Decreasing the restoring force

At low frequencies and in the absence of an external force, Eq. (25) can be simplified to $S z = - M \ddot x$and read as follows: a relative displacement of the seismic mass by $-\Delta z$ indicates a ground acceleration of magnitude

$$\ddot x = (S/M) \Delta z = {\omega_0}^2 \Delta z = (2 \pi / T_0)^2 \Delta z$$ (34)

where $\omega_0$ is the angular eigenfrequency of the pendulum, and T₀ its eigenperiod. If $\Delta z$ is the smallest mechanical displacement that can be measured electronically, then the formula determines the smallest ground acceleration that can be observed at low frequencies. For a given transducer, it is inversely proportional to the square of the free period of the suspension. A sensitive long-period seismometer therefore requires either a pendulum with a low eigenfrequency or a very sensitive transducer. Since the eigenfrequency of an ordinary pendulum is essentially determined by its size, and seismometers must be reasonably small, astatic suspensions have been invented that combine small overall size with a long free period.

  

Figure 7: a: Garden-gate suspension; b: Inverted pendulum

The simplest astatic suspension is the ``garden-gate'' pendulum used in horizontal seismometers (Fig. 7a). The mass moves in a nearly horizontal plane around a nearly vertical axis. Its free period is the same as that of a mass suspended from the point where the plumb line through the mass intersects the axis of rotation (Fig. 8a). The eigenperiod $T_0=2 \pi \sqrt{\ell/g \sin \alpha}$ is infinite when the axis of rotation is vertical ($\alpha=0$), and is usually adjusted by tilting the whole instrument. This is one of the earliest designs for long-period horizontal seismometers.

Another early design is the inverted pendulum held in stable equilibrium by springs or by a stiff hinge (Fig. 7b); a famous example is Wiechert's horizontal pendulum built around 1905.

  

Figure:(a) Equivalence between a tilted ``garden-gate'' pendulum and a string pendulum. For a free period of 20 sec, the string pendulum must be 100 m long. The tilt angle $\alpha$ of a garden-gate pendulum with the same free period and a length of 30 cm is about 0.2^o. The longer the period is made, the less stable it will be under the influence of small tilt changes. (b) Period-lengthening with an auxiliary compressed spring.

An astatic spring geometry for vertical seismometers invented by [LaCoste 1934] is shown in Fig. 9a. The mass is in neutral equilibrium and has therefore an infinite free period when three conditions are met: the spring is prestressed to zero length (i.e. the spring force is proportional to the total length of the spring), its end points are seen under a right angle from the hinge, and the mass is balanced in the horizontal position of the boom. A finite free period is obtained by making the angle of the spring slightly smaller than 90^o, or by tilting the frame accordingly. By simply rotating the pendulum, astatic suspensions with a horizontal or oblique (Fig. 9b) axis of sensitivity can be constructed as well.

  

Figure 9: Lacoste suspensions

The astatic leaf-spring suspension (Fig. 10; [Wielandt 1975]) is in a limited range around its equilibrium position comparable to a LaCoste suspension but is much simpler to manufacture. A similar spring geometry is also used in the triaxial seismometer Streckeisen STS2 (Fig. 10b).

The delicate equilibrium of forces in astatic suspensions makes them susceptible to external disturbances such as changes in temperature; they are difficult to operate without a stabilizing feedback system.

  

Figure 10: Leaf-spring astatic suspensions

Apart from genuinely astatic designs, almost any seismic suspension can be made astatic with an auxiliar spring acting normal to the line of motion of the mass and pushing the mass away from its equilibrium (Fig. 8b). The long-period performance of such suspensions is however quite limited. Neither the restoring force of the original suspension nor the destabilizing force of the auxiliary spring can be made perfectly linear (i.e. proportional to the displacement). While the linear components of the forces may cancel, the nonlinear terms remain and cause the oscillation to become anharmonic and unstable at large amplitudes. Viscous and hysteretic behaviour of the springs may also cause problems. The additional spring (which has to be soft) may introduce parasitic resonances. Modern seismometers do not use this concept and rely either on a genuinely astatic spring geometry or on the sensitivity of electronic transducers.

  

Figure 11: The relative motion of the seismic mass is the same when the ground is accelerated to the left as when it is tilted to the right.