  We present a procedure for the design and construction of a passive,
multipole, mechanical high-stop vibration isolator. The isolator, consisting of
a stack of metal disks connected by thin wires, attenuates frequencies in the
kilohertz range, and is suited to both vacuum and cryogenic environments. We
derive an approximate analytical model and compare its predictions for the
frequencies of the normal modes to those of a finite element analysis. The
analytical model is exact for the modes involving only motion along and
rotation about the longitudinal axis, and it gives a good approximate
description of the transverse modes. These results show that the high-frequency
behavior of a multi-stage isolator is well characterized by the natural
frequencies of a single stage. From the single-stage frequency formulae, we
derive relationships among the various geometrical parameters of the isolator
to guarantee equal attenuation in all degrees of freedom. We then derive
expressions for the attenuation attainable with a given isolator length, and
find that the most important limiting factor is the elastic limit of the spring
wire material. For our application, which requires attenuations of 250 dB at 1
kHz, our model specifies a six-stage design using brass disks of approximately
2 cm in both radius and thickness, connected by 3 cm steel wires of diameters
ranging from 25 to 75 microns. We describe the construction of this isolator in
detail, and compare measurements of the natural frequencies of a single stage
with calculations from the analytical model and the finite element package. For
translations along and rotations about the longitudinal axes, all three results
are in agreement to within 10% accuracy.
