Instability attenuation and bifurcation studies of a non-ideal rotor involving time-delayed feedback

In non-ideal vibratory system, the excitation is a nonlinear function of system response. The dynamic behavior of such system is often characterized by an energy source with limited power. The study of instability phenomena in non-ideal rotor driven through a non-ideal energy source is of considerable current interest. The non-ideal rotor system often gets destabilized on exceeding a critical input power near the resonance. This kind of instability is termed as Sommerfeld effect marked with nonlinear jump phenomena. This paper investigates the attenuation of nonlinear jump phenomena and numerical study of bifurcations of a non-ideal unbalanced rotor system with internal damping using time-delayed feedback via active magnetic bearings. The results show that the time delay indeed plays a critical role on the suppression of the jump phenomena. Following, some new insights are also revealed through a numerical study of saddle node, Hopf and trans-critical bifurcations with time delay as a bifurcation parameter. The transient analysis confirms the results obtained analytically through the steady-state consideration.


Introduction
The control of instability of high-speed non-ideal rotor system has been considered a great challenge to many researchers in recent times. These rotors are coupled with an energy source with limited power (i.e., DC motor, induction motor etc.) which also gets affected by the response of the system contrary to its ideal counterpart in which the source is not influenced by the system response [1]. Mathematical modelling of such non-ideal systems involves an additional degree of freedom which describes the nonlinear interaction of the rotor system with the electric motor. The interaction between the motor and the rotor system is highly complex and nonlinear. During the coast up operation, close to the system resonance, the rotor speed suddenly jumps to a very high value and the corresponding amplitude simultaneously jumps to a very low value on exceeding a critical power input of the motor. A similar kind of jump in the reverse manner is also observed during the coast down operation wherein the spin speed jumps to a very low value and the corresponding amplitude jumps to a very high value. This jump phenomena marked with a S. S. Dasgupta (&) School of Mechanical Engineering, Vellore Institute of Technology, Vellore, Tamil Nadu, India e-mail: sovan@vit.ac.in curve of discontinuity are observed in the frequency response which reveals that certain spin speeds could not be achieved during the steady state when the input power exceeds a critical value. This kind of jump phenomena is well known as Sommerfeld effect [2] which often destabilizes the system near the resonance. The non-conservative circulatory forces due to intrinsic material damping also cause the system unstable beyond a certain threshold spin speed which is referred to as a second kind of Sommerfeld effect in rotor dynamics literature [3].
A few studies are reported so far dealing with the attenuation of instability for non-ideal systems. A tuned liquid damper is proposed by Felix et al. [4] as a passive control strategy which successfully attenuated the instability of a structural frame excited through a non-ideal excitation. A kind of friction model is employed to mitigate the Sommerfeld effect of a nonideal structure using a vibration absorber [5]. Semiactive control technique is also employed to study the Sommerfeld effect mitigation such as using MR damper on a non-ideal vibratory system powered through a DC motor is reported by Castao et al. [6]. Another important work on the suppression of nonlinear jump phenomena using a magnetorheological damper is reported in [7]. In 2018, Balthazar and his co-authors proposed an experimental technique using shape memory element to mitigate the jump phenomena exhibited by a non-ideal cantilever [8]. In the recent past, a novel approach [9] based on fractional order external damping is employed to study the attenuation of the jump phenomena of a non-ideal internally damped shaft-disk system. Subsequently, a semi-active control scheme through a switched stiffness device was also proposed by the same authors [10].
Active Magnetic bearing (AMB) has many advantages over its passive counterpart such as noncontacting nature of suspension over the convention fluid film and anti-friction bearings, elimination of lubricant, very low friction, no wear, high rotor speed and high load carrying capacity at extreme conditions such as very high or very low temperature and vacuum. Many scientific articles dealing with control of rotating machinery via AMB system have been published [11][12][13][14][15][16][17][18][19][20][21][22][23]. But AMB-based active control strategy dealing with suppression of the Sommerfeld effect in non-ideal internally damped rotor system was not reported before 2019. In 2019, Jha and Dasgupta proposed an active control scheme based on a linearized model of AMB system to attenuate the Sommerfeld effect in a non-ideal rotor disk with material damping successfully [24].
