99
Chapter 7
7.
EXPERIMENTAL PROCEDURE
The study of the rotordynamic issue is currently carried out in experimental tests in which an additional precession circular motion of the impeller is imposed in order to study the effects. In real turbopumps, however, the whirling orbit actually followed by the center of the impeller is far from being a perfect circle.
From a practical point of view, if a circular whirl motion of the impeller is imposed, it is possible to correlate the forces measured by the dynamometer to the position and speed of the rotor along the orbit. Moreover it is also possible, at least theoretically, to completely determine the radial forces and the rotordynamic matrix force relative to the particular operating conditions. The enforcement of a circular orbit corresponds to the execution of a forced vibration experiment in a mechanical system. It follows that the obtained rotordynamic coefficients can also be used for a dynamic analysis of a more general system, aimed for example to the determination of the critical speeds and the onset speed of instability.
Alternatively, the approach of free vibration can be exploited, which consists in measuring the forces induced by the whirl motion directly on the turbomachine that is affected from the phenomena or on a scaled model. Without the imposition on the rotor’s motion, the approach of the free vibration provides more truthful informations whereas on the other hand it does not allow to keep under control the position of the rotor, so as to relate it to the corresponding values of the forces.
7.1.
BACKGROUND THEORY
Through the theoretical approach described in chapter 4 it is possible to correlate the rotordynamic forces and moments to the position of the rotor and the forces measured by the dynamometer in rotating reference frame. However a practical expression for the equations given in chapter 4.2 are needed. Recalling the expressions 4.15 and 4.16 for F0X and F0Y respectively:
0 0 0
0
0 0
1 cos( t) (t)cos( ) (t)sin( ) sin( t) (t)sin( ) (t)cos( ) T X x y x y F F F T F F dt = Ω Ω − Ω + − Ω Ω + Ω
∫
(7.1)100
0 0 0
0
0 0
1 cos( t) (t)sin( ) (t)cos( ) sin( t) (t)cos( ) (t)sin( ) T Y x y x y F F F T F F dt = Ω Ω + Ω + + Ω Ω − Ω
∫
(7.2)Figure 7.1 Schematic representation of the forces in rotating reference frame (x,y) with respect to the absolute reference frame (X,Y). L. Pecorari [1]
and from 4.17 and 4.18 the components of the rotordynamic force matrix are:
(
)
{
}
{
(
)
{
}
(
)
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 02 (t)cos( )cos( ) (t)sin( )cos( )
2 1 cos (t)cos( ) (t)sin( )
2
sin (t)sin( ) (t)cos( )
cos (t)cos( ) (t)sin(
T XX x y T x y x y x y A F t t F t t dt T t F F T t F F t F F ε ω ω ω ω ω ω ω ω ω ω ω ω = Ω + Ω + − Ω + Ω + = = Ω − Ω − − Ω − + − Ω − Ω − + Ω − + + Ω + Ω + −
∫
∫
{
}
(
)
{
}
}
0 0 0 0 0 0 )sin t Fx(t)sin( ) Fy(t)cos( ) dt
ω ω ω ω Ω + + − Ω + Ω + + Ω + (7.3)
(
)
{
}
{
(
)
{
}
(
)
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 02 (t)sin( )cos( ) (t)cos( )cos( )
2 1 cos (t)sin( ) (t)cos( )
2
sin (t)cos( ) (t)sin( )
cos (t)sin( ) (t)cos(
T YX x y T x y x y x y A F t t F t t dt T t F F T t F F t F F ε ω ω ω ω ω ω ω ω ω ω ω ω = Ω + Ω + + Ω + Ω + = = Ω − Ω − + Ω − + + Ω − Ω − − Ω − + + Ω + Ω + +
∫
∫
{
}
(
)
{
}
}
0 0 0 0 0 0 )sin t Fx(t)cos( ) Fy(t)sin( ) dt
ω
ω ω ω
Ω + +
+ Ω + Ω + − Ω +
101
(
)
{
}
{
(
)
{
}
(
)
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 02 (t)cos( )sin( ) (t)sin( )sin( )
2 1 cos (t)sin( ) (t)cos( )
2
sin (t)cos( ) (t)sin( )
cos (t)sin( ) (t)cos
T XY x y T x y x y x y A F t t F t t dt T t F F T t F F t F F ε ω ω ω ω ω ω ω ω ω ω ω ω = Ω + Ω + − Ω + Ω + = = − Ω − Ω − + Ω − + − Ω − Ω − − Ω − + + Ω + Ω + +
∫
∫
{
}
(
)
{
}
}
0 0 0 0 0 0 ( )sin t Fx(t)cos( ) Fy(t)sin( ) dt
ω ω ω ω Ω + + + Ω + Ω + − Ω + (7.5)
(
)
{
}
{
(
)
{
}
(
)
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 02 (t)sin( )sin( ) (t)cos( )sin( )
2 1 cos (t)cos( ) (t)sin( )
2
sin (t)sin( ) (t)cos( )
cos (t)cos( ) (t)sin(
T YY x y T x y x y x y A F t t F t t dt T t F F T t F F t F F ε ω ω ω ω ω ω ω ω ω ω ω ω = Ω + Ω + + Ω + Ω + = = Ω − Ω − − Ω − + − Ω − Ω − + Ω − + − Ω + Ω + −
∫
∫
{
}
(
)
{
}
}
0 0 0 0 0 0 )sin t Fx(t)sin( ) Fy(t)cos( ) dt
ω
ω ω ω
Ω + +
+ Ω + Ω + + Ω +
(7.6)
where the period T can be expressed as a multiple of the time needed to complete the fundamental cycle (Tc) that is the time at which the conditions return to be as at the initial time
