Dissipation in pendulum wires

In order for the mirrors in a gravitational wave interferometer to approximate freely-falling test masses, they must be suspended with a resonant frequency that is low compared with the signal frequencies of interest. The universal choice of a pendulum for this suspension comes from a special feature of pendulums that should make thermal noise lower than in a suspension with purely elastic restoring forces. The dominant restoring force in a pendulum comes from the horizontal component of the constant tension in the wires due to the weight of the test mass, with only a weak additional elastic restoring force from flexing of the wires. While there is internal friction associated with the latter elastic effect, there is none (to first order) associated with the tension restoring force; its horizontal component arises purely from the change in angle of the wire as it moves. Thus the dissipation from internal friction is diluted by a factor representing the latter's fractional contribution to the total restoring force.

The validity of this ``dissipation dilution'' effect has been assumed in all models of the strength of thermal noise in gravitational wave interferometers. Since in typical pendulum designs the elastic restoring force is of order times the tension restoring force, this feature of a pendulum suspension is responsible for a factor of reduction in thermal noise amplitude, compared with the case of a purely elastic suspension.

While pendulums are in fact generally seen to exhibit high quality factors, there has been little careful checking of the validity of the model discussed above. Part of the reason is that recoil of the structure that supports the pendulum may diminish the quality factor (or Q) that it will exhibit, without invalidating the model or thermal noise predictions based on it.

The Ph.D. thesis research of Yinglei Huang (Ph.D. 1996, now at Zoom Telephonics in the Boston area) was devoted to the sort of detailed checks that this central feature of interferometer designs deserves. The motivations of his work were twofold: to obtain as clean a check of the theory as is possible, and also to use detailed comparisons of theory and experiments as a tool to hunt for excess loss mechanisms in addition to internal friction in the wires.

Huang measured quality factors of the violin modes of wires in a variety of styles of pendulums. Violin modes exhibit the same dissipation dilution effect as the pendulum mode itself. (This has been known for some time, but the most thorough theoretical treatment of the problem is a paper published by González and Saulson in 1994.[7]) The advantages of study of violin modes come from our ability to 1) isolate the modes from recoil in the structure by interposing a compact upper mass in a double pendulum arrangement, 2) check the frequency dependence of various effects by comparisons of the Q's of various modes, and 3) compare Q's under tension with zero-tension dissipation measured in the same band of frequencies, in the manner of Kovalik and Saulson.[9]

The last element of the list given above is the key to the precise test of theory Huang has carried out, and is responsible for several of his discoveries. A measurement series begins by hanging a sample wire by means of a clamp at its top, under no tension other than the weight of the wire itself. (Spooled wire can require straightening by stretching or annealing; more recent measurements have been made on wires that have never been spooled.) Except for the lowest mode of such a wire, the spectrum of transverse modes accurately matches that expected from a cantilever clamped at one end; in that case the modal quality factors are related to the loss factor of the wire by the simple relation

Since these modes are closely spaced, we obtain excellent information on the frequency dependence of the internal friction of the wire.

Because this measurement of the internal friction of the wire is obtained from the very same sample about to be placed under tension in a pendulum, held in the same way and excited into the same sort of transverse oscillation, we have the right to expect that a correct theory of the ``dissipation dilution'' effect would give a close match to the quality factors measured under tension. Repeatability of such measurements is typically at the 10% level or better; this sets the quality of our confrontation between theory and experiment.

Experiments have been carried out primarily on wires of stainless steel and of tungsten, in thicknesses close to those appropriate for suspending 10 kg masses in single or double slings. Wires have been subjected to wide ranges of tension up to their breaking strength, and been attached at their ends in several different ways. The result of a large number of measurements is summarized below. A paper summarizing his thesis has recently been submitted to the Review of Scientific Instruments.

1. ``Dissipation dilution'' works as advertised in a large number of cases. Many modes in many wires, under tensions close to the breaking strength, show the improvement in Q predicted by the theory. The highest Q observed in agreement with theory is .
2. Typically, the best agreement between theory and measurement is seen at the highest tension. The only indication of internal friction enhancement at high tensile stress is a 40% degradation of Q in tungsten wires loaded to 160% of the tabulated breaking stress. Larger disagreements with theory at low tension appear to be an indication of sliding friction, since they occurred only with prism contacts, not in bolted clamps (see next item).
3. When wires are attached by being held by tension against sharp prisms, sliding friction can be an important source of excess loss. Sliding friction has an intrinsically non-linear dependence on the tangential force between the wire and its contact, so can be diagnosed by measurements at large amplitude. It can take on a linear dependence at small amplitudes, in the presence of a bias force, such as that caused by misaligment of wires, or when low frequency modes (e.g., pendulum, rocking, torsional) are excited to large levels. Sliding friction can be substantially reduced when wires are held between machined steel plates clamped together by screws; under these circumstances, we almost always see losses within a factor of two of the theoretically predicted level. Sliding friction is large when wires are held over glass prisms.
4. Thermoelastic damping, the dominant internal friction mechanism in steel wires of the appropriate thickness, has a different frequency dependence when the wires are under high tension than that predicted by the classic theory of Zener.[] The change is caused by non-negligible conduction of heat along the length of the wire, instead of purely transverse conduction. This occurs when tension is large enough that the length scale of elastic bending becomes almost as short as the diameter of the wire. The effect has been seen in steel wires, although comparison with theory at better than the factor of two level has not been carried out; accurate calculation will require thermal analysis using finite element techniques.
5. No wire yet tested has shown Q in agreement with theory at the 10% level for all modes, so we suspect at least one undiagnosed loss mechanism is present. Mechanisms tested or calculated and shown to be negligible at the level include: recoil in support structure, excitation of test mass internal motion, eddy currents, and residual gas damping.

In a real sense, even our discrepant measurements represent a substantial accomplishment of our research program. Others have only been able to trust comparisons between theory and experiment at the factor of two level. At this looser tolerance, all of our careful measurements agree with theoretical predictions.