It is well known that the time delay often destabilizes the system. Thus, the analysis of the dynamic behaviour of AMB with time delay is extremely important. The source of the intrinsic time delay is due to the hardware of the digitally controlled AMB system which comprises digital signal processor (DSP), analog-to-digital (A/D) and digital-to-analog (D/A) converters, memory, filters, interfacing components, etc. The sample and hold device introduces a delay of half a sampling period. The second source of delay is the controller computation. The amount of computation delay depends upon how the inputs and outputs are synchronized in the controller implementation algorithm. Another source of time delay is the reaction time of the electromagnetic actuators to the control decisions. Usually, a multiplexer topology is employed for multiple A/D conversion channels for sampling. Improper synchronization of A/D converters with sample and hold circuit, delay in A/D and D/A conversion, computational dead time may have adverse effect on the phase margin of the system leading to instability. The infinite-dimensional nature of the delay-differential equation (DDE) is due to the presence of the past effects, described by functions in the time domain. Computational time delays and sampling delay caused by the zero-order hold (ZOH) element in the A/D converter are not the same. Computational time delays become especially important in the case of MIMO control in conjunction with sequential A/D conversion based on a multiplexer architecture. A semi-discretization control algorithm [25] is implemented to discretize the DDE with the delayed term which may be represented as t iÀr ¼ i À r ð Þh; using the first discretization step t i ¼ ih with discretization step h, r is a positive integer. If r ! 1 and h ! 0, then the time periodic time delay tends to constant time delay. Thus, the analogous time delay is used instead of sampling time.
The AMB actuated rotor-disk system is a digitally controlled mechanical system. As a consequence, the digital system samples the position and the velocity at discrete time instances. Since some time is needed to process the measured signal, the force exerted by the actuator is bit after the sampling instant and this processing time is often same as the sampling time. As a consequence of the digital effects such as sampling, the delay and the finite resolution of the digital analogue converter caused by the round-off error, the behaviour of the system can be approximated by a linear map, known as the micro-chaos map [26]. However, this map often leads to micro-chaotic vibration.
A few works discussed the problem of the timedelays in AMB-system [20,[27][28][29]. Influence of time delay is reported in [27] to study the stability of a Jeffcott rotor-AMB system based on Hopf bifurcations. In a similar kind of study, it has been shown that on exceeding a critical time delay, instability occurs in a rotor system coupled with AMB [28]. Following, occurrence of period-doubling bifurcations is investigated in a system with time-delayed magnetic levitation marked with soft spring characteristics [29]. A study on vibration suppression of a time-delayed AMB system subjected to multi-parametric excitation is reported in [20]. This study also reveals that the efficacy of the controller may be enhanced if the time delay is kept within a stable range. Eissa et al. [30] studied the effectiveness of a time-delayed PD controller to attenuate the instability of a nonlinear system coupled with a magnetic bearing. Balthazar et al. [31] studied the Hopf bifurcation on a centrifugal vibrator actuated by a non-ideal motor. However, influence of time delay in attenuating the Sommerfeld effect of non-ideal eccentric rotor system with internal damping is not yet reported so far. The bifurcation studies of such non-ideal systems are also not available in literature.
This paper aims to present the influence of intrinsic time delay in attenuating the nonlinear jumps of a nonideal rotor-AMB system driven through a brushed DC motor. Moreover, it also reveals the complex dynamics of this non-ideal system through different bifurcation studies (such as saddle node, Hopf and transcritical) with the help of root-loci plots. The rest of the paper organised as follows. Section 2 gives the mathematical modelling of the proposed system with time delay. Section 3 discusses the influence of time delay in attenuating the Sommerfeld effect. Following a transient analysis is also performed which confirms the analytical results given in Sect. 3. Numerical studies of bifurcations through a set of root-locus plots are presented in Sect. 4 which reveal the complex dynamics of instability influenced by the time delay based on certain conditions. Section 5 gives the conclusions and future work. The steady-state amplitude frequency responses close to the system resonance are obtained for several time delays. Following, a transient analysis is found to be in good agreement with the steady-state results.

Model description and equations of motion
A typical internally damped non-ideal eccentric rotor-AMB system with time delay driven through a brushed DC motor is considered for the present work as shown in Fig. 1. The system equations of the proposed rotordisk model required to perform analytical and transient analysis are presented in the following subsections.

