Keywords

1 Introduction

Modeling for rotor-bearing-foundation systems is one of the key bases for efficient dynamic analysis in the field of turbomachinery. The rapid development of computing devices and appearance of new computational techniques lead to the requirement of high precision during simulations for the manufacturers and designers. However it should be noted that the term foundation structure should be interpreted depending on the purpose of the rotating machine. Thus in heavy power machinery it is usually a massive fixed base structure made of steel frames and concrete e.g. for power generating turbine units, but in aerospace it is relatively light and flexible beam type wing for aircraft engines. The feature of latter one is that the wing during normal flights may vibrate with low frequencies. Ordinarily, rotor system of modern aero-engine together with the case mounted on the wing represents a uniform system that is supposed to be considered as a rotor-bearings-foundation structure. At the same time this system is subjected to complex loads, those from them that are connected with the rotating structure could be designated as internal excitation sources, Fig. 1, while the others such as airborne loads, landing loads, foreign object damage and etc., acting first on foundation structure could be called external excitation sources. Existence of multiple vibration sources increases the risk of natural frequencies excitation for the full rotor-foundation system which in turn can damage the engine and its components. Hence currently there is a tendency among aerospace engineers for improved calculation and development of reliable rotor-bearing-foundation structure models. Since rotating machinery such as modern aircraft engines usually designed, marketed and sold for the most part based on analytical and numerical predictions, methods to incorporate the foundation influence effect in rotordynamic calculations are very important.

Fig. 1
figure 1

Sources of vibration for aircraft engine

Full size image

1.1 Recent Methods for Foundation Structure Modeling

All available literature in the field of foundation structure modeling for rotordynamic calculation could be generally divided on discrete approach and continues. The first one mainly includes simplified presentation of the foundation structure as single-degree of freedom models (SDOF) or lumped concentrated masse, while the second one include multi-degree of freedom models (MDOF). Basic simplified support structure models could be found in works of Lund [1] and Gunter [2, 3] mainly aimed on instability problems with some examples of Jeffcott rotor with flexible supports. SDOF model was used by Nicholas et al. [4, 5] to model flexible support structure for steam turbine. Han et al. [6] built a test rig with elastically supported base in order to study resonance capture phenomenon, but used simplified two-mass SDOF model for simulation purposes. They showed that intensive vibration of the foundation base structure can absorb much of the total energy what in turn can lead to situation when rotor is not able to get enough energy and cannot successful pass through resonance region. The SDOF approach is quite advantageous with respect to its simplicity and its applicability in standard rotordynamic software. However, one can say that simplicity of SDOF models is benefit and drawback at the same time, since physically it’s application is quite limited due to it is reasonable only as long as only one natural mode of the substructure interacts with the rotor in the operating frequency range or if this interaction is comparably small.

If several natural modes are presented in the operating speed range when there is also high interaction with the rotor, SDOF model might be not informative and do not describe the real dynamic behavior of the rotor-bearing-foundation system. What brings to decision for application of MDOF models. Vasquez et al. [7, 8] proposed a method to include the dynamic characteristics of bearing supports using transfer function, where it could be extracted from modal information of the supports or from experimentally obtained forced response functions of the substructures at the bearing locations. Edwards et al. based on experimental equipment proposed an identification method for the flexible foundation parameters using run-up/run-down method [9–11]. Cavalca et al. [12–14] developed a methodology of using mixed co-ordinate method to analyze the influence of the foundation structure on the rotor-bearing system on the base of experimental test rig with plate type foundation on flexible springs. The calculation model used by the authors was theoretical for the rotor and supports but for the foundation structure it was improved based on experimental results. Obtained results showed significant influence of the foundation on rotor dynamic response and operational modes. Bonello and Brenn showed in [15] a mechanical impedance technique for modeling of the dynamic behavior of a rotor on flexible foundations, concluding that sometimes it’s cheaper and easier to measure dynamics of the foundations rather than to model them, since further they could be combined with theoretical model in so-called hybrid model. Chen et al. [16] described a pseudo mode shape method based on use of frequency response function for establishing equivalent mass, damping and stiffness matrices of the foundation without having to build the physical model. A combination of finite elements and transfer matrix approach to add foundation effect in rotordynamic calculations was proposed in [17, 18] by Feng et al. with further experimental validation [19]. Differences between the dynamic behavior of the system with foundation structure and without for a light small turbocharger rotor were investigated, modeled and discussed in [20] by Ying et al.

