This thesis studied the vibrational and rotational aspects of the Cryo I Helsinki cryostat using Fourier analysis methods. Extensive Fourier analysis was performed to model the vibrations mathematically. The goal was to lower noise levels to improve cryostat operations. The second part of the work tested the homogeneity of a superconducting magnet. Significant preparation was required to build testing equipment for evaluating the magnet.

1. Abstract of Master’s Thesis Author: Kulvik Aleksanteri Name of the
thesis: ibrational Study of Cryo I Helsinki, and Testing of Homogeneity of a
Superconducting Magnet Date: 23.4.2008 Number of pages: 60
Department: Department of Engineering Physics and Mathematics Pro-
fessorship: Tfy-3. Material Physics
Supervisor & Instructor: Matti Krusius
In the thesis, rotational aspects and vibrational analysis of Cryo I Helsinki
cryostat have been studied.
These studies were done extensively using Fourier methods. The mathe-
matics of Fourier methods have been presented, and also mathematical mod-
els of vibrations have been studied.
The main goal is to lower the noise due to various aspects that affect the
operations of the cryostat.
The second part of the work considers a superconducting magnet, and
the testing of how homogenous it was. There was a great preparation made
into building of the testing equipment.
1

2. Master’s Thesis
Kaarle Aleksanteri Kulvik
52664T
24th April 2008
Vibrational Study of Cryo I Helsinki, and Testing of Homogeneity of a
Superconducting Magnet
Supervisor M. Krusius
2

3. Contents
1 Preface 2
2 Introduction 3
3 Fourier Methods 3
4 Piezoelectric Accelerator 12
5 Low-pass filter and measurement setup 23
6 Vibrations 23
7 Torsional vibrations 40
8 Balancing rotation 49
9 Studies of the Noise of Old Rota I Cryostat 60
10 Superconducting high-homogeneity magnet for NMR mea-
surements 67
11 Superconductivity 68
12 Superconductor quenching 71
13 Nuclear Magnetic Resonance, and Imaging 73
14 Cooling of the Cryostat 75
15 Preparing the NMR measurements 79
16 Conclusions 80
1

4. 1 Preface
Dedicated to Alexiel, Pulla and those who do not dwell amongst us anymore...
Albert, Kustaa, and Rosiel.
Solitary trees, if they grow at all, grow strong. -Winston Churchill
2

5. 2 Introduction
This master thesis will study the vibrational, and rotational aspects of Cryo I
Helsinki, which has now been dismantled. Suggestions for bettering rotaional
cryostat have been given in length, and the methods for study have been
made clear for even a student reading this work. The second part consist
of testing a superconducting magnet. The preparations are also included, as
quite a lot of work needed to be done in order to test the magnet.
3 Fourier Methods
Fourier transformation can be seen as a tool that converts a ”signal”, or as in
this case, an output voltage into a sum of sinusoids of different frequencies,
amplitudes and phases. In general, both input and output of the fourier
transform are complex vectors, which have the same length. A frequent
difficulty in understanding Fourier transformation lies in the comprehension
of the physical meaning of the results.
The voltage versus time representation becomes magnitude versus fre-
quency in the Fourier transform.
The one dimensional Fourier series is given by the following formula
f(t) = a0 +
∞
n=1
an cos(nωt) +
∞
n=1
bn sin(nωt), (1)
where t is an independent variable which in our case represents time, and Tp
is the repetition period of the waveform. ω = 2π/Tp is the angular frequency
related to the fundamental frequency ff , by ω = 2πff . The constant a0 is
given by the formula
a0 =
1
Tp
Tp
2
−Tp
2
f(t)dt, (2)
an by
an =
2
Tp
Tp
2
−Tp
2
f(t) cos(nωt)dt, (3)
and bn by
3

6. bn =
2
Tp
Tp
2
−Tp
2
f(t) sin(nωt)dt. (4)
The frequencies nω are known as the nth harmonics of ω. The series may be
written in exponential form
f(t) =
∞
n=−∞
dneinωt
(5)
in which
dn =
1
Tp
Tp
2
−Tp
2
f(t)e(−inωt)
dt. (6)
is complex and |dn| has the units of voltage in our case. Negative frequencies
do not have any physical meaning rather being purely mathematical.
The two conditions for f(t) are:
1. The integral of |f(t)| from −∞ to ∞ exists 2. Any discontinuities in
f(t) are finite.
The squared modulus of a transform is referred as the energy spectrum.
|F(ω)|2
is the energy spectrum of f(t). Usually the graphs are given as the
energy spectrum versus ω.
The complex and trigonometric forms are related by the following
dn = |dn|eiφn
, (7)
where
|dn| = (a2
n + b2
n)
1/2
(8)
and
φn = − tan(bn/an), (9)
where φn is the phase angle of the nth harmonic component.
4

7. The series approach has to be abandoned when the waveform is not pe-
riodic for example when Tp becomes infinite. As Tp increases the spacing
between 1/Tp = ω/2π decreases to dω/2π eventually becoming zero. The
discrete variable nω becomes continuos ω, and the amplitude and phase
spectra become continuos. This means that dn → d(ω) and Tp → ∞. With
these modifications we get the normalized formula [1],
dn = F(iω) =
1
2π
−∞
∞
f(t)e(−inωt)
dt. (10)
F(iω) is the complex Fourier integral,
F(iω) = Re(iω) + iIm(iω) = |F(iω)|eiφ(ω)
, (11)
where the amplitude is given by,
|F(iω)| = (Re(iω)2
+ Im(iω)2
)
1
2
(12)
and the phase by,
φ(w) = arctan[Im(iω)/Re(iω)]. (13)
|F(iω)| has the units of V Hz−1
.
The Fourier transform (FT) has very useful properties [2]. If f(x) has the
Fourier transform F(s), then f(ax) has the Fourier transform |a|−1
F(s/a).
Its application to waveforms and spectra is well known as compression of
the time scale corrensponds to expansion of the frequency scale. If f(x)
and g(x) have the Fourier transforms F(s) and G(s), then f(x) + g(x) has
F(s) + G(s) as the FT. The FT of f(x) is F(s), then f(x − a) has the FT
e−2πias
F(s). If f(x) has the FT F(s), then f(x) cos ωx has the FT 1
2
F(s −
ω/2π) + 1
2
F(s + ω/2π). If f(x) and g(x) have FTs F(s) and G(s), then
convolution f(x) ∗ g(x) has the FT F(s)G(s). The squarred modulus of a
function versus the squarred modulus of a spectrum yields;
∞
−∞
|f(x)|2
dx =
∞
−∞
|F(s)|2
ds. (14)
If f(x) has the FT F(s) then f′
(x) has the FT i2πF(s).
It can be seen from:
5

8. ∞
−∞
f′
(x)e−i2πxs
dx =
∞
−∞
lim
f(x + ∆x) − f(x)
∆x
e−i2πxs
dx = lim
∞
−∞
f(x + ∆x)
∆
e−i2πxs
dx − lim
∞
−∞
With these interesting properties let us turn to the discrete Fourier trans-
form (DFT) and the fast Fourier transform (FFT).
The digitalization of the analogue data requires the Fourier transforms
to be discrete. The analogue values are sampled at regular intervals and
then converted to binary representation. The operational viewpoint is that
it is irrelevant to talk about existence of values other than those given, and
those computed namely the input, and the output. Therefore we need the
mathematical theory to manipulate the actual quantified measurements.
Discreteness arises in connection with periodic functions. Discrete in-
tervals describing a periodic function may be viewed as a special case of
continuous frequency. This transform is thus regarded as equally spaced
deltafunctions multlipied by coeffients to determine their strengths.
A typical fuction x(t) of the measurement is fed through an analogue to
digital converter. It samples x(t) at a series of regularly spaced times as
seen in Figure 1. Taking the sampling interval as ∆, then the discrete value
of x(t) = xr at time t is t = r∆, and can be written as a discrete time
sequence {xr}, r = . . . , −1, 0, 1, 2, 3, . . .. We are interested in the frequency
composition of sequence {xr} by analysis obtained from a finite length of
samples.
The historical method to estimate spectra from measured data was to
estimate an appropriate correlation function first and then to FT this func-
tion to obtain the required spectrum. This approach was until late 1960’s,
and practical calculations follewed the mathematical route of spectra defined
as FTs of correlation functions. The classical methods are studied in great
detail, and there is extensive literature ( [3], [4] and [5] on this subject.
The position was changed when fast Fourier transforms (FFT) came
along. This way of calculating the FT is much more efficient and faster.
Instead of determining a correlation function, and then calculating the FT,
FFT directly estimates the original FT of the time series.
If x(t) is a periodic function with period T, then it can be written:
x(t) = a0 + 2
α
k=1
ak cos(
2πkt
T
) + bk sin(
2πkt
T
) (16)
where k ≥ 0 is an integer, and
6

9. Figure 1: Sampling a continuous function at regular intervals.
ak =
1
T
T
0
x(t) cos(
2πkt
T
dtbk =
1
T
T
0
x(t) sin(
2πkt
T
dt. (17)
The previous can be combined into a single equation:
Xk = ak − ibk (18)
and putting
e−i2πkt/T
= cos
2πkt
T
− i sin
2πkt
T
(19)
from we get
Xk =
1
T
T
0
x(t)e−i2πkt/T
dt. (20)
Knowing only the equally spaced samples of the continuous time series
x(t) represented by the discrete series {xr}, r = 0, 1, . . ., (N − 1), where
7

10. Figure 2: Calculating Fourier coefficients from a discrete series using approximation.
t = r∆, and ∆ = T/N, then the integral may be replaced approzimately by
the summation
Xk =
1
T
N−1
r=0
xre(−i2πk/T)(r∆)
∆. (21)
This is just assuming the total area under the curve in Figure 2. Putting
T = N∆ gives
Xk =
1
N r=0
N − 1xre−i2πkr/N
. (22)
This may be regarded as approximation for calculating the Fourier series.
The inverse formula for the series {xr} is
xr =
N−1
k=0
Xke2iπkr/N
. (23)
8

11. This can be seen
N−1
k=0
Xke2iπkr/N
=
N−1
k=0
1
N
N−1
s=0
xse−2iπks/N
e2iπkr/N
=
N−1
k=0
N−1
s=0
1
N
xse(−2iπk/N)(s−r)
(24)
and by interchanging the summation
=
N−1
s=0
N−1
k=0
e−2i(πk/N)(s−r) 1
N
xs (25)
and the exponentials all sum to zero unless s = r when the summation
equals N and hence
N−1
s=0
N−1
k=0
e−2i(πk/N)(s−r) 1
N
xs = xr. (26)
The components Xk are limited to k = 0 to N − 1 corresponding to
frequencies ωk = 2πk/T = 2πk/N∆.
DFT of the series {xr}, r = 0, 1, . . ., N − 1 is defined as
Xk =
1
N
N−1
r=0
xre−2iπkr/N
(27)
for k = 0, 1, . . ., N − 1. Calculating values of Xk for k is greater than
N − 1. Letting k = N + l then
XN+l =
1
N
N−1
r=0
xre−(2iπr/N)(N+l)
=
1
N
N−1
r=0
xre−2iπr/N
e−2iπr
= Xl. (28)
The coefficients Xk just repeat for k > N −1, so plotting | Xk | along the
frequency axis ωk = 2πk/N∆, the graph repeats periodically. It is also easy
to see that X−l = Xl (the complex conjugate) and hence | X−l |=| Xl | is
symmetrical about the zero frequency position. The unique frequency range
is | ω |≤ π/∆ rad/s. The higher frequencies are repetitions of those which
apply below π/∆ rad/s. The coefficients Xk calculated by the DFT are
correct for frequencies up to
9