Equations of motion
A flexible internally damped shaft driven through a brushed DC motor supported by a pair of identical rigid bearings is shown in Fig. 1. A small disk is mounted with small eccentricity at the mid-span of the shaft. No gyroscopic term is involved as the disk is centrally mounted. Additionally, two identical AMBs with a small time delay s ð Þ are employed as shown in Fig. 1. The proposed model offers one more degree of freedom (DOF) to describe the nonlinear complex interaction between the system and non-ideal DC motor. Using the Lagrange-differential equations, the generalized vector differential equation may be obtained as follows: Equation (1) represents a delay differential equation (DDE) [32] which is typical representative of infinite-dimensional system as time delay s ð Þ is exclusively present in the RHS of Eq. (1). This equation also represents a generalized discretized model of a linear rotor which is axially symmetric about its spin axis considering the dynamic equilibrium [35].
The nonlinear system with the time delays has been an active topic of research over the past decades [19,20,33,34]. In the present system, the main sources of instability are due to non-conservative internal damping and time delay induced source loading term. The nonlinear interaction between the DC motor and the system may be represented by the following equation: M; D and K are symmetric and positive definite diagonal matrices, H is an anti-symmetric circulatory matrix [35] i:e:; H ¼ ÀH T ð Þwhich may destabilize the system involving material damping R i ð Þ [36], q is a 2 Â 1 column vector of generalized coordinates. The material damping in rotor dynamics plays a key role in determining the stability of the system. It is considered to be a function of past history of loading, strain rate and temperature of the system. Owing to the presence of R i ; the follower types circulatory forces F CH ð Þ appear in the source loading (see Eq. 4) is nonconservative in nature as r Â F CH 6 ¼ 0; causes selfexcited vibration, h represents the angular rotation of the eccentric disk. On the right-hand side of Eq. (1), the second term, F c t À s ð Þ represents control force provided by the AMB actuator with time delay s ð Þ: In the RHS of Eq. (2), T m and T L represent the torque supplied by the DC motor, and the nonlinear source Phase corresponding to initial position of mass centre of the disk  1) and (2) are presented in Table 1.

AMB model with time delay
AMB system suspends rotor through electromagnetic force, thereby eliminating physical contact with the bearing. A schematic diagram of AMB system with time delay is shown in Fig. 2. A Hall-effect-based proximity sensor senses the rotor displacement and sends the signal as a negative-feedback to the controller. The controller, based on a control law, generates the control current which then passes through a time delay block to incorporate a delay. Then the time-delayed signal gets amplified by the power amplifier. Following, control force is built up in the electromagnetic actuator which applies on the rotor shaft.
In the present study, the control currents given by the AMB system along the x and y direction according to Proportional-Derivative (PD) control scheme are expressed as: Þmay be expressed as: where K AMB ¼ ÀK u þ K i K a K p K s and C AMB ¼ The AMB parameters are given in Table 2. Details of the derivation are given in [35].
Considering a small time delay present in the digitally controlled AMB actuators and using Taylor series expansion as given in [37], the displacement and the velocity feedback terms given in Eq. (3) may be approximated as: Making use of Eq. (4), Eq. (1) & (2) makes Eq. (1) into a two-dimensional ODE as follows: where The RHS terms in Eq. (2) i.e., the torque supplied T m ð Þ by the DC motor, and the nonlinear source loading torque T L ð Þ given in Eq.
(2) may be further expressed as:   (2)) symmetry [38,39]. Hence a self-symmetric solution under SO(2) may be expressed as The steady-state synchronous whirl amplitude F 0 ð Þ may be expressed as: where C d ¼ R e À sK AMB þ C AMB ð Þand the phase angle, b ¼ tan À1 y=x ð ÞÀx r t ð Þ : Here, synchronous whirl amplitude is not influenced by the material damping as the speed of the non-ideal energy source (i.e., DC motor) is assumed to be constant. It is well known that for a critical circulatory parameter i:e:; c crit ¼ x r R i ; c crit [ 0 ð Þ [40] the energy supplied to system W m ð Þ equals to the energy dissipated by the system W d ð Þ. Applying this condition on the previous equations [41], a polynomial of 5 th order is obtained as follows: It is obvious that not all the roots of Eq. (9) give stable solutions. Using the approach given in [42] which states that a stable energy state exists if and only if W m À W d ð Þhappens to be a decreasing function for a critical value of x r say x rc . This criterion ensures possible existence of a locally minimum energy state and every root satisfying the above criterion must lie on the stable branch. Moreover, the term given in Eq. (9) 2MK À C 2 d À Á [ 0 implies the number of  3 Influence of time delay and bias current on attenuation of the Sommerfeld effect Using the system parameters [24] (see Table 3) and making use of Eq. (9) along with the minimum energy state criterion as mentioned above, the following numerical simulations are carried out. The result presented in Fig. 3 is in good agreement with [24] in which during coast up phase, the nondimensional whirl amplitude b ¼ Bx 2 n =g À Á jumps from a higher value ('b') to a lower value ('d') nearer to resonance, upon exceeding a critical input power while the non-dimensional rotor spin speed Þjumps simultaneously from a lower ('b') to higher value ('d') with no AMB and no time   delay i.e., i a ¼ 0 s ¼ 0: Similarly, during coast down, the amplitude jumps to a higher value ('a') and from 'c', whereas the rotor speed goes to a lower value at ('a'). These jumps are also illustrated in b vs V * and x Ã vs V * as shown in Fig. 3b, c, respectively, where V * is non-dimensional voltage, V * = V s = l m x n ð Þ: The blue and red branches as shown in the following figures represent the stable and unstable branches, respectively. Figure 4a, b illustrates the attenuation of the Sommerfeld effect as time delay decreases for some fixed bias current. The similar phenomena are observed when the non-dimensional spin speed amplitude and versus non-dimensional supply voltage as shown in Figs. 5 and 6, respectively, for several time delays with a given bias current. The transient analysis shown in Fig. 8a confirms the steady results illustrated in Fig. 4a, b. On the other hand, if the time delay is fixed, the Sommerfeld effect is found to be attenuated as the bias current decreases (see Fig. 7a, b) which is contrary to the results reported in [24] wherein the Sommerfeld effect is found to be reduced gradually as the bias current increases for the same non-ideal system without time delay. This result suggests that the AMB system with time delay is not only more realistic but also advantageous as it requires very small current to suppress the Sommerfeld effect. The same result is also true for the rotor speed and amplitude versus supply voltage graphs. The transient analysis illustrated in Fig. 8b confirms the above result. The transient study shown in Fig. 9 is found to be in good agreement with the steady-state result.