cyc c
T N T= (7.7)
For Tc it is necessary that the period of rotations of the two motions satisfy the following condition
2 2
c J I
T = π = ωπ
Ω (7.8)
where I and J are integer numbers and therefore
I r
J
ω = =
Ω (7.9)
that corresponds to the whirl frequency ratio. Moreover, assuming the following notation
10 x 0 y 0 20 x 0 y 0 1 x 0 0 y 0 0 2 x 0 0 y 0 0 1 x 0 0 y 0 0 2 x 0 0 (t) (t)cos( ) (t)sin( ) (t) (t)sin( ) (t)cos( ) (t) (t)cos( ) (t)sin( ) (t) (t)sin( ) (t)cos( ) (t) (t)cos( ) (t)sin( ) (t) (t)sin( ) F F F F F F F F F F F F F F F F F ω ω ω ω ω ω ω − − + + = Ω − Ω = Ω + Ω = Ω − − Ω − = Ω − + Ω − = Ω + − Ω + = Ω + +Fy(t)cos(Ω +0 ω0) (7.10)
102
(
)
(
)
(
)
(
)
(
)
(
)
0 10 20 0 10 20 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 J J X C S J J Y S C J I J I J I J I XX C S C S J I J I J I J I XY S C S C J I J I J I J I YX S C S C J I J I J I J I YY C S C S F F F F F F A F F F F A F F F F A F F F F A F F F F ε ε ε ε − − + + − − + + − − + + − − + + − − + + − − + + − − + + − − + + = − = + = − + − = − − + + = + + + = − − + (7.11)where for example 𝐹2+𝑆𝐽+𝐼 is the Fourier coefficient with frequency J+I times the fundamental frequency whereas the S or C stands for sine or cosine component respectively.
From equations 7.11, the values of FN and FT can be obtained through equations 4.11 and 4.12
reported here for simplicity
(
)
(
)
1 2 1 2 N XX YY T XY YX F A A F A A ε ε = + = − + (7.12)The two components form the unsteady hydrodynamic force (rotordynamic force) that arise as a consequence of the presence of an eccentricity (Figure 7.2).
103
7.2.
INITIAL ANGLES
The angles at initial time, Ω0 and ω0, acts an important role in the evaluation of the forces. These
angles are defined through the data obtained from the contrast sensors that are placed at the primary and secondary shafts (see Figure 5.32).
The angle Ω0 is the angle between the x axis of the rotating reference frame (that coincides with the
dynamometer’s reference frame) and the X axis of the absolute reference frame. It can be determined through the knowledge of the contrast sensor position which has a counterclockwise angular distance of θ from the X axis (see Figure 7.3). Hence the evaluation of this angle can be done through the following expression
0 α0 ϑ 90
Ω = + − ° (7.13)
where α0 is the angular distance between the contrast sensor and the y axis at initial time. However,
the position of the y axis is considered as coincident with the notch even if this is not true due to the fact that wherever will be placed with respect to the axis, the contrast sensor will be taken as unknown and determined before the tests through a specific offset test. The reason of this approach is that it is impossible to place the notch at the exact position of the y axis because the dynamometer is not visible and even if this was true an error could occur, affecting the results of the experimental campaign in a significant way. In the next sections it will be shown the offset test with which it is possible to define the angle θ and the values needed for the calibration of the dynamometer.
Moreover, α0 is known through the speed of rotation Ω (supposed constant) and the elapsed time
between the initial time and the first passage of the notch.
Figure 7.3 Fundamental angles for main shaft.
The angle ω0 is the angle between the eccentricity vector and the X axis of the absolute reference
frame (Figure 7.4). The problem of its evaluation is simple to solve as soon as the secondary notch is supposed to be placed at the eccentricity vector. In real situation the position of the notch with respect to the eccentricity vector is unknown but it does not affect the measurement since that once the contrast sensor and the notch are fixed, the important information is the angular distance between
104
them, β. The contrast sensor position, δ, is then taken as unknown and it will change anytime the secondary notch is moved or the eccentricity of the experiment is changed. The evaluation of the angle δ is therefore obtained from a specific test, at the same time of when the eccentricity vector is found (see next sections).
Figure 7.4 Fundamental angles for whirl motion.
7.3.
SINUSOIDAL SIGNALS
In experiments, the errors in measurements may alter a signal that is sinusoidal leading to a situation as in Figure 7.5.
