Experimental and theoretical investigations of the lateral vibrations of an unbalanced Jeffcott rotor

Abstract

The current work experimentally explores and then theoretically examines the lateral vibrations of an unbalanced Jeffcott rotor-system working at several unbalance conditions. To this end, three conditions of eccentric masses are considered by using a Bently Nevada RK-4 rotor kit. Measurements of the steady-state as well as the startup data at rigid and flexible rotor states are captured by conducting a setup that mimics the vibration monitoring industrial practices. The linear governing equation of the considered rotor is extracted by adopting the Lagrange method on the basis of rigid rotor assumptions to theoretically predict the lateral vibrations. The dynamic features of the rotor system such as the linearized bearing induced stiffness are exclusively acquired from startup data. It is demonstrated that, with an error of less than 5%, the proposed two-degrees-of-freedom model can predict the flexural vibrations at rigid condition. While at flexible condition, it fails to accurately predict the dynamic response. In contrast to the other works where nonlinear mathematical models with some complexities are proposed to mathematically model the real systems, the present study illustrates the applicability of employing simple models to predict the dynamic response of a real rotor-system with an acceptable accuracy.

Keywords

rotor-dynamics / Jeffcott rotor / lateral vibrations / Lagrange method / system-equivalent stiffness / sleeve bearings / oil film induced stiffness

Introduction

The literature presents good efforts on mathematically simulating the unbalance response of rotor systems with high accuracy. Krodkiewski et al. [¹] presented a nonlinear mathematical model to simulate the unbalance response of a rotor considering the nonlinear effect of its support system. Sanches and Pederiva [²] worked on identifying the unbalance response in a rotor using correlation matrices and modal order reduction method. Pjadatare and Pratiher [³] used the strain energy and the kinetic energy equations to study the effect of nonlinear geometric coupling on the dynamics of an unbalanced rotor. Gunter [⁴] illustrated the use of the single mass flexible model to industrial machines based on the simple Jeffcott rotor dynamics and by employing some eigenvalue analyses [⁵]. Taplak et al. [⁶] designed a neural network predictor [⁷] to analyze and measure the vibration parameters of the flexible rotating system at the bearing points. Malenovský [⁸] investigated the computational modeling of nonlinear rotor dynamic systems and presented the theoretical basis of the method of dynamic compliances and the modal method, supplemented by the method of trigonometric collocation [⁹]. Hong et al. [¹⁰] studied the mechanism leading to non-continuity in rotor joints to clarify how joint stiffness deteriorates upon applying a heavy load to rotor joints. They combined a numerical model of joint stiffness with a model of rotor stiffness to examine the dynamic characteristics of a rotor system. Weder et al. [¹¹] experimentally investigated the vibrational behavior of a rotor-stator system coupled by a viscous liquid using a test bench with rotor and stator simplified as disks. They exhibited the mode shapes, eigen-frequencies and damping factors measured over a large number of parameter combinations of the stator thickness, the rotor thickness, the clearance between rotor and stator as well as the rotation speed of the rotor. Chipato et al. [¹²] explored an overhung rotor model to determine the effect of friction during contact between the rotor and stator. Their model had two degrees of freedom with rotor stator contact and the equations of motion were non-dimensionalised. They also conducted a parametric study of the friction coefficient and eccentricity and displayed the results on three dimensional bifurcation plots, orbit plots, Poincaré maps and spectral intensity plots to classify the solutions. Saxena et al. [¹³] conducted an experimental study to find out the effect of gear pair contact on the modal behavior of an actual geared rotor system mounted on rolling element bearings. They measured the frequency response functions of the uncoupled and coupled geared rotor systems to explore the effect of gear pair contact on the natural frequencies of the system.

Those studies and many more, such as Refs. [¹⁴–¹⁶], were motivated mainly by the demand of machinery design firms to model the unbalance response of rotor systems with high accuracy. However, the case is different in machinery operation facilities where the need to such high accuracy is not often required, especially considering the testing requirement and the complexity of applying such techniques on running machines. Thus, this study comes with the intention to investigate the feasibility of using an analytical, linear and simple-to-use mathematical model to simulate the unbalance response of a rotor using only a startup data collected with conventional vibration monitoring. If proven feasible, this model will provide a great advantage in helping the decision-making process in the operation and maintenance of rotating equipment.