12. ωk =
2πk
N∆
=
π
∆
(29)
where k is in the range k = 0, 1, . . ., N/2. The frequencies above π/∆
rad/s, which are present in the original signal, introduce a distortion called
aliasing. The high frequencies contribute to {xr}, and therefore distorts the
DFT coefficients for frequencies below π/∆ rad/s. When ω0 is the maximun
frequency present in x(t), then the problem can be avoided by taking the
sampling ∆ small enough so
π
∆
> ω0 (30)
or f0 = ω0/2π giving
1
2∆
> f0. (31)
This 1/2∆ Hz is called the Nyquist frequency, which is also the maximum
frequency that can be detected with particular time spacing ∆ (seconds).
Aliasing is most important when analysing measured data, and to ensure
that DFT is good the sampling frequency 1/2∆ must be high enough to
cover the full frequency range that the continuous time series operates in. If
this is not satisfied the the spectrum from equally spaced samples will differ
from the true spectrum because of aliasing. One way to ensure this is to filter
all frequencies above the frequency components above 1/2∆ before making
the DFT.
FFT is an algorithm for calculating the DFTs. For working out values of
Xk by directly calculating from the basic DFT definition for each N values
requires N2
multiplications. The aim of the FFT is to reduce the number of
operations to the order of N log2 N. The FFT therefore offers great amount
of reduction in the prosessing time, and accuracy increases as fewer round-off
errors is reduced.
FFT partitions the sequence {xr} in shorter sequences, and then combines
these to yield the full DFT. Suppose {xr}, r = 0, 1, . . ., N − 1 is a sequence
where N is an even number and this is partitioned to two shorter sequences
{yr} and {zr} where yr = x2r and zr = x2r+1, r = 0, 1, . . ., (N/2 − 1). The
DFT’s of these are
Yk =
1
N/2
N/2−1
r=0
yre−i2πkr
N/2 Zk =
1
N/2
N/2−1
r=0
zre−i2πkr
N/2 , k = 0, 1, . . ., N/2 − 1.(32)
10

13. Firstly we separate the odd and the even terms in {xr} getting
Xk =
1
N
N−1
r=0
xre−2iπkr/N
=
1
N
N/2−1
r=0
x2re−i2π(2r)k
N +
N/2−1
r=0
x2r+1e−i2π(2r+1)k
N =
1
N
N/2−1
r=0
yre−i2πrk
N/2 + e
from which
Xk =
1
2
Yk + e−i2πk/N
Zk , k = 0, 1, . . ., N/2 − 1. (34)
The original DFT can be obtained from Yk and Zk. If the original number
of samples is a power of 2, then the half-sequences {yr} and {zr} can be
partitioned into quarter-sequences, and so on, until the last sub-sequences
have only one term each. Yk and Zk are periodic and repeat themselves with
period N/2 so that Yk−N/2 = Yk and Zk−N/2 = Zk.
Calculating
Xk =
1
2
Yk + e−2iπk/N
Zk , k = 0, 1, . . ., N/2 − 1Xk =
1
2
Yk−N/2 + e−2iπk/N
Zk−N/2 , k = N/2, N/2 + 1
or
Xk =
1
2
Yk + e−2iπk/N
Zk Xk+N/2 =
1
2
Yk + e−2iπ(k+N/2)/N
Zk =
1
2
Yk − e−2iπk/N
Zk , k = 0, 1, . . ., N/
These are the formulas occuring in most FFT programs, and defining
W = e−2iπ/N
we obtain what is called coputational butterfly [6]. The FFT
changed the approach to digital spectral analysis when it was implemented
in 1965 ( [7] and [8]).
For general purposes Matlab’s FFT is used. It is based upon FFTW-
libraries [9]. FFTW uses several combinations of algorithms, including vari-
ation of the Cooley-Tukey algorithm, a prime factor algorithm [10], and a
split-radix algorithm [11]. The split-radix FFT requires N to be a power of
2 so the original sequence can be partitioned into two half-sequences of equal
length, and so on.
With these methods one is able to study the frequency depedence of the
input data. One should make the number of samples taken to be in the
form 2n
, where n is an integer. Even with number of samples the FFT
works quite some faster, for example a sequence that has N = 1048576 =
220
samples calculated directly with DFT compared to FFT has the ratio
11

14. N2
/(N log2 N) = 52428, 8. Also one should make the sampling time interval
so small that the largest frequency that can be measured is well withing the
Nyquist frequency to avoid distortion due to aliasing. Otherwise if this is
not possible then filtering should be used in the experimental setup to cut
the frequency components above the Nyquist frequency to avoid aliasing.
4 Piezoelectric Accelerator
Piezoelectric effect was found in 1880 by Jaques and Pierre Curie in crys-
talline minerals, when subjected to a mechanical force the crystal became
electrically polarized. Compression and tension generated oppositely po-
larized voltages in proportion to the force. In converse if voltage-genarating
crystal was exposed to a electric field it contracted or expanded in accordance
with the polarity and field strength. These effects were called piezoelectric
effect and inverse piezoelectric effect. Quartz and other natural crystals
are widely used today in microphones, accelerometers, and ultrasonic trans-
ducers. Their applications include smart materials for vibration control,
aerospace, and astronautical applications of flexible surfaces, and vibration
reduction in sports equipment.
Consistent with the IEEE standards of piezoelectricity [12], the transduc-
ers are made of piezoelectric materials that are linear devices whose prop-
erties are governed by a set of tensor equations. To better understand the
workings of piezoelectricity we firstly turn to making of a piezoelectric ce-
ramic crystal. A piezoelectric ceramic is a perovskite crystal composed of
a small tetravalent metal ion placed inside a larger lattice of divalent metal
ions and O2 (see Figure 3). Preparing such a ceramic, fine powders of the
component metal oxides are mixed in very specific proportions, and heated
to form a uniform powder, which is then mixed with an organic binder. The
powder turns into dense crystalline structure via specific process of heating,
and cooling.
Above the Curie temperature, each perovskite crystal exibit no dipole
moment (see Figure 4). Just below the Curie temperature each crystal has
tetragonal symmetry, and a dipole moment. Alaining these dipoles using
electrodes on the appropriate surfaces to create a strong DC electric field,
gives a net polarization. This is called poling process. After removing the
electric field most of the dipoles remain in a locked place, creating permanent
polarization and permanent elongation. The length increase of the element
is usually within the micrometer range. Tension or mechanical compression
changes the dipole moment associated with the particular element creating
a voltage. Tension perpendicular to direction of polarization or compression
12

15. Figure 3: Crystalline structure of a piezoelectric crystal, before and after polarization.
along the direction of polarization generates voltage of the same polarity
as the poling voltalge. Tension along the polarization or compression per-
pendicular to the direction or polarization generates an opposite voltage to
that of the poling voltage. The voltage and the compressive stress generated
applying stress to the piezoelectric crystal are linearly proportional up to a
specific stress. In this way the crystal works as a sensor. The piezoelec-
tric crystal expands and contracts when poling voltages are applied, and in
this way the use is an actuator. This way electric energy is converted into
mechanical energy. When electric fields are low, and small mechanical stress
the piezoelectric materials have a linear profile. Under high stresses and elec-
tric fields this breaks into very nonlinear behavior. Straining mechanically a
poled piezoelectric crystal makes it electrically polarized, producing an elec-
tric charge on the surface of the material. This is the direct piezoelectric
effect and it is the basis of sensory use.
The electromechanical equations for a linear piezoelectric crystal are (
[12], [13]):
εi = SE
ij σj + dmiEm (37)
13

16. Figure 4: Poling process: (i) Before polarization; (ii) Polarization is gained using a very
large DC electric field; (iii) The remnant polarization after removing the field.
14

17. Figure 5: Axis for linear piezoelectric material describing the electromechanical
equations.
Dm = dmiσi + ξσ
ikEk, (38)
where i, j = 1, 2, . . ., 6 and m, k = 1, 2, 3 are different directions in the
coordinate system shown in Figure 5. The equations are usually written in
another form when the appications involve sensory actions:
εi = §D
ij σj + gmiDm (39)
Ei = gmiσi + βσ
ikDk (40)
where σ . . . stress vector (N/m2
) ε . . . strain vector (m/m) ξ . . . (F/m) E
. . . vector of applied electric field (V/m) d . . . matrix of piezoelectric strain
constants (m/V ) S . . . matrix of compliance coefficients (m2
/N) g . . . matrix
of piezoelectric constants (m2
/C) β . . . impermitivity component (m/F) D
. . . vector of electric displacement (C/m2
)
The asumption here is that measurements of D, E, and σ are taken at
constant electric displacement, constant stress and constant electric field.
Usually the crystal is poled along axis 3, and piezoelectric crystals are
transversely isotropic. Thus the equations simplify as S11 = S22, S13 = S31 =
15

18. S23 = S32, S12 = S21, S44 = S55, S66 = 2(S11 −S12), and others are zero. The
non-zero piezo-electric strain constants are d31 = d32, and d15 = d24. Also
the non-zero dielectric coefficients are eσ
11 = eσ
22, and eσ
33. One can write these
in matrix form to give:








ε1
ε2
ε3
ε4
ε5
ε6








=








S11 S12 S13 0 0 0
S12 S11 S13 0 0 0
S13 S13 S33 0 0 0
0 0 0 S44
0 0 0 0 S44 0
0 0 0 0 0 2(S11 − S12)
















σ1
σ2
σ3
σ4
σ5
σ6








+








0 0 d31
0 0 d31
0 0 d33
0 d15 0
d15 0 0
0 0 0










E1
E2
E3

(41)
and


D1
D2
D3

 =


0 0 0 0 d15 0
0 0 0 d15 0 0
d31 d31 d33 0 0 0










σ1
σ2
σ3
σ4
σ5
σ6








. (42)
dij is the ratio of the strain in the j-axis to the electric field along the
i-axis, taking external stresses constant. Voltage V is apllied as Figure 6 with
the crystal beign polarized in the z-direction, generates electric field:
E3 =
V
t
. (43)
This strains the transducer, e.g.
ε1 =
∆l
l
(44)
and
∆l =
d31V l
t
. (45)
Constant d31 is usually negative as the positive electric field generates a
positive strain in z-direction. dij can also be interpreted as the ratio of short
16

19. circuit charge per unit area flowing between connected electrodes perpendic-
ular j to the stress applied along i. A force Fk is applied to the transducer
generates the stress
σ3 =
F
lw
(46)
resulting in the elecric charge
q = d33F (47)
flowing through the short circuit. The constant gij denotes the electric
field along i when the material is stressed along j. Force F applied in the
positive i, resulting in the voltage:
V =
g31F
w
. (48)
The other way to intepret gij is to take the ratio of strain along j to the
charge per unit area deposited on electrodes perpendicular to i. Placing an
electric charge Q on the surface electrodes (plates are on top and bottom
perpendicular to k) changes the thickness by:
∆l =
g31Q
w
. (49)
Constant Sij is the ratio of the strain in i-direction to the stress in j-
direction, given that there’s no stress along the other two directions. Direct
strains and stresses have indeces 1 to 3, and shear stresses and strains have
indeces 4 to 6. E is used to mark elastic compliace SE
ij measured with the
electrodes short-circuited, and similarly D denotes that the measurements
were done with electrods left open-circuited. SE
ij is maller than SD
ij as me-
chanical stress results in an electrical responce that can increase the resultant
strain meaning that a short circuited piezo has a smaller Young’s modulus
of elasticity than open-ciruited.
The dielectric coefficient eij is the charge per unit area along x-axis due to
applied field applied in the y-axis. Relative dielectric constant K is defined
as the ratio of the absolute permitivity of the material by the permittivity of
free space. σ in eσ
11 is the permittivity for a field applied along x-axis, when
the material is not restrained.
17

20. Ability of a piezoceramic to transform electrical energy to mechanical
energy and vice verca is denoted by kij. The stress or strain is j-oriented
and electrodes are perpendicular to i. Applying a force F to the crystal,
while leaving the terminals open circuited makes the device deflect like a
spring. This deflection is ∆z, and the mechanical work is:
WM =
F∆z
2
. (50)
Electric charges accumulate on the electrodes due to piezoelectric effect
amounting to the elecrical energy in the piezoelectric capacitor:
WE =
Q2
2CP
. (51)
From this we get
k33 =
WE
WM
=
Q
F∆zCp
. (52)
The coupling can be written otherwise as
k2
ij =
d2
ij
Sij
E
eσ
ij = gijdijEp, (53)
where Ep is the Young’s modulus of elasticity. Now we turn to see how the
piezoelectric sensor works on these basis, as described above. Piezoelectric
sensors offer superior signal to noise ratio, and better high-frequency noise
rejection, thus they are quite suitable for applications that involve measur-
ing low strain levels. When a piezoelectric crystal sensor is subjected to a
stress field, assuming the applied electric field is zero, the resultind electrical
displacement vector is:


