Vibration Attenuation in Rotating Machines Using Smart Spring Mechanism

This paper proposes a semiactive vibration control technique dedicated to a rotating machine passing by its critical speed during the transient rotation, by using a Smart Spring Mechanism SSM . SSM is a patented concept that, using an indirect piezoelectric PZT stack actuation, changes the stiffness characteristics of one or more rotating machine bearings to suppress high vibration amplitudes. A Genetic Algorithm GA optimization technique is used to determine the best design of the SSM parameters with respect to performance indexes associated with the control efficiency. Additionally, the concept of ecologically correct systems is incorporated to this work including the PZT stack energy consumption in the indexes considered for the optimization process. Simulation carried out on Finite Element Method FEM model suggested the feasibility of the SSM for vibration attenuation of rotors for different operating conditions and demonstrated the possibility of incorporating SSM devices to develop high-performance ecologic control systems.


INTRODUCTION
Commonly, rotating machines cross critical speeds during their transient rotation leading the system to undesirable vibration amplitudes.Under this condition, the occurrence of catastrophic failures caused by crack propagation due to the fatigue process is intensified.Therefore, significant research effort has been devoted to the development and improvement of mechanisms capable to attenuate undesirable vibrations in rotating machines [1 -3].
The control approaches for rotating machines are clustered into three main categories, namely passive, active and semi-active techniques.Passive techniques are normally performed by devices known as absorbers or isolators.These techniques perform typically over a limited frequency bandwidth and, consequently, are unable to adapt their characteristics to changes in the system.Differently, active approaches promise vibration suppression over a broadband of frequencies in which the suppression is performed by incorporating active actuators, such as PZT stacks, magnetic bearings and electromagnetic actuators to the machine to act directly against the vibratory loads.Unfortunately, successful implementation of these approaches has been limited by displacement capabilities of the piezoelectric actuators and the expensive costs of magnetic bearings [4].The semi-active techniques represent an alternative solution to these problems.In semi-active approaches, the vibration is attenuated through an indirect manner by changing the structural parameters of the machine, such as damping and/or stiffness.In our days the implementation of this technique in rotor dynamics is made possible by techniques such as magneto-rheological and electro-rheological dampers and SSM.
SSM uses an indirect PZT stack actuation to change the stiffness characteristics of the rotating machine to attenuate high vibration amplitudes.This mechanism can be implemented to attenuate a wide range of vibratory loads such as axial, bending, torsion, or a combination of these ones [4].For bending control purpose, which is the case studied in the present contribution, the SSM must be orthogonally arranged in a plane located at the one or more bearings of the rotating machine.
However, this paper presents a numerical simulation to evaluate the effectiveness of the SSM to attenuate high vibration amplitudes of a rotating machine passing by a critical speed during its transient rotation.The system is composed by horizontal flexible shaft with two rigid discs and three bearings.The SSM parameters are optimized using a GA multi objective optimization technique so that the objective function includes the simultaneous minimization of the norm and the maximum absolute value of the outputs measured at all the bearings together with the PZT stack energy consumption, in order to improve the control performance.The inclusion of the energy consumption in the minimization process aims at considering a new worldwide trend in machine design, i.e., to obtain control systems that are ecologically correct.This relevant concept has attracted the attention of many researches, being Skladanek, Der Hagopian and Mahfoud (2009); Richard, Guyomar and Mohammadi (2007); Matichard and Gaudiller (2005); Maslen, Allaire, Noh and Sortore (1996), the examples of recently published papers involving ecologically correct control systems design, in some of them, for rotating machines, namely Eco-Rotors.

ROTOR MODEL
The FEM model of a flexible rotor in transient motion is represented by a matrix differential equation that describes the dynamic behavior of the system: Proceedings of the 9th Brazilian Conference on Dynamics Control and their Applications Serra Negra, SP -ISSN 2178-3667 where M is the inertia matrix, D is the damping matrix, G is the gyroscopic matrix, K is the stiffness matrix, K st is a stiffness matrix resulting from the transient motion, x is the generalized displacement vector, F ext is the external forces vector and ∅ is the angular velocity of the rotor [9].In this paper, the shaft was modeled by using the Timoshenko's beam element with two nodes and four degrees of freedom per node, two displacements and two rotations.Due to the size of the matrices involved in the equation of motion, the pseudo-modal method was used to reduce the dimension of the FEM model.For this aim, the reduction is achieved by changing from the physical coordinates x to modal coordinates q as follows: where Ф is the modal matrix containing the m first vibration modes of the non-gyroscopic, symmetric and undamped associated rotor.Substituting the equation (2) into equation ( 1) and multiplying the resulting expression by Ф  the reduced equation of motion of the rotor is given by: where The solution of the equation (3) results in a response vector described in modal coordinates.By applying the equation (2) it is possible to convert the dynamic response to physical coordinates.