To avoid these errors an ideal signal can be found from the method of least squares which is an approximate solution of overdetermined systems. The solution will therefore minimize the sum of the squares of the errors made in any of the equations. If the signal (like a force) can be approximated by a sinusoid, it can be represented as follows:
( )= cos( + )+ = sin( )+ cos( )+
x i xm i i i
F t F ft ϕ c a⋅ ft b⋅ ft c (7.14) where ti = iΔt is the discrete time and f is its frequency. The signal is therefore defined by means of the
three parameters:
• The mean amplitude Fxm;
• The phase φ that is the angle between the x axis of the dynamometer and the eccentricity vector;
• The offset c;
For the fitting of the experimental data, the ideal model can be compared to real results and the mean squared error can be therefore obtained from the following equation
[
]
2[
]
2 2 1 1 1 N ( ) 1 N sin( )+ cos( )+ x i xi i i xi i i F t F a ft b ft c F N N σ = = =∑
− =∑
⋅ ⋅ − (7.15)105
Figure 7.5 Sinusoidal force during offset test.
Finding the minimum value of σ2 corresponds to impose a zero value to its first derivatives. Since
it depends from a,b and c, a linear system of three equations can be set:
[
]
[
]
2 1 2 1 1 1 1 2 1 1sin( )+ cos( )+ sin( )=
sin ( ) sin( ) cos( )+ sin( ) sin( ) = 0 sin( )+ cos( )+ cos( )=
sin( )cos( ) cos N i i xi i i N N N N i i i i xi i i i i i N i i xi i i N i i i a ft b ft c F ft a a ft b ft ft c ft F ft a ft b ft c F ft b a ft ft b σ σ = = = = = = = ∂ ⋅ ⋅ − ∂ = + − ∂ ⋅ ⋅ − ∂ = +
∑
∑
∑
∑
∑
∑
∑
[
]
2 1 1 1 2 1 1 1 1 ( )+ cos( ) cos( ) = 0 sin( )+ cos( )+ = sin( ) cos( )+ = 0 N N N i i xi i i i i N i i xi i N N N i i xi i i i ft c ft F ft a ft b ft c F c a ft b ft cN F σ = = = = = = = − ∂ ⋅ ⋅ − ∂ = + −∑
∑
∑
∑
∑
∑
∑
(7.16)Once the values of a,b and c are found, the parameters that define the ideal signal are given from: 2 2 arctan xm F a b a b c c ϕ = + = − = (7.17)
106
7.4.
EXPERIMENTAL PROCEDURE
The experimental campaign is based on a series of tests performed on different impeller configurations as described in chapter 6. All the tests consist in the acquisition of the data obtained from a turbopump rotating at a speed of 1750 rpm with a whirl frequency ratio within a range of -0.7 and +0.7.
Fundamental parameters as contrast sensor angles (chapter 7.2), eccentricity vector and offset values for the calibration of the dynamometer are found during special tests at the beginning of each series.
For the evaluation of the other parameters as the efficiency and the rotordynamic forces and moments, two type of tests have been performed: discrete and continuous. The former consists in a series of experiments where the whirl frequency ratio is varied discretely in a range of -0.7 and +0.7 with a gap of 0.2 between each test (clearly ω/Ω = 0 is performed only for cavitating and noncavitating performance). Some tests were performed with a gap of 0.1 only as a proof of the good results of the continuous tests.
The continuous test is performed to have a better understanding of the particular transitions that arise at certain values of whirl frequency ratio. It consists in two separated tests where ω/Ω varies from 0 to -0.7 and from 0 to +0.7. This is done by means of a constant acceleration of 0.2 round/s2 of
the whirl motor imposed by the control panel of the DACS2000.
The imposed nominal eccentricity is of 2 mm but the value obtained from the kinematic mechanism shown in chapter 5.19.1 is different and therefore an eccentricity test is needed to find its real value whenever the eccentricity is changed.
The experimental campaign provided also the comparison between hot and cold tests that have been performed at 70 °C and 20 °C respectively.
Another parameter that needs to be set before the tests is the cavitation number. Three tests have been performed
• Noncavitating test • Slightly cavitating test • Highly cavitating test
Any series is therefore determined by [in brackets the number of values are indicated] • the cavitation number [3]
• the flow coefficient [3] • the temperature [2]
• the whirl frequency ratio [up to 14 in discrete tests] • the eccentricity [1]
Buoyancy forces, tare forces, centrifugal forces and weight forces have been found from tests in water and in air. The latter are needed to calibrate the forces so the results take into account the hydrodynamic forces only.
In particular, the buoyancy force is obtained as “weight difference” by means of the measurement of the force along the Y axis of absolute reference frame during a test performed at 2 rpm (with whirl motor at rest) in water where the corresponding value in a test in air is subtracted:
0 0 w a w a buoyancy p p buoyancy Y Y F =F −F → F =F −F (7.18) where w stands for water and a stands for air.