A Jeffcott rotor, which was described first by Jeffcott in 1919 [¹⁷], was selected in the study as it is asimple enough to ease the development of the mathematical model using conventional methods/laws: Newton’s Laws, Energy Principle, etc. Nevertheless, this kind of rotor also could be complex enough to mimic real dynamic response of rotating equipment.

The following section of the paper details the work of the study. Section 2 presents the test setup and procedure used to carry out the experiment. Section 3 discusses the solution approach to mathematically model the system and predict the lateral vibrations of the tested rotor. Section 4 presents the results of the experimental study as well as the mathematical model simulated results and discusses the findings. Finally, Section 5 summarizes the main outcome of the study and presents concluding remarks.

Experimental setup and procedure

The testing apparatus for this investigation, Fig. 1, consists of Bently Nevada RK-4 single disk rotor kit driven by a direct current motor speed control device, a stainless-steel rotating disk (800 g of mass, 7.5 cm in diameter and 2.5 cm in thickness) mounted midway between two sleeve bearings with a clearance at each bearing of 5 mils and stainless-steel cylindrical shaft of 1.0 cm diameter and 50.0 cm between-bearing-length, Fig. 1(a). The disk attached to the rotating shaft is equipped with holes 30.0 mm away from the disk center that allow attachment of eccentric masses in the shape of screws to simulate different mass unbalances states, Fig. 1(b). More details of the rotor-kit specifications are available in Ref. [¹⁸]. The measurements were collected using ADRE 408 data acquisition system of 24-bit resolution with 3300XL8 mm proximity transducer sensors installed 18.0 cm away from each bearing to avoid nodal points. The sensors were powered and connected to ADRE 408 by a proximitor assembly.

Two tests were conducted in this study: a startup test to evaluate the system equivalent stiffness and damping, and a time-wave response test at constant speeds (1000 RPM and 3600 RPM) to check the validity of the mathematical model at a practical operation range. Three different unbalance states were simulated using 0.4, 0.8, and 1.2 g eccentric masses.

Theory and solution approach

Assumptions and system degrees-of-freedom

To sufficiently model the rotor used in the experiment without adding excessive complexity, appropriate initial identification of the system degrees-of-freedom needs to be performed. This is viable through a proper analysis of the system constrains and by making the right assumption. Considering the rotor kit seen in Fig. 1(a), we can make the following set of assumptions.

1) ‚The mass of the shaft is negligible compared to the mass of the disk (lumped-parameter system).

2) ‚The shaft is elastic radially and rigid axially.

3)‚ The bearings are soft support.

4)‚ The dynamic properties of the two bearings are identical.

5)‚ The rotor rotation around axes perpendicular to the axis of rotation is negligible (static unbalance).

6) ‚The lateral vibrations of the system are small.

7)‚ Damping is present in the form of viscous damping.

The assumptions were either verified by inspection or made to simplify the development of the mathematical model. They all support the rigid rotor assumption neglecting the shear forces and the gyroscopic moment effects. We shall see the implications of making such assumptions on the accuracy of the mathematical model in the results section.

Rigid bodies undergoing spatial motion can have generally a maximum of six degrees-of-freedom, three rotations and three displacements [¹⁹]. Assumptions #2 and #5, which are essentially constrains, reduce the number of degree-of-freedom for this model from six into three, spin rotation and two lateral displacements. Specifying the rotational speed as a user-controlled parameter, the degrees-of-freedom get further reduced into two degrees of freedom only.

Generalized coordinates and resulting equations of motion

A two degrees-of-freedom system require two resulting equations of motion, one along each generalized coordinate, to fully describe the motion of the system. The experimental rotor can be modeled as seen in Fig. 2. Points A and B represent the points of measurement were the proximity probes were installed, which are the inboard and outboard bearings respectively. Since only one rotation is considered, a fixed-inertial reference frame, labeled as \overrightarrow{x}, \overrightarrow{y}, \overrightarrow{z}, is sufficient to fully describe the motion of the two points. In addition, assumption #2, #4, and #5 suggest that the displacement at point A is identical to that at point B since only spin rotation is considered. Therefore, the displacement along the \overrightarrow{x} and \overrightarrow{y} direction at point A, corresponding to the inboard bearing, are chosen as the generalized coordinates of the entire rotor-system.

