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Tuning transient dynamics by induced modal interaction in mechatronic systems
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Tuning transient dynamics by induced modal interaction in mechatronic systemsR Fadi Dohnal Department of Biomedical Computer Science and Mechatronics, Division for Mechatronics Lienz, Private University for Health Sciences, Medical Informatics and Technology (UMIT), Lienz, 9900, Austria
a r t i c l e
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Article history: Received 1 September 2015 Revised 17 March 2017 Accepted 27 May 2017 Available online xxx Keywords: Semi-active vibration control Modal energy transfer Parametric anti-resonance
a b s t r a c t Introducing time-periodicity in one or more system parameters may lead, in general, to a dangerous and well-known parametric resonance. In contrast to such a resonance, a properly tuned time-periodicity is capable of transferring energy between vibration modes. Time-periodicity in combination with system damping is capable of efficiently extracting vibrational energy from the system and of amplifying the existing damping affecting transient vibrations. Operating the system at such a specific time-periodicity, the system is tuned at a parametric anti-resonance. The present contribution outlines the basic physical interpretation of this concept and summarises the experimental validation for different mechatronic systems. Starting with a theoretical performance measure, all experiments related to this concept are compared qualitatively and the two most successful implementations are discussed in more detail. © 2017 Elsevier Ltd. All rights reserved.
1. Introduction Transient vibrations in mechanical systems are a common problem in engineering, see e.g. [1–3]. All concepts to tackle this problem can be classified as passive, semi-active, active or hybrid means. For passive concepts, the system parameters are designed once and do not change during operation of the vibrating structure. Design parameters for handling transient vibrations are damping as well as mass and stiffness parameters. One class of passive concepts is re-designing the mass and stiffness distribution in order to shift the natural frequencies outside from the frequency range of operation or to tune corresponding vibration modes to acceptable deformation shapes. Once the main system design is set, the last passive possibility to affect the transient vibration is the choice of a proper damping. Semi-active concepts summarise concepts that change system parameters actively. For example, utilising electro- or magnetorheological fluids, the force transferred between the fluid and the system depends mainly on the effective damping which is set by an external electric or magnetic field. Active concepts incorporate real-time control of the force acting on the system directly. Finally, hybrid concepts combine features of all concepts. This contribution discusses a semi-active concept, the parametric vibration absorption or parametric anti-resonance.
R This paper is dedicated to Aleš Tondl, the pioneer of parametric anti-resonance. The content of this paper was presented as a keynote during the 3rd International ´ Conference MECHATRONICS – Ideas for Industrial Applications in Gdansk, Poland, May 11–13, 2015. E-mail address: [email protected]
In contrast to the classic passive dynamic vibration absorber, the parametric vibration absorption does not need an additional subsystem consisting of mass, stiffness and damping elements. Knowing only the natural frequencies of the system of focus, a parametric vibration absorption is implemented by varying one or more system parameters periodically in time. Tuned properly, a parametric anti -resonance is created which couples the vibration modes of the underlying Hamiltonian system and enables an energy transfer between a lightly and a strongly damped vibration mode. This targeted energy transfer has been observed only for strongly nonlinear vibration absorbers outlined in [4] so far. However, parametric vibration absorption does not need nonlinearity and works for purely linear systems, too. Systems of differential equations with time-periodic coefficients, also termed parametrically excited systems, have been the focus of research since many decades. Parametrically excited vibrations occur if one or more parameters of the equations of motion are not constant but are described explicitly by a function of time; periodic and independent of the system motion. Classic examples are the pendulum with periodically varying length or periodically moving pivot point leading to the famous Mathieu, Hill or Meissner equation. Parametrically excited systems and structures have been studied extensively in the past because of the interesting phenomena that occur in such systems. A parametrically excited system may exhibit a destabilising parametric resonance if at least one system parameter is varied close to a parametric excitation frequency (see e.g. [5–7])
ν kl,n =
|ωk ∓ ωl | n
,
k, l, n = 1, 2, . . .
(1)
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Please cite this article as: F. Dohnal, Tuning transient dynamics by induced modal interaction in mechatronic systems, Mechatronics (2017), http://dx.doi.org/10.1016/j.mechatronics.2017.05.010
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Herein ωk and ωl denote the k-th and l-th angular frequency of the underlying undamped system with constant coefficients (Hamiltonian system). The denominator n represents the order of the parametric resonance. This frequency is called a principal parametric resonance for k = l, and a parametric combination resonance for k = l. The systems analysed so far showed that the first order resonances, n = 1, are most significant, see e.g. [5–7]. Fig. 1. Linear example system possessing four degrees of freedom xj .
