Method and device to investigate the behavior of large rotors under continuously adjustable foundation stiffness

2.1. Device for adjustable foundation stiffness

In this design, the rotor was mounted on a horizontally free and vertically and axially rigid bed, illustrated in Fig. 1. This was achieved by supporting the rotor bed cradle with plate springs mounted to the rest of the body from the other end. Thus, the cradle was suspended, and buckling could be prevented. The plate springs enabled almost free horizontal movement while holding the rotor rigidly in respect to the axial and vertical directions. However, as the horizontal movement of the cradle increases, gravity exerts a larger effect. Nevertheless, the movement of the cradle was small, and thus the effect of gravity could be considered negligible.

Fig. 1Horizontally, the rotor bed is supported in a very flexible manner. The cradle hangs on thin plates, which provide support mainly in the vertical and axial directions

The horizontally free rotor bed could be converted to offer adjustable stiffness by attaching an external device to control the horizontal stiffness of the cradle (rotor bed). This external device is presented in Fig. 2. The principle of the device is simple: the position of the beam support (horizontal stiffness adjuster, HSA) can be varied, thus changing the stiffness of the beam end.

Fig. 2Adjustable stiffness device: a) the appearance of the device and b) the supporting structure inside the device. The beam, which determines the horizontal stiffness of the cradle, is colored blue

a)

b)

To enable continuous, in-process stiffness adjustment, the HSA position was controlled with a ball screw and a servo motor. The HSA included several different parts, including the frame, bearings, axles and supports. It was tightened into one stiff structure with screws, which eliminated clearances from the system. Finally, the external device was added to the horizontally free rotor bed. The cradle was directly mounted to the end of the beam with a connection bar, and thus the stiffness of the system was determined by the external device. The complete device with sensors is presented in Fig. 3.

Fig. 3The rotor bed with horizontally adjustable stiffness. The length of the beam can be changed by moving the HSA device along the beam. In addition, the positions of the force and acceleration sensors can be seen

2.2. Stiffness characterization of the system

2.2.1. Analytical stiffness characterization

The stiffness of the device in a horizontal direction can be determined with an analytical solution. Bedford and Liechti [19] have extensively presented the effects of loads on beam deflection and have demonstrated different strategies for solving various beam configurations. Their calculations are based on Euler-Bernoulli beam theory, which assumes beam deflections to be small. The theory is a simplification of the linear theory of elasticity, which has been widely used in the field of engineering. The theory derives a static relationship between the distributed load $q$ and the deflection $v$, which can be now exploited to solve the stiffness of the device:

1

$\frac{d^{4}v\left(x\right)}{{dx}^4}=\frac{q}{EI},$

where $E$ is the elastic modulus and $I$ is the second moment of area. Both variables are in this case constant. Because in the equation the deflection is in the form of the fourth derivative, the final deflection of the beam can be solved by integrals:

2

$\frac{d^{3}v\left(x\right)}{{dx}^3}=\frac{Q}{EI},$

3

$\frac{d^{2}v\left(x\right)}{{dx}^2}=\frac{M}{EI},$

4

$\frac{dv\left(x\right)}{dx}=\theta ,$

5

$v\left(x\right)=\delta ,$

where $Q$ is shear force, $M$ is moment, $\theta$ is the slope of a beam and $\delta$ is the deflection of a beam.

Frequently, a tabulated expression is available for the required beam configuration. However, in this case, the beam deflection and stiffness must be derived using the Euler-Bernoulli equations presented above. Before the equations can be applied, the configuration of the beam must be determined. The free body diagram in Fig. 4 illustrates the forces, their relations and distances in a device where the beam is supported rigidly from its lower end, the movement in $y$ direction is supported by intermediate beam support (HSA) and the other end is connected to the rotor foundation. The connection to the rotor foundation can be now modeled with force $F$.

