21.5 Driven Oscillators with Damping

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This chapter has been mainly concerned with oscillators in free-decay. It is also possible to make

quantitative predictions from measurements at steady state. Confidence in predictions, however, depends

on the nature of the damping. Such data are of limited value for most nonlinear systems, unless

supplemented with free-decay data.

21.5.1 MUL Apparatus

Some of the techniques applicable to driven systems are illustrated by the multipurpose undergraduate

laboratory (MUL) apparatus shown in Figure 21.19, that has been used by students in the physics

department at Mercer University.

For the purpose of measuring the Lorenz force (basis for defining the current unit, the ampere) a

constant current is supplied through the posts to the pivoted-on-points brass wire on which a weight, W ;

is shown hanging on one of the horizontal arms of the wire. Current enters the wire through one post via

the banana plug inserted into a drilled hole. It thereafter travels through the lower (invisible) shorter

Box 21.2

METHODS FOR QUANTIFYING DAMPING

Damping (Q Estimation) Techniques (Q ¼ p=bT; T ¼ Period)

Logarithmic

Decrement

(Full-Cycle, N)

Turning

Points

(Half-Cycle, n)

Nonlinear

Fit to

Envelope

Time t

to l=e

ð0:3679x0 Þ

Short Time

Fourier

Transform

Bandwidth,

Magnification Factor,

Hysteresis Loop,

Step-Response

bT ¼

1

N

ln

x0

xN

bT ¼

2 2 ln

􀀒

1 2 ðxn21 2 xnþ1 Þ

ðxn21 2 xnþ1 Þ

􀀓

2y_ ¼ ay2 þ by

þ c

bT ¼ T=t 2bT ¼

T ln 10

20

[slope(dB/s)]

de Silva (2000),

p. 379

Experimental Techniques in Damping 21-19

© 2005 by Taylor & Francis Group, LLC

straight segment of the wire located between the

poles of the drive magnet; and it finally exits

through the banana plug on the opposite post.

When carrying a current, the force on the wire

from the part inside the magnet causes vertical

deflection, the direction up or down being

determined by the direction of the current.

This results in a rotation about the pivot points

(indented tops of the posts). The position is

measured by the capacitive sensor, S (one of

several variants of the SDC patent).

The sensitivity of this current balance

depends on the location of the center of

mass of the oscillatory wire, which is determined in part by the position of the rare earth magnet,

M, which hangs from a steel nut on the threaded part of the heavier brass rod having a 908 bend.

The upper end of this threaded rod is held by a plexiglass member that also holds the ends of the

oscillatory wire.

21.5.2 Driven Harmonic Oscillator

The MUL becomes a driven harmonic oscillator when the excitation current is a.c. rather than the d.c.

used for the Lorenz force study. The damping is determined primarily by eddy currents in the aluminum

ring, R, that lies on the wooden base underneath and in close proximity to magnet M.

The apparatus is useful for studying both free-decay and driven oscillation. Engineering students

Brandon R. Bowden and James D. Sipe have programmed LabVIEW to generate both free-decay curves

and resonances.

An example Lorentzian (resonance response) is given in Figure 21.20. (Additional information is

found in a laboratory writeup (Peters, 1998).)

FIGURE 21.20 Screens from the LabVIEW program used with the MUL to study both transient and resonance

phenomena.

FIGURE 21.19 Apparatus for studying resonance and

the Lorenz force law.

21-20 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

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