2. Parametric anti-resonance Almost all investigations on the dynamics of a single or of coupled differential equations having time-periodic coefficients are focusing on the resonant behaviour of parametric excitation; see e.g. [5–9]. The main focus there was to investigate the destabilising effect of parametric excitation, i.e. the instability boundary curves in the domain of system parameters. The non-resonant cases were not considered since they do not compromise the operation of a machine or do not amplify the system vibration in sensor applications. The benefit of introducing a non-resonant parametric excitation in an unstable coupled system was first highlighted in the pioneering work by Tondl in [10]. He showed that an unstable self-excited system can be stabilised by introducing a timeharmonic stiffness coefficient, a parametric excitation, at a specific parametric combination resonance frequency. Since this occurs at the frequency of a parametric resonance, the mechanism was named parametric anti-resonance. Tondl’s work initiated theoretical studies of several self-excited two-mass systems with time-harmonic stiffness using numerical simulation in [11,12] and analytical studies in [13,14]. The concept was enhanced to systems with time-harmonic damping [15] or inertia coefficients [16]. Almost all these investigations considered the variation of a single system parameter. Comprehensive analytical and numerical investigations for time-harmonic variations of a single as well as multiple physical parameters - a simultaneous variation of stiffness, damping and inertia coefficients - were carried out in [12,17]. This is especially relevant for implementations because in general it is rather difficult to control a single physical property without changing other properties in a specific system. For a long time it had been believed that parametric antiresonance can occur only in combination with self-excitation, see [11] and the review article [19]. However, the stabilising mechanism of a parametric anti-resonance was identified as the coupling of eigenvalues (vibration modes) of the underlying system with constant coefficients, see [20]. This qualitative interpretation drastically enlarged the applicability of the concept of damping by parametric excitation, since a properly chosen parametric antiresonance not only stabilises an already unstable system (stabilisation by parametric excitation) but is also capable of enhancing the already existing damping; independent of self-excitation being present or not. Furthermore, introducing a parametric antiresonance in a system with multiple degrees of freedom offers the unique possibility of coupling only two of the many vibration modes of the original system and induces an energy transfer between these selected vibration modes while the remaining vibration modes stay decoupled, at least in first order approximation [17,20]. The main theoretical contributions with respect to parametric anti-resonances in this context can be found in [10–12,17]. Recent reviews on the parametric anti-resonance can be found in [17,18] from which some findings are summarised in the following.
3. Basic physical interpretation The physical interpretation of a parametric anti-resonance as an induced modal interaction, which incorporates a modal energy
Fig. 2. Time series of modal displacements for the undamped system with cascaded time-periodicity (taken from [21]).
transfer, was proposed in [19] and the energy flow has been considered in [12,21]. This physical interpretation was rediscovered in [22]. In the following, the basic working principle of a parametric anti-resonance is outlined on a simple chain mass system with four degrees of freedom for which different parametric antiresonances are realised by a harmonic stiffness variation, see Fig. 1. The physical interpretation proposed in [19] of coupling two of the many vibration modes of the underlying constant system is discussed in means of physical and modal time histories according to [17]. This interpretation leads intuitively to the calculation of the energy flow of each vibration mode and leads to clear physical insight of how parametric anti-resonances work. Employing a properly tuned, time-periodic variation of at least one system parameter enables two things: 1. Inducing an energy transfer between only two of the many vibration modes of the underlying system with constant coefficients. 2. Increasing the energy dissipation due to improved action of the already existing damping element. It has to be highlighted that vibration mitigation (energy dissipation) is only achieved due to the interaction of a parametric anti-resonance and the existing system damping. Both components are needed in order to realise a desired vibration mitigation. It was shown in [17,20,21] that choosing specific parametric anti-resonance frequencies induces an energy transfer between different modes. This knowledge is exploited by introducing a cascaded time-periodic change of the stiffness element k1 (t) in Fig. 1 as shown in Fig. 2 on the top. For the present study only n = 1 is considered which in general induces the strongest modal interaction. Starting with the undamped system with deactivated time-periodicity and initial conditions corresponding to the first mode, the stiffness coefficient is kept constant during the first 0.5 s. Then, the parametric anti-resonance frequency ν 12 in (1) is
Please cite this article as: F. Dohnal, Tuning transient dynamics by induced modal interaction in mechatronic systems, Mechatronics (2017), http://dx.doi.org/10.1016/j.mechatronics.2017.05.010
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Fig. 3. Time series of physical displacements for the undamped system with cascaded time-periodicity (taken from [21]).
activated with a 30% amplitude variation of the stiffness parameter. This induces an energy transfer between the first and the second vibration mode. At the time instant of 2.2 s most of the vibration energy of the first mode has been transferred to the second mode. Now, the parametric anti-resonance is instantly switched to another frequency ν 23 in (1). This induces an energy transfer between the second and the third mode as clearly visualised in the modal displacements in Fig. 2. At the time instant of 2.5 s most of the vibration energy has been transferred to the third mode and the time-periodicity is switched off again. Such a cascade of parametric anti-resonances enables the transfer of low-frequency vibration energy into high frequency vibration energy. The corresponding physical displacements are shown in Fig. 3. Evaluating the time history of the vibrational energy for this example shows that the system gains additional energy through the time-dependent stiffness element. The maximum damping that is achievable by induced parametric anti-resonance can be approximated by perturbation theory as, see [17] for more details, 0 0 DPA max = Dk + Dl
ωl , ωk
ωk < ωl .
(2)
Herein, D0i are the modal damping coefficient and k, l the indices of the angular frequencies of the modes k and l of the underlying system with constant coefficients.
