INTRODUCTION
Acting as one of the core devices of fracturing equipment group, the function of sand blender was to mix, stir and deliver the high pressure fracturing fluid. The structure character of the sand blender was that the onboard devices including mixing tank, reciprocating pump and water tank were installed on the auxiliary beam while the auxiliary beam was fixed on the main beam of a heavy truck. During the operation of the sand blender, due to the periodic motion of the motors, reciprocating pump and mixing propeller, the auxiliary beam might develop a serious periodical vibration which had an adverse effect on normal operation of sand blender, moreover, the periodical vibration could result in microcracks in the auxiliary beam, putting great hidden troubles to fracturing work and endangering the safety of onboard devices. To avoid this situation, it is necessary to master the dynamic characteristics of the auxiliary beam for its possibly needed structure modification and for the proper setting of working parameters of sand blender.
At present, the study on the dynamic characteristics of the auxiliary beam
is seldom, but modal analysis provides us an effective way to study mechanical
dynamics (Zhu et al., 2009). There are two methods
of modal analysis: Computational modal analysis and experimental modal analysis.
The former one mainly adopts CAE software and finite element method to calculate
the structure modal parameters (Wang et al., 2008;
Eritenel and Parker, 2009; Kiracofe
and Parker, 2007) while the later one calculates the structure modal parameters
by indentifying input and output signals obtained by test devices (Guan
et al., 2005; Takatsu, 1991; Liu
et al., 2009). Comparing to computational modal analysis, experimental
modal analysis has higher accuracy, it has already widely used in the fields
of car, bridge and petroleum equipments, etc. In this study, based on related
modal test theory, the experimental modal test for the auxiliary beam of sand
blender was carried out by using uTekl dynamic signal analysis system, the testing
data were analyzed, the former six order natural frequencies and vibration modes
were obtained, then comparision between natural frequencies of the auxiliary
beam and working frequencies of onboard devices were made, the vibration mechanism
of the auxiliary was founded which could work as the basis of structure modification
of the auxiliary beam and proper setting of working parameters of the sand blender.
EXPERIMENTAL MODAL TEST OF AUXILIARY BEAM
Theorem of experimental modal test: The vibration of auxiliary beam
can be assumed as the movement of a linear elastic physical system with n degree
of freedom whose vibration differential equation is as follows (Fu
and Hua, 2000):
where, [M] is mass matrix, [C] is damping matrix, [K] is Rigidity matrix, [M],
[C], [K] are real square matrixes with n factorials; {x(t)} is displacement
vector
is velocity vector,
is acceleration vector {x(t),
and
are n dimensional vectors and {f(t)} is exciting force vector with n dimensions.
By applying Fourier transform to Eq. 1, we get following equation:
where, ω is natural frequencies of vibration system, i is unit of imaginary component; by setting H[ω] = [ω^{2}[M]+iω[C]+[K]]^{1} and placing it into Eq. 2, it is obtained that:
where, H[ω] is frequency response matrix.
If we excite the vibration system at point P and pick up the response signal
at point L, then the transfer function between point P and L can be expressed
as (Chen et al., 2007):
where, M_{i} is modal mass of order i, C_{i} is modal damping of order i, K_{i} is modal rigidity of order i, Φ_{Li}, Φ_{Pi} are vibration modes of order i at point L and P.
According to dynamic BettiRayleigh Theorem (Achenbach,
1984), the whole frequency response matrix can be obtained by testing just
one column or one row of the transfer function. In real test, the input force
signal as well as its response signal can be tested and identified by testing
system. Then the auto power spectrum of both force pulse signal and response
signal and the crosspower spectrum between the two signals can be calculated.
Thus we have the frequency response matrix as following:
where, G_{yy} is auto power spectrum of response signal, G_{xx} is auto power spectrum of force pulse signal.
Coherency function γ(ω) is used to judge the confidence of the test whose expression is:
where, Gxy is crosspower spectrum between input force signal and response signal. The closer the value of γ(ω) is to 1, the greater correlation between input signal and response signal will have which means the frequency response is more believable, so coherency function can be used to judge whether a certain exciting is reliable.
The method of averaging is often employed in order to reduce the influence caused by undesirable signal and increase the signaltonoise ratio. In this test practice, repeating exciting at intervals was used. To avoid the interference between two pre and post signals, the time interval between two excitings should be large enough.
Model establishing for modal test of auxiliary beam: Figure
1 was the 3D model of the auxiliary beam established in SolidWorks software.
As can be seen in Fig. 1, overall, the auxiliary beam was
welded by profile steel, forming a frame structure whose length, width and height
were respectively 8.77, 2.48 and 0.58 m.