107
The resulting radial and rotordynamic forces are therefore given by subtracting the values obtained in air tests from the corresponding values obtained in water whereas for the radial force along Y the buoyancy has to be taken into account as follows:
0 0 0 0 0 0 w a X X X w a Y Y Y buoyancy w a XX XX XX w a XY XY XY w a YX YX YX w a YY YY YY F F F F F F F A A A A A A A A A A A A = − = − − = − = − = − = − (7.19)
Similar considerations can be done for rotordynamic and radial moments but in this case the buoyancy force produce a moment along X axis that can be obtained as a difference between buoyancy moment in water and in air
0w 0a buoyancy X X M =M −M (7.20) 0 0 0 0 0 0 w a X X X buoyancy w a Y Y Y w a XX XX XX w a XY XY XY w a YX YX YX w a YY YY YY M M M M M M M B B B B B B B B B B B B = − − = − = − = − = − = − (7.21)
Finally the normal and tangential forces are found from equations 7.12. The general procedure is shown in Table 7.1.
Many of the parameters that are set as input during the tests are measured from transducers. Indeed a feedback is necessary to obtain the real value of rotational speed of the main motor from the evaluation of the data obtained from the contrast sensor. This is also done for inlet pressure whereas for the volumetric flow rate and the temperature a manual detection of data is needed.
All the data are recorded at a frequency of 5000 samples per second and for the calculations of the rotordynamic forces and moments a value of Ncyc = 250 has been used.
108
Test Ω [rpm] ω [rpm] Time [s] samples Total Output Eccentricity 0 1000 5 25000 Eccentricity εAngle δ
1
Residual
eccentricity 1000 0 5 25000 Residual eccentricity ε2 Air offset 2 0 120 600000
Angle θa
Mass m
Air offset vector (Fxa, Fya, Fza, Mxa, Mya, Mza)
F0Ya M0Xa
Discrete air test 1750
±1225 ±1050 ±875 ±700 ±525 ±350 ±175 120 600000 F0Xa, F0Ya AXXa, AXYa, AYXa, AYYa
M0Xa, M0Ya BXXa, BXYa, BYXa, BYYa
Continuous air
test 1750
0 / -1225
0 / +1225 120 600000 Same as discrete air tests
Water offset 2 0 120 600000
Angle θw
Water offset vector (Fxw, Fyw, Fzw, Mxw, Myw, Mzw)
F0Yw M0Xw
Motor electric torque
Discrete water test 1750 ±1225 ±1050 ±875 ±700 ±525 ±350 ±175 120 600000 F0Xw, F0Yw AXXw, AXYw, AYXw, AYYw
M0Xw, M0Yw BXXw, BXYw, BYXw, BYYw
Mzw
Continuous
water test 1750
0 / -1225
0 / +1225 120 600000 Same as discrete water tests
Table 7.1 Experimental procedure with tests performed with CPRTF facility.
7.5.
OFFSET TEST
The offset test is the first step and it is performed before any set of experiment whenever the operative condition changes. The aim of it is to:
1. calibrate the dynamometer removing the permanent deformations that are not re-absorbed when the payload is removed.
2. Evaluate the mass of the pump.
3. Measure the angular position θ of the main contrast sensor with respect to the X axis. The test consists in a rotating main motor at a speed of 2 rpm whereas the whirl motor is at rest with an acquisition time of 120 seconds and an acquisition frequency of 5000 samples per second. The
109
low rotating speed is intended to avoid centrifugal forces that thus can be neglected. The result is that the only force that is measured is the weight force as shown in Figure 7.6.
Figure 7.6 Forces during offset test.
Since the dynamometer rotates at 2 rpm and the only force is the weight force, the signal measured along x and y will be sinusoidal. Hence the mean values of all the forces and moments obtained from the dynamometer will be the mean components that are present in the bridges as a result of permanent deformations and are here referred to as offset values:
The weight of the pump can be evaluated as the maximum of the sinusoidal signal from which the offset value is subtracted:
max max ( ) ( ) 1 2 x x y y F c F c m g − + − = (7.22)
where the mean value is used to reduce the errors.
Another parameter that it is possible to obtain is the angle θ between the main contrast sensor and the X axis. There are different ways to find it and one of them can be schematically indicated as:
• Find the angle α between the secondary notch at initial time and the main contrast sensor; • Find the angle Ω0 from the knowledge of the instant when the force evaluated along y axis
is at its minimum and thus when the y and Y axis are parallel; • Find the angle θ by means of equation 7.13.
This test is repeated whenever the operative conditions like temperature or cavitation number are changed or if the elapsed time between one set of experiments and another is sufficiently large. This is done to obtain results with a high accuracy since any error in this test can affect all the results that are evaluated in the next phases.
From the offset in air and in water it is possible to find the buoyancy force exploiting the relations 7.18 and 7.20.
110
7.6.
ECCENTRICITY TEST
The eccentricity test is performed in air at the beginning of the experimental campaign whenever the pump is replaced or the secondary notch is moved. The aim of this test is to measure the eccentricity imposed by means of the kinematic mechanism.