Fig. 4The structure of the device converted into a free body diagram: a) original beam and its support, b) free body diagram, c) free body diagram with point loads in relation to force F

a)

b)

c)

As the free body diagram indicates, only point loads are acting in the system. In this study, the point loads are derived for the shear forces of the beam, and then the shear forces can be directly applied with the Eq. (2). The shear forces form two different sections on the beam; thus, two equations must be created according to these sections:

6

$EI\frac{d^{3}v\left(x\right)}{{dx}^3}=-F\left(\frac{L-R}{R}\right),\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\left(0<x<R\right),$

7

$EI\frac{d^{3}v\left(x\right)}{{dx}^3}=F,\left(R<x<L\right).$

The final equation for the deflection can now be solved by integrating these equations three times:

8

$EIv\left(x\right)=-\frac{F\left(L-R\right)x^3}{6R}+\frac{C_1{x}^2}{2}+C_{2}x+C_3,\left(0<x<R\right),$

9

$EIv\left(x\right)=\frac{F{x}^3}{6}+\frac{C_4{x}^2}{2}+C_{5}x+C_6,\left(R<x<L\right).$

The integration constants can be solved using boundary conditions:

10

$v\left(x\right)=\frac{F{x}^2\left(L-R\right)\left(R-x\right)}{6REI},\left(0<x<R\right),$

11

$v\left(x\right)=\frac{F\left(R-x\right)\left(Rx-R^2+2LR+x^2-3Lx\right)}{6EI},\left(R<x<L\right).$

Finally, the stiffness of the beam can be solved when the applied force and the deflection of the beam end is known:

12

$\mathrm{S}\mathrm{t}\mathrm{i}\mathrm{f}\mathrm{f}\mathrm{n}\mathrm{e}\mathrm{s}\mathrm{s}=\frac{v\left(L\right)}{F}.$

By substituting the structure and material parameters from the actual device with Eq. (11), the deflection of the beam end can be calculated as a function of the intermediate support position ($R$). This value can be divided by the applied force $F$ as in Eq. (12), which leads to the developed stiffness of the beam [N/m]. The parameters used in the calculations are presented in Table 1. The force used can be arbitrarily decided when solving the stiffness. The other parameters are determined by the structure and materials of the device.

Table 1Structure and material parameters used in the calculations

Parameter	Deflection position $x$ [mm]	Force $F$ [N]	Elastic modulus $E$ [GPa]	Second moment of area $I$ [mm4]	Length of the beam $L$ [mm]
Value	491.75 (=$L$)	2000	210	202667	491.75

2.2.2. Simulated stiffness characterization

The stiffness of the device can also be determined with a simulation utilizing the Finite Element Method (FEM). This is achieved by loading the beam as in the analytical solution and then observing the beam end movement in relation to the force used. Only the essential parts of the structure were left for static FEM analysis that virtually meant the analysis of the beam. Siemens NX was used for the modelling and analysis.

Fig. 5Simulation process. On the left-hand side there is meshed beam with constraints and force. The beam end has all 6 DoF (Degrees of Freedom) constrained and different HSA positions have only the horizontal one. On the right-hand side there is a result of the first loading case in which the force F was applied, and HSA was in position 0 mm. The result describes the absolute displacement of the beam end in mm. h= 85 mm and H= 385 mm

The beam was first divided into suitable sections to enable correct positioning of the intermediate support and hence also analysis of the corresponding stiffness range. Then a 3D element meshes with a maximum element size of 5 mm was created according to the actual beam. The static FEM analysis also demanded the determination of the forces and constraints of the model. The beam was constrained rigidly from the lower end and laterally from the HSA positions. The movement of the HSA in the simulation corresponded the HSA movement in the actual device. The Fig. 5 illustrates the simulation process and the used HSA positions. The beam was loaded from the free end with a force of 2000 N. The parameters used in the simulation are presented in Table 2.

Table 2Structure and material parameters used in the simulation

Parameter	Element size [mm]	Deflection position $x$ [mm]	Force $F$ [N]	Elastic modulus $E$ [GPa]	Second moment of area $I$ [mm4]	Length of the beam $L$ [mm]
Value	5	491.75 ($=L$)	2000	206.94	Not a constant	491.75

2.2.3. Experimentally measured stiffness characterization

The stiffness of the rotor-mounting cradle in a horizontal direction could be determined by measuring the displacement of the cradle under a known force. Because the stiffness of the complete system must be known, the measurement setup was arranged in the following way. The cradle was connected to the stiffness-adjusting beam, which was displaced with a screw through a force sensor. The displacement of the cradle and the force that developed the displacement were measured. The measurement setup is presented in Fig. 6. A non-linear change in stiffness was expected during the movement of the HSA; thus, it was necessary to repeat the measurement several times at different points so that the full stiffness range could be properly determined. The utilized sensors and their specifications are introduced in Table 3.