Bifurcation analysis
The study of instability phenomena characterized with nonlinear jumps seems to be more realistic and convincing when time delay is considered. Delayinduced bifurcation studies in a complex dynamical system reveal a couple of interesting facts. In the following sub-sections, numerical experimentations of saddle-node, Hopf and trans-critical bifurcations are carried out systematically with the help of root loci technique [45].

Root locus method
Using a fifth order polynomial of x r given in Eq. (9), one may write the following transfer function between a fictitious input and output pair as: where s is a complex variable,N s ð Þ and D s ð Þ are numerator and denominator, respectively, and The root-loci plot (see Fig. 10a) clearly shows how the closed loop poles vary with the system gain parameter i.e., supply voltage V s ð Þ in the complex splane. This diagram also confirms the existence of the Sommerfeld effect through the presence of multiple roots between the break-away marked as 'b' and break-in marked as 'a' point with gain parameter V s ð Þ 463 V and 168 V, respectively, for i 0 ¼ 0.6 A and s ¼ 0:001 s. It is well known from the linear control theory that the break-away (with maximum gain) and break-in (with minimum gain) can be obtained using the condition dV s =ds ¼ 0 ð Þ : The points 'b 0 ' and 'a 0 ' in the root loci plot (as shown in Fig. 10a) correspond to the points 'b'and 'a', respectively, as shown in Fig. 10b where the jumps are shown by the dash line during the coast up and coast down, respectively.

Saddle-node bifurcation
Considering the bias current i 0 ð Þ as a bifurcation parameter with fixed time delay, the distance between the break-away and break-in point found to be reduced. Finally, these two points are found to be merged into a single point known as bifurcation point by annihilating each other when bias current i 0 ð Þ attains 4.598 A with the same time delay (i.e., s ¼ 0.001 s) with supply voltage V s ð Þ is 437 V shown Fig. 11a. This kind of bifurcation is referred to as cyclic saddle node bifurcation [46]. The complete cessation of the Sommerfeld effect is observed when i 0 [ 4.598 A as shown in Fig. 11b.
On the other hand, considering the time delay as bifurcation parameter, the saddle node bifurcation occurs when time delay attains its critical value i:e:; s c ¼ 0:000393 s ð Þ for fixed bias current shown in Fig. 12a. The Sommerfeld effect would disappear completely when s\0:000393 s: Figure 12b illustrates the same phenomena for a much lower value of bias current, the saddle-node appears when the critical time delay is found to be much lower i.e., s c ¼ 0:000052 s: These critical values of time delays are negligibly small for a digitally controlled AMB system and not feasible. The above-mentioned numerical experimentation reveals that the saddle-node bifurcation is not practically achievable considering the time delay as a bifurcation parameter because the critical time delays are found to be negligibly small and hence not practicable for relatively smaller fixed bias current.