D1
D2
D3

 =


0 0 0 0 d150
0 0 0 d15 0 0
d31 d31 d33 0 0 0










σ1
σ2
σ3
σ4
σ5
σ6








(54)
18

21. The charge generated can be determined
q = D1 D2 D3


dA1
dA2
dA3

 , (55)
where dA1, dA2 and dA3 are the differential electrode areas in the y − z,
x − z and x − y planes. The voltage generated VP is related to charge
VP =
q
CP
, (56)
where CP is the capacitator of the sensor. By measuring VP , the strain can
be calculated from the integral above. A calibrated piezoelectric accelerom-
eter is a sensor, and the voltage measured can then be used as measure or
acceleration thus this is very useful for very precise frequency analysis. Fig-
ure 6 shows a typical configuration of a piezoelectric accelerometer mounted.
Small size, and rigidness generally means that stucture’s vibrational charac-
teristics will be minimal, but the structure does often affect the vibrational
characteristics of the attached accelerometer. Accelerometer’s sensitivity is
defined as the ratio of the output signal (voltage in our case) to the acceler-
ation of its base. The major resonant frequency of the accelerometer is the
lowest frequency for which the sensitivity has a maximum. The frequency
range of use is generally taken as that region in which sensitivity does not
change significantly from the value found near 100 Hz [14] when calibrated
on a conventional shaker table. The upper limit of an accelerometer is lower
than the determined by resonance of the accelerometer alone, when the mass
of accelerometer affects the motion of the structure.
To estimate the largest frequency to be measured can be calculated using
the following model. We assume the system consisting of masses (m1, m2 and
m3), and springs (assumed massless, and their spring constants ka and ks)
as in Figure 6. Let us suppose that a sinusoidally varying mechanical force
F cos ωt, is imposed on m3 from outside the system taking the position x3 is
A3 cos ωt. The resultant motion of m1, and m2 or the varying positions x1
and x2 (Figure 7) are considered as the frequency of the drive force is varied.
After transient effects die away, the equations describing the motion of m1
and m2 under the dynamic forces are:
m1 ¨x1 + ka(x1 − x2) = 0 (57)
m2 ¨x2 + ka(x2 − x1) + ks(x2 − x3) = 0. (58)
19

22. Figure 6: A typical configuration of a piezoelectric accelerometer.
Figure 7: Applied force and displacements.
20

23. The resonance sought is the lowest value of ω at which a maximum of
(x1 − x2) occurs by varying ω. When the system is in dynamic equilibrium
and resonance is approached from below, the motions will be at the drive
frequency and in phase.
x1 = A1 cos ωt (59)
x2 = A2 cos ωt (60)
¨x1 = −A1ω2
cos ωt (61)
From this we obtain
(ka − m1ω2
)A1 − kaA2 = 0 (62)
−kaA1 + (ks + ka − m2ω2
)A2 = ksA3 (63)
Resonance occurs when A1 − A2 has a maximum value. There’s no prob-
lem due to phase considerations because resonance is approached from below
and, with no damping, x1 and x2 can be considered to be in phase with the
motion of the driving element i.e. with x3. The solution for A1 and A2 each
has the determinant of its coefficients in the denominator. Thus maximum
values of A1 and A2 occur when this determinant vanishes. An equation in
the resonant frequency ω results:
(ka − m1ω2
)(ka + ks − m2ω2
) − k2
a = 0. (64)
This can be written as quadratic in ω2
,
ω4
− [ka(
1
m1
+
1
m2
) + ks
1
m2
]ω2
+
kaks
m1m2
= 0 (65)
Now we simplify the calculations by taking new constants a = m2/m1,
r = ks/ka, and ω2
0 = ka/m1. Substituting these into the above equation and
calculating the frequency:
ω = ω2
0
[1 + 1
a
(1 + r)] − [1 + 1
a
(1 + r)]2 − 4r
a
2
. (66)
21

24. From this we can get the upper limit of the usable frequency. Usually this
value is given, and sometimes the relative frequency is given as ω/ω0. If this
value is not given, then one can estimate. Now in our case the measurements
on the cryostat imply that the mass of the accelerometer does not alter
the results, and the accelerometers base m2 is rigidly attached (in our case
a very strong magnet and straps) to a very large mass m3 (cryostat’s or
the supporting frame’s mass) so that m1 and ka are the only resonance-
determined parameters. These parameters are usually given, but if not it
can be relatively easy to approximate those. In this case of rigid attachment
masses m2 and m3 are combined, and taking m3 = ∞ we get
ω = ka/m1. (67)
The Nyquist frequency should be set little below this frequncy, and the all
frequencies above the Nyquist frequency should be filtered out (see next sec-
tion). Or if it is know what frequencies are to be looked for then the highest
frequency should be set according to that. Accelerometers are also subjected
to thermal-transient stimuli from stronger vibrations. Certain properties
of piezoelectric accelerometers can cause them to generate spurious output
signals in response to such thermal transients, leading to significant measure-
ment errors. Many piezoelectric crystalline materials are also pyroelectric [15]
that is, a change of temperature causes a change in the polarization charges
in the material. Pyroelectric output signals can result from a uniform or
non-uniform distribution of thermal charges within the material. In addi-
tion, mechanical strain within the piezoelectric element, resulting differential
thermal expansion of the components of an accelerometer subjected to ther-
mal transients, may generate spurious output signals. In conditions where
the accelerometer is exposed to blasts, non-uniform heating is propable. The
resultant output signal will thus include pyroelectrically generated charges
and charges produced by changes in the mechanical loading of the crystal
resulting from differential expansion of accelerometer components. This is
the reason why after rigid attaching of the piezoelectric accelerometer on the
cryostat or its frame one should wait for some time for normal conditions to
reappear. Usually the pyroelectric effect under normal stated conditions for
most piezoelectric crystals are not significant under frequencies of 3000 Hz
and amplitudes of 5 g [16]. So there should not be any problem with normal
measurements with the setup of the cryostat. However care should be taken
as to where the accelerometer is placed, e.g. it should not be placed in close
proximity of electronic devices that give strong electric fields or directly leak
heat into the surroundings.
22

25. 5 Low-pass filter and measurement setup
Taking into account the different frequency aspects namely the Nyquist fre-
quency, and the maximum frequency of the piezoelectric accelerometer under
which it operates, one needs to have a low-pass filter. Using this wisely will
eliminate almost all coputed aliasing, and bad signals from the crystal. Usu-
ally the window of interest lies somewhere in the region of 0 to 100 Hz, which
is easily attained by the machinery used. Ideally low-pass filters completely
eliminate all frequencies above the cut-off frequency while passing those be-
low unchanged. Real time filters approximate the ideal filter by windowing
the infinite impulse response to make a finete impulse response. Digital filter-
ing in our case is not the best solution, better is to use an electronic low-pass
filter. A second-order filter does a better job of attenuating higher frequen-
cies. There are many different types of filter circuits, with different responses
to changing frequency. A first-order filter will reduce the signal amplitude by
half every time the frequency doubles (goes up one octave). As the frequency
reach of the equipment used is so large, the low-pass filter can be relatively
simple one. One could use or build easily an active low-pass filter. In the
operational amplifier shown in the Figure 8, the cutoff frequency is defined
as
fc =
1
2πR2C
(68)
or equivalently in radians per second
ωc =
1
R2C
. (69)
The gain in the passband is −R2/R1, and the stopband drops off at −6dB
per octave, as it is a first-order filter.
If this doesn’t work as wished one can easily build a second order (or
higher) Butterworth filter (see Figure 9 [19]), which decreases −12dB per
octave. Also the frequency responce of the Butterworth filter is maximally
flat [18] in the passband compared to Chebyschev Type I / Type II or an
elliptic filter [17].
6 Vibrations
The types of vibrations in our case can be divided into two main categories:
the unbalanced rotation of the cryostat creates harmonic oscillations and
23

26. Figure 8: An active low-pass filter.
Figure 9: Butterworth low-pass filter in a circuit used to obtain vibration spectra [19].
24

27. other noisy vibrations, and other is resulting from external vibrations e.g.
electronic devices on the cryostat, the pumps and vibrations from ground
or foundation vibrations. Torsional vibration analysis is vital for ensuring
reliable machine operation, especially as very precise measurements are made
on the large cryostat. If rotating component failures occur on the cryostat
as a result of torsional oscillations, the consequences can be catastrophic.
In the worst case, the entire machine can be wrecked as a result of large
unbalancing forces, and worse injury to human beigns might be inflicted.
The foundation and electronics vibations are easier to allocate, but are big
enough to cause problems as vibrations could affect the nuclear stage and
the demagnetization solenoid creating a heat leak there as suggested by [19].
The level of vibration in a structure can be attenuated by reducing either the
excitation or the response of the structure to that excitation or both. These
could be relocating equipment, or isolating the structure from the exciting
force. The torsional vibrations can be reduced by balancing the load on the
rotating machinery. Real structures consist of an infinite number of elestically
connected masses and have infinite number of degrees of freedom. In reality
the motion is often such that only a few coordinates is needed to describe
the motion. The vibration of some structures can be analysed using a sigle
degree of freedom. Other motions may occur, but in our case for instance
for analysing the electric and foundation vibrations other vibrations can be
dimished, and electrical devices can be measured one at the time (to see more
comprehensive study [20]. A body of mass m is free to move along a fixed
horizontal surface attached to a spring k one end fixed. Displacement of the
mass is denoted by x, so giving this initial displacement x0, and letting go
we get:
¨x +
k
m
x = 0 (70)
giving
x = A cos ωt + B sin ωt, (71)
where A and B are constants, and ω is the circular frequency. Now with
x = x0 and t = 0 gives A = x0, and ˙x = 0 and t = 0 gives
x = x0 cos
k
m
t. (72)
25

28. When springs are in series the total spring constant can be calculated as
the deflection at the free end, δ, experienced applying the force F is to be
the same in both cases,
δ = F/ke =
i
F/ki (73)
so that
1/ke =
i
1/ki. (74)
Similarly parallel springs give
ke =
i
ki. (75)
Let us consider a beam with m as the mass unit length, and y is the
amplitude of the deflection curve (see Figure 10) then
Tmax =
1
2
˙y2
maxdm =
1
2
ω2
y2
dm, (76)
where ω is the natural circular frequency of the beam, and Tmax is the
maximum kinetic energy.
The strain energy of the beam is the work done on the beam which is
stored as elastic energy. If the bending moment is M, and the slope of the
elastic curve is θ, the potential energy is
V =
1
2
Mdθ. (77)
Assuming the deflection of beams small
Rdθ = dx, (78)
thus
1
R
=
dθ
dx
=
d2
y
dx2
. (79)
26

29. Figure 10: Beam deflection.
From beam theory [21], M/I = E/R, where R is the radius of curvature
and EI is the flexural rigidity:
V =
1
2
M
R
dx =
1
2
EI
d2
y
dx2
2
dx. (80)
Now Tmax = Vmax;
ω2
=
EI d2y
dx2
2
dx
y2dm
. (81)
This expression gives the lowest natural frequency of transverse vibration
of a beam. It can be seen that to analyse the transverse vibration of a partic-
ular beam by this method requires y to be known as a function of x. In the
case of the cryostat’s frame this method can prove to be quite cumbersome.
Real structures dissipate vibration energy, so damping sometimes becomes
significant. Damping is difficult to model exactly because the mechanisms
of the structures. Using simplified models usually gain quite good results,
and can give insight to the problem. Viscous damping is a common form of
27