Table 3Specifications of sensors

Measured unit	Sensor	Resolution	Repeatability
Force	HBM S9M	1 N (accuracy class 0.02)	1 N (accuracy class 0.02)
Displacement	Sylvac S229	1 μm	2 μm

Fig. 6Measurement setup for the stiffness characterization. The displacement (force) on the beam was produced with a screw through the force sensor and cradle. The displacement of the cradle was measured with the dial gauge. The blue color indicates the parts that acted with the force between the sensor and the beam

The experimental measurement was conducted as follows:

1) The horizontal stiffness adjuster (HSA) was set its highest position.

2) A force of 250 N was applied to the system.

3) 100 corresponding displacement and force samples were taken.

4) The force was increased by increments of 250 N.

5) Steps 3 and 4 were repeated until the force was 2500 N.

6) The HSA was moved downwards in increments of 30 mm.

7) Steps 2 to 6 were repeated until the full movement (300 mm) was reached and measured.

Thus, the stiffness of the system was measured in 11 different HSA locations. The measurement described above was performed for both devices at both ends of the rotor. The averaged results of the measurement points were employed to solve the stiffness of the devices.

2.3. Subcritical rotor behavior measurement

The device presented in the previous sections was mounted at the both ends of the test rotor. Changes in subcritical vibration behavior were monitored when the horizontal stiffness of the foundation was decreased. Monitoring was conducted by measuring the response (center point movement) at the middle cross-section of the rotor, acceleration in the bearing housings, and radial bearing forces. The complete test setup is presented in Fig. 7.

Fig. 7Complete test setup. The accelerometers and force sensors were placed at the rotor ends, and displacement measurement was performed with laser sensors attached to the yellow arc

2.3.1. Test setup

The test setup was constructed on a rigid grinding machine bed, with the test rotor fixed to the device. The test rotor was a 735 kg paper machine roll, and its main dimensions are presented in Fig. 8. The rotor speed and position of the HSA were controlled with a CNC. The test rotor was connected to the motor drive through a gearbox with a universal joint.

Fig. 8Main dimensions of the test rotor. All dimensions are in mm

Fig. 3 shows the measurement points of the acceleration and force sensors and Fig. 7 the displacement measurement. The specifications of the sensors are presented in Table 4. The accelerometers were attached to the bearing housings, and only the vertical and horizontal directions were measured. Measurement of the radial bearing forces was performed by integrating the force sensors into the device. The principle of radial bearing force measurement is straightforward and has been presented in a previous study [18]. In the present study, the center point movement was measured by exploiting the four-point method developed by Kuosmanen and Väänänen [20] and further investigated by Viitala et al. [21]. The method is a combination of two different methods: the Ozono three-point method [22], developed for roundness measurements, and the two-point method, which is a straightforward method to measure the diameter variation of the rotor. The method was applied with four reflective laser sensors that were arranged around the rotor at certain angles. The four-point method enabled the separation of the center point movement and the roundness of the rotor in the dynamic measurement.

Table 4Sensors used in rotor behavior measurement

Measured unit	Sensor	Sensitivity	Range
Acceleration	Brüel & Kjær type 4381	10.0 pC/ms-2	0.1 to 4800 Hz
Force (horizontal)	Kistler 901A	–4.3 pC/N	0 to 7.5 kN
Force (vertical)	HBM PaceLine CFW	–4.3 pC/N	0 to 100 kN
Displacement	Matsusita NAIS LM 300	1 V/mm	27 to 33 mm

3.1. Beam stiffness characterization

Beam stiffness was determined with analytical, simulation and experimental methods. The analytically determined deflection and stiffness of the beam end can be seen below in Fig. 9 as a function of the HSA position. In the calculations, a force of 2000 N was used, which is the same force as in the measurements. The results were obtained from the range of 0-491.75 mm, which corresponds to the total length of the beam. The HSA can move in a range of 106.75-406.75 mm.

Fig. 9Analytic results for the deflection and stiffness of the beam. 0 mm corresponds to the upper end of the beam

As can be observed from the stiffness diagram, theoretically stiffness increases towards infinity when the rigid beam support approaches the beam upper end (position 0 mm). However, in reality, the rigid support at the beam end and HSA (Fig. 4) are not completely rigid, which limits the stiffness.