Fig. 1: 
3D model of auxiliary beam 

Fig. 2: 
Nodeline model of the auxiliary beam 
The structure along the length was constructed by four sections with 2 rectangular
steel tubes on the center and 1 channel steels on each side. The upper structure
along the width consisted of 9 channel steels and rectangular steel tubes welded
on the structure along the length which was functioned as the platform for onboard
devices. The lower structure of along the width was formed by welding 7 steel
plates on the structure along the length which was used to connect the auxiliary
beam and the main beam of the truck. Also there were dozens of mounting bases
and lugs welded on the auxiliary beam.
The auxiliary beam was a continuous elastomer. When it vibrates under the external stimulation, displacement will occur between any two points. Thus as in finite element analysis, in experimental modal test, the continuous elastomer should be simplified as a structure in which some neighbouring nodes have geometric relationship. By connecting all neighbouring nodes with lines, the continuous elastomer was turned into a nodeline model.
Because harmful vibration was mainly along the height while the height was much less than the length and width, when establishing the nodeline model, the height size was neglected, the auxiliary was therefore simplified as a 2D model by changing the crossed lengthwidth structure into the intersected one. Also some small structures such as mounting bases and lugs were neglected. The completed nodeline model of the auxiliary beam was shown as Fig. 2 which consisted of 94 nodes. Here node 1 was not located on the entity of auxiliary beam, it was just a reference point used to define the coordinates.
Modal test method: As shown in Fig. 3, the modal testing system consisted of 1 signal acquisition unit, 1 multichannel charge amplifier, 1 impact hammer, 1 piezoelectric force sensor, 15 ICP piezoelectric acceleration sensors, data lines, notebook computer and modal analysis software.
Since the purpose of modal test was to study the resonant frequency of the auxiliary beam, a preinstalled auxiliary beam was used to carry out the modal test, on which all the onboard devices were mounted and the auxiliary beam was fixed on the main beam of a trailer.
The method of “single point pulsing and multiple points receiving”
was used to conduct the modal test. All test points on the auxiliary beam were
marked with numbers which are the same as those on the nodeline model. Based
on the observation of the structure of the auxiliary beam, the exciting point
was selected at the middle position on the rear of the auxiliary beam which
was located at the opposition of point 46. When collecting the data, the exciting
point was hammered vertically downward.

Fig. 3: 
Structure diagram of modal test system 

Fig. 4: 
Time history of exciting force signal of test point 76 
Totally there were 16 channels on the signal acquisition unit, one of which
was connected to the force hammer, thus at a time it was impossible to test
all 93 points, at most 15 points could be tested. For this reason, batch test
was employed to conduct the test, that was, the 93 test point were divided into
several groups, after one group was tested, moving acceleration sensors to next
group of test points, continue exciting and testing until response signals were
picked up from all test points. Among all 93 test points, acceleration sensors
could not be mounted on 14 points because of the block of onboard devices,
so they could not be tested directly with acceleration sensors, constraint equations
were used to obtain the vibration parameters of these points. The rest 79 test
points were tested in 7 batches, each exciting point was hammered 4 times. To
reduce the interference from external environment, the force window was used
to process the force signal; to improve the signaltonoise ratio, speed up
the attenuation of vibration signal and avoid the leak of frequency response
function, response signals were processed by rectangle window.
Data collection and analysis: By using the method introduced above,
the input force signals and response signals of all test points were collected.
For example, Fig. 47 were the curves of
time history of exciting force signal, time history of response signal, auto
power spectrum of response signal and coherency function between exciting force
signal and response signal of test point 76.