One should not forget that a lot of research work for foundation structure modeling was made in the field of ground power engines. Experimental approach to include foundation effect using dynamic stiffness matrices first was shown in 1976 by Gasch [21]. Procedure of generating matrices for foundation structure based on the test data was described by Stephensone and Rouch [22]. Murphy and Vance [23] revealed how to use transfer matrix method for making models for rotordynamic calculations taking into account the connections in order to calculate critical speeds for rotor-support system. Kramer [24] observed different types of foundation structures such as direct mounting in building structures, frame foundations, block and table rigid foundations. Examples of practical cases for incorporation of foundation elasticities in rotordynamic calculation could be found in [25–27]. Detailed work about foundation vibrations identification for power steam turbine units and further recommendations for reliable design of new units and refinement of existing were made by Russian scientist Runov [28], Kalmens [29], Savinov [30], Shatochin and Zimmerman [31].

1.2 Recent Methods for Foundation Structure Experimental Testing

Creation of adequate mathematic model is an important stage for every research since accurate model can help significantly in prediction of dynamic behavior for real machine and save a lot of time and money instead creation of experimental scale model. In most of the abovementioned papers it is revealed that even with usage of advanced sophisticated computer programs, it is still difficult and time-consuming to establish reliable rotor-foundation system model using finite element methods. In such a way, rotor-bearing systems are usually modeled with FEM while foundation structure model is tuned and corrected with help of experimental results.

Generally foundation identification methods could be classified in two main categories: methods based on data from the frequency response function (FRF) obtained from experimental modal testing (e.g. Nicolas et al. [4, 5], Cavalca et al. [12–14], Gasch [21], Stephenson and Rouch [22], Vance and Murphy [23]) and methods which use vibration data from the rotor or foundation structure during run-down or run-up of the whole system (e.g. Edwards et al. [9–11], Feng and Hahn [17, 18]). For the first method modal parameters and parameters for the most significant modes are estimated from FRF experimental data obtained for foundation structure and then introduced into equation of motion for the overall system to perform foundation behavior. For the second method transient vibration data from the rotor journals or bearing supports are used to establish foundation model and modal testing is not necessary. Nevertheless, main benefits of the FRF method are that it is more precise when system has such features as easy component disassemble, and hence modal testing can be performed for separate components, but the number of modes identified must be equal to the number of measured degrees of freedom. Thus in order to obtain more accurate dynamic properties for the full system, more measurement points should be added to substructure. In such a way, run-up/run down method is probably more convenient in practice, but it requires more correct rotor model and bearing characteristics. At the same time noise and modeling errors are the headache for the both approaches.

2 Theoretical Descriptions

2.1 Influence of Foundation Structure on Rotordynamic Behavior of the Full System

One of the main issues in the field of foundation structure analysis is an objective prediction of rotating machine safe and adequate behavior when foundation structure is subjected to excitation. Another question is how the foundation parameters such as mass and stiffness will influence on the dynamic behavior of the full system. Meanwhile the question is how to build a model which will be simple, fast and reliable. Kramer [24] described one of the simplest models for rotor-foundation structure, Fig. 2. Equation of motion for this system is given below in matrix form:

Fig. 2
figure 2

Natural frequencies for Kramer’s rotor-foundation model

Full size image
$$ \left[ {\begin{array}{*{20}c} {m_{R} } & 0 \\ 0 & {m_{F} } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\mathop u\limits^{..}_{R} } \\ {\mathop {u_{F} }\limits^{..} } \\ \end{array} } \right\} + \left[ {\begin{array}{*{20}c} {k_{R} } & { - k_{F} } \\ { - k_{R} } & {k_{R} + k_{F} } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {u_{R} } \\ {u_{F} } \\ \end{array} } \right\} = \left\{ {\begin{array}{*{20}c} 0 \\ F \\ \end{array} } \right\}, $$
(1)

where m R , m F —are rotor and foundation masses; u R , u F , ü R , ü F —are their displacement and acceleration; k R , k F —are rotor bearing stiffness and foundation suspension stiffness respectively. Natural frequencies for the Kramer’s model were calculated in MATLAB and results are shown in Fig. 2 for different mass ratio.

As a result Kramer’s model shows that natural frequencies of the coupled system are almost equal to the natural frequencies of the rotor and foundation for the large mass ratios. At the same time situation when foundation structure natural frequency is equal to rotor’s natural frequency is not so dangerous, since modes coupling produce a split where first natural frequency is less than rotor’s natural frequency, but second conversely is higher.