SMART SPRING MECHANISM
Several applications of the SSM have been explored for vibration suppression in the last years.Daley, Johnsonb, Pearsonc and Dixond (2004) developed a system based upon an electromagnet combined in parallel with passive elements for vibration suppression in marine structures, namely the Smart Spring Mounting System.The results demonstrated that only the rigid body modes of the machinery were controlled.Yong, Zimcik, Wickramasinghe and Nitzsche (2004) obtained successful results using a different SSM patented concept based on two springs arranged in parallel.
In their paper, it was demonstrated the concept ability for multiple harmonics vibration control of helicopter blades.Nitzsche, Harold, Wichramasinghe, Yong and Zimcik (2005) presented the control efficiency of the same concept through numerical and experimental investigations.Furthermore, a feedback control system was used to improve the control efficiency.The architecture of the SSM used in the last two presented papers is conceptually shown in Figure 1.
If the friction between the PZT stack and the host structure is disregarded (normal force N(t) = infinite and the friction f(t) = 0), the SSM behavior can be represented by two distinctly dynamic systems, according to the PZT stack status.The first one is active when the PZT is turned off and the equivalent SSM stiffness is given by k ps + k as .The second status is obtained when the PZT is turned on.In this situation, the active spring is unattached leading the SSM to operate only through the primary spring k ps .Note that if the friction is considered, the SSM becomes able to vary combinations of stiffness and damping at a particular location of the structure.Thus, the SSM mechanisms can actuate over a broadband of frequencies with low voltage requirement and displacement capability of the piezoelectric actuator [4].It is worth mentioning that most semi-active vibration control approaches are capable to change only a single structural property.

APPLICATIONS
The proposed methodology was numerically applied to a rotor system composed of a horizontal flexible steel shaft, represented by 20 Timoshenko's beam elements, two rigid steel discs (D 1 and D 2 ) and three asymmetric bearings (Figure 2).The physics and geometric properties of the shaft, discs and bearings are given in Table 1.The matrix equation of motion of the studied rotor was solved by using a MATLAB/SIMULINK ® code.The obtained responses were compared with the commercial software ROTORINSA for validation purposes.In all analyses performed in this contribution, only the displacement responses generated by the first ten vibration modes of the rotor measured along the x and z directions at the bearings positions were considered.
The proposed SSM control strategy is based on instantly reductions in the stiffness of the bearings when the rotor reaches a critical speed during its transient rotation.This procedure is performed in order to change the position of the critical speed in the Campbell diagram and, consequently, to attenuate the vibration amplitudes generated at that particular rotation speed.For this aim, in this paper the SSM Proceedings of the 9th Brazilian Conference on Dynamics Control and their Applications Serra Negra, SP -ISSN 2178-3667  was installed along the z direction at the location of the bearing B 3 (Figure 3)the highest output amplitude measured at the z direction was obtained in B 3to control the vibration generated by the rotor in transient rotation over 0 to 4200 RPM, during 2 seconds.In this application, the target section of the SSM (see Fig. 1) was attached to the ground and only the stiffness of the bearing was used as a varying parameter, being the friction between the PZT stack and the host structure disregarded.Additionally, for design purposes the equivalent SSM stiffness k ps + k as is equal to the stiffness k zz of the bearing B 3 .Figure 4 shows the regions where the PZT stack is either turned off, thus producing an equivalent stiffness associated with the bearing B 3 (k ps + k as ), or turned on, so that the equivalent stiffness of the bearing B 3 is simply k ps .Note that for the output amplitudes between 0.5 x 10 -3 m and -0.5 x 10 -3 m (output measured along the z direction of the bearing B 3 ) the PZT stack is in the off status.Outside of this range, the region above the green line and below the pink one, the PZT stack remains turned on.An important remark is that the choice of the SSM operation region is defined by the user, considering some key points such as, for instance, the desired vibration attenuation and the available electric power for the PZT stack.The best choice of the SSM parameters k ps and k as , which are associated with the control performance, was determined by using a multi-objective optimization procedure, as performed by the so-called compromise programming (CP).The unconstrained minimization of the scalar objective function was achieved by using a GA technique with an initial population of 100 individuals and 10 generations.
According to Vanderplaats (2005), CP is able to combine various objective functions in order to obtain a reasonable compromise solution to many objectives.The compromise objective function is given by: was determined by the optimization of each objective function considered independently, and     was associated to the worst configuration.CP was performed combining the norm and the maximum absolute value of the responses measured at the bearings positions, measured along x and z directions, and the total number of times, N b , that the PZT stack is turned on.These indexes were minimized considering the design variable P (0 < P < 1), being k as = P k zz .Table (2) presents the values of k ps and k as as found in the minimization process.Note that the results for CP show that the stiffness reduction provided by the SSM was as big as 84.44%, an abrupt reduction that could generate instability in the rotor motion.However, Figure 5 shows through the outputs measured along the x and z directions at the position of the bearings that the rotor keeps stable despite the large stiffness reduction imposed to the rotor.This behavior can be explained by the amount of damping in the system.Additionally, Figure 5 compares the dynamic responses for the rotor without and with SSM.The control performance appears to be efficient along the z direction at all the positions where a bearing is placed to support the rotor.This means that the SSM is efficient not only at the point to which the SSM was attached (B 3 ).Observe that the outputs measured along the x direction were only modified by the SSM since it was installed along the z direction.
In order to demonstrate quantitatively the SSM control efficiency, Table 3 shows the RMS value (Root Mean Square) of the outputs obtained for the rotor without and with the SSM.One can note that the vibration attenuation is efficient along the z direction (greater than 20%), as shown in the Figure 5. Vibration attenuation is observed also at the bearing B 1 along the x direction (around 12% of reduction).However, also along the x direction, at the positions of the bearings B 2 and B 3 a small increase in the amplitude is observed (around 5%).Another aspect of the SSM technology presented in this paper is related with the design of ecologically correct machines.Removing N b from the optimization process, the PZT is turned on 43 times.When the energy consumption is considered in the process (case presented in the Figure 5 and Table 3) this number decreases to 42 (approximately 2% of reduction).Additionally, supporting the adoption of the ecorotor concept in the design of machines and systems, Table 4 shows the RMS values of the response.This means that by including the energy consumption in the optimization process do not alter significantly the efficiency of the control.However, the amount of energy consumed by the system is reduced for this configuration.Previous results showed the efficiency of the SSM with respect to vibration reduction of a rotor that crosses critical speeds during its transient rotation.However, rotating machines are also susceptible to external disturbances, such as impact.Therefore, it is necessary to evaluate the SSM influence for this kind of excitation.Figure 6 shows the behavior of the rotor subjected to impact (100 N) applied along the z direction at the position of the disc D 1 when t = 1.0 sec.The dynamic responses of the rotor with SSM are then presented in the Figure 6.The rotor is accelerating from 0 to 4200 RPM, similarly to the previous case.It is possible to observe significant differences between the outputs obtained with and without impact (refer to the Figure 5).However, one can see that the rotor motion is kept stable when the SSM is operating, which means that the behavior of the system with SSM is robust to external disturbances during the transient motion of the rotor.The same evaluation was performed for the rotor under steady state condition at 4200 RPM as depicted in the Figure 7.Here again, an impact (100 N) was applied along the z direction at the position of the disc D 1 when t = 1.0 sec.It is possible to observe significant differences between the outputs obtained with and without the impact disturbance.However, as in the previous cases, the rotor motion remains stable when the SSM is operating due to the amount of damping in the system.This indicates that the system with SSM operating under steady state condition is robust to external disturbances.Finally, the influence of the SSM was evaluated for the rotor at rest. Figure 8 shows the output measured when the rotor was excited by impact (100 N) applied along the z direction at the position of the disc D 1 when t = 1.0 sec.Note that when the SSM is operating the maximum output amplitude is 2.5 x 10 -3 m (Figure 8f).This high value was not reached in the other applications.