The beam end deflection and stiffness derived by the simulation are presented below in Fig. 10. The results were obtained from 11 different positions from a range of 106.75-406.75 mm measured from the beam upper end in which the HSA was able to move.

The results of the experimental measurement are presented in Fig. 11. The stiffness was measured with the complete assembly, in which the force was conducted through the cradle, instead of measuring only the stiffness of the beam. This is likely to be a more accurate method for describing the real stiffness of the existing system. The interpretation of the results is straightforward except for the slope that can be seen in the tending-end results. The slope is a result of the small clearance between the frame and the cradle; the cradle collided with its frame when using high loads. However, this was not a problem when the device was used in rotor measurements, since excessive forces were not produced. Moreover, the problem could be eliminated by increasing the clearance between the frame and the cradle. The results indicate that the clearance between the cradle and the frame was approximately 0.8 mm at the tending end when the collision occurred. At the driving end, the maximal movement of the cradle was approximately 1 mm when a 2000 N force was applied.

Fig. 10Simulated results for the deflection and stiffness of the beam. HSA position 0 mm corresponds to the upper position of the HSA and 300 mm to the lower position of the HSA

Fig. 11The horizontal displacement of the cradle at a constant 2000 N force in different HSA positions. HSA position 0 mm corresponds to the upper position of the HSA and 300 mm to the lower position of the HAS

The final stiffnesses of the devices were averages calculated from the 10 measuring points shown in Fig. 12. At each point, the measured force is divided by the corresponding displacement, which provides the stiffness at that point. This procedure was repeated for 11 different HSA positions. However, it was necessary to eliminate some points from the averaging process due to collision and to non-linearity when using low forces. After these issues were controlled for, the stiffnesses were linear in the force range used for each HSA position. The averaged stiffnesses of both HSA devices and their combined average are presented in Table 5. The stiffness average of both HSA devices was used as the horizontal stiffness of the complete system.

Fig. 12Stiffness measurements under different loads in 11 different HSA positions

A regression curve was fitted to the averaged stiffness points to achieve a continuous stiffness curve that acts as a function of the HSA position. This enables accurate control of the horizontal stiffness between the minimum and the maximum. The derived regression curve is presented together with the analytical and simulation stiffness curves in Fig. 13.

Fig. 13Measured, analytical and simulated stiffness curves as a function of HSA position. HSA position 0 mm corresponds to the upper position of the HSA and 300 mm to the lower position of the HSA

As can be observed from the Fig. 13, the measured results differ from the other results, particularly at the stiff end. According to the analytical results, the stiffness of the device varies between 1.72 and 37.44 MN/m, depending on the HSA position, whereas the simulated stiffness varies between 1.52 and 26.18 MN/m and the measured stiffness between 2.04 and 18.32 MN/m. The difference is mainly due to errors in the analytical and simulation models. The models assume that the beam supports are completely rigid. Therefore, the stiffness curve increases exponentially when the beam support approaches the beam end. These models could be further developed by replacing the rigid supports with spring supports.

The analytical and simulated solutions developed for this study are applicable for the dimensioning of the beam to achieve a certain stiffness range and thus also desired foundation stiffness in the horizontal direction. This can be achieved by changing the beam material ($E$), length ($L$) or beam profile ($I$).

3.2. Rotor behavior measurement

The results were obtained by measuring the acceleration of the bearing housings, the radial bearing forces and the displacement of the rotor in the middle cross-section. The horizontal results form the main part of the study, since the effects of varied stiffness were expected only on that axis. The effect of unbalance and eccentricity (the first harmonic component) was excluded from the results.

The behavior of the rotor was analyzed in two different ways utilizing the data gathered. The first approach presents the peak values of the amplitudes at a certain stiffness and rotor rotation frequency. These results are separated into horizontal and vertical components, since different natural frequencies in these directions were expected. Figs. 14-16 illustrate the horizontal results and Fig. 17 presents the vertical results.

The second approach presents the subcritical harmonic components of the vibration at a certain stiffness and rotating frequency after the FFT analysis. Only the horizontal results are presented in Fig. 18, since the vibration behavior of the rotor in the vertical direction did not change according to the results presented in Fig. 17.