M_{\mathrm{D}} is the mass of the disk, m is the mass of the added weight, C_{\mathrm{eq}x} and C_{\mathrm{eq}y} are the equivalent viscous damping coefficient along the \overrightarrow{x} and \overrightarrow{y} direction, K_{\mathrm{eq}x} and K_{\mathrm{eq}y} are the equivalent stiffness along the \overrightarrow{x} and \overrightarrow{y} direction. Each of the stiffness terms is composed of two stiffnesses in-series, one imposed by the elasticity of the shaft and one imposed by the bearing along each generalized coordinate. The abbreviation ‘IB’ and ‘OB’ correspond to the In-Board and Out-Board bearings, respectively. \omega is the rotational speed of the disk as defined by the user. The used fixed-inertial reference frame in the model, labeled as \overrightarrow{x}, \overrightarrow{y}, \overrightarrow{z} is shown at the left-hand side of the figure.

By properly identifying the forces acting on the system along the generalized coordinates, we arrive to the following equations of motion using Lagrange operator,

(1)M_{\mathrm{D}}{\ddot{x}}_{\mathrm{A}}+C_{\mathrm{eq}x}{\dot{x}}_{\mathrm{A}}+K_{\mathrm{eq}x}{x}_{\mathrm{A}}=me{\omega }^{\mathrm{2}}\cos \left(\omega t\right),

(2)M_{\mathrm{D}}{\ddot{y}}_{\mathrm{A}}+C_{\mathrm{eq}y}{\dot{y}}_{\mathrm{A}}+K_{\mathrm{eq}y}{y}_{\mathrm{A}}=me{\omega }^{\mathrm{2}}\sin \left(\omega t\right),

where e is the added weights eccentricity and t is the time. The above equations of motion are in correspondence with what was developed by Yukio and Toshio [²⁰] of a similar 2-degrees-of-freedom rigid rotor-system. They are both describing a system of two second-order, uncoupled, ordinary, linear differential equations that can be solved independently and analytically for the two generalized coordinates x_{\mathrm{A}}\left(t\right) and y_{\mathrm{A}}\left(t\right).

Dynamic system characteristics evaluation

The unknown dynamic characteristics in Eqs. (1) and (2) are the system-equivalent damping and stiffness, K_{\mathrm{eq}} and C_{\mathrm{eq}}, respectively, along each generalized coordinate. A practical method of evaluating the equivalent dynamic characteristics of a rotor system without getting into the details of system components (shaft, disk, and bearings) is to analyze startup data. In a bode plot, which is a typical presentation of startup data, the equivalent damping ratio (\mathit{\xi }) can be evaluated using the half-power bandwidth method. Even though this method is strictly applicable to lightly damped materials (), it is still widely used with none lightly damped materials (Some modification needs to be considered) [²¹].

As illustrated in Fig. 3, the equivalent damping ratio can be calculated using the following equations,

(3)\mathit{\xi }={\displaystyle \frac{{\omega }_{\mathrm{2}}-{\omega }_{\mathrm{1}}}{\mathrm{2}{\omega }_{\mathrm{natural}}}},

(4)\mathit{\xi }=\sqrt{\mathrm{0.5}-\sqrt{\mathrm{0.25}-\mathrm{0.0625}{\left({\displaystyle \frac{{\omega }_{\mathrm{2}}-{\omega }_{\mathrm{1}}}{\mathrm{2}{\omega }_{\mathrm{natural}}}}\right)}^{\mathrm{2}}{\left({\displaystyle \frac{{\omega }_{\mathrm{2}}+{\omega }_{\mathrm{1}}}{\mathrm{2}{\omega }_{\mathrm{natural}}}}\right)}^{\mathrm{2}}}},

where {\omega }_{\mathrm{2}} and {\omega }_{\mathrm{1}} correspond to the two half-power frequencies and {\omega }_{\mathrm{natural}} is the single maximum frequency reached by the system at that bandwidth, also known as the system-equivalent natural frequency. The system-equivalent viscous damping coefficient is related to the damping ratio by the following equation,

