Plate 3.2 A photograph of the vibrating reed system and the auxiliary electronic equipment for the measurement of
3.2 The E lectro n ic D rive and D etection System
Figure 3.3 show s a schem atic representation o f the vibrating reed electro n ic circu it. The system is a more sophisticated version o f that developed by Briscoe e t a l .
(1985) used in conjunction with a vibrating reed system for the measurement o f gas densities at high pressures.
0.10 0.08 - (0
o
>
0.06 - 0.04 - Q.E
<
0.02?
0.00 0.20 0 .2 5 0 .3 0 0.00 0 .0 5 0.10 0 .1 5Displacement (mm)
Figure 3.2 The calib ration plot representing the variation o f amplitude with displacement
for the vibrating reed.
r
Sensor Electronics
bios
Power supply
Magnetic core amp
Pre amplifier Reed Power amplifier Optical transducer Frequency meter u ____ I
There are basically two modes o f operation for this type o f system. An open loop manually driven system in which resonant con d ition s are determ ined by m anually changing the drive frequency or an automatically tuned system. In the former case , the sinusoidal drive signal is generated using a Philips (type PM 5190) oscillator with ±1 x 10~^ Hz resolution. This signal is fed to the electrom agnet driven by a pulsating alternating current supplied from an oscillator (Philips type PM 5190; ±1%1 0 Hz
resolution) incorporated in a 30 watts pow er am plifier. The pulsed drive signal attracts the reed towards the electrom agnet for only part o f a full cycle thus m inim ising the degree o f interaction between the drive and the oscillator (Bak 1986) . This is manifested in a more stable and efficient mode o f operation as compared to a continuously driven system. The duration and the phase lag o f the drive pulse relative to the reed signal may be adjusted at w ill via suitable arrangements incorporated in the power amplifier.
The system is manually driven into resonance by sweeping through the drive frequency. Resonance corresponds to the drive frequency producing a 90^ phase lag between the drive and the detected signals (Beards, 1983). A Racal Dana (type 1990) universal counter is used to measure both the resonant frequency and phase lag. Figure 3.4 shows a typical response for the relationship between the phase lag and amplitude o f vibration. It may be noted, that the 9 0 ° phase lag also corresponds to the maximum amplitude o f vibration. As the later parameter may be more conveniently and accurately monitored, it is used in this study as a means o f identifying resonance.
"c 0) “O 3
3.0
2.0
1.0
« - —PI0.0
40
45
50
55
Frequency (Hz)60
cc O) ■73 3 J= cu200
150
100
50
0
-50
40
45
50
55
Frequency (Hz)60
F ig u r e 3 .4 T h e variation o f a m p litu d e a n d p h a s e lag with f r e q u e n c y .
Although the manually controlled m ode o f operation is useful for studying the characteristics o f the vibrating reed system itself, it is o f limited value for automatic monitoring o f response as requires a delicate tuning procedure. An automatic tuned system operating on the basis o f a regenerative feed back loop (Seely, 1958) overcomes these problems. In this case, the detected signal is simply fed back to the electromagnet via the power amplifier, thus by - passing the oscillator. The system then autom atically tunes onto the resonant frequency. A variable bandpass frequency filter incorporated w ithin the detection electronic circuit insures that the system only tunes onto the first harm onic resonant freq u en cy. T h is is a ch iev ed by elim inating other undesirable frequencies. In the present study, the automatic tuning system is adopted for the measurement o f the added mass.
3 .3 T h e T h e o r e t ic a l P r e d ic t io n o f T h e R e s o n a n t F req u en cy: O p tim isa tio n o f T he S y stem s R esp o n se
The ayailability o f a theoretical model for the prediction o f the resonant frequency is both important and useful. Such a m odel elim in ates the need for the production o f resonant frequency versus mass calibration curves for the subsequent determination o f the added mass. It also identifies the role o f various factors which affect the system's sensitivity in response and hence identifies the optimal design criteria.
