Randall D. Peters

Physics Department

Mercer University

Macon, Georgia 31207

| (1) |

The angle q_{D} is determined by the amount of static twist
imparted to the boom,
when the large perturbing masses are `swung' from their `zero' (centered)
position to their maximum displacement (touching the glass plates of the
housing). In moving from zero, the otherwise balanced
boom is disturbed from its static equilibrium starting position q_{e}.
Because the masses are typically swung in roughly 2 s, and because the
period of oscillation of the boom is roughly 250 s, the disturbance
represents essentially a Heaviside step function. The resulting response
of the boom is a simple harmonic free-decay, in which the mean position of
the decay is displaced from the starting position by
the angle q_{D}. If the masses had been swung the opposite direction
by the same amount, then the final displacement (after transient decay)
would be -q_{D}.

The example of fig. 1 was generated by modeling the well known case from classical mechanics, of a viscous damped simple harmonic oscillator. Parameters were chosen to approximate the manual case illustrated in Figure B1. In both that document and the present document, all position data are expressed in terms of sensor voltage rather than angle in radians. Because the sensor is essentially linear, the two are proportional through a calibration constant, with a representative value being 45 V/rad.

**Fig. 1** Simulation of a viscous damped simple harmonic oscillator.

The only output from the sensor is a voltage corresponding to boom position. Also shown in Fig. 1 is a ``velocity'' curve that has been labeled with quotation marks to draw attention to the fact that it was scaled in magnitude to yield a curve of comparable magnitude to the position curve. It is provided to illustrate phase differences that are important for understanding how to drive the boom at its resonance frequency (later discussion).

The damping coefficient b can be determined from any three adjacent
turning points,
such as q_{1}, q_{2}, and q_{3} shown in fig. 1 as follows:

| (2) |

Because of measurement errors, accuracy is improved by working with more than
three turning points. The total number N employed must be odd, and
a near-optimum case for the Cavendish balance has been shown to be
N = 11. (Details along with the expression required for using arbitrary N
to determine x = e^{(-bT/2)} are
provided in the manual.)

| (3) |

**Fig. 2** Illustration of `magnification' that results from driving
at the resonance frequency.

| (4) |

| (5) |

| (6) |

If the phase were instead lagging the motion by 90 degrees,
it would correspond to an antiresonance drive. This would cause the
sign of the 2q_{D}/A term in eq. (4) to be negative,
with a resulting motion that dies out
faster than the free decay case. Accomplished by swinging the masses
180 degrees different in phase from the case shown in Fig. 2, it can
be an effective
means for bringing the balance more quickly to an operational status.

Since the fundamental component of a square wave is 4/p times the amplitude of the square wave, one sees that equation (5) is consistent with the better known case of a harmonic drive; i.e., the steady state amplitude at resonance in that case is simply Q times the amplitude of the drive.

File translated from T

On 9 Aug 2005, 08:28.