It consists in the evaluation of the centrifugal force in a test that lasts 5 seconds with an acquisition frequency of 5000 samples per second and where the whirl motor operates at 1000 rpm whereas the main motor is at rest. In these conditions the only forces measured by the dynamometer, which does not rotate, are the weight force and the centrifugal force (see Figure 7.7):
(
)
P C
F F= +F =mg m+ ω ω ε× ×
(7.23) Since the dynamometer does not rotate and the weight force is a constant force, the centrifugal force is the magnitude of the sinusoidal force observed for Fx or Fy (Figure 7.8) from which the mean
value has to be subtracted. To reduce the errors the mean value of the resulting magnitudes can be evaluated and the eccentricity is given by
2 C M mF ε ω = (7.24)
Moreover the angular distance δ between the X axis and the secondary contrast sensor can be obtained from the first maximum of the force modulus. Indeed, when the weight force and the centrifugal force are parallel with the same direction, the eccentricity vector is vertical with an angle of 270° with respect to the X axis. Hence from the first maximum of the force modulus
2
max M
F =mg m+
ε ω
(7.25) It is possible to obtain δ from the knowledge of the first passage of the secondary notch from the contrast sensor.
111
Figure 7.8 Force measured by the dynamometer along y axis during eccentricity test of VAMPIRE.
7.7.
RESIDUAL ECCENTRICITY TEST
Although the theory would limit the evaluation of the eccentricity to the previous test, it has been observed that a residual eccentricity vector can be present as a consequence of tolerances or mounting errors that lead to the misalignment of the center of mass (Figure 7.9). Therefore another test, which lasts 5 seconds with an acquisition frequency of 5000 samples per second, is needed, in which the whirl motor is at rest and the main motor rotates at 1000 rpm. In these conditions, if a misalignment is present, a centrifugal force will arise and the relation between them will be similar to equation 7.24. Hence: 2 C R mF ε = Ω (7.26)
Since the secondary motor is at rest, and the dynamometer rotates at 1000 rpm like the main motor, the weight force is a sinusoidal signal whereas the centrifugal force is a constant that can be determined from the modulus of the measured constant vector.
Once the residual eccentricity is determined, the eccentricity is given by the subtraction of εR from
εM.
M R
ε ε= −ε (7.27)
The experimental results for the eccentricity in the three turbopump configurations are • DAPROT3: 1.063 mm
• VAMPIRE: 1.130 mm • VAMPDAP: 1.160 mm
112
Figure 7.9 Residual eccentricity vector.
7.8.
DISCRETE PROCEDURE
Once the offset and the eccentricity tests are performed, the discrete experiments can begin. They consist in a set of tests where all the parameters are fixed except the whirl frequency ratio that is varied discretely in the range -0.7 / +0.7 between each single test by means of the secondary motor controller (DACS2000). Each of these experiments are performed with a total acquisition time of 120 seconds and the record of the data starts when the desired volumetric flow rate and inlet pressure are reached.
The acquisition frequency is 5000 samples per second and the main motor rotates at 1750 rpm for all the tests of the experimental campaign.
In this case both of the motors are running when the acquisition starts and since their velocity is constant, the initial angle can be easily obtained from the knowledge of the elapsed time between the initial time and the first passage of the notch (main or secondary).
For the evaluation of the results, the exact theory described in chapter 7.1 is exploited and as previously seen in chapter 7.4, tests in water and in air are performed in order to apply the experimental procedure shown by means of equations 7.19 and 7.21. In case of air, the set is made by a group of 14 experiments whereas in case of water any set is composed by 8 or 14 tests depending on the decided gap between the whirl frequency ratios.
At the end of each test the resulting parameters are saved into a text file: • Nondimensional radial forces F0X, F0Y
• Nondimensional rotordynamic force 𝐹����⃗, and normal and tangential components F𝑅 N, FT
• Nondimensional rotordynamic force matrix [A] • Nondimensional radial moments M0X, M0Y
• Nondimensional normal and tangential moments MN, MT
• Nondimensional rotordynamic moment matrix [B] • Hydraulic efficiency ηM
113 • Actual flow coefficient Φ
• Actual cavitation number σ • Head coefficient Ψ
• Total pressure rise across the pump Δptot
• Temperature T • Eccentricity ε
The results of air tests will be saved on different files for any whirl frequency ratio whereas for water tests a single file will contains the results of the same set for identical operating conditions and different whirl frequency ratios.
In Figure 7.10 it is possible to observe the nondimensional normal force resulting from a set of water tests performed on the DAPROT3 with flow coefficient of 0.065, cavitation number of 0.976 and water temperature of 20 °C.
It is important to note that in the air tests the results of the normal force are parabolic due to the centrifugal force. This result can also be used as a countercheck for the correct execution of the test.
Figure 7.10 Nondimensional normal force for a single set of discrete experiments for DAPROT3 with σN = 1.015, Φ = 0.065, and T = 20 °C.
7.9.
