Vibration Rectification Errors

The vibration rectification error (VRE) is a measure of the DC offset in null-position when a test-mass is exposed to vibration, measured in units of g/g2. The DC output of the sensor changes as a function of the peak acceleration, even though the total time-averaged acceleration is zero. There are two main causes of vibration rectification; asymmetric damping, and DC scaling non-linearity [235].

Asymmetric damping occurs if the motion of the test-mass is more strongly damped in one direction, i.e. +g direction, than the opposite, i.e. -g. The deflection of the test- mass as it responds to vibration will therefore be biased towards the negative direction. If the signal is time-averaged it will give a net DC output that is shifted to the negative g direction compared to when no vibration is present.

The other mechanism that leads to vibration rectification is a non-linearity in the sensing range due to the test-mass, where the time averaged response in one direction is not equal to that in the other direction. This is often related to a geometric asymmetry, or cross-coupling to another axis direction. This tends to be the more dominant effect over asymmetric damping.

test-mass deflection equation in response to an acceleration a, such that the deflection D(t) is written as [235]:

D(t) = Ca + βca2, (6.31)

where C, βc are constants. The constant C is the linear scale factor, measured in fig. 6.6,

and βc is referred to as the co-efficient of the second order non-linearity. If the test-mass

is now subjected to a sinusoidal acceleration of A cos Ωdt, eq. 6.31 becomes:

D(t) = CD0 cos (Ωdt) + βcD02cos2(Ωdt), (6.32)

where D0 is the scale factor. The time average of the first term in eq. 6.32 is zero, whereas

the time average of the squared term is proportional to R_tcos2(Ωdt) dt = 1₂, therefore the

time averaged deflection is:

¯ D0=

βcD20

2 . (6.33)

The measurement of ¯D0 (the 0 g bias level) reports a false acceleration that is pro-

portional to acceleration squared, and is governed by the constant β2/2, more commonly

referred to as the VRE:

VRE = βc 2 = ¯ D0 D2 0 . (6.34)

Therefore, the deflection can be written as:

D0 = C g + 2g2× VRE. (6.35)

The VRE is measured for the ADXL327 and the WGM sensor (using microsphere- cantilever[E ]) in two ways and then compared to investigate if the non-linearity of the WGM sensor response can account for the reduced sensing range found in fig. 6.6 b).

First, the 0 g bias level is measured by averaging the modulated WGM transduction signal/ADXL327 signal when the system is driven by a sinusoidal acceleration with in- creasing amplitude g, and plotted as a function of g2. The results for the ADXL327and the WGM sensor are shown in fig. 6.9 and fig. 6.10 respectively. Since the data presented here corresponds to the same data used to deduce the sensing range in figs. 6.6 a)& b), a direct comparison can be made using a second method to find the VRE by fitting the previously measured sensing range with eq. 6.31.

Figure 6.9: The VRE for the ADXL327 is calculated by measuring the DC bias shift of the output signal, at increasing acceleration, plotted as a function of g2. The gradient gives a VRE of −23.5 ± 0.3 mg/g2, found by using the scale factor in fig. 6.6 a). This data is the same as that used in fig. 6.6 a).

Figure 6.10: The VRE for the WGM sensor using microsphere-cantilever[E ] is calculated by measuring the DC bias shift of the output signal, at increasing acceleration, plotted as a function of g2. This data is the same as that used in fig. 6.6 b). A linear fit is applied, but only up to the linear sensing limit found in fig. 6.6 b) (i.e. g < 1.7). The gradient of the fit gives a VRE of −14.6 ± 0.5 mg/g2, using the scale factor in fig. 6.6 b). At g2 > 22, the WGM transmission is non-linear with respect to larger acceleration (see fig. 6.7), such that the transduced signal is no longer equal to the actual motion of the microsphere-cantilever. The VRE will cause d0 to further decrease, but the time averaged

WGM measures d0 non-linearly.

are converted into acceleration units using the scale factors obtained for the ADXL327 (fig. 6.6 a)) and the WGM sensor (fig. 6.6 b)) respectively, such that figs. 6.9 & 6.10 repre- sent D¯0

D2 0

. Therefore, using eq. 6.34 the VRE for the ADXL327 is −23.5 ± 0.3 mg/g2 and for the WGM sensor, −14.6 ± 0.5 mg/g2. Note that the plot of the 0 g bias level for the WGM sensor is only fitted up to 2 g2. This is because the WGM does not transduce large ∆d around d0 linearly, as described by fig. 6.7, and the sinusoidal modulation of the WGM T

with respect to the sinusoidal acceleration becomes distorted such that the 0 g level is not equal to half the peak-to-peak amplitude.

