Simulation of Probe Motion Effects on Harmonics

II\. Torsional Vibrations

H. Glass

11/13/1998

In the previous note, I gave a general description of how this study is
being done \[1\], so read that first. Here I examine the effect of
torsional vibrations, in which the angular orientation of the rotating
probe deviates from nominal by an amount T(θ), so that the position of
each vertex z on the probe has a motion described by z(θ) = z0 exp(i θ +
iT(θ)). An excellent analytical treatment of this effect is given in
\[2\]. His approach is to calculate the flux generated in the probe in a
nearly pure 2n-pole magnet, keeping only first-order terms in T(θ). The
resulting flux consists of terms generated by the magnet's true
harmonics plus some extra terms. These extra terms are the result of
spurious harmonics caused by the torsional vibration:

$C^{'}(m)e^{{- \text{im \{}\alpha}_{m}^{'}} \approx \text{in}\left\lbrack \left( \frac{K_{n}}{K_{m}} \right)T_{m - n}C(n)e^{- \text{in}\alpha_{n}} - \left( \frac{K_{n}^{}}{K_{m}} \right)T_{- m - n}^{}C(n)e^{- \text{in}\alpha_{n}} \right\rbrack$
(1)

where the K~i~ are the probe sensitivities and the T~i~ are the Fourier
components of T(θ). Several comments: the overall factor of *i* results
in the spurious harmonic having a phase shift of 90º with respect to the
vibration, e.g., if T(θ) = T~0~ cos(*p* θ), then only skew spurious
harmonics will be induced. The factor K~n~ means that the spurious
harmonics are proportional to the probe's sensitivity to the principal
harmonic *n*. It is therefore essential to construct a probe that bucks
it out (in our case we have quadrupole bucking, which as we'll see below
works rather well).

In the special case of single mode vibrations (only one value of *p*),
then at most two spurious harmonics are induced, in which m = p + n and
m = \| p -- n \|. For a quadrupole magnet (n=2), the following table
shows what happens.

Table 1. Spurious harmonics induced by torsional vibrations.

  -------------------------------- --------------------------------------
  *vibration mode p*               *spurious harmonics m*

  1                                1, 3

  2                                4

  3                                1, 5

  4                                2, 6

  5                                3, 7
  -------------------------------- --------------------------------------

As an example, let's calculate the effect of mode p = 2 on the octupole
(m=4). In the case of an unbucked tangential coil, the ratio of probe
sensitivities is

$\frac{K_{n}}{K_{m}} = \frac{m}{n}\left( \frac{R_{c}}{R_{\text{ref}}} \right)^{n - m}\frac{\text{sin}(\frac{n\Delta}{2})}{\text{sin}(\frac{m\Delta}{2})}$
(2)

where ∆ is the tangential coil opening angle, R~c~ is the coil radius,
and R~ref~ is the reference radius of 17 mm. Choosing a vibration
amplitude T~0~ = 1.0 mrad, we obtain a spurious skew octupole a~4~ = 8.8
units. I ran this in the Probesim simulation \[3\], for the two cases of
a pure quadrupole field and the "rich harmonic" field described in Part
I. In both cases, the result was ∆a~4~ = 8.8 units, in agreement with
calculations.

Now when bucking is included, K~2~ becomes zero and one should get ∆a~4~
= 0. This is indeed what one gets when the probe is used in the pure
quadrupole field. In the rich harmonic field the bucking is not perfect,
although the result is ∆a~4~ = 0.002 units, which is awfully good.

###### Simulation results

*Spurious gradient strength:*

Unlike the case of transverse vibrations, which showed a negligible
effect on the quadrupole strength measurement (\<0.1 unit error for all
cases studied), certain torsional vibration frequencies will result in
large errors. From Table I one sees that for mode p=4, the quadrupole is
affected. This is a concern because, unlike the higher order harmonics
measurements, the quadrupole is measured without bucking (by the Q1 or
Q2 coils). Figure 1 shows the measurement error of the strength for a 10
mrad vibration as a function of different frequencies. At p=4, the
strength is underestimated by 0.9%. Vibration frequencies need not be
integer (periodic over one rotation), and a substantial error will be
made in the strength measurement between 3.0 \< p \< 5.0. Away from this
region, the error is typically \< 1 unit.