CONTINUOUS PROCEDURE
Another testing procedure, capable to attain the continuous spectrum of the rotordynamic force has been developed at Alta S.p.A.. A continuous spectrum can be obtained from a test in which ω/Ω is varied in time by means of a constant acceleration of the whirl motor, allowing a better understanding of the actual trend of the parameters with respect to the whirl frequency ratio. This method allows also to identify the minima and maxima and to obtain all the values in two separated experiments, one for
114
The procedure applied to obtain the continuous spectrum consists in the exploit of the rotordynamic forces in the rotating frame. Indeed the forces in the absolute reference frame fixed with the laboratory and the forces in the rotating frame are related to each other from the following:
0 0 0 0 ( ) ( )cos( ) ( )sin( ) ( ) ( )sin( ) ( )cos( ) X x y Y x y F t F t t F t t F t F t t F t t = Ω + Ω − Ω + Ω = Ω + Ω + Ω + Ω (7.28)
where the angular position of the rotating axis x with respect to the stationary axis X at certain instant t is given by Ωt+Ω0=Ω�. From the combination of the equation 7.30 with the definition of the
instantaneous resulting force, the following equations can be found: 0
0
( )cos ( )sin cos sin
( )sin ( )cos cos sin
x y X XX XY x y Y YX YY F t F t F A A F t F t F A A ε ω ε ω ε ω ε ω Ω − Ω = + + Ω + Ω = + + (7.29)
where 𝜔𝑡 + 𝜔0= 𝜔� is the angle at instant t between the eccentricity vector and the absolute axis X. From the previous equations the general expression for the force measured by the dynamometer in the rotating reference frame is
0 0
0 0
( ) sin cos sin sin sin
cos cos cos sin cos
( ) cos cos cos sin cos
[ sin cos sin sin sin ]
x Y YX YY X XX XY y Y YX YY X XX XY F t F A A F A A F t F A A F A A ε ω ε ω ε ω ε ω ε ω ε ω ε ω ε ω = Ω + Ω + Ω + + Ω + Ω + Ω = Ω + Ω + Ω + − Ω + Ω + Ω (7.30)
At this point the analysis is the same of the rotordynamic theory shown in chapter 4. From equation 7.29 it is possible to observe that there is a linear relationship between the lateral force in the rotating reference frame (Fx,Fy), the radial force in the stationary reference frame (F0X,F0Y) and the
rotordynamic matrix [A]. The same equation 7.30 can be expressed in a matrix form as follow:
D M R= ⋅ (7.31)
0 0
cos sin
sin cos
cos cos cos sin
; ;
sin cos sin sin
cos sin cos cos sin sin sin cos X Y x XX T y XY YX YY F F F A D F R M A A A ε ω ε ω ε ω ε ω ε ω ε ω ε ω ε ω Ω − Ω Ω Ω Ω − Ω = = = Ω − Ω Ω Ω Ω Ω (7.32)
This new approach allows to consider the forces measured by the dynamometer as a set of data that can be predicted from a linear model. The vector R represents the radial and rotordynamic forces and it can be obtained as a result of a least-squares procedure where the experimental data, Fx and Fy, are
exploited. Since R depends, in general, on the whirl frequency ratio, the reduction procedure can be applied only if in the time interval in which the least-squares method is applied the whirl frequency ratio is considered constant or if the vector R can be considered constant in the range of ω/Ω considered in the data sample.
The tests have been performed at a constant rotational speed of the main motor and the angular position of the main shaft Ω� is given by the knowledge of the initial angle exploiting the same method of the discrete tests.
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Conversely, the whirl speed has been controlled imposing a linear chirp. Hence a constant acceleration of the secondary motor, 𝜔�̈, is set. The acceleration phase starts after a certain amount of time to allow a correct evaluation of the initial angle ω0 and it ends before the end of the test allowing
a constant velocity final phase. The three phases performed within the test can be indicated as follows: 0 1 1 1 1 2 0 ( - ) fin t t t t t t t t t t ω ω ω ≤ ≤ = < ≤ > (7.33)
where 𝜔�̇𝑓𝑖𝑛 is the final velocity of the whirl motor, t1 is the time at which the whirl motor starts
running and t2 is the time at which the acceleration is set to zero. The dependence in time of the
angular position and the velocity, are respectively shown in Figure 7.11 and Figure 7.12. From the two diagrams the parabolic trend of the angular position and the linear trend of the velocity can be observed when t1<t<t2.
Figure 7.11 Parabolic trend of the angular position as a function of the time.
The data reduction procedure of continuous test is very sensitive to the absolute position of the eccentricity vector and therefore the most important parameters are the time at which the ramp starts and ends since a small angle error may affect the results in a significant way. The integration of the time-evolution of the whirl speed yields to the following relations for the eccentricity anomaly:
0 0 1 2 1 2 2 2 2 2 2 ( ) fin( ) t t t At Bt C t t t At Bt C t t t t ω ω ω ≤ ≤ = + + < ≤ + + + − > (7.34)
where 𝜔�0 is the anomaly at initial time, which is constant for t ≤ t1. From the relation of the anomaly
of the eccentricity when the acceleration, 𝜔�̈, is different from zero (t1 < t ≤ t2), the following
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Figure 7.12 Linear trend of the whirl rotation speed as a function of time.