Kang et al. [32] focused on computation of simple models in order to include foundations in rotordynamics calculations. As a result several types of foundation structures were observed and described in details: modeling with the discrete lumped masses, modeling foundation with a continuous beam and continuous solid plate-type foundation. Previously it was pointed out that simple lumped mass foundation structure model will not give accurate solution since real structures are continuous and inherent to have continuous distribution of mass and stiffness, and hence may possess many natural frequencies. Nevertheless it is easy and fast in modeling and can be used for general understanding of foundation structure parameters influence on dynamic behavior of rotating structure. In such a way Jeffcott rotor model with lumped foundation masses was built in rotordynamic software Dyrobes, Fig. 3. Rotor model was made of steel with total length L = 0.6 m and shaft diameter Ds = 0.010 m. Disk outer diameter was Ddisk = 0.75 m and axial length of the disk Ld = 0.25 m. Mass of the disk Mdisk = 0.85 kg. The shaft was modeled using beam elements with 4 degrees of freedom (d.o.f.) at each node. The equation of motion for the full rotor-foundation system could be written in short matrix form as:

Fig. 3
figure 3

Jeffcott rotor model with lumped mass foundations

Full size image
$$ \left[ {\begin{array}{*{20}c} {M_{rr} } & 0 \\ 0 & {M_{ff} } \\ \end{array} } \right]_{{}} \left\{ {\begin{array}{*{20}c} {\mathop x\limits^{..}_{r} } \\ {\mathop x\limits^{..}_{f} } \\ \end{array} } \right\} + \left[ {\begin{array}{*{20}c} {C_{rr} } & {C_{rf} } \\ {C_{fr} } & {C_{ff} } \\ \end{array} } \right]_{{}} \left\{ {\begin{array}{*{20}c} {\mathop x\limits^{.}_{r} } \\ {\mathop x\limits^{.}_{f} } \\ \end{array} } \right\} + \left[ {\begin{array}{*{20}c} {K_{rr} } & {K_{rf} } \\ {K_{fr} } & {K_{ff} } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {x_{r} } \\ {x_{f} } \\ \end{array} } \right\} = \left\{ {\begin{array}{*{20}c} {F_{R} } \\ {F_{F} } \\ \end{array} } \right\}, $$
(2)

where M rr , C rr , K rr —are the mass, damping and stiffness matrix of the rotor; M ff , C ff , K ff —are the mass, damping and stiffness matrix of the foundation; C rf , C fr , K rf , K fr —are cross-coupled damping and stiffness matrix of rotor and foundation interaction; x r —is the d.o.f. vector for the rotor structure displacements, while x f —is the vector of foundation physical coordinates; F R , F F —are the vectors of external forces acting on rotor and foundation respectively.

Results of multiple analyses were combined together to get the undamped critical speeds depending from foundation mass variation, Fig. 4. Since operating speed range for this kind of rotor was defined as 200 Hz, only first four ordinal lateral modes were considered. Mode shapes which were identified as foundation structure modes were highlighted with the asterisk marker. Obtained numerical results revealed, that for high foundation structure stiffness increase of foundation mass do not influence on the change of critical speeds, and hence they are mainly regulated by rotor critical speeds. As a result there are no foundation modes in the operating speed range or nearby, Fig. 4a, b. Meanwhile, for low foundation structure suspension stiffness critical speeds for each mode decrease with increase of foundation mass, Fig. 4c, d. In case of low and high mass ratio, critical speeds varies slightly, but in case of intermediate mass ratio they may rapidly decrease. In the same manner critical speeds for first five modes of the rotor model on rigid bearings with lumped foundation masses were calculated in order to estimate influence of foundation suspension stiffness, Fig. 5. Mass ration MF/MR = 0.5, 1, 25 and 100 were chosen for analysis.

Fig. 4
figure 4

Undamped critical speeds versus foundation mass, Kb = 109 N/m: a Kf = 107 N/m; b Kf = 106 N/m; c Kf = 105 N/m; d Kf = 104 N/m

Full size image
Fig. 5
figure 5

Undamped critical speeds versus foundation structure suspension stiffness, Kb = 109 N/m: a MF/MR = 0.5; b MF/MR = 1; c MF/MR = 25; d MF/MR = 100

Full size image

Results have shown that increase of foundation stiffness tends to increase critical speeds for each mode. At the same time in case of very big and very small stiffness ratio critical speeds do not increase rapidly, except for the first two modes when the rigid modes are acting. However for moderate stiffness ratio foundation suspension stiffness increase brings to quick increase of critical speeds for almost all modes.