Tab. 4. SSM control efficiency of the
However, one can conclude that the difference founded between the cases with the rotor in rotation and the one with the rotor at rest is associated with the damping.In this paper a vibration attenuation technique based on SSM for rotating machines that cross critical speed during its transient rotation was evaluated.The proposed approach presented promising results, leading to reductions as large as 50% for the case in which the parameters of the SSM were optimized.The optimization process was conducted using a GA technique that was applied to a multi-objective problem.Despite the stiffness reduction (around 84%) imposed to the rotor by the SSM mathematical simulation results indicate that the system did not become unstable during the test.The robustness of the approach with respect to external disturbances was analysed.First, an impact force was applied to the rotor disc during the transient rotation.In the second case, the rotor was evaluated for steady state conditions (constant rotation) and, finally, to system vibration was analysed for the rotor at rest.For the three cases considered the results showed that the control technique is robust to external disturbances.Additionally, it was possible conclude that the inclusion of ecologically correct concept in the design of control systems did not reduce significantly the SSM efficiency.The proposed approach seems to be very effective to reduce amplitude vibration for different operating conditions of the rotor.Further work includes the modelling of the friction of the PZT stack inside the SSM so that the stiffness and damping of the system can be altered at the same time.Also, the technique will be implemented experimentally.

Fig. 4 .
Fig. 4. Equivalent stiffness of the SSM related with the output amplitude.
is the CP objective function,   is the weighting factor assigned to the k-th objective function,    is the k-th objective function,   *  is the k-th objective function target and     is the worst known value of the k-th objective function.In this paper the weighting factors   were all chosen as equal to 1.0;   *  of the 9th Brazilian Conference on Dynamics Control and their Applications Serra Negra, SP -ISSN 2178-3667 4

( a )Fig. 8 . 9 5
Fig. 8. Output responses measured for the rotor at rest excited by an impact force.