To verify if the source of the vibration rectification is due to the non-linearity of the sensor, the second method to find the VRE is applied. Here, eq. 6.35 is fitted to each sensor’s full measured range (i.e. including the non-linearity in figs. 6.6 a) & b)), to obtain the second order non-linearity co-efficient βc = 2 × VRE. Using this method, a VRE of

−104 ± 4 mg/g2 and −25.0 ± 0.9 mg/g2 is obtained for the WGM sensor and the ADXL327 respectively. The value for the ADXL327 VRE from fitting βc is in good agreement with

the value obtained directly from measurement. However, the VRE of the WGM sensor from measurement of βc is significantly higher. If eq. 6.35 is now fitted over the linear

sensing range of the WGM sensor in fig. 6.6 b), a value of βc = −2 × (21 ± 7) mg/g2 is

obtained, which is closer to the measured VRE in fig. 6.10.

Therefore, a possible explanation for the lower than expected absolute WGM sensing range (i.e. until the microsphere touches the taper) is as follows; at a peak acceleration of√3 g, a mechanical misalignment of the supporting mounts or stages occurred, pushing the null position towards the taper (in fig. 6.10, at g2 = 3, the WGM DC transmission increases, signifying a jump over the turning point in the WGM power coupling of fig. 6.7). A misalignment of 100 nm is feasible, especially as the system is exposed to continuously high vibrations. Currently the tapered fibre and the microsphere-cantilever are held by mounts on separate supports, which are then fixed to a common base. If the mechanical shift in d0 is related to these supporting mounts, the relative stability can be improved by

mounting both the microsphere-cantilever and a tensioned taper onto the same frame. The sensing range is ultimately limited by the microsphere-cantilever material proper- ties. The proportionality limit defines when stress is equal to strain, governed by a material specific constant (Young’s modulus). This is the basis of Hooke’s law, and relates applied stress to the linear deflection of the cantilever. The elastic limit of silica determines when the cantilever becomes permanently deformed, and does not return to its original state once the applied force is removed. The stress required for silica fibre to break is approximately 6 GPa [236]. A cylindrical cantilever with rc= 60 µm, m = 20 × 10−9kg, and L = 5 mm,

can therefore withstand a load of Fmax = 0.2 N acting on the free-end before rupture12.

This is equivalent to an applied acceleration of over 1×106g, and implies the cantilever

12_{Here eqs. 6.18 & 6.17 are used to form the relationship σ = Mh/I, where h is the distance from the}

could withstand a car crashing into a wall whilst travelling at a velocity of 100 km/hr (approx 100 g deceleration) [237]. However, the deflection of the cantilever in response to a = 100 g is 10 µm, over 100 times larger than the coupling distance between the sphere and the taper. In comparison, the ADXL327 is specified to survive a shock of 10,000 g, and would be more suitable for high g environments unless the microsphere-cantilever is actively prevented from touching the tapered fibre i.e. in a closed-loop scheme.

Clearly, the WGM sensor studied here requires further work to survive 10,000 g, in- cluding the need for re-calibration if the microsphere-cantilever must be seperated from the taper after touching. This may not be an issue if one considers the other end of the sensing scale; the minimum resolvable acceleration. The exquisite transduction sensitivity obtained in chapter 3 has already proven that the WGM is able to detect the thermal motion of the microsphere-cantilever, enabling feedback cooling in chapter 4. The next section derives the minimum noise equivalent acceleration, limited by the thermal mo- tion of the test-mass, and sets the absolute sensitivity for detecting acceleration. The sensitivity is then experimentally deduced and tested.