30. damping, and the viscous damping force is proportional to the first power of
the velocity across the damper, and it is always opposed to motion, so that
damping force is linearly continuous function of the velocity. Simple model
can be imagined taking a horizontal m mass attached to a spring k and a
damper c (damping force is proportional to velocity), which are both fixed.
As before we get for the equation of motion:
m¨x + c ˙x + kx = 0. (82)
Assuming solution of the form x = Xest
= 0, and substituting for roots:
s1,2 = −
c
2m
±
(c2 − 4mk)
2m
, (83)
hence
x = X1es1t
+ X2es2t
, (84)
where X1 and X2 are arbitrary constants found from initial conditions.
The dynamic behavious of the system depends opon the numerical value of
the radical, so defining critical damping cc = 2
√
km making the radical zero,
and undamped natural frequency ω = cc/2m. Defining damping ratio by
ζ = c/cc, (85)
and
s1,2 = (−ζ ± (ζ2 − 1))ω. (86)
When damping is less critical ζ < 1
s1,2 = −ζω ± i (1 − ζ2)ω (87)
so
x = Xe−ζωt
sin ( (1 − ζ2)ωt + φ). (88)
28

31. When ζ = 1 the damping is critical and equation x = (A + Bt)e−ωt
is
valid. Finally damping greater than critical ζ > 1 gives two negative real
values of s so that x = X1es1t
+ X2es2t
. Substituting for damping constant
a constant friction force Fd that represents dry friction (Coulomb damping)
applicaple in many mechanisms:
m¨x + kx = Fd. (89)
Getting a solution
x = A sin ωt + B cos ωt +
Fd
k
. (90)
Hence
x = (x0 +
Fd
k
cos ωt +
Fd
k
, (91)
where the oscillation ceases with | x |≤ Fd/k, and the zone x = ±Fd/k is
called the dead zone. Many real structures have both viscous and Coulomb
damping. The two damping actions are sometimes dependent of amplitude,
and if the two cannot be separated a mixture of linear and exponential decay
functions have to be found by trial and error. In most real structures separat-
ing stiffness and damping effects is often not possible. This can be modeled
using complex stiffness k∗
= k(1 + iη), where k is the static stiffness, and
η is the hysteric damping loss factor. A range of values for η can be found
for common engineering materials in basic literature ( [22]). The electronic
devices on the cryostat behave as external excitation forces usually periodic.
From previous we construct a model as taking mass m connected to a fixed
spring and viscous damper, whilst a harmonic force of circular frequency ν
and amplitude F:
m¨x + c ˙x + kx = F sin(νt). (92)
Solution can be taken as x = X sin(νt − φ), where motion lags the force
by vector φ, so substituting and using cos − sin relations we get
mXν2
sin(νt − φ + π) + cXν sin(νt − φ + π/2) + kX sin(νt − φ) = F sin(νt).
(93)
29

32. From this
F2
= (kX − mXν2
)2
+ (cXν)2
, (94)
or
X = F/ ((k − mν2)2 + (cν)2), (95)
and
tan(φ) = cXν/(kX − mXν2
). (96)
The steady state solution
x =
F
((k − mν2)2 + (cν)2)
sin(νt − φ), (97)
where
φ = tan−1 cν
k − mν2
. (98)
The complete solution includes the transient motion given by the com-
plementary function:
x = Ae−ζωt
sin(ω (1 − ζ2)t + α), (99)
where ω = k/m and Xs = F/k so that
X
Xs
=
1
(1 − (ν/ω)2)2 + (2ζν/ω)2
, (100)
and
φ = tan−1 2ζ(ν/ω)
1 − (ν/ω)2
. (101)
30

33. Figure 11: Isolating vibrating machine.
X/Xs is known as the dynamic magnification factor, where Xs is static
deflection of the system under a steady force F, and X is the dynamic ampli-
tude. The mechanical vibration arises from the large values of X/Xs, when
ν/ω has a value near unity, meaning that a small harmonic force can produce
a large amplitude of vibration. Resonance occurs when the forcing frequency
is equal to natural frequency e.g. ν/ω = 1. The max of X/Xs can be attained
from differentiating to get:
(ν/ω)(X/Xs)max = 1 − 2ζ2) ≃ 1, ζ ≈ 0, (102)
and
(X/Xs) = 1/(2ζ 1 − ζ2). (103)
For small ζ, (X/Xs)max ≃ 1/2ζ is a measure of the damping and is known
as the Q factor. The force transmitted to the foundation or supporting struc-
ture can be reduced by using flexible mountings with the correct properties.
Figure 11 shows a model of such a system.
31

34. The force transmitted to the foundation is the sum of the spring force
and the damper force. Thus the transmitted force is given by
FT = (kX)2 + (cνX)2. (104)
The transmissibility is given by
TR =
FT
F
=
X k2 + (cν)2
F
(105)
since
X =
F/k
(1 − ν
ω
2
)2 + 2ζ ν
ω
)2
, (106)
TR =
1 + (2ζ ν
ω
)2
1 − (ν
ω
)2 + 2ζ ν
ω
2
. (107)
Therefore the force and motion transmissibilities are the same. It can be
seen that for good isolation ( [21]) ν/ω >
√
2, hence for a low value of ω
is required which implies a low stiffness, that is a flexible mounting. In the
cryostat it is particularly important to isolate vibration sources e.g. the elec-
trical devices because vibrations transmitted to structure radiate well, and
serious heat leak problems can occur. Theoretically low stiffness isolators
are desirable to gice a low natural frequency. There are four types of spring
material commonly used for resilient mountings and vibration isolation: air,
metal, rubber, and cork. Air springs can be used for very low-frequency
suspensions: resonance frequencies as low as 1 Hz can be achieved whereas
metal springs can only be used for resonance frequencies greater than about
1.3 Hz. Metal springs can transmit high frequencies, however, so rubber
or felt pads are often used to prohibit metal-to-metal contact between the
spring and the structure. Different forms of spring element can be used as
coil, torsion, cantilever and beam. Rubber can be used in shear or compres-
sion but rarely in tension. It is important to determine the dynamic stiffness
of a rubber isolator because this is generally much greater than the static
stiffness. Rubber also possesses some inherent damping although this may
be sensitive to amplitude, frequency and temperature. Natural frequencies
32

35. from 5 Hz upwards can be achieved. Cork is one of the oldest materials
used for vibration isolation. It is usually used in compression and natural
frequencies of 25 Hz upwards are typical. For precise isolation systems, and
materials please refer to [23]. The above analysis is more for analysing the
vibrations due to electrical devices and the pumps, as they are consistent
usually having some periodicity giving only specific frequency peaks in fre-
quency spectrum. For external noise from the surroundings are more likely
to be random processes (as in [19]) possibly due to heavy traffic on a nearby
road or other large machinery used nearby. Collection of sample functions
x1(t), x2(t), . . . , xn(t), which make up the ensemble x(t). Normal or Gaussian
process is the most important of random processes because a wide range of
physically observed random waveforms represented by Gaussian process:
p(x) =
1
√
2πσ
e− 1
2
x−x
σ
2
, (108)
is the density function of x(t), where σ is the standard deviation of x,
and x is the mean of x. The values of σ and x may vary with time for a non-
stationary process but are independent of time if the process is stationary.
x(t) lies between −λσ and λσ, where λǫR+
taking x with probability
Prob{−λσ ≤ x(t) ≤ λσ} =
λσ
−λσ
1
√
2πσ
e(− 1
2
x2
σ2 )
dx. (109)
Probabilities with varying λ can be found for example in [24]. We now
turn to actual methods of damping the unwanted vibrations. Some reduction
can be achieved by changing the machinery generating the vibration, for ex-
ample removing the fans from electrical devices, and using static heat sinks
commercially available noting problems involved. It is desirable for the cryo-
stat and the framework to possess sufficient damping so that the response
to the expected excitation is acceptable. If damping in the structure is in-
creased the vibrations and noise, and the dynamic stresses will be reduced
directly resulting in lowered heatleak. However increasing damping might be
expensive and may require big changes in already existing buildings. Good
vibration isolation can be achvieved by supporting the vibration generator
on a flexible low-frequency mounting. Air bags or bellows are sometimes
used for very low-frequency mountings where some swaying of the supported
system is allowed. Approximate analysis shows that the natural frequency of
a body supported on bellows filled with air under pressure is inversely pro-
portional to the square root of the volume of the bellows, so that a change
33

36. in natural frequency can simply be affected by change in the volume of the
bellows. Greater attenuation of the exciting force at high frequencies can be
achieved by using a two-stage mounting. In this arrangement the machine
is set on flexible mountings on an inertia block, which is itself supported by
flexible mountings. This may not be expensive to install since for example
the cryostat can be used as the inertia block. Naturally, techniques used for
isolating structures from exciting forces arising in machinery and plant can
also be used for isolating delicate equipment from vibrations in the struc-
ture. Normal solution to vibrational problems is to place the cryostat on a
heavy block supported by air springs, and rotating motors placed on bellows
wrapped with isolating tape [25]. Of course increasing the mass of the block
increases the resonant frequency decreases, this might be a problem with ro-
tation. There are also active isolation systems in which the exiciting force or
moment is applied by an externally powered force or couple. The opposing
force or moment is applied by an externally powered force. The opposing
force can be produced by means such as hydraulic rams. All materials dis-
sipate energy during cyclic deformation due to molecular dislocations and
stress changes at grain boundaries. Such damping effects are non-linear and
variable within material. Some particular materials such as damping alloys
have a certain enhanced damping mechanisms. The load extension hysteresis
loops for linear materials and structures are elliptical under sinusoidal load-
ing, and increase in area according to the square of the extension. Although
the loss factor η of a material depends upon its composition, temperature,
stress and the type of loading mechanism used, an approximate value for η
can be given [26]. Pure aluminium has loss factor of 0.00002 − 0.002, and
hard rubber has 1.0. In a single or multi degree of freedom system mode
is excited into resonance, and the excitation frequency nor the natural fre-
quency can be altereded then adding a single degree of freedom can be of
use. One can consider this using a model such as in Figure 12, where K and
M are the effective stiffness and mass of the primary system.
The absorber is represented by the system with parameters k and m. The
equations of motion for the primary system:
M ¨X = −KX − k(X − x) + F sin νt (110)
and for the vibration absorber
m¨x = k(X − x), (111)
where X = X0 sin νt and x = x0 sin νt.
34

37. Figure 12: System with undamped vibration absorber.
It can be easily seen that
X0 =
F(k − mν2
)
∆
, (112)
and
x0 =
Fk
∆
, (113)
where ∆ = (k − mν2
)(K + k − Mν2
) − k2
, and ∆ = 0 is the frequency
equation. Now the system possess two natural frequencies, Ω1 and Ω2, but by
arranging k − mν2
= 0, X0 can be made zero. Now if (k/m) = (K/M),
the response of the primary system at its original resonance frequency can
be made zero. This is the usual tuning arrangement for undamped absorber
because the resonance problem in the primary system is only severe when
ν ⋍ K/M. When X0, x0 = −F/k, so that the force in the absorber spring,
kx0 is −F thus the absorber applies a force to the primary system which
is equal and opposite to the exciting force. Hence the body in the primary
system has a net zero exciting force acting on it and therefore zero vibration
35

38. amplitude. If correctly tuned ω2
= K/M = k/m, and if the mass ratio
µ = m/M, the frequency equation ∆ = 0 is ( [21], p.196)
ν
ω
4
− (2 + µ)
ν
ω
2
+ 1 = 0, (114)
hence
Ω1,2
ω
= 1 +
µ
2
± µ +
µ2
4
1/2
. (115)
For a small µ, Ω1 and Ω2 are very close to each other, and near to ω,
increasing µ gives better separation between Ω1 and Ω2. This is of impor-
tance in systems where the excitation frequency may vary e.g. µ is small,
resonances at Ω1 or Ω2 may be excited. Now:
Ω1
ω
2
= 1 +
µ
2
− µ +
µ2
4
(116)
and
Ω2
ω
2
= 1 +
µ
2
+ µ +
µ2
4
(117)
then multiplying gives
Ω1Ω2 = ω2
(118)
and
Ω1
ω
2
+
Ω2
ω
2
= 2 + µ. (119)
One can use these relations to desing an absorber, and can be used for
instance for a pump having mass of mp rotating at constant speed of ωp
rev/min, giving large unbalance vibrations. Fitting an undamped absorber
so that the natural frequency of the system is removed by 20%. We model
the pump as in Figure 13, so we get the equation of motion:
36