(5)\mathit{\xi }={\displaystyle \frac{C_{\mathrm{eq}}}{\mathrm{2}\sqrt{K_{\mathrm{eq}}{M}_{\mathrm{eq}}}}},

where M_{\mathrm{eq}} is the system equivalent mass. The identified system natural frequency, as shown in Fig. 3, can be further used to evaluate the system-equivalent stiffness using the following equation,

(6){\omega }_{\mathrm{natural}}=\sqrt{{\displaystyle \frac{K_{\mathrm{eq}}}{M_{\mathrm{eq}}}}}\mathrm{.}

ƒThe obtained characteristics are sufficient to fully develop the mathematical model. Furthermore, the analysis of the startup data can also lead to the evaluation of the bearing induced stiffness, which is often a desired outcome in the research field of rotor-dynamics due to their nonlinear effects [²³]. The overall stiffness of the system is induced by the shaft and the bearings. Since both stiffnesses are subjected to the same loading, they can be think of as two stiffnesses in series as seen in Fig. 2. The equivalent stiffness of two stiffnesses in series can be evaluated as the following,

(7){\displaystyle \frac{\mathrm{1}}{K_{\mathrm{eq}}}}={\displaystyle \frac{\mathrm{1}}{K_{\mathrm{shaft}}}}+{\displaystyle \frac{\mathrm{1}}{K_{\mathrm{bearing}}}}\mathrm{.}

Shaft stiffness can be evaluated using the theory of strength of materials [²⁴]. The theory suggests that elastic elements like beams [²⁵–²⁸] or shafts exhibits a static deflection when loaded. Therefore, they can be modeled as ideal springs. The stiffness of the shaft at x-distance away from the bearing can then be approximated by [²⁹]:

(8)K_{\mathrm{shaft}}={\displaystyle \frac{P}{{\delta }_{\mathrm{static}}}}\cdot {\displaystyle \frac{\mathrm{48}EI}{\left[\mathrm{3}{L}^{\mathrm{2}}x-\mathrm{4}{x}^{\mathrm{3}}\right]}},

where P is the loading applied on the shaft center, {\delta }_{\mathrm{static}} is the shaft’s static deflection, E is the modulus of elasticity for the shaft, I is the cross-sectional moment of inertia of the shaft, L is the total length of the shaft between bearings and x is an arbitrary distance at which the load was applied measured from shaft end. E of steel shaft is \mathrm{210}\mathrm{G}\mathrm{N}/{\mathrm{m}}^{\mathrm{2}}, l of the shaft used in this study is \mathrm{50}\mathrm{c}\mathrm{m} and x is the 25 cm. Having access to the system-equivalent stiffness, we can evaluate the stiffness at the bearings using Eq. (4) as it will be the only unknown.

Results and discussions

A 1X bandpass filter was applied to all experimentally collected data to filter out the synchronous lateral vibrations. That is because the interest is to study the unbalance induced vibrations only. Figure 4 shows the bode plots of the startup data collected at each unbalance state for the inboard bearing (point A in Fig. 2).

Several trends can be noted from Fig. 4. The first being that the amplitude of vibrations, particularly at resonance, increases with the increase of mass eccentricity added to the system. This is expected and well explained by Eqs. (1) and (2) where the mass unbalance magnitude is directly proportional to the amplitude of vibration. It is noted that even when no eccentric mass was attached to the rotating disk, the dynamic behavior of the system was similar to the other states with active mass unbalance. This implies that rotor is inherently unbalanced. Another noted trend is the fact that resonance frequency for the three unbalance states seems to slightly decrease with increasing amplitude. This phenomenon is normal and has been observed before in similar studies, for example by Littler [²⁴].

It is also observed that there is a variance in the amplitude of the vertical (\overrightarrow{y}) and horizontal (\overrightarrow{x} ) vibrations at all states, where the vertical vibrations seem to be always higher. This supports the fact that bearings contribute significantly to the stiffness of the system. That is because, ideally a shaft of isotropic properties, if considered as the only source of stiffness in the system should lead to identical amplitude of vibrations along the lateral direction (\overrightarrow{x} and \overrightarrow{y} ). This realization was owed by Kelm to the oil film variation in sleeve bearings [³⁰]. According to Ref. [³⁰], a variant amplitude of vibrations along the lateral direction is commonly observed in machinery bearings and it is due to the variant oil pressure around the bearings. Variant oil pressure leads to variant stiffness properties along the lateral direction of the rotor. This notion was also supported by the work of Nicholas and Barrett [³¹] and was taken into consideration when developing the mathematical model.