2.2 Foundation Modes Prediction

In order to study dynamic of foundation structure on the base of abovementioned simple model, rotor bearings stiffness was set as 2E7 N/m, while stiffness ratio Kf/Kb = 1.5E-02 was used for support structure. Mass ratio MF/MR = 25 was used to establish foundation mass. Undamped critical speeds and mode shapes are summarized in Table 1. Comparison of the Table 1 results and potential energy diagram (Fig. 6) had shown that first two modes are foundation rigid cylindrical and pivotal modes, due to concentration of energy in region of the supports, while the other modes were identified as pure rotor bending modes with more than 90 % of potential energy on the shaft.

Table 1 Predicted critical speeds and mode shapes for rotor with lumped foundations
Full size table
Fig. 6
figure 6

Potential energy distribution diagram, Kb = 2e7 N/m; Kf = 3e5 N/m

Full size image

More advanced model was build for the same type of rotor but with plate type foundation structure (table size: 850 mm × 150 mm × 30 mm) using solid elements in ANSYS software, Fig. 7. Foundation base was made of steel with approximate weight ≈30 kg. In this design, rotor-bearing model was located on foundation which had flexible beam suspension system. Meanwhile rotor model was modeled both by solid and beam elements (with disk modeled as a lumped mass). Natural frequencies for the full system are summarized in Table 2. Comparison of Campbell diagrams for models with different foundations is shown in Fig. 8 Critical speeds are highlighted with asterisk. Mode shapes associated with rotor and foundation structure for both models in ANSYS have similar values of natural frequencies, but results are hardly converged with simple lumped mass model, Fig. 8. Nevertheless, it could be seen that simplified lumped mass model predicted appearance of two foundation rigid modes (cylindrical and pivotal) which are close to each other before the first bending lateral mode of the rotor, while for the solid foundation model rigid modes are separated from each other. For lumped mass model node point for mode 2 was located in the region of the disk (Table 1) hence unbalance located on the disk will not excite it. In such a way it could be said that in the range of interest (50 Hz) only one foundation mode is significant for all three models.

Fig. 7
figure 7

Plate type foundation model in ANSYS

Full size image
Table 2 Natural frequencies and mode shapes for models in ANSYS
Full size table
Fig. 8
figure 8

Comparison of Campbell diagram: a lumped mass foundation structure model, b solid plate type foundation/beam rotor model

Full size image

3 Foundation Dynamic Analysis

3.1 Influence of Unbalance—Lumped Mass Foundation Model

At the same time, force response analysis was made for model with lumped foundation masses in order to confirm predicted rotor-foundation structure reaction due to applied unbalance (5.0E-5 kg mm @ 45 deg on the disk) and estimate approximate influence of damping for the full system. Obtained results have shown that in operating range (0–50 Hz) there are two critical speeds (Fig. 9) where the first one is foundation rigid cylindrical mode while the second one is rotor first midspan bending mode. It is interesting to note that in region around the first peak (foundation mode) phase is changing twice, pointing out that there is another foundation mode, but resonance is not excited.

Fig. 9
figure 9

Bode plots for unbalance response: a measurements on the disk, b on the support

Full size image

3.2 Harmonic Excitation of the Foundation Structure

Meanwhile in order to understand how foundation structure vibration can influence on rotor structure, the base of the previously observed model with solid foundations built in ANSYS was subjected to harmonic force excitation in horizontal direction (along z axis). Obtained results have shown that in case of in phase central excitation there are also only two peaks in the range of interest, where the first one is inherent to foundation mode and second to rotor first midspan mode shape, Fig. 10.

Fig. 10
figure 10

Amplitude versus frequency for beam rotor/solid foundation model

Full size image

4 Conclusion

It could be said in the conclusion that due to current demand on more precise calculations foundation modes are quite important and hence should be taken into account while rotordynamic modeling. To sum up, increase of foundation mass influence significant on decrease of critical speeds for the full rotor-foundation system especially for moderate and low suspension foundation stiffness. At the same time increase of foundation suspension stiffness leads to increase of critical speeds for the full system. Foundation modes could be efficiently identified by inspection of potential energy plots. Concentration of potential energy on the support structure (more than 70 %) will highlight them. Analysis of forced vibration due to unbalance can further reveal which foundation modes might be significant in the operating range of the machine. Applied harmonic excitation on the foundation base showed that not only foundation modes could be excited but also rotor modes, hence there is interaction between foundation and rotor structure what confirm that there will be influence on rotordynamics. Finally it could be said that neglect of efficient foundation structure modeling can bring to result when foundation modes could appear on real machine in operating range. Foundation excitation can lead to rotor structure unpredicted motion when perfectly balanced rotor will deviate from its steady motion what in turn can lead to its damage.