39. Figure 13: Model of a pump.
mp ¨x1 + kpx1 = F sin νt, (120)
gettin
x1 = X1
F
k1 − mpν2
, (121)
where k1 can estimated or deviced. When X1 = ∞ then ν = k1/mp,
that is resonance occurs when ν = ω = k1/m1. Assuming x2 > x1 (Figure
14) we get:
m2 ¨x2 = −k2(x2 − x1) (122)
and
mp ¨x1 = k2(x2 − x1) − k1x1 + F sin νt. (123)
37

40. Figure 14: Adding the absorber on the pump.
Taking x1,2 = X1,2 sin νt giving
X1(k1 + k2 − m1ν2
) − X2k2 = F (124)
and
−X1k2 + X2(k2 − m2ν2
) = 0. (125)
From this
X1 =
F(k2 − m2ν2
(k2 + k1 − mpν2)(k2 − mmν2) − k2
2
. (126)
If ν2
= k2/m2, X1 = 0 then the frequency equation is (k1+k2−mpν2
)(k2−
m2ν2
) − k2
2 = 0. Putting µ = m2/mp = k2/k1 and Ω = k2/m2 = k1/mp,
giving
ν
Ω
2
=
2 + µ
2
±
µ2 + 4µ
4
. (127)
38

41. From this and smallest absorber mass ν1/Ω = 0.8 as then ν2/Ω = 1.25,
which is acceptable. Thus µ = 0.2 and hence
m2 = µmp = 0.2mp, (128)
and
k2 = 2πωp. (129)
One good system is a ciscous damped absorber such as in Figure 15.
Equations of motion are:
M ¨X = F sin νt − KX − k(X − x) − c( ˙X − ˙x) (130)
and
m¨x = k(X − x) + c( ˙X − ˙x). (131)
Substituting X = X0 sin νt and x = x0 sin (νt − φ) gives, after some
manipulation,
X0 =
F (k − mν2)2 + (cν)2
((k − mν2)(K + k − Mν2) − k2)2 + (cν(K − Mν2 − mν2))2
.(132)
When c = 0 this reduces to the undamped vibration absorber. If c is
large then
X0 =
F
K − ν2(M + m)
. (133)
Response of the primary system is minimized over a wide frequency range
by choosing different c. If k = 0,
X0 =
F
√
m2ν4 + c2ν2
((K − Mν2)mν2)2 + (cν(K − Mν2 − mν2))
. (134)
When c = 0,
39

42. Figure 15: System with damped vibration absorber.
X0 =
F
K − Mν2
, (135)
and when c is very large,
X0 =
F
K − (M + m)ν2
. (136)
7 Torsional vibrations
Historically torsional modes in machinery were always the first to consider
and analyze, in order to avoid extreme stresses. Today torsional vibration
analysis is routinely done throughout design of rotating machines. Their ex-
istence can be discovered when using dedicated instruments [27] to measure
torsional vibrations. Torsional vibration is an oscillatory angular motion
causing twisting in the shaft of a system. Motion is rarely a concern with
torsional vibration unless it affects the function of a system. It is stresses
that affect the structural integrity and life of components and thus determine
40

43. the allowable magnitude of the torsional vibration. In our case the determin-
ing factor is the heat leak into the nuclear stage. The complicated system of
the cryostat can be crudely modeled to gain insight into the problem. How-
ever the torsional vibration is a complex vibration having many different
frequency components. The cryostat can be crudely taken as cylinder ro-
tating a perpendicular axis. The polar moment of inertia can be calculated
from the general fromula J = r2
dm, where r is the instanteneous radius,
and dm is the differential mass. The formula for the polar moment of inertia
of a cylinder rotating about a perpendicular axis is
J =
πd4
lγ
32g
, (137)
where J is the polar moment of inertia, γ material density, d diameter of
cylinder, l is axial length of cylinder, and g acceleration due to gravity. The
torsional stiffness is (πd4
G)/(32l), where G is rigidity modulus, and substi-
tuting d2−d1 for d gives you formula for an annulus with outer-inner diameter
d2-d1. Taking the cryostat as a circular shaft (Figure 16) is made of material
of mass density ρ and shear modulus G and has a length L, cross-sectional
area A, and polar moment of inertia as above. Let x be the coordinate along
the axis of the shaft. The shaft is subject to a time-dependent torque per
unit lenght, T(x, t). Let θ(x, t) measure the resulting torsional oscillations
where θ is chosen positive clockwise. Figure 17 shows free-body diagrams
of a differential element of the shaft at an arbitrary instant of time. The
element is of infinitesimal thickness dx and its left face is a distance x from
the left end of the shaft.
The free-body diagram of the external forces shows the time-dependent
torque loading as well as the internal torques developed in the cross sections.
The internal resisting torques are the resultant moments of the shear stress
distributions. If Tr(x, t) is the resisting torque acting on the left face of the
element, then a Taylor seris expansion truncated after the linear terms gives:
Tr(x + dx, t) = Tr(x, t) +
δTr(x, t)
δt
dx. (138)
The directions of the torques shown on the free-body diagram are consis-
tent with the choise of θ positive clockwise. Since the disk is infinitesimal,
the angular acceleration is assumed constant across the thickness. Thus the
free-body diagram of the effective forces simply shows a moment equal to the
mass moment of inertia of the disk times its angular acceleration. Summation
of moments about the center of the disk
41

44. Figure 16: Circular shaft subject to torsional loading.
Figure 17: Free-body diagram of the differential element of shaft at arbitrary instant.
42

45. M
ext
= M
eff
(139)
gives
T(x, t)dx − Tr(x, t) + Tr(x, t) +
δTr(x, t)
δx
dx = ρJdx
δ2
θ(x, t)
δt2
(140)
or
T(x, t) +
δTr(x, t)
δx
= ρJ
δ2
θ
δt2
. (141)
From mechanics of materials,
Tr(x, t) = JG
δθ(x, t)
δx
(142)
which leads to
T(x, t) + JG
δ2
θ
δx2
= ρJ
δ2
δ
δt2
. (143)
Using the following to simplify x∗
= x/L, t∗
= G/ρ(t/L), and T∗
(x∗
, t∗
) =
T(x, t)/Tm, where Tm is the maximum value of T. From these
L2
Tm
JG
T(x, t) +
δ2
θ
δx2
=
δ2
θ
δt2
, (144)
where the ∗ has been dropped from nondimensional variables. The prob-
lem formulation is completed by specifying appropriate initial conditions of
the form
θ(x, 0) = g1(x) (145)
and
δθ(x, 0)
δt
= g2(x). (146)
43

46. Consider
δ2
θ
δx2
=
δ2
θ
δt2
. (147)
Let us look at some cases to analyse the cryostat with the above analysis.
Firstly let us make x the length of the cryostat a unilength so that free
end is x = 1 and taking the fixed end at x = 0. The boundary condition
is θ(0, t) = 0, and ˙θ(1, t) = 0 (the derivate is in terms of x). Applying a
moment M is statically applied to the end of the shaft leading to the initial
condition θ(x, 0) = Mx/(JG) = γx. Since the shaft is released from rest
a second initial condition is ˙θ(x, 0) = 0 (the derivative is in terms of t). A
separation of variables is assumed θ(x, t) = X(x)T(t), which gives
1
X(x)
d2
X
dx2
=
1
T(t)
d2
T
dt2
. (148)
leading to
d2
T
dt2
+ λT = 0, (149)
and
d2
X
dx2
+ λX = 0, (150)
where λ is the separation constant. The solution is
T(t) = A cos
√
λt + B sin
√
λt, (151)
where A and B are arbitrary constants of integration. Similarly
T(t) = C cos
√
λx + D sin
√
λx (152)
The initial conditions give C = 0, B = 0 and the only reasonable solution
λk = [(2k − 1)
π
2
]2
k = 1, 2, . . . (153)
44

47. Now infinity of solutions arise corresponding to
Xk(x) = Dk sin (2k − 1)
π
2
x (154)
for any Dk. The modes are orthogonal giving
(Xk(x), Xj(x)) =
1
0
DjDk sin (2k − 1)
π
2
x sin (2j − 1)
π
2
xdx = 0 (155)
for j = k, but when k = j we get
1 = (Xk, Xk) =
D2
k
2
(156)
leading to
θ(x, t) =
∞
k=1
√
2 sin (2k − 1)
π
2
x[Ak cos (2k − 1)
π
2
t]. (157)
From the initial conditions we get the last
Ak =
4γ
√
2(−1)k+1
π2(2k − 1)2
(158)
yielding in total
θ(x, t) =
8γ
π2
∞
k=1
(−1)k+1 1
(2k − 1)2
sin((2k − 1)
π
2
x) cos((2k − 1)
π
2
t). (159)
Let us now consider a circular shaft fixed at x = 0 and has a thin disk
of mass moment of inertia I, similar to the electronics above the cryosta,
attached at x = 1. The partial differential equation governing [28]
δθ(1, t)
δx
= −β
δ2
θ(1, t)
δt2
(160)
where β = I/(ρJL). Separation of variables give:
45

48. dX(1)
dx
= βλX(1). (161)
The solution is
X(x) = D sin
√
λx (162)
giving
tan
√
λ =
1
β
√
λ
. (163)
There are countable but infinite values of λ, and at large k, λk approaches
((k + 1)π)2
. Let λi and λj are distinct solutions with corresponding mode
shape Xi(x) and Xj(x) respectively. The mode shapes satisfy the boundary
conditions Xi(0) = 0, Xj(0) = 0, ˙Xi(1) = βλiXi(1), and ˙Xj(1) = βλjXj(1):
d2
Xi
dx2
+ λiXi = 0 (164)
and
d2
Xj
dx2
+ λjXj = 0. (165)
Multplying the first of these by Xj(x) and integrating from 0 to 1 gives
1
0
d2
Xi
dx2
Xjdx + λi
1
0
XiXjdx = 0 (166)
and integrating by parts leads to:
Xj(1)
dXi
dx
(1) − Xj(0)
dXi
dx
(0) −
1
0
dXi
dx
dXj
dx
dx + λi
1
0
XiXjdx = 0. (167)
so that
βλiXi(1)Xj(1) −
1
0
dXi
dx
dXj
dx
dx + λi
1
0
XjXidx = 0. (168)
46

49. Integrating the second equation, after the multiplication by Xi(x), from
0 to 1:
βλjXj(1)Xi(1) −
1
0
dXi
dx
dXj
dx
dx + λj
1
0
XiXjdx = 0 (169)
and subtracting the last two equations leads to
(λi − λj) βXi(1)Xj(1) +
1
0
XiXjdx = 0. (170)
This implies since λi = λj
βXi(1)Xj(1) +
1
0
XiXjdx = 0 (171)
and defining scalar product of g and f by
(f, g) =
1
0
f(x)g(x)dx + βf(1)g(1) (172)
then (Xj, Xk) = 0.
The mode shape is normalized
1 = (Xk, Xk) =
1
0
D2
k sin2
( λkx)dx + D2
kβ sin2
λk = D2
k
1
0
1
2
(1 − cos(2 λkx))dx + β sin2
λk =
Little manipulation produces
Dk =
√
2(1 + β sin2
√
λ)−1/2
(174)
where λk is the kth solution.
Let us see a forced vibration example. This is similar to the problems
with the motor and the belt rotating the cryostat. We use the above model
as the cryostat and subject the thin disk to harmonic torque,
T(t) = T0 sin ωt. (175)
47

50. The torsional oscillations, in terms of nondimensional variables, with
θ(0, t) = 0 are
δθ
δx
(1, t) = −β
δ2
θ
δt2
(1, t) +
T0L
JG
sin ˜ωt (176)
where ˜ω = L ρ/Gω. Since the external excitation is harmonic, the
steady state response is assumed as
θ(x, t) = u(x) sin ˜ωt. (177)
This leads to
d2
u
dx2
sin ˜ωt = −˜ω2
u sin ˜ωt (178)
or
d2
dx2
+ ˜ω2
u = 0. (179)
From the boundary conditions given in the previous example and u(0) = 0
leads to
du
dx
(1) − β˜ω2
u(1) =
T0L
JG
. (180)
The solution is
u(x) =
T0L
(˜ω cos ˜ω − β˜ω2 sin ˜ω)JG
sin ˜ωx. (181)
Note that if ˜ω is equal to any of the system’s natural frequencies, the
denominator vanishes. The assumed for the solution must be modified to
account for this resonance condition. The total solution is the steady-state
solution plus the homogeneous solution, which is a summation over all free-
vibration modes. Initial conditions can then be applied to determine the
constants in the linear combination.
48