The theoretical m odelling o f the response o f the vibrating reed system has been performed on the basis o f finite element an alysis. This technique is e x te n siv e ly em p loyed for the modelling o f response o f structures to external vibrations (Davies, 1980 and Fenner, 1975) and hence only a brief account is given h ere.
In the present case, the vibrating system is assumed to be composed o f a number o f elements, laid on two supports which represent the clamping arrangement as shown in figure 3.5.
The distance between the two supports is the length over which the compression fitting is in contact with the reed. Each elem ent is assum ed to be o f a constant mass and bending stiffness, connected to the neighbouring elem ent via a m ass-less spring. By d efin ition , the supports allow rotational but no translational displacement (Bishop and Johnson, 1960). The model considers the lateral inertia and elastic forces caused by bending deflection only; shear deflection and rotary inertia are assumed to be n egligib le. The transverse vibration o f each elem ent is analysed in terms o f the Euler - Bernoulli equation ( Bickley and Talbot, 1961 ) given by
5 x ' El 5,^
3.1
where E is the modulus o f elasticity, I is the area moment o f inertia o f the cross-sectional about the neutral axis, is the cross-section al area, Pr is the reed density and y^ is the
B
F igure 3.5 A schematic representation (not to scale) o f a typical beam divided into a number o f finite elements. The distance between the suppoerts A and B represents the clamped element.
transverse displacement at time t, at a distance x, along each e le m en t.
Further analysis o f the transverse displacem ent for each elem ent ,incorporating both kinetic and potential energies leads to the following matrices.
[m] = ml 420 1 5 6 221 5 4 221 4 1 ' 1 31 5 4 - 1 3 1 1 3 1 - 3 l ' 1 5 6 - 2 2 1 2 -131 - 3 - 2 2 1 41 3.2 ■ 12 61 -1 2 61 ■ El 61 41^ -61 21 ' I' -1 2 -61 12 -61 . 6 1 2^ -61 4 l \ 3.3
where [k] and [mj are the stiffness and mass matrices respectively and 1 is the length o f each element.
The two matrices k and m are square and similar for all elem ents. The final form o f these equations for the beam is produced by appropriate superposition o f these m atrices to produce the global mass matrix G and stiffness matrix J.
The equation o f motion for free vibration is
( J - coG) .Q (t) = 0 3.4
where co is the angular resonant frequency and Q(t) is the global displacem ent coordinates.
The natural frequencies o f the reed are obtained by solving equation 3.4 using a computer programme (appendix A).
The proceeding sections are designated to the optimisation o f the system 's response on the basis o f the finite elem ent method described above.
3 .3 .1 O p tim is a tio n
There are a number o f design and operating parameters which are expected to affect the system's sensitivity and stability in response. The important factors include:
i) Reed clamping ratio and system mode o f operation
ii) Ratio o f the mass attached to that o f the reed (i.e mass ratio) iii) The geometric size o f the attached mass
There are other factors which are shown to affect the perform ance o f the system . T hese include reed material o f construction and shape, w all effects and the am plitude o f vibration. Wall and amplitude o f vibration effects are discussed in more detail in the proceeding sections. On the other hand,
finite element analysis indicates that tapered shaped reeds with the largest ratio o f Young's modulus to density produce the most sen sitive response in terms o f mass measurement. A ceramic reed fulfils the second requirement but significant difficulties in machining even sim ple shapes were encountered.