Fig. 14Peak values of the displacement measurement (center point movement) at a certain rotating frequency and stiffness. Different modes are color-coded, and each ridge denotes a harmonic component

Fig. 15Peak values of the radial bearing force measurement at a certain rotating frequency and stiffness. Different modes are color-coded, and each ridge denotes a harmonic component

The results presented in Figs. 14-16 reveal large changes in the horizontal natural frequencies of the rotor system at different stiffnesses. No natural frequencies were crossed, since the study focused on subcritical vibrations.

The radial bearing force measurement in Fig. 15 allows the first three horizontal modes of the rotor to be distinguished from the vibration. The identification of the subcritical resonance curves was straightforward, since they appear at fractions of their natural frequency, such as 1/2 times the natural frequency (2$H$) and 1/3 times the natural frequency (3$H$). This is related to harmonic excitations that can occur multiple times per revolution, and therefore coincidence between the excitation frequency and the natural frequency is possible below the critical speed. In the results, the harmonic frequencies were given as a function of the rotating frequency, and thus 2$H$ subcritical resonance was visible when the rotating speed was half of the corresponding critical speed.

Fig. 16Peak values of the acceleration measurement at a certain rotating frequency and stiffness. Different modes are color-coded, and each ridge denotes a harmonic component

Fig. 17Demonstration of the effect of varied stiffness on the vertical natural frequencies. The vertical harmonic components of the displacement measurement are presented. The vertical mode is illustrated as red lines and each ridge denotes a harmonic component

However, as the results show, the measurements contain major differences in their ability to distinguish different modes. For instance, in the displacement measurement displayed in Fig. 14, the second mode cannot be seen at all. This is a result of the test setup, as the displacement measurement was performed at the middle cross-section of the rotor. As already shown, the node of the second mode is in the middle of the rotor, which makes detection of the second mode impossible. However, this could be avoided by selecting another measurement point along the rotor. By contrast, the acceleration measurement presented in Fig. 16 detects all the horizontal modes, but it is also mixed with vertical modes: the first and second harmonic components can be clearly seen in the results. Furthermore, its ability to separate the horizontal harmonic components was low compared to the other measurements.

The results indicate the significance of low harmonic components. For example, the fifth harmonic component of the first mode produces high amplitudes at low frequencies in the displacement and force diagrams, Fig. 14 and Fig. 15 respectively. Furthermore, very low harmonic frequencies still exert a significant effect on the behavior of the rotor, as demonstrated by the eighth harmonic component of the third mode in Fig. 14 and Fig. 16. In turn, Fig. 17 shows that the stiffness variation in the horizontal direction did not affect the vertical natural frequency as expected.

Fig. 18The effect of stiffness variation on the harmonic components. Stiffness decreases when moving downwards. In each measurement, the last presented harmonic component is the last significant one

a) Center point movement (Middle cross-section of rotor)

b) Radial bearing force (Bearing housing, tending end)

c) Acceleration (Bearing housing, tending end)

The second analysis was performed by applying the FFT algorithm to the measured data. This method allowed the harmonic components to be investigated more effectively. Fig. 18 presents the results in the frequency domain, where each harmonic component can be separately observed. Only three different stiffness points are shown; the remainder are compressed and are available in gifs (appendix in the electronic form of this article). The diagrams display the same natural frequencies and their decrease as in the previous representation. Utilizing this method, the higher harmonic components could be distinguished. In the diagrams below, the first harmonic component is removed, and the highest presented harmonic component is the last that has significant amplitude.

As the measurement results show, the natural frequencies of the modes responded differently to varying levels of stiffness. Here, the frequencies of the harmonic components changed in an identical manner, as they are fractions of the natural frequency. Thus, the natural frequencies could be determined and calculated by multiplying the frequencies of the harmonic components presented in the results. Fig. 19 demonstrates the effect of varying stiffness on the natural frequencies and their harmonic components.

Fig. 19The effect of stiffness on natural frequencies

As the diagram indicates, the largest change was to the natural frequency of the second mode, where the natural frequency decreased by 60.7 % from 34.59 Hz to 13.61 Hz when the stiffness was decreased 88.9 % from 18.32 MN/m to 2.04 MN/m. The natural frequencies of the first and third modes decreased 53.6 % and 21.1 % respectively. The effects of varying stiffness on the natural frequencies can be seen in Table 6.