51. 8 Balancing rotation
The unbalance of rotating machinery is the most common malfunction, even
so that any lateral vibrations are usually wrongly thought to be due to un-
balances. In our case the unbalance of the cryostat is obvious as the electric
devices, and pumping systems mounted are not symmetric, and there are
restrictions in placing them. Quite frequently, balancing procedures per-
formed on the machine, which another type of malfunction, worsens the
situation. These unbalances have been recognized for over 100 years. Bal-
ancing procedures are equally old. However during the last 25 years they
have experienced substantial improvements due to implementation of vibra-
tion measuring electronic instruments and application of computers for data
acquisition and processing. For over a century researchers have published
hundreds of papers on how to balance machines. For more advanced methods
one should consult [29]. The problem due to unbalance is easiest to identify
and correct. The unbalance causes vibrations and alternating or variable
stress in the cryostat it self and the supporting structure elements. These
vibrations are directly linked to heat leak, and thus again should be minized
as possible. The balancing problem is solved by either, relocating electronic
equipment or adding masses. After proper balancing, rotating vibrations
should be reduced in the entire range of rotational speeds, including the op-
timal operating speeds, as well as the resonance speed range. The latter is
especially important when the cryostat is operated in hihg speeds exceed-
ing the first, the second, or even higher natural modes of resocance. As the
unbalance force is proportional to the rotational frequency squared [30], the
unbalance-related grows considerably with increasing rotational speed. The
plane that is rigidly attached on the cryostat carrying the electronics can be
thought of as beign a symmetric rotor with the axis of rotation directly in
the middle. The unblance condition changes the rotor mass centerline not to
coincide with the axis of rotation. Unbalance is due to the restricted placing
of the electronic devices mounted on the plane. During rotation, the rotor
unbalance generates a centrifugal force perpendicular to the axis of rotation.
This force excites, rotor lateral vibration e.g. rotor fundamental response.
In the following presentation, the modal approach to the rotor system as a
mechanical structure, has been adopted. At the beginning, the first lateral
mode of the rotor is considered only. The lateral mode can either be rotor
bending mode or susceptibility mode. Conventionally fundamental the vibra-
tion response of the rotor at its lateral mode is due to the inertial centrifugal
exciting force, generated by unbalance. In the modal approach, limited to
the first lateral mode, the unbalance-retaled exciting force is discrete, i.e. an
average integral, lumped effect of the axially distributed unbalance in the
49

52. first mode. The average unbalance angular force location will be referred
to as a heavy spot. Rotors are usually similarly constrained in all lateral
directions. Therefore, they exhibit lateral vibrations in space, with two in-
separable components of motion at each specific axial section of the rotor.
These two components result in a two-dimensional orbiting motion of each
axial section. Typically, two displacement proximity transducers, mounted in
XY orthogonal configuration, will measure the lateral vibrations of the rotor
in one axial section plane. The isotropic rotor lateral synchronous motion,
as seen by the displacement transducers 90 degrees apart, will differ by 90
degrees phase angle. The rotor lateral vibrations can be observed on an oscil-
loscope in the time-base mode, and in orbital mode. The latter represents a
magnified image of the actual rotor centerline path in this section. Figure 18
illustrates the waveforms and an orbit of a slightly anisotropic rotor funda-
mental response. The angular position of the force and response vectors are
vital parameters for the balancing procedure. In practical applications, the
response phase is measured by the Keyphasor transducer ( [31], and [32]).
Keyphasor is a transducer generating a signal used in rotating for observing
a once-per-revolution event. A notch is made on the rotor, which during
rotor rotation causes the Keyphasor displacement transducer to produce an
output impulse, every time the Keyphasor notch passes under the trans-
ducer. The one-per-turn impulse signal is simultaneously received, together
with the signals from the rotor lateral displacement-observing transducers.
The Keyphasor signal is usually superimposed on the rotor lateral vibration
response time-base waveform presentation and on rotor orbits. On the os-
cilloscope display, the Keyphasor pulse is connected to the beam intensity
input (the z-axis of the oscilloscope; while the screen displays x and y axis).
The Keyphasor pulse causes modulation of the beam intensity, displaying
a bright dot, followed by a blank spot on the time-base and/or orbit plots.
The sequence bright/blank may vary for different oscilloscopes and for rotor
notch/projection routine, but is always consistent and constant for a partic-
ular oscilloscope and rotor configuration; this sequence should be checked on
rotor waveform time-base responses when the oscilloscope is first used.
The unbalance force at a constant rotational speed, Ω as seen in Figure
19 can be characterized in the following way. There’s a fixed relation to the
rotating system. The nature of the rotating period is strictly harmonic time-
base, expressed by sin Ωt, cos Ωt or eiΩt
, where t is time. When the frequency
is equal to the actual rotational speed the unbalance is rotating at the same
rate in sync with the rotor rotation. The force F is proportional to three
physical parameters namely: unbalance average, modal mass m, and square
of the rotational speed.
50

53. Figure 18: Rotor lateral motion measured by two displacement proximity transducers in
orthogonal orientation and the Keyphasor phase reference transducer.
51

54. Figure 19: Time-base waveforms of rotor response to unbalance inertia force. Note that
response lags the force by the phase difference.
52

55. F = mrΩ2
(182)
Force phase that is the angular orientation δ mesured in degrees or radians
from a reference angle zero marked on the rotor circumference. The unbal-
ance force causes rotor response in a form of two-dimensional orbital motion.
The harmonic time-base is expressed by a similar harmonic function as the
unbalance force and the frequency is equal to the actual rotational speed
Ω. Amplitude B is directly proportional to the amplitude of the unbalance
force, and F is inversely proportional to the rotor synchronous dynamic stiff-
ness [33]. Phase lag β represents the angle between the unbalance force vector
and response vector plus the original force phase, δ. The response always lags
the force, thus the phase moves in the direction opposite to rotation. Both
unbalance force and rotor response are characterized by the single frequency
equal to the frequency of the rotational motion. The vibrational signal read
by a pair of XY displacement transducers should, therefore, be filtered to
frequency Ω, or what is the same, to frequency describing synchronous fre-
quency of the rotor response as a multiple of one. There may exist other
frequency components in the rotor response. These possible components of
the vibrational signals are not directly useful for rotor balancing. A vector
filter can, for instance, be used for filtering of the measured signal to the first
component only. In the characterization of both the force and response, the
amplitude and phase were emphasized as two equally important parameters.
Using the complex number formalism, these two parameters can be lumped
into one the force vector and response vector correspondingly. The amplitude
will represent the length of the vector, the phase its angular orientation in
the polar plot, coordinate format. The unbalance force and rotor response
are therefore, described in a very simple way. Unbalance force is:
Fei(Ωt+δ)
= mrΩ2
ei(Ωt+δ)
(183)
and rotor fundamental response
RF = Bei(Ωt+β)
. (184)
The corresponding vectors are obtained when the periodic function of
time, eiΩt
is eliminated:
−→
F = Feiδ
= mrΩ2
ejδ
(185)
53

56. Figure 20: Unbalance force vector and rotor fundamental response vector in polar
coordinate format. Note conventional direction of response angle, β, lagging the force
vector in direction opposite to rotation.
and (Figure 20)
−→
RF =
−→
B = Beiβ
. (186)
The keyphasor transducer provides a very important measurement of the
rotor response phase. Since the keyphasor notch is attached to the rotor,
the keyphasor signal dot superimposed on the response waveform, represents
the meaningful angular reference system. A useful convention of coordinates
describes the angles. When the notch on the rotor is exactly under the
keyphasor transducer, the rotor section under the chosen lateral transducer
has the angle zero. In order to locate the heavy spot, looking from the driving
end of the rotor, rotate the rotor in the direction of the rotation by the angle
β. The heavy spot will then be found under the chosen lateral transducer.
This way there is no angular ambiguity, independently of the lateral probe
positions.
The rotor orbit displayed on the oscilloscope is a magnified picture of
the rotor centerline motion. The rotor fundamental response orbit, as can
be observed on the oscilloscope screen in orbital mode. Elliptical orbits are
54

57. due to anisotropy of the rotor support system, which is the most common
case in machinery. One Keyphasor dot on the orbit is at a constant posi-
tion, when the rotational speed is constant. It means that during its one
rotation cycle the rotor makes exactly one lateral vibration orbiting cycle.
Direction of orbiting is the same as direction of rotation called forward or-
biting. For a constant rotational speed, the orbit exhibits a stable shape
and the Keyphasor dot appears on the orbit at the same constant angular
position. The phase of the rotor fundamental response is often referred to
as the high spot. It corresponds to the location, on the rotor circumference,
which experiences the largest deflections and stretching deformations at a
specific rotational speed. Although just One-plane balancing does not have
many practical applications in machinery, it provides a meaningful general
scheme for balancing procedures. Basic equation for one plane balancing of
the rotor at any rotational speed is represented by the one mode isotropic
rotor relationship between input force vector, Feiδ
, rotor response vector,
Beiβ
and complex dynamic stiffness [34],
−→
k (Ω):
−→
k (Ω)Beiβ
= Feiδ
(187)
where
−→
k (Ω) = K − MΩ2
+ iDSΩ. (188)
The complex dynamic Stiffness represents a vector with the direct part,
kD = K − MΩ2
and quadrature part kQ = DΩ.
The rotor in Figure 18 is now defined more precisely it contains the rotor
Transfer Function [35], which is an inverse matrix of the Complex Dynamic
Stiffness,
−→
k . The inverse of the complex dynamic dtiffness is also known
receptance. The objective of balancing is to introduce to the rotor a corrective
weight of mass, mc, which would create the inertia centrifugal force vector
equal in magnitude and opposite in phase to the initial unbalance force vector.
This way, the rotor input theoretically becomes nullified and the vibrational
output results also as a zero. In practical balancing procedures, the input
vector force of the initial unbalance has therefore to be identified. Using
again the block diagram formalism, the one-plane balancing at a constant
rotational speed is illustrated in Figure 22.
Introduce the vectorial notation:
−→
F = Feiδ
and
−→
B = Beiβ
, for the
unbalance force vector and response respectively, as well as,
−→
H = 1/
−→
k for
the rotor transfer function vector. The original unbalance response at a
constant speed, Ω is:
55

58. Figure 21: Angular positions of unbalance force vector (heavy spot) and response vectors
at two rotational speeds and two directions of rotor rotation (a), (c), and (b), (d) and for
two locations of lateral transducers (a), (b), and (c), (d). Note that minus signs for the
angles are most often omitted and replaced by lag.
56

59. Figure 22: Balancing in one-plane.
−→
F
−→
H =
−→
B . (189)
In this relationship, there are two unknown vectors,
−→
H and
−→
F . The
response vector,
−→
B is measured thus its amplitude and phase are known. In
order to identify the initial unbalance force vector, it is sufficient to stop the
rotor and introduce a calibration weight of a known mass mτ at a known
radial rτ and angular δτ position into the balancing plane. When the rotor is
run again at the same constant speed, Ω, the mass mτ generates an additional
input force vector,
−→
F = mτ rτ Ω2
eiδτ
. This run is called a calibration run.
The measured rotor response vector is now
−→
B 1 = B1eiβ1
, which is different
from the response vector. For this second run, the following input/output
relationship holds true:
(
−→
F +
−→
F τ )
−→
H =
−→
B1. (190)
In the above the unknown vectors are
−→
H and
−→
F , and the others are
known. The last two equations are suffiecient to solve the one-place balancing
57

