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2138
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Theoretical investigation of cylindrical
piezoceramic transducers′ vibrations (we suggested
the principles for the development of precision
piezoelectric motors with several degrees of freedom, thus realizing high precision in space and in
plane positioning systems)
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[1, 2]
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and dynamic
analysis of their components have indicated that an
increase in the loading force and initial tension
decreases the harmonic components of fluctuations.
This article analyzes the working cylindrical
piezoceramics transducers (Figure 1), when a control signal is sent only to one active control cylindrical transducer of the mechatronic system [3–5].
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2512
 Prefix

systems) [1, 2] and dynamic
analysis of their components have indicated that an
increase in the loading force and initial tension
decreases the harmonic components of fluctuations.
This article analyzes the working cylindrical
piezoceramics transducers (Figure 1), when a control signal is sent only to one active control cylindrical transducer of the mechatronic system
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[3–5]
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.
In this case, even deformation of the working part
surface in the operation area of the active cylindrical piezoceramics transducers′ vibrations is observed. Slight surface deformation in the operation
area of other active piezostack emerges due to
conditions of their fixing onto the surface of the
working part of the mechatronic system.
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3821
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Upon having illuminated the hologram after two
exposures with a copy of the cylindrical piezoceramic transducers vibrations, both transducers
reflected by the object surface before and after the
deformation, are restored at the same time
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[6–8]
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.
The results of their interference – the system of interferential fringes is observed against
the background of the surface of the object image, which provides information about changes
in the object’s status having occurred in the period between showings.
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4698
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The ineffective electrical energy is stored as electrostatic
energy in the piezoceramics material and reverts
it to the power supply in the final process of an
operating cycle. The analyzed criteria have made
possible to choose the piezomaterial for an optimal construction having a maximum displacement
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[9, 10]
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.
The experimental investigation of precision
vibrosystems by means of holographic interferometry enables one to obtain appreciably larger
amounts of information about the vibrating surface
in comparison with traditional methods.
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5308
 Prefix

The paper
deals with the consideration of methods for the
determination of the vibrational characteristics of
precision mechanical systems from the holographic interferograms of linked analysis of these characteristics by using numerical techniques based on
the theories of mechanical system vibration and
holographic interferometry
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[11, 12]
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.
Method of holographic visualization of cylindrical piezoceramic transducers vibrations
When wave properties are defined by a travelling wave, quantitative analysis of interferograms
shall be performed in the following method.
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8577
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The characteristic function of the distribution of
interference bands on the surface of the piezoceramic cylinder, when holographic interferogram is fixed at timeaverage and harmonic vibration is present, is defined as
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[12]
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:
.exp
1
0
kdt
T
Miix
(9)
Inserting expressions (7) into the equation
(6) we get the value of Ωi that should be inserted
into the characteristic function of band distribution (9).
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8982
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We obtain the following
sin),
(cos
2
exp
1
2
0
1
dt
i
T
Mix
(10)
where
()cos,
cos()cos
0
100
i
i
r
i
i
i
t
i
i
i
z
i
WK
UKVK
(11)
i
i
r
i
i
i
t
i
i
i
z
i
VKWK
UK
()sin()sin
sin
00
20
.
(12)
After the use of equation
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[4]
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equation (10)
will look like:
,2
1
2
2
2
01
MxiinJ (13)
where J0 – Bessel function of the first kind of
order zero.
Using (7), (11), (12) and (13) values of point
i located on the surface of piezoceramic cylinder,
we will obtain the following equation of the distribution of the interference bands on the surface
of the vibrating piezoceramic cy
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10772
 Prefix

FWij , FVij , FUij are calculated using analytical expressions of selfexcited vibration types related to
the geometrical shapes of transducers and the conditions of their fixing in various structures.
Aiming to calculate coefficients AjW, AjV, AjU
and angles αi, βi, γi we need to minimize the equation formed on a basis of the equation (14) in a
method described
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[12]
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:
222
11
1
2
1
11
4
sinsin
cossin
coscos
i
i
i
z
n
u
ij
u
ij
i
t
n
v
ij
v
j
i
i
r
n
w
ij
w
ij
i
z
n
u
ij
u
j
i
i
t
n
v
ij
v
ij
i
r
n
w
ij
w
ij
AFKAFK
AFKAFK
FAFKAFK
.(16)
Function Fi shall be differentiated with respect to each unknown component o
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