![](media/image1.wmf){width="5.995138888888889in"
height="4.0993055555555555in"}

Figure 1. Gradient strength error as a function of vibration frequency

The error grows linearly with the vibration amplitude, as seen in Figure
2, which shows the amplitude dependence for the p=4 mode. A linear fit
gives a slope of 9.2 units/mrad error. So, if we wish to limit the size
of the strength error to \< 1 unit (a reasonable goal), then we must the
p=4 vibrations below 100 μrad. In the study done using VMTF probe data,
observed amplitudes for all frequencies were between 20 -- 60 μrad
\[4\].

![](media/image2.wmf){width="5.995138888888889in"
height="4.0993055555555555in"}

Figure 2. Strength error as a function of vibration amplitude at p=4.

There is also a significant dependence on the phase of the vibration, as
seen in Figure 3. This is shown for a vibration amplitude of 15 mrad.
The maximum errors are in the neighborhood of 20º and 200º. There are
some frequencies where the error is zero, but it would be really good
luck to encounter those; better to try to dampen as much as possible any
p=4 vibration, regardless of phase.

![](media/image3.wmf){width="5.995138888888889in"
height="4.0993055555555555in"}

Figure 3. Strength error asa function of vibration phase at p=4.

*Spurious harmonics:*

The situation with regard to harmonics is much better than for strength.
The simulation proceeded in the usual way by studying the behavior of
spuriously generated harmonics as a function of vibration amplitude,
frequency, and phase. Figures 4 and 5 show the amplitude dependence for
modes p=2 and p=4, both with zero phase. For all harmonics the behavior
is linear, but the slope (and sign) varies among all the harmonics.
Considering Fig. 4, one expects from Table I to find a~4~ to be large
(remember that phase=0 should affect the skews), and indeed that is the
case; however, it is not significantly worse then some other harmonics,
e.g., a6, a7, b6 and b7. And in Fig. 5 one sees that a6 is large as
expected, but other harmonics are also nearly as large. So one may
conclude that bucking suppresses the potential problems listed in Table
I for reasonable vibration amplitudes. Inspection of the two figures
shows that all the harmonics are \< 0.01 units when the amplitude is
limited below about 5 mrad.

![](media/image4.wmf){width="5.995138888888889in"
height="4.0993055555555555in"}

Figure 4. Amplitude dependence of torsional vibrations on harmonics for
mode p=2.

![](media/image5.wmf){width="5.995138888888889in"
height="4.0993055555555555in"}

Figure 5. Amplitude dependence for mode p=4.

The phase dependence is shown in Fig. 6, where one sees the p=4
vibration mode with a 15 mrad amplitude. All of the harmonics exhibit a
one period oscillation over 360º, but with varying amplitudes and
phases. The most extreme is a6, expected, but not all that much worse
than some others.Curiously, some harmonics appear to be grouped
together, having a common response to the vibration phase. The deviation
of a6 is worst at about 120º and 300º, and is about a factor of 2 worse
than at 0º. Comparing this observation to the slope seen for a6 in Fig.
5, one concludes that at the worst case phases, one needs to get the
vibration amplitude below about 3 mrad in order to limit the a6 error to
0.01 units.

![](media/image6.wmf){width="5.995138888888889in"
height="4.0993055555555555in"}

Figure 6. Spurious harmonics as a function of vibration phase\
for p=4, amplitude =15 mrad.

Next let's consider the effect of vibration frequency. Figure 7 shows
what happens when the probe is vibrated with a 10 mrad amplitude and a
phase of 0º with modes in the range 1 \< p \< 9. It's a spaghetti plot,
but the fortunate thing is that the damage appears limited to ±0.03
units. Over this range of frequencies, then, it seems that one can limit
the errors to 0.01 units by restraining the vibration amplitudes below
about 3 mrad.

![](media/image7.wmf){width="5.995138888888889in"
height="4.0993055555555555in"}

Figure 7. Spurious harmonics generated by a 10 mrad amplitude, phase =0
over a range of vibration frequencies.

*Conclusions*: From this limited study one can conclude a few useful
things. First, one must limit torsional vibrations for mode p=4 below
0.1 mrad to prevent a significant error in the quadrupole strength
measurement. Then, for all other frequencies, one can relax this limit
to 3 mrad to prevent significant errors in the harmonics measurements.

####### References

\[1\] H. Glass, Simulation etc, Part I: Transverse vibrations

\[2\] A. Jain, Harmonic coils, CERN 98-05

\[3\] J. DiMarco, Probesim,
http://tsmtf.fnal.gov/\~dimarco/probesim/probesim.html

\[4\] J. DiMarco,
HTTP://tsmtf.fnal.gov/\~dimarco/hgqs02/mag_meas/cold/var_anal
/results/sum_concl.html