2 2 2
0 12 (t t1) 12 t t t1 0 12 t1
ω ω = + ω − → ω= ω −ω +ω + ω (7.35) Hence the generic constants of the parabola A, B, and C, can be specified as follows:
2
1 0 1
1 ; ; 1
2 2
A= ω B= −ωt C=ω + ω t (7.36) Finally, the time t2 can be obtained imposing the coincidence of the slope for the ending linear
phase and the parabolic curve:
2 2 2 fin B At B t A ω ω= + → = − (7.37)
The time at which the chirp ends, t2, can be found from the previous expression but the last sample
considered for the evaluation of the parameters of the parabola is different and it will be lower. The maximum error of the contrast sensor is given by the following:
2 W err sps ω = Ω (7.38)
where W is the rotational speed of the main engine in rad/s and sps stands for “sample per seconds”. Hence the maximum velocity error is given by:
max err2
err ωW
π
∆ =
Ω (7.39)
Subtracting this value to the final hypothetical whirl speed, the final sample point can be found. This procedure is needed due to the fact that during the final phase, the whirl speed oscillates between the theoretical operating speed due to the experimental errors and thus the point has to be lower to provide a safe distance from the constant-velocity phase. This provides a higher accuracy in the evaluation of A, B, and C. Moreover, multiplying Δerrmax by a safe factor and adding it to the final
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velocity, the actual value of whirl speed in the third phase can be obtained. The initial sample at which the parabola starts is clearly the time at which the secondary notch passes from the contrast sensor for the first time.
The experimental data have been recorded with a sample rate of 5000 sps for an acquisition time of 120 seconds. The rotational speed of main engine has been set to 1750 rpm. The acceleration imposed to the secondary engine in the central phase has been fixed to 0.2 round/s2 and due to the restrictive
conditions on the R vector, the number of samples for the application of the reduction procedure has been chosen as equal to 5000 samples, which corresponds to 1 second. Hence the six parameters contained in R have been considered constant in the range of whirl frequency ratio equal to 0.0069 which is an acceptable quantity. The reduction procedure is applied during the central phase and for each reduction, the parameters are associated to the mean whirl frequency ratio of the sample data. As discrete tests, the data reduction procedure is applied in air and water tests, and the experimental procedure shown in detail in chapter 7.4 is exploited to obtain the fluid induced forces at a given operating condition. In Figure 7.13 the experimental curves obtained for the DAPROT3 inducer at 20 °C, noncavitating condition, and Φ = ΦD = 0.065, are presented.
Figure 7.13 Results for the nondimensional normal force in continuous experiments for DAPROT3 with σN = 1.015, Φ = 0.065, and T = 20 °C.
The reduction procedure produces unreliable values at very low whirl speeds due to the deviation from the actual value of the initial angle found from the least square method and due to fact that the whirl speed at the begin is so slow that in the first data samples the shaft does not complete a whirl orbit in less than a second. As a consequence of this, the experimental curve has been truncated for ω ≈ 0. This is done comparing the first whirl frequency obtained between air and water tests, and eliminating the values of whirl frequency ratios that are not in common. This procedure allows to avoid the results obtained in normal procedure as shown in Figure 7.14.
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Figure 7.14 Experimental curve without elimination of the low whirl speed results.
7.10. TEST MATRICES
A set of tests can be defined as an array of experiments where the only parameter that is changed is the whirl frequency ratio. Hence a set is composed by 8 discrete tests (or 14 in case of tests performed with a gap of whirl frequency ratio of r = 0.1) and 2 continuous tests (with their offset and eccentricity tests). Once the other fundamental variables (eccentricity, main motor speed of rotation) are selected, the parameters that can be varied between a set and another are: the flow coefficient, the cavitation number, and the temperature. Any combination between them have been performed. In Table 7.2, the imposed values are reported for the three performed configurations.
Hence, in the three cases, the tests are performed:
• in hot and cold environment (70 °C and 20 °C respectively),
• in presence of noncavitating, moderate, and highly cavitating conditions,
• at three different values of flow coefficient which include the design condition and two flow rates, one higher and one lower than the design.
It can be noted that the three configurations present different values of flow coefficient but its variation is in such a way that the volumetric flow rate remains the same. Indeed the inducer and the radial pump are designed to operate at the same volumetric flow rate. Moreover the geometry for the evaluation of the flow coefficient (see equation 2.9), in VAMPDAP pump takes into account the impeller of the VAMPIRE and thus the two pumps present the same values. Since the volumetric flow rate and the angular speed are fixed in the three configurations, the area and the tip radius will define the flow coefficient used in the experimental campaign. The correspondence between nominal flow coefficient parameter and the experimental values measured by the flowmeter placed on the discharge line is reported in Table 7.3.
