# Long Period Pendulums

Long Period Pendulums find use for measurements of gravity, designs of seismometers, and for vibration isolation.

The longest "mass on a string" pendulum was made in 1901 by faculty members of the Michigan College of Mines. They were helping to survey a local vertical mine shaft using plumb bobs which were as long as 4440 ft. Being physicists, they also pushed the bobs aside and measured the period of oscillation (70 sec). Michigan College of Mines is now known as Michigan Technological University (MTU).

Ref: see B H Suits, "Long Pendulums in gravitational gradients," European Journal of Physics 27, L7-L11 (2006).
Additional historical details can be found here.

The remaining pendulums below are physically small pendulums which use some mechanical means to achieve a long period. Remembering that the basic oscillator frequency is proportional to the square root of

Return force per unit displacement
Inertia

you'll see that in each case mechanical forces are provided which modify this ratio. In most cases, the return force per unit displacement is made small by a mechanical constraint.

Except where noted, the list below is restricted to those where the return force is provided by gravity alone but where the motion is subject to mechanical constraints. This seems a reasonable, though not overly restrictive, definition of a "pure" (physical) pendulum. The references given are recent/interesting applications and are not necessarily primary references.

If a figure is not clear due to finite screen resolution, right click on it and choose "view image".

### Balance-like physical pendulum

Similar to a balance used for comparing weights, a physical pendulum supported slightly away from the center of mass, but symmetrically from either end, will have a large inertia but little restoring force. For this configuration, the equilibrium position is horizontal. The closer the pivot point is to the center of mass, the longer the period.

Ref: I P Krylov, "A low-frequency pendulum for studying vortex lattice melting," Supercond. Sci. Technol. 9, 583-588 (1996).

### Off-center or Nearly balanced physical pendulum

A physical pendulum which is almost perfectly balanced is a simple way to make a long period pendulum. Similar to the balance-like pendulum except the pivot point is displaced from the center of mass along the length of the pendulum. It can be regarded as a particular coupling between an (unstable) inverted pendulum and a stable pendulum. The equilibrium position is vertical.

Refs: R. D. Peters, "Metastable states of a low-frequency mesodynamic pendulum," Applied Physics Letters 57, 1825-1827 (1990).

### Folded Pendulum

A folded pendulum is, in some sense, a generalization of the "nearly balanced physical pendulum." Here a pendulum and an inverted pendulum are also coupled. The return force for the inverted pendulum is negative, so it will cancel the return force for the pendulum. As long as the cancellation isn't complete, some oscillation can result. The motion of the mass M is almost horizontal.

An equation for the oscillation frequency (for a slightly different set-up than what is shown) is contained in the reference.

Ref: See A. Bertolini, et al., "Monolithic Folded Pendulum Accelerometers for Seismic Monitoring and Active Isolation Systems," IEEE Trans. Geoscience and Remote Sensing 44, 271-276 (2006).

### Lehman Seismometer Pendulum

The Lehman Seismometer is based on a rotational system which is slightly tilted from vertical. This is somtimes also called the Garden Gate configuration. If theta were 90 degrees, it would be a simple physical pendulum. The support structure counteracts much of the gravitational force, and the component of the gravitational force which is left can be very small. The Zollner suspension system gives rise to the same type of motion.

If the mass of the supporting structure can be neglected compared to M, the effective pull of gravity becomes Mg sin(theta). Putting that in the equation for a physical pendulum yields the frequency (or period) of oscillation.

Ref: J. Walker, "The Amateur Scientist," Scientific American Magazine, July 1979. (There are many discussions of the Lehman seismometer on the World Wide Web - use your search engine).

### Zero Length Spring Pendulum

A spring which obeys Hooke's law and where the force extrapolates to zero when the spring has zero length is called a zero length spring. Of course the spring will never reach zero length in practice. When balanced against gravity a very long period oscillation in the vertical direction can be obtained. This is really a combination of a mass on a spring and a physical pendulum but is included here for historical reasons.

For a typical spiral wound steel spring, a zero-length spring will be pretensioned during manufacturing so that with no force applied, the windings are already in contact.

A mathematical analysis for such a pendulum is available here (pdf).

Refs: See U.S. Patents 2293437 and 2674887 or use your search engine to look for Worden and/or La Coste Gravitometers or search for "zero length spring."

### Crossed Wire Pendulum

The crossed wire pendulum or "X-pendulum" tends to minimize the return force while maintaining the inertia. The motion is adjusted to be horizontal near the center of mass of the hanging portion. Thus there is a plenty of inertia and little return force.

In the figure only two wires are visible. In practice, there will be at least two in parallel for stability, making a minimum of three wires. Four wires (two parallel pairs) is common.

A computation of the period of oscillation and a good description of the motion is contained in the first reference.

Refs: See
M. A. Barton and K. Kuroda, "Ultralow frequency oscillator using a pendulum with crossed suspension wires," Review of Scientific Instruments 65, 3775-3779 (1994);
and N. Kanda, M. A. Barton, and K. Kuroda, "Transfer function of a crossed wire pendulum isolation system," Review of Scientific Instruments 65, 3780-3783 (1994).

The Scott-Russel Linkage constrains the suspension point for a mass to be along a path similar to that of a very long mass-on-a-string pendulum. Hence, provided the mechanism itself doesn't add much mass, the return force per unit displacement is similar to a very long pendulum and hence a long period results. Referring to the figure, the constraints include a string connected to a point near the middle of a rigid rod, forcing that point to move in a circular arc, while the bottom of the rod is constrained to move vertically. The geometry is chosen so that the motion of the center of mass is almost horizontal.

In the figure a horizontal wire under minimal tension is shown as the mechanism to constrain the base of the rigid rod so that it can only move vertically. An additional wire perpendicular to this can be used to restrict sideways motion if needed. Other mechanical means can be used to provide this constraint. If horizontal wires are used, the extra spring force in the vertical direction due to these wires should be kept minimal or will also need to be taken into account.

A computation of the oscillation frequency is contained in the reference.

Ref: See J. Winterflood and D. G. Blair, "A long-period conical pendulum for vibration isolation," Physics Letters A 222, 141-147 (1996).

### Magnetic Levitation Pendulum

Magnetic forces can be used to counteract the force due to gravity. Hence, the total return force per unit displacement can be minimized. This "pendulum" is an exception to the definition used above, that the return force should be only due to gravity subject to mechanical constraints, but it is an interesting application. The basic physics is similar to the zero-length spring pendulum, except magnetic forces are used instead of springs.

A computation of the oscillation frequency is contained in the reference.

Refs:
Y. Otake, A. Araya, and K. Hidano, "Seismometer using a vertical long natural-period rotational pendulum with magnetic levitation," Review of Scientific Instruments 76, 054501 (2005).
R. Gilman, "Report on some experimental long-period seismographs," Bull. of the Seismological Soc. of America 50, 553-559 (1960).

While not really a pendulum, another interesting magnetic suspension system uses strong magnets and a diamagnetic material (graphite), with the magnetic field profile designed to provide an appropriate return force.

See: I. Simon, A. G. Emsilie, P. F. Strong, and R. K. McConnell, Jr, "Sensitive Tiltmeter Utilizing a Diamagnetic Suspension," Review of Scientific Instruments 39, 1666-1671 (1968).
[This tiltmeter is sensitive enough to clearly see Earth tides!]
(or put diamagnetic suspension seismometer into your search engine).

(C) 2006,2007,2012 B H Suits, Physics Dept, MTU