to the problem of high output reactance would be to increase the frequency of the oscillator supply; however, real systems derive limited benefit from increased frequency because of stray capacitance effects. Both these and edge effects become less significant for sensors of increased symmetry, such as the new full-bridge or symmetric differential capacitive (SDC) transducers (6). Additionally, sensitivity is greater by a factor of 2 compared to the half-bridge (traditional) sensor with the same electrode area. Viewed in relationship to the general importance of symmetry in physics, these results should not be surprising. What is a matter of surprise is how long it has taken for their discovery.

The SDC sensor strengths are realized when supported by an instrumentation amplifier and a phase sensitive detector (PSD = lock-in amplifier), which takes advantage of the p shift in phase of the output as the bridge voltage passes through zero. In addition to the outstanding linearity that is thus achieved, these packages are characterized by an outstanding signal to noise ratio, since the vast majority of electronic noise is not correlated with the reference (drive source) input to the PSD. Finally, it should be noted that they can be down-scaled in size without severe penalties, because of an ``invariance to scaling" property of their output. This can provide a dramatic improvement in vibration isolation, and additionally it makes them attractive candidates for inclusion in micro-electro-mechanical systems (7).


Shown in Fig.1 is an instrument designed for measuring small tilts (8). In some respects it is similar to the Wood-Anderson torsion seismometer (9); except that in the present instrument the rotor center of gravity is much closer to the torsion fiber. Thus it can be made sensitive to tilt but insensitive to linear acceleration. Using torsion and gravity in opposition, it is an instrument having no friction due to bearings (hinges); and it is not severely hampered by internal friction, if material for the torsion fiber is properly selected. The period is mechanically adjustable, and sensitivity is proportional to the square of the period.  Thus operation near the upper limit (approaching instability) yields a dramatic mechanical amplification of any small tilt, g, perpendicular to the axis defining b. In practice, it is difficult to operate with periods greater than 30 s because of instabilities that derive from (i) differences in thermal expansion coefficients of components, and (ii) mechanical creep which, although small, can be very important near the critical point.

In a comparison of figure 1 with a conventional tiltmeter, the instruments being functionally similar; a significant difference should be noted. The distance from axis to center of gravity is dramatically smaller in the torsion-gravity instrument. Whereas both would respond similarly to a change only in the direction of g; the torsion-gravity instrument is much less sensitive to ground accelerations.

Studies performed with instruments that are similar to the one described have clearly demonstrated that friction is simultaneously the single most important factor and also the least understood feature of such instruments. Without mechanical amplification, discontinuous phenomena of complex character, related to the Portevin Le Chatelier effect (10), mask the majority of low-level processes that would otherwise be interesting candidates for study. In the very low-level regime, mesoanelastic complexity frequently dominates the nonlinear dynamics of mechanical oscillation. It has been known for centuries that defects largely regulate the material properties of solids in the macro-regime. Likewise, they are of critical importance to meso-properties, especially as influenced by surfaces. Apparently because of this complexity, common tiltmeters are unable to observe the hypothesized tidal force excited free earth vibrations. This could be due to their purposely large damping (usually near critical).

Rotor motion in the tiltmeter of Figure 1 is detected most effectively with a full-bridge capacitive detector, such as illustrated in Figure 2. Linearity is typically better than 0.1% over 90% of full-range, and resolution (independent of mechanical amplification) is about 0.1 mrad. The overall system (tilt) sensitivity is >1 nrad with a dynamic range > 150 dB, and its low frequency performance has typically been limited by electronics stability.

One can understand the physics of this detector by considering how areas of the static plates ``communicate" with each other as the rotor angle is altered. Any portion of an intervening rotor ``blade" constitutes a ``shield"; since charge cannot be induced through a ground plane. In its equivalent circuit, adjacent bridge components change, through area variation, by an equal amount but with opposite sign.

The use of arrays is attractive not only for reason of improved sensitivity, but also for overall size reduction. There is great advantage in mechanical compactness, because the frequencies of undesirable mechanical modes can then be shifted farther from noises of the environment. This provides an increased immunity to noise as compared to larger instruments, just as was noted with scanning tunneling microscopes subsequent to their size reduction. Not only does this increase the signal to noise ratio of the instrument, but there can be improvements in user friendliness and versatility as well.


1. D. C. Agnew, ``Strainmeters and tiltmeters", Rev. Geophys. 24 (3), 611 (1986).

2. M. H. Kwon, ``Refinements of a new balance for measuring small force changes", PhD dissertation, Texas Tech University, 71 (1990).

3. Methods of Experimental Physics, Vol 1, Classical Methods, ed. Estermann, 56 (1959).

4. L. N. Mertz, ``Interferometric angle encoder", Rev. Sci. Instrum. 62 (5), 1356-1360 (1991).

5. R. Pallas-Areny and J. Webster, Sensors and Signal Conditioning, Wiley, New York (1991).

6. R. Peters, ``Capacitive angle sensor with infinite range", Rev. Sci. Instrum. 64 (3), 810-813 (1993).

7. L. O'Connor, ``Mems: microelectromechanical systems", Mech. Engr. 114 (2), 40 (1992).

8. R. Peters, ``Mechanically adjustable balance and sensitive tiltmeter", Meas. Sci. Technol. 1, 1131 (1990).

9. T. Teng, ``Seismic instrumentation", Methods of Experimental Physics, Vol. 24 part B, GEOPHYSICS Field

Measurements, 55 (1987).

10. A. Portevin and F. Le Chatelier, ``Tensile tests of alloys undergoing transformation", Comptes Rendus Acad. Sci. 176, 507 (1923).


Figure 1. Example of  a tiltmeter with adjustable sensitivity, that incorporates properties of a torsion balance.

Figure 2. Example of a symmetric differential capacitive (full-bridge) detector for measuring rotor position in the tiltmeter of Figure 1. The detector uses an array of eight identical sensors, connected in parallel, to increase sensitivity.