60. problem and calculate the unknown parameters. The unkonwn vector −
−→
F
and the corrective mass, mc are calculated, therefore, as follows:
−
−→
F = mcrcΩ2
eiδc
=
−→
F τ
−→
B
−→
B −
−→
B1
(191)
or
−
F
Ω2
= mcrceiδc
=
−→
F τ
−→
B
(
−→
B −
−→
B 1)Ω2
(192)
where rc and δc are radial and angular positions of the corrective weight
with mass mc. Note that the corrective weight is supposed to be inserted
at the same axial location on the rotor, as the calibration weight. Note also
that if the radii for the calibration and corrective weights are equal (rτ = rc)
and the original and calibration run measurements are taken at the same
rotational speed giving:
mceiδc
= mτ eiδτ
−→
B
−→
B −
−→
B1
. (193)
Finally, note that the balancing procedure does not require calculation of
the second unknown parameter, the rotor transfer function vector
−→
H provides
this vector as well:
−→
k =
1
−→
H
=
−→
F τ
−→
B 1 −
−→
B
. (194)
This synchronous dynamic stiffness vector, totally overlooked in balancing
procedures, represents a meaningful characteristic of the rotor. It should be
calculated, stored and reused, if balancing is required in the future. During
the next balancing, the old and new rotor dynamic stiffness vectors should be
compared. For a constant speed balancing, the synchronous dynamic stiffness
vector is often used in the form
−→
k /Ω2
and is known as the sensitivity vector.
The analytical solution for the corrective mass and its radius can be obtained
by splitting it into real and imaginary parts:
mcrceiδc
= mτ rτ eiδτ
Beiβ
Beiβ − B1eiβ1
= mτ rτ eiδτ
B
B − B1ei(β1−β)
(195)
58

61. from where
mcrc(cos δc + i sin δc) = mτ rτ (cos δτ + i sin δτ )
B
B − B1(cos (β1 − β) + i sin (β1 − β))
= mτ rτ (cos δτ + i
Now the real and imaginary parts are:
mcrc cos δc = mτ rτ B
(B − B1 cos (β − β1)) cos δτ − B1 sin (β − β1) sin δτ
B2 + B2
1 − 2BB1 cos (β − β1)
(197)
and
mcrc sin δc = mτ rτ B
(B − B1 cos (β − β1)) sin δτ + B1 sin (β − β1) cos δτ
B2 + B2
1 − BB1 cos (β − β1)
.(198)
From these we get:
mcrc = mτ rτ
B
B2 + B2
1 − 2BB1 cos (β − β1)
(199)
and
δc = δτ + arctan
B1 sin (β − β1)
B − B1 cos(β − β1)
. (200)
These represent the analytic result for the one plane balancing. This can
be done first choosing a rotational speed Ω for balancing. Next is to run the
rotor and measure its original synchronous response vector,
−→
B = Beiβ
, at the
rotational speed Ω. After this stop the rotor and choose a radial and angular
scale for plotting vectors. Draw the vector
−→
B = Beiβ
in the polar plot.
Now introduce a known calibration weight into the rotor at a convenient,
known axial, radial and angular position. Convenience consists in installing
the calibration weight in the rotor in the opposite half-plane to the original
unbalance. On the polar plot draw the corresponding calibration force vector
−→
F τ = Fτ eiβτ
= mτ rτ Ω2
eiδτ
. Run the rotor at the same speed Ω. Measure
the new rotor synchronous response vector
−→
B1 = B1eiβ1
and draw it in the
polar plot using the same scale, and then stop the rotor. Subtract vectorially
−→
B1 from
−→
B in the plot; draw a vector
−→
B −
−→
B1. Find the corrective weight
angular position as δc = δτ + θ. The angle θ is between the vectors
−→
B and
59

62. Figure 23: One plane balancing of a rotor using polar plot.
−→
B −
−→
B1. Since the response is proportional to the input force, the triangles
−→
B ,
−→
B1 ,
−→
B1 −
−→
B , and
−→
F ,
−→
F +
−→
F τ ,
−→
F τ are similar; they have
the same angles. Now measure on the plot the length of the vector
−→
B −
−→
B1
using the assigned scale. Next calculate the corrective mass, mc, applying
the formula:
mc = mτ
rτ B
rc
−→
B −
−→
B 1
. (201)
Finally introduce the correct weight with mass mc at the angle δc, and
radius rc to the same plane rotor as the calibration weight. This procedure
can be iterated when new equipment is added to the plane carrying the
electronics.
9 Studies of the Noise of Old Rota I Cryostat
As Rota I Cryostat was to be modified, and also partly rebuilt, rotational
characteristics were measured focusing on noise. The measurements were
60

63. done with all the cryostat’s electronics in place, and other set of measure-
ments were done when all the electronics were removed. The measurements
were done in the same way as previous vibrational noise measurements [36].
The measurements were done in the normal, and tangential directions from
the axis of rotation. The rotation speeds were 250, 1000, 2000, 3000 mrad/s.
The accelerometer used is a Bruel and Kjaer Accelerometer Type 4370 [37],
and Bruel and Kjaer preamplifier was used before collecting the data with
Agilent 54641 Oscilloscope [38]. The data aquired was done using TCL, and
transferred by GPIB.
The results were similar to previously done measurements [36]. The nois-
iest direction by far was the tangent of the rotation vector, this means that
the noise was rather torsional rather than back and forth swaying of the
cryostat. The 250 mrad/s speed is a problematic speed as it showed some
risen noise levels. This was propably due that some part of the cryostat
experiences its first harmonic.
In Figure 24. at 250 mrad/s some peaks in the fourier transform seemed
to be quite visible and large compared to the 1000 mrad/s case. Some of
the peaks have lowered and also disappeared at 1000 mrad/s, but the overall
noise seems to have increased at all frequencies. This is taken with all the
electronics on, and in the tangential direction. In the normal direction the
largest peaks are half to that of tangential and the overall noise is somewhat
lower at all frequencies.
As the speed was increased the noise-levels rose, but not all frequencies
seemed to be affected by this. In Figure 25. the noise-levels are additively
integrated over region up to 55 Hz at different speeds. In this tangential
noise-plot one is able to see the unbalanced increases at different frequencies.
The removal of electronics had an unexpected effect on the noiselevels.
Vibrational noise fell sharply in the measurements after the removal of the
electronics. This was probably due to the fact that load from the air bearings
was lifted, and thus removing some friction. This means that the air bearings
are experiencing their maximum load, and when the electonics are mounted
on the cryostat the friction rises.
The integration without the electronics reveals that the noise at different
speeds in the tangential directions does not vary as expected. Rather the
250 mrad/s case seems to dominate in this situation as can be seen in Figure
28. The moment of inertia has been changed, so the high noise-level at 250
mrad/s is not necessarily a first harmonic that the whole cryostat or the belt
of the rotating motor is experiencing. It could also be that motor is not
functioning perfectly at this speed.
61

64. 0 5 10 15 20 25 30 35 40 45 50 55
0
0.5
1
1.5
2
2.5
3
3.5
x 10
4
V(10
−3
)
Tangential Noise with Electronics at 250 mrad/s
0 5 10 15 20 25 30 35 40 45 50 55
0
1
2
3
4
x 10
4
Hz
V(10
−3
)
Tangential Noise with Electronics at 1000 mrad/s
Figure 24: Two fourier spectra show measured tangential noise at rotating speeds 250
and 1000 mrad/s.
62

65. 0 5 10 15 20 25 30 35 40 45 50
0
1
2
3
4
5
6
7
8
x 10
4
Hz
VHz(10
−3
)
Tangential Additive Noise Integral with Electronics
250
1000
2000
3000
Figure 25: The integrated additive plot shows the complexity of rising noise level at
different speeds in tangential direction.
63

66. 0 5 10 15 20 25 30 35 40 45 50
0
0.5
1
1.5
2
2.5
x 10
4
Hz
VHz(10
−3
)
Normal Noise Additive Integral Plot with Electronics
250
1000
2000
3000
Figure 26: The integrated additive plot shows the complexity of rising noiselevel at
different speeds in normal direction.
64

67. 0 5 10 15 20 25 30 35 40 45 50 55
0
2000
4000
6000
8000
V(10
−3
)
Tangential Noise without Electronics at 250 mrad/s
0 5 10 15 20 25 30 35 40 45 50 55
0
2000
4000
6000
8000
Hz
V(10
−3
)
Normal Noise without Electronics at 250 mrad/s
Figure 27: This plot shows how the normal and tangential directions differ taken at 250
mrad/s without electronics.
65

68. 0 5 10 15 20 25 30 35 40 45 50 55
0
1
2
3
4
x 10
4
V(10
−3
)
Tangential Noise With Electronics at 3000 mrad/s
0 5 10 15 20 25 30 35 40 45 50 55
0
1
2
3
4
x 10
4
Hz
V(10
−3
)
Tangential Noise Without Electronics at 3000 mrad/s
Figure 28: This plot shows how the tangential noise varies with and without the
electronics taken at 3000 mrad/s.
66

69. 0 5 10 15 20 25 30 35 40 45 50
0
2000
4000
6000
8000
10000
12000
14000
Hz
VHz(10
−3
)
Tangential Additive Noise Integration without Electronics
250
1000
2000
3000
Figure 28: Additive integration is taken without electronics in the tangential direction.
10 Superconducting high-homogeneity mag-
net for NMR measurements
Chemical analysis, and imaging of biological samples are commonly probed
using nuclear magnetic resenance methods. The samples in a magnetic field
are studied using continuous wave NMR causing the samples’s magnetic mo-
ments of the atomic nuclei to arrange in a fashion minimizing the magnetic
potential energy. The NMR frequency of the sample is changed according to
the dependent magnitude, thus the polarization field strenght is sweeped [39]
and the resulting spectrum measured gives information about the structure
Getting a good NMR signal requires the magnetic moments to be as uniform
as possible. A very homogenous field is thus a must, which requires that
the polarizing magnet is constructed with great care. The homogeneity is
defined as the relative deviation from the center point value B0:
|
∆B
B0
| = |
B − B0
B0
|. (202)
One must also take into account the space where the measurements are
to be made. They are in volume very constricted, and the sample as in our
67

70. case is helium is to be measured in a cylindrical cell. Solenoid magnet is
thus a very good choice for this experiment, and the field can be represented
in closed form ( [40], [41]). Increasing the coil length and optimizing the
diameter of the solenoid improves the homogeneity, which in general means
decreasing the diameter. The homogeneity is reduces when one moves axially
away from the center, and trying to reduce this short compensation coils at
both ends are placed. The most common of the compensation methods is
the sixth order end compensation [42], in which the correct choice of magnet
dimensions cancels the first five derivative terms in the expansion series of
the field. The homogeneity can also be improved using the Meissner effect,
which occurs as superconducting material is placed in a magnetic field. As
in the case of perfect diamgnet the superconductor sets up surface currents
cancelling the field within the material by opposing magnetization. The
solenoid is surrounded by superconducting material in order to force the
field lines to concentrate in center of the solenoid center [43] and also acts as
a shield for external unwanted fields. The magnet used for the measurements
was already provided [44]. The wire consist of quite fragile Super Con Inc. 18
filament NbTi inside CuNi alloy matrix [45]. As the measurements are done
notably under the critical 4.2 K temperature, there should be no problem of
proper thermalization. The dimensions of the magnet can be seen in Figure
29. The magnet was previously tested merged in liquid helium with maximun
current load of 12.5 A without quenching. The absolute value of the magnetic
field strength for the magnet was not accurately measured as the Hall probe
used by Vesa Lammela was designated to work at temperature range of −10
to 125o
C. The homogeneity of the magnet was therefore to be tested with
actual NMR-measurements.
11 Superconductivity
The phenomenon of superconductivity occurs in specific materials at ex-
tremely low temperatures. The characterization includes exactly zero elec-
trical resistance, and the Meissner effect [46]. The resistivity of the metallic
conductor gradually decreases as temperature is lowered, and drops abruptly
to zero when the material is cooled below the critical temperature. It is a
quantum mechanical phenomenon. The physical properties vary from mate-
rial to material, especially the heat capacity and the critical temperatures.
Regular conductors have a fluid of electrons moving across a heavy ionic
lattice, where the electrons constantly collide with the ions, thus dissipat-
ing phonon energy to the lattice converting into heat. The superconductor
situation differs in that the electronic fluid cannot be distinguished into in-
68