To quantify the contribution of the bearing stiffness as well as the shaft stiffness to the overall system stiffness, startup data along with Eqs. (7) and (8) were used as illustrated in the theory section. The evaluated dynamic properties of the system are listed in Table 1.

This practical method of gives easy access to the linearized stiffness coefficient as induced by the bearings, which is often calculated in a nonlinear manner in literature. The values show that the bearing stiffness is higher than that of the shaft for both directions. They also show that the stiffness along the horizontal direction (\overrightarrow{x}) is higher than that at the vertical direction (\overrightarrow{y}). This outcome matches that of studies like Ref. [³²].

Based on the depicted system-equivalent properties, the mathematical model was developed. Figure 5 shows a sample of the simulated data at the 0.8 g unbalance state as compared to the experimental data.

Figure 5 shows that the developed mathematical model is able to follow the general behavior of the experimental results and match the amplitude accurately for speeds below the critical speeds. To have a deeper insight on the accuracy of the mathematical model, the steady-state response of the system for the same unbalance state at around 1000 RPM is considered as shown in Fig. 6.

The response of the system, as seen in Fig. 6, is a simple harmonic response. This is explained by Eqs. (1) and (2) where the excitation is a sinusoidal function, and it is evident that the dynamic response of a linear system would follow the input excitation. The error present in the simulated results is about 4% on both directions, taking the bearing clearances as a reference and not the experimental values. The reason behind choosing this error calculation method is that alarm and trip limits of machinery protection systems in the industrial practice are usually based on the mechanical clearances [²⁴]. Thus, it is more representative to base the accuracy of the modal on the available bearings clearances and not the experimental values. This outcome and accuracy of the mathematical model is expected at a speed below the critical speed due to the fact that the rotor is considered rigid at that speed range, which is in line with the assumptions made. It is desired to investigate the accuracy of the mathematical model at the flexible condition where the rotor is operated at speeds higher than the critical speed.

Figure 7 shows the simulated response of the mathematical model at around 3600 RPM for the 0.8 g unbalance state. As explained by Eqs. (1) and (2), the amplitude of vibration increases and so does the error in the simulated results (maximum of 32% on both directions). This proves that the mathematical model does not capture the full dynamics of the system when the rotor is presumed flexible. This outcome is expected as the assumptions made did not account for the dynamic effects associated with the flexible rotor condition, such as the effect of gyroscopic moment. Similar trends were captured for the other unbalance states and are summarized in Table 2.

In line with the assumption earlier made of a rigid body undergoing one spin rotation, the inboard bearing lateral vibrations are expected to match those of the outboard bearing. The values of the experimental peak-to-peak vibration amplitude at the outboard bearing for each unbalance state are tabulated in Table 3.

The values in Table 3 validates the assumption of identical vibrations amplitude at both bearings for rigid rotors. Certainly, there will be some inconsistency in the measurement between bearings in reals systems, and that explains the slight variation in error values obtained for both bearings. As for the flexible rotor condition and similar to the inboard bearing results, the mathematical model also fails to capture the vibration amplitude with a maximum error of 51%.

Conclusions

This study investigated the feasibility of mathematically modeling a single mass Jeffcott rotor-kit to simulate the system lateral vibrations when running at various unbalance states. Experimental tests were conducted Using Bently Nevada RK4 between-bearing rotor kit. Three states of mass eccentric unbalance were used in the experiment, 0.4, 0.8, and 1.2 g masses all attached 30 mm away from the shaft centerline. Measurements showing the startup data, and steady-state data at the rigid and flexible rotor condition. The startup data were used to extract the system equivalent dynamic characteristics, and also quantify, linearly, the contribution of the sleeve bearings to the overall system stiffness. Those data were used to construct the mathematical model, which is based on the Lagrange method. The model was found valid and accurate to predict the response of the real system in the rigid condition with a maximum error of 5%. On the other hand, the model failed to predict the response of the real system when flexible rotor is presumed.

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