119 DAPROT3 Φ σN T [°C] 0.052 (0.8ΦD) 1.015 20 70 0.143 20 70 0.088 20 70 0.065 (ΦD) 1.015 20 70 0.143 20 70 0.088 20 70 0.078 (1.2ΦD) 1.015 20 70 0.143 20 70 0.088 20 70 VAMPIRE Φ σN T [°C] 0.074 (0.8ΦD) 0.604 20 70 0.110 20 70 0.080 20 70 0.092 (ΦD) 0.604 20 70 0.110 20 70 0.080 20 70 0.111 (1.2ΦD) 0.604 20 70 0.110 20 70 0.080 20 70 VAMPDAP Φ σN T [°C] 0.074 (0.8ΦD) 0.604 20 70 0.081 20 70 0.053 20 70 0.092 (ΦD) 0.604 20 70 0.081 20 70 0.053 20 70 0.111 (1.2ΦD) 0.604 20 70 0.081 20 70 0.053 20 70 Table 7.2 Test matrices of DAPROT3 (left), VAMPIRE (center), and VAMPDAP (right) configurations.
Φ 𝑸̇ [lt/s] DAPROT3 0.052 15.93 0.065 19.95 0.078 23.90 VAMPIRE 0.074 15.62 0.092 19.65 0.111 23.55 VAMPDAP 0.074 15.62 0.092 19.65 0.111 23.55
Table 7.3 Correspondence between nominal flow coefficient and experimental volumetric flow rate in liters per second measured by the flowmeter for the three pump configurations.
7.11. TEST WITH TURBOPUMP TEST
For the noncavitating pumping performance tests, a different offset test is necessary. Indeed the torque needed for the correct evaluation of the hydraulic efficiency of the turbopump, has to be calibrated with the permanent component of the moment registered from the dynamometer. The offset test previously explained in detail in chapter 7.5 is performed with a 2 rpm speed of the main engine and therefore the torque if affected by this rotation.
Hence this test consists in an acquisition time of 5 seconds during which the turbopumpdoes not rotate. The parameters obtained are the pressure difference measured by the differential pressure transducer (which may be different from zero)
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7.12. TRANSDUCERS CONFIGURATION
For rotordynamic tests, the same configuration of performance tests, explained in detail for DAPROT3 inducer in chapter 6.1.1, for transducers and instruments have been exploited. Hence the CPRTF configuration and the position of the pressure transducers exploited for the tests performed in the present experimental campaign are shown in Figure 7.15. For what concern the VAMPIRE and VAMPDAP pumps, the CPRTF has been modified by replacing the differential pressure transducer of 1 bar FS by another transducer with 5 bar FS.
The acquisition system (whether for cavitating or noncavitating performance, or for whirling tests) record the data in a text file into a matrix with 16 columns and n rows, where n indicates the total number of acquisitions. As shown in chapter 7.4, for rotordynamic tests a total of 120 seconds for 5000 acquisitions per second have been performed and therefore the number of rows is equal to 600000, whereas for noncavitating performance tests a 5 seconds for 5000 acquisitions per seconds have been used, thus reducing the number of rows to 25000. The columns of the resulting matrix recorded in this kind of configuration will be as shown in Table 7.4.
Figure 7.15 CPRTF configuration and pressure transducers position for DAPROT3 inducer.
Column Information recorded
1 Main notch
2 Secondary Notch
3 Recessed inlet pressure (1 bar FS) 4 Inlet pressure (2 bar FS)
5-14 Tension measured by Wheatstone bridges 15 Torque (of engine, measured in Volts) 16 Differential pressure (1 bar FS) Table 7.4 Columns order of the matrix saved into the file results during the tests.
For the inlet pressure, it has been taken into account the measurement made by the recessed pressure transducer, with the aim of avoiding inlet prerotation. The negative side of this selection is
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that the differential pressure, which exploit the same tap of the recessed pressure transducer, includes the losses between the pressure tap and the blade leading edges. Hence a ceratain amount of losses are present even if they can be considered as negligible. Moreover, the presence of the divergent duct needed for the size matching between pipes and test chamber increases the vorticity of the inlet prerotation and may enlarge the phenomena to a greater distance (vortex stretching) with respect to the presence of a duct with constant cross section. Also in this case the effect of these head losses are considered as negligible.
Since the pressure transducers are placed at the floor level, a pressure correction is needed to take into account the additional pressure measured due to the height difference between pipes and floor which is approximately equal to 60 cm. This means that the pressure difference is about 0.06x105 Pa.
The correction is applied to the inlet pressure transducer only, since the differential pressure transducer make the difference between the two taps, which are both at the same height.
7.13. REFERENCES
[1] L. Pecorari, Studio delle prestazioni cavitanti e delle forze rotodinamiche su induttori per uso spaziale, Tesi di Laurea in Ingegneria Aerospaziale, Università degli studi di Pisa, 2009.
[2] Pasini, A., Torre, L., Cervone, A., d’Agostino, L., 2011, Continuous Spectrum of the Rotordynamic Forces on a Four Bladed Inducer, ASME Journal of Fluids Engineering, Vol. 133, Is. 12.