71. Figure 29: Number of turns on each layer, and the dimensions of the magnet.
dividual electrons, but it consists of Cooper pairs [47] caused by electrons
exchanging phonons. The Cooper pair fluid energy spectrum possesses an
energy gap amounting to a minimum energy ∆E that is needed to excite the
fluid. When ∆E is larger than the lattice thermal energy kT, there will be
no scattering by the lattice as the Cooper pari fluid is a superfluid without
phonon dissipation. The critical temperature Tc varies with the material,
and convetionally are in the range of 20K to less than 1K. The transition to
superconductivity is accompanied by changes in various physical properties.
In the normal regime the heat capacity is proportional to the temperature,
but at the transition it suffers a discontinous jump, and linearity is impaired.
At the low temperature range it varies as e−α/T
where α is some constant
varying with the material. The transition as indicated by experimental data
is of second-order, but the recent theretical improvements (which are ongo-
ing) show that within the type II regime transition is of second order and
within the type I regime first order, and the two regions are separated by a
tricritical point [48]. Considering the Gibbs free energy per unit volume g,
which is related to internal energy per unit volume u and entropy s:
g = u − Ts (203)
69

72. where the volume term has been neglected. The associated magnetic
energy in the presence of an applied magnetic field ¯Ba is − ¯M ·d ¯Ba, where ¯M
is the magnetic dipole moment per unit volume. A change in the free energy
is given by
dg = − ¯M · d ¯Ba − sdT. (204)
Integrating yields:
g(Ba, T) = g(0, T) −
Ba
0
¯M · d ¯Ba. (205)
For type-I-superconductor ¯M = − ¯H [50] due to the Meisness effect, and
thus the previous can be written as:
gS(Ba, T) = gS(0, T) +
Ba
0
BadBa
µ0
= gS(0, T) +
B2
a
2µ0
. (206)
We see B2
a/2µ0 as the extra magnetic energy stored in the field as resulting
from the exclusion from the superconductor. At the transition between the
superconductivity and normal state we have gS(Bc, T) = gN (0, T), where
Bc is the critical magnetic field and the magnetization of the normal state
has been neglected. The entropy difference ∆s < 0 between normal and
superconducting states can be obtained from this using s = −(δg)/δT. Now
the heat capacity per unit volume c = Tδs/δT is:
∆c =
T
µ0
Bc
d2
Bc
dT2
+
dBc
dT
2
, (207)
which shows that there is a discontinuous jump in ∆c even for Bc = 0.
As our superconducting filament behaves as convetional superconductiv-
ity the pairing can be explained by the microscopic BCS theory. The assump-
tion of the BSC theory [49] is from the assupmtion that electrons have some
attration between them overcoming the Coulomb repulsion. This attraction
in most materials is brought indirectly by the interaction between the elec-
trons and the vibrating crystal lattice as the opposite spins becoming paired.
As electron moves through a conductor nearby a positive lattice point causes
another electron with opposite spin to move into the region of higher positive
charge density held together by binding energy Eb. When Eb is higher than
70

73. phonon energy from oscillating atoms in the lattice, then the electron pair
will stick together, thus not experiencing resistance and describing an s-wave
superconducting state. Let us look more closely to the electron-phonon in-
teraction. Electron in a crystal with wavevector ¯k1 scatters to ¯k′
1 emitting a
phonon ¯q. Then this phonon is absorbed by a second electron ¯k2 to ¯k′
2 hence
the conservation of crystal momentum:
k1 + k2 = k′
1 + k′
2 = k0. (208)
States in the ¯k-space can interact, but are restricted by the Pauli exclusion
principle corresponding to electron energies between EF and EF + ωD, where
EF is the Fermi energy and ωD is the Debye frequency. The number of allowed
states occur when ¯k1 = −¯k2, and they are called the Cooper pairs [51]. The
transition temperature Tc is given by BSC model [52]
kBTc = 1.14 ωDe−1/V0g(EF )
, (209)
where V0 is the phonon-electron interaction strenght. In the BCS ground
state (T = 0), there is a binding energy 2∆ to the first allowed one-electron
state, in which the Cooper pairing is broken. The gap energy in conventional
superconductors are in the range of 0.2-3 meV, thus very much smaller than
EF . Next we will consider quenching mechanisms for the breaking up of the
superconducting state to normal state. It is exactly this that the Cooper
pairing is broken as some disturbance overcomes this binding energy.
12 Superconductor quenching
We will deal with the matter of quenching here, as it will be quite a impor-
tant factor affecting the work. Quenching occurs when a superconductive
filament goes to normal resistive state. There are three critical parameters
namely temperature, current density, and magnetic field affecting this behav-
ior. When one of these parameters’ critical value 1is exceeded by some phys-
ical process, superconductor becomes normal-conducting. When the cooling
power for the filament is not sufficient the zone of the normal conductor ex-
pands. Usually in a quench case the entire energy stored in the magnet is
dissipated as heat, even burning the filament. There are two main distur-
bances namaley transient and continuous, which can be again divided into
two more causes point and distributed. In the case of continuous disturbance
the steady power becomes a problem due to e.g. bad joint, or soldering. The
71

74. distributed disturbances are usually caused by heat leak to the cryogenic
environment. Now transient disturbances are sudden, and can be for exam-
ple caused by a breaking of a turn in a magnet due to excess Lorenz force
moving the turn by δ. In this case the work done by the magnetic field is
BJδ, where J is the current density, and this energy will heat the magnet to
normal state. Our filament is embedded in a copper matrix, which is a good
absorber and distributor of heat, thus if the copper gets good enough contact
where to dissipate the energy, the magnet can stay in superconducting state
even thought some problems persist. For example cooling of the filament by
helium is given by the adiabatic heat balance equation [53]:
ρcu(T(x, t))
I(t)2
Acu(x)
= c(T(x, t))A(x)
dT(x, t)
dt
, (210)
where ρcu is the resistivity of copper, Acu is the copper cross-section of
the composite, I is the current in the filament, T is the temperature of the
filament, and c is the heat capacity. Voltage V (t) is a function of time in the
magnet coil
V (t) = I(t)R(t) + L(I)
dI(t)
dt
−
i
Mi
dI
dt
− UP C, (211)
where I(t) is the current, R(t) is the resistance, L(I) is the self inductance,
Mi is the mutual induction of a neighboring turn or coil, and Upc is the voltage
of the power converter. Now with a good power converter, the quenching
voltage can be given:
VQt) = I(t)R(t) + LQ(t)
dI
dt
, (212)
where LQ is the partial inductance and R(t) is the resistance of the
quenching zone. Taking mutual inductance zero, we get [54]:
VQ(t) = I(t)R(t)(1 − LQ(t)/L). (213)
The quench zone thus expands as resistance and partial inductance grow.
A good power supply can detect this, and switches itself off.
72

75. 13 Nuclear Magnetic Resonance, and Imag-
ing
Quantum mechanical magnetic proterties of atom’s nucleus can be studied
with the nuclear magnetic resonance (NMR). The pehomenon was first dis-
covered by Isidor Rabi in 1938 [?]. Neutrons and protons have spin, and the
overall spin is determined by the spin quantum number I, and non-zero spin
associates with a non-zero magnetic moment µ by
µ = γI, (214)
where, γ, is the gyromagnetic ratio. Angular momentum quantization,
and orientation is also quantized. The associated quantum number is known
as the magnetic quantum number m, and can only have values from integral
steps of I to −I thus there are 2I + 1 angular momentum states. Taking the
z component Iz:
Iz = m . (215)
The z-component of the magnetic moment is
µz = γIz = mγ . (216)
¯I2
has eigenvalues I that are either integer or half integer. This can be
meaning
(Im µx′ Im′
) = γ (Im Ix′ Im′
), (217)
where µx′ and Ix′ are components of the operators ¯µ and ¯I along the
arbitrary x′
-direction. This is based on the Wigner-Eckart equation [56]. For
simplicity we consider system of two m states +1/2 and −1/2 by numbers N+
and N−, where the total number N of spins is constant. With the propability
for transition W the absorption of energy is given:
dE
dt
= N+W ω − NW ω = ωWn, (218)
where ω is the angular frequency of the time dependent interaction (or
the frequency of an alternating field driving the transitions), and n has to be
73

76. zero for a net absorption. Now in a similar fashion using a alternating field
we get for the absorption energy [57]
dE
dt
= n ωW = n0 ω
W
1 + 2WT1
(219)
where T1 is the spin-lattice relaxation time, and as long as 2WT1 ≪
1 we can increase the power absorbed by increasing the amplitude of the
alternating field. Taking the alternating magnetic field as Hx(t) = Hx0 cos ωt,
and it can be broken into two rotating components with amplitude H1 in
opposite directions. They are denoted by:
¯Hr = H1(¯i cos ωt + ¯j sin tωt) (220)
and
¯HL = H1(¯i cos ωt − ¯j sin ωt). (221)
We consider only ¯HR as ¯HL is just same with negative ω. Taking ωz as
component of ω along z-axis:
¯H1 = H1(¯i cos ωzt + ¯j sin ωzt). (222)
Now the equation of motion with the static field ¯H0 = ¯kH0 is
d¯µ
dt
= ¯µ × γ[ ¯H0 + ¯H1(t)]. (223)
Now moving to a coordinate system such that the system rotates about
the z-direction at frequency ωz then:
δ¯µ
δt
= ¯µ × [¯k(ωz + γH0) +¯iγH1]. (224)
Now near resonance ωz + γH0 ≃ and by setting ωz = −ω states that in
the rotating frame moment acts as though it experiences a static magnetic
field:
¯Heff = ¯k(H0 −
ω
γ
) + H1
¯i. (225)
74

77. The effective field exactly when the resonance condition is fullfilled the
effective field is ¯iH1, and a magnetic moment parallel to the static field will
then precess in the y − z plane. Turning H1 on for a wave train of duration
tw so that moment precesses through an agle θ = γH1tw = π so inverting the
moment. Now if θ = π/2 the magnetic moment is turned from z-direction to
the y-direction. Turning H1 off then would make the moment remain at rest
in the rotating frame. Simple method for observing magnetic resonance can
be done with these remarks. Putting a sample in a coil, the axis of which
is perpendicular to ¯H0. Alternating field applied to the coil produces an
alternating magnetic field. Adjusting tw and H1 we can apply a π/2 pulse,
after which the excess magnetization will be perpendicular to ¯H0 precessing
at angular frequency γH0. Now the moments make alternating flux through
the coil inducing emf that may be observed. Now variations of the similar
principle can be used to make measurements using NMR. The method in
our experiment is using continuous wave method. The receiver coils are set
perpendicular to the magnetic field from the magnet. The current is sweeped
in the magnet, and the resonance signal should pick-up in our coil. The
magnitude of nmr resonace signals is proportional to the molar concentration
of the sample [58].
In our experiment the magnet is placed within a vacuum, and in the
middle of the magnet one places the sample to be measured with the pick-up
coils in perpendicular direction to the magnetic field (Figure 30)
The resonance circuit in Figure 30 is tuned for both excitation and de-
tection. At first there was no cooled preamplifier in the setup. The lock-in
amplifier measured the signal from the pick-up coils, which was locked to the
frequency of the oscillator. The oscillator was used to drive the excitation
frequency of the circuit. The magnet was supposed to be swept across to
observe NMR spectrum from the 3
He sample inside the magnet. The test-
ing was supposed to be done for 25 turns per each pick-up coil, and then
50 turns. The pick-up coils were very carefully coiled on glass piece that fit
directly on the sample tube right in the middle. The sample was place in
the center of the magnet, and thus the magnetic field from the magnet and
the pick-up coils were perpendicular. The gyromagnetic ratio of 3
He is 32.43
MHz/T [?]
14 Cooling of the Cryostat
The main purpose in our measurement is to measure the homogeneity of
the superconducting magnet, by doing NMR measurements of normal liquid
3
He, which requires temperatures below 3.19 K. The experiment is done
75