Hopf bifurcation
Using the transfer function given in Eq. (10) with supply voltage V s ð Þ as a gain parameter, the following fifth-order polynomial can be constructed as: where According to the Routh-Hurwitz stability criteria [45], the vibratory response is stable if all nonzero a i in Eq. (11) have the same positive or negative sign (necessary condition) and the corresponding all the Hurwitz determinants must be positive D i ; i ¼ 1; 2; ::; 5 ð Þ (sufficient condition) which implies, As the time delay s ð Þ exceeds a critical value s c ð Þ; a pair of complex conjugate roots will cross the imaginary axis which leads to Hopf bifurcation with multiplicity of two. On substituting s ¼ jx in Eq. (11) and  Fig. 13a, the Hopf bifurcation is observed as two complex conjugate pairs are seen to cross the imaginary axis simultaneously when the time delay becomes 0.4361 s for a fixed bias current 0.45 A. The corresponding phase plot as shown in Fig. 13b confirms the existence of Hopf as a typical wire framed paraboloid [47,48] is formed. This paraboloid represents a limit cycle with time delay as a bifurcation parameter representing the third dimension. This unstable limit cycle confirms the presence of Hopf as the solution goes to infinity. A couple of similar kind of Hopf bifurcations may be obtained for different critical time delays as illustrated in Fig. 14a, b.

Trans-critical bifurcation
Another kind of bifurcation is observed through the root locus as shown in Fig. 15a. A critical time delay s c ð Þ ¼ 0.002961 s is obtained numerically with a given bias current i 0 ð Þ = 1.2 A for a certain condition, C 2 d [ 4MK: At critical value of the time delay, a discontinuous line is seen in the left portion of the frequency response curve as given in Fig. 15b which clearly a sign of inception of instability occurs at s c owing to the presence of a pair of complex conjugate poles with positive real parts lying on the right half plane of the root locus as shown in Fig. 15a. For s\s c ; the Sommerfeld effect is seen to be present in the form of jump phenomena where in near to the critical speed, the amplitude jumps to a lower value and the rotor speed jumps to a much higher value as shown Fig. 16b corresponds to a root locus plot shown in Fig. 16a. A similar kind of jump is discussed in Fig. 10 in which the break-in and break-away points are illustrated corresponding to minimum and maximum gain, respectively. On the contrary, when s [ s c ; the jump phenomena still exist but in the reverse manner i.e., at the very outset, the jump occurs leading to a discontinuous region marked in red color in the frequency response diagram as shown in Fig. 17b where the steady state solution does not exist and the value of damping is found to be negative causing self-excited vibration. Figure 17a shows the root-locus diagram corresponds to Fig. 17b also confirms the switching of stability as the nature of the root loci is just appeared in a reverse manner relative to Fig. 16a. Owing to presence of negative damping at the beginning of run up, the rotor system attains its instability threshold at the very outset. Moreover, the break-in and breakaway points are also found to be swapped with each other. This kind of switching of stability refers to as trans-critical bifurcation [49] as the corresponding break-in and break-away points will never destroy each other but switching the stability as the time delay (i.e., bifurcation parameter) becomes greater or lesser than its critical value s c ð Þ.

Conclusion and future work
A novel approach is presented in this paper which reveals the influence of time delay on the complex dynamics of instability marked with nonlinear jumps exhibited by a non-ideal eccentric rotor with timedelayed AMB system. The analytical solution obtained through an energy balance confirms that the time delay indeed influences the attenuation of the Sommerfeld (i.e., nonlinear jumps) effect with fixed bias current. It is evident from the frequency response diagram that the Sommerfeld effect starts attenuating as the time delay decreases. On fixing the time delay, the instability effect is also found to be attenuated as the bias current decreases. This observation reveals that the time-delayed AMB system is more economic compared to an AMB system without time delay aiming to suppress the Sommerfeld effect as in the latter case, the instability starts attenuating only when the bias current increases [24]. The transient analyses confirm the above-mentioned results obtained analytically. In addition, three different bifurcations are studied numerically using root-loci technique with time delay as a bifurcation parameter. The saddle node bifurcation occurs when the bias current attains a critical value for a given time delay at which the Sommerfeld effect is found to be disappeared completely. On the other hand, saddle-node also occurs at a specific critical time-delay with a given bias current. But the critical time delay is very small and hence not practicable for a digitally controlled AMB system. Following, Hopf bifurcation study reveals the existence of multiple Hopf points with the formation of a wire framed paraboloid in the phase-portrait which destabilizes the system further. Switching of stability is also observed at a critical time delay obtained with a specified condition. This switching of stability confirms the existence of trans-critical bifurcation. The present model of the rotor system with the inclusion of fractional order AMB with variable time delay along with the prediction of chaotic motion can be envisioned as a future work.
Funding This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.
Data availability Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

Declarations
Conflict of interest The author declares no potential conflict of interest concerning the research, authorship, and/or publication of this article.