3.3.1.1 : O ptim um C lam ping R atio and M ode o f O p eration
The clamping ratio is defined as the ratio o f the drive length to the total length o f the reed. Its magnitude affects both the sensitivity and the stability in response. Figure 3.6 shows the variation o f the resonant frequency for the first curve A and the second curve B modes o f vibration versus clamping ratio as p red icted from theory. The corresp on d in g ex p erim en ta lly measured values for the first mode o f operation are included for comparison and validation o f the theory. The data in figure 3.6 refer to a 224 mm long and 3.2 mm diameter silver steel rod. Referring to curve A , the results indicate that the rate o f change in the resonant frequency with the clamping ratio dramatically increases as the clamping ratio approaches 0.5. In this region , a finite change in the clamping ratio, say, due to variation in reed's length would give rise to a relatively large change in the resonant frequency. This situation may arise in practice, if the remote span o f the reed is for example subjected to different temperatures. A lso, any uncertainty associated with the determination o f the clamping ratio around 0.5 gives rise to a relatively large error in the prediction o f the resonant frequency. This is in view o f the fact that, in the present studies, the clamping ratio can be only measured with a lim ited accuracy (ca 95% certainty) as the
600 S 500 >» ç 400
cr
300 G 200 0.0 0.2 0 .4 0.6 0.8 1.0Clamping Ratio
Figure 3.6 The variation o f the resonant frequency with clamping ratio: curve A , first mode; curve B , second mode^ and # experiment.
precise length over which the reed is clam ped is unknown. Operation at a clamping ratio o f around 0.5 would therefore typically lead to ±5 Hz error in the prediction o f the resonant frequency. This w ould in turn translate into an unacceptable error in the predicted added m ass, (figure 3 .8 ). A further justification for not operating at clamping ratios approaching 0.5 is the strong possibility o f interaction o f the first and the second modes o f vibration: c .f curves A and B in figure 3.6. This phenomenon is often observed in practice and leads to large instabilities in response .
On the other hand, results o f experiments with different clamping ratios have shown that for clamping ratios o f less than 0.5 and operation at the first modal resonant frequency, the vibration is readily transmitted from the drive to the remote sectio n through the com p ression fittin g . Furtherm ore the response o f the system is primarily dictated by the conditions at the rem ote end. T his is illu strated from the resu lts o f experiments shown in figure 3.7. The data show the effect o f changing the magnitude o f the attached mass at the remote (curve A) and the drive sides (curve B) o f the reed on the norm alised resonant frequency at a clam ping ratio o f 0.3. Norm alised frequency is the ratio o f the resonant frequency corresponding to zero attached mass to that for a given attached mass. The magnitude o f the attached mass was varied by uniformly embeding pre-weighed lengths o f ca. 2mm diameter steel rod into a 25mm dia. polystyrene sphere threaded into the end o f the reed and measuring the corresponding resonant
1.04 1.03 o c Ü 3 cr 2 a B u O Z 0.99 3 0 0
200
100 0Attached Mass (g)
Figure 3.7 The effect o f added mass on the remote (curve A) and the drive (curve B) spans o f the reed
frequency. The rational for adopting this siz e o f sphere is explained in section 3.3.1.3.
As it may be observed from the figure, the system 's response is predominantly governed by the conditions at the remote span and hence this clamping ratio was employed for the proceeding experiments. In addition, the response is linear (0.998 correlation coefficient) in the range tested.
3 .3 .1 .2 O ptim um M ass R atio
The ratio o f the attached mass to that o f the reed, here defined as the mass ratio, also affects the system's sensitivity in terms o f mass measurement.
Figure 3.8 show s the theoretical variation o f resonant frequency versus mass ratio (m i/m2) . m% represents the mass
attached to the remote end and m2 is the total mass of the reed.
The sensitivity o f the system is expressed in terms o f the rate o f change in the resonant frequency with the mass ratio. From the figure, it may be observed that the sensitivity o f the system decreases with increase in mass ratio, hence heavy attached m asses g iv e rise to lower sensitivity. H owever, low attached masses on the other hand give rise to a non-linear response.
Figure 3.9 shows the same data but in the attached mass ratio range adopted in the p roceed in g exp erim en ts. The corresponding experimentally obtained results are also included for comparison. A typical system sensitivity is in the range 3.5 -
200 150 -