Fig. 5: 
Time history of response signal of test point 76 

Fig. 6: 
Auto power spectrum of response signal of test point 76 

Fig. 7: 
Coherency function between exciting force signal and response
signal of test point 76 
As can be seen in Fig. 7, in the range of 01000 Hz, the coherency function values of most points were larger than 0.9 while in the range of 0500 Hz, larger than 0.95 which strongly demonstrated that the test data was creditable.
Modal analysis was conducted after all points were tested. In this study, modal
calculation was on the basis of real modal theory, modal type was selected as
normal density, data fitting method was integral fitting.

Fig. 8(af): 
(a) 1st order modal shape of the auxiliary beam, (b) 2nd order
modal shape of the auxiliary beam, (c) 3rd order modal shape of the auxiliary
beam, (d) 4th order modal shape of the auxiliary beam, (e) 5th order modal
shape of the auxiliary beam and (f) 6th order modal shape of the auxiliary
beam 
Table 1: 
Condition of the experiment 

According to the peak values of ensemble average curve of frequency response
function, the initial estimated values of modal frequencies were selected and
fitted, then the completed modal parameters were obtained by integrated processing
of vibration modes, including processing of measuring direction, processing
of constraint equations and unitary processing of modal shape by maximum freedom
degrees. The first six order modal shapes of the auxiliary beam were shown in
Fig. 8. The related modal frequency and damping ratio were
listed in Table 1.
By analyzing the first 6 order vibration modal shapes of auxiliary beam shown
in Fig. 8 and Table 1, it could be found
that the first order modal shape was mainly bending vibration with natural frequency
was 12.45 Hz while the second to six order modal shapes were mainly torsional
vibration accompanied by local vibration with natural frequencies from 27.51
to 72.33 Hz.
Table 2: 
Working frequencies of truck mounted equipments 

The bending vibration frequency was smaller than the torsional vibration frequency,
that was because the length of the auxiliary beam was much larger than the width,
the rigidity along the length was smaller than that along the width.
From the damping ratio listed in Table 1, it was found that
the damping ratios of all first 6 order vibration modals were between 0.97~3.38%
which were small values and near to that of passenger car with similar frame
structure (Cai et al., 2007), thus the result
of experimental modal test was proved to be reliable.
The working frequencies of onboard devices were listed in Table
2. By contrasting the related data of Table 1 and 2,
we found the rotation speed of onboard engine, circulating centrifugal pump
and charge centrifugal pump was 2100 rpm, the corresponding working frequency
was 35 Hz which was near to the third modal frequency (34.22 Hz) of auxiliary
beam, also, the rotation speed of clean water centrifugal pump was 2900 rpm
with working frequency was 48.33 Hz which was near to the fourth modal frequency
(48.60) of auxiliary beam. Thus, during the operation of sand blender, the third
and fourth modals of the auxiliary beam would be excited and serious vibration
would be generated, therefore proper measures such as structural dynamic modification
of the auxiliary beam and adjustment of the working condition of the sand blender
and so on should be taken to avoid the vibration.
CONCLUSION
The structure of auxiliary beam of sand blender was analyzed in this study, the principles and method of modal experiment was explained, the experimental modal test for auxiliary beam was carried out, obtaining the first six order modal shapes and vibration frequencies, precisely demonstrating the dynamic characteristic of auxiliary beam.
The results of modal experiment showed that for the current structure of auxiliary beam and operating condition of sand blender, because the third and fourth modal frequencies were near to the working frequencies of onboard engine and clean water centrifugal pump, during the operation of the sand blender, the auxiliary beam would develop a serious vibration.
To avoid the serious structural vibration and possible fatigue failure, proper measures of structure optimization and operation parameters adjustment of truck mounted equipments should be taken. This study provided scientific references for such measures. Further work will focus on studying the method of structural dynamic modification and adjustment of working condition of the sand blender.