3
System 2, 2006: Artificial two-mass system: Air track with two gliding rigid bodies connected by helical springs. One body is selfexcited electromagnetically while the second body is attached to a string with time-periodic tension [23]. System 3, 2006: Artificial two-mass system: two rigid bodies connected by helical springs and hanging on long inextensible wires. One body is connected to a long-stroke (up to 40 mm) electromagnetic actuator driven by a variable current. The electromagnetic device was designed by the author in [24]. Successful tests were reported in [25]. System 4, 2008: Flexible cantilever under transversal force realised by a semi-active electromagnetic mount [26]. System 5, 2009: Flexible cantilever under mainly axial force realised by a string which tension is controlled by a piezoelectric actuator [27]. This device was built for testing an active control concept [30] and was reused to resemble the system in [31]. System 6, 2010: Flexible rotor with multiple disks supported by active magnetic bearings [28]. System 7, 2012: Jeffcott rotor supported by two active magnetic bearings [17]. All these experiments are benchmarked in [17] (extended compared to [18]) and listed in Table 1 that is based on an analytically derived performance index
αkl = max(ε )
Qkl Qlk
4ωk ωl kk ll
(3)
Herein, max(ε ) expresses the maximum achievable time-periodic variation of a stiffness parameter in terms of amplitude/constant value, ii are related to the modal damping coefficients in (2) and Qkl are the stiffness-related cross-couplings between different modes. This performance measure is derived from an analytical approximation using a perturbation technique, see [18] for more details, and enables an objective comparison between the effectiveness of different experimental configurations. This comparison helps to identify potentials and limitations in the application field of this concept and identifies clearly which of the presently existing system configurations need to be redesigned. The most successful configurations (System 4 and System 7) are well designed since they are close to the optimally tuned System 1. These configurations were realised by electromagnetic variablestiffness actuators driven by a periodic open-loop control at a single frequency, the parametric anti-resonance frequency in (1). These are the most promising configurations for potential applications. The remaining system configurations could perform better and a redesign should be taken into consideration. In the following, more details are given for the successful configurations System 4 [26] and System 7 [17].
4.1. Details on System 4 [26] 4. Experimental verification In recent years several attempts were undertaken to verify the existence of parametric anti-resonances experimentally. Starting with analogue computer calculations [10], followed by discrete two-mass systems of an artificial nature [23–25], the concept of damping by parametric excitation was confirmed for simple continuous flexible cantilevers [26,27] or on flexible rotor supported by two active magnetic bearings [17,28]. Recently, a piezoactuated journal bearing was built to test the influence on the onset of bearing instability [29]. A chronological order of experimental verification of a parametric anti-resonance is listed here [17]: System 1, 1998: Analogue computer calculation by Tondl [10] of a nonlinear self-excited two-mass system exhibiting parametric anti-resonance.
Fig. 4 shows the experimental realisation of the system, a highly flexible aluminium cantilever (390 × 10 × 1 mm) with an additional electromagnetic mount, see Fig. 4 for more details. The cantilever performs lateral vibrations in the horizontal plane such that gravity is not effective. The parametric anti-resonance is implemented by an electromagnetic actuator that consists of two pairs of permanent magnets and electromagnets. Different to System 3, the electromagnetic actuator consists of permanent magnets moving between two fixed electromagnets and no ferro-magnetic housing is used. The distance between the electromagnets can be adjusted in order to achieve an almost linear force characteristic within the deflection range of the beam. The permanent magnets are attached to the beam while the electromagnets were fixed. The magnetisation was chosen such that repulsive forces, i.e. forces that simulate a spring, were generated between the permanent
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F. Dohnal / Mechatronics 000 (2017) 1–7 Table 1 Performance measure (3) of all existing experiments on parametric anti-resonance (taken from [17], extended from [18]). System configuration
Performance index
System 1, 1998: Tondl [10]: Analog computer calculation of self-excited two-mass system System 2, 2006: Dohnal et al. [23] System 3, 2006: Dohnal et al. [24,25] System 4, 2008: Dohnal and Mace [26] System 5, 2009: Ecker and Pumhössel [27] System 6, 2010: Dohnal and Markert [28] System 7, 2012: Dohnal [17]
α lk
αlk /αlkTondl
40.0 1.6 1.0 35.0 5.3 6.7 25.3
(100%)% (4%) (2%) (88%) (13%) (17%) (63%)
Fig. 5. Transient responses of System 4 (taken from [18]): (top) effective damping at a parametric excitation amplitude of ε = 80%, (bottom) comparison of the nominal (unactuated) system and actuated mount at 73 1/s.
Fig. 4. Configuration of System 4: Cantilever with additional electromagnetic mount (taken from [18]): (top) sketch, (bottom) setup, view from top.
magnets and electromagnets driving the beam to its equilibrium position. These forces depend on the beam deflection and the magnetic field. The magnetic field can be changed in an arbitrary manner by the current provided to the electromagnets. A parametric anti-resonance is realised by driving the electromagnets by a predefined periodic current in an open-loop manner. Both coils are fed by a single amplifier such that exactly the same current flows through both coils. An electrically operated latch was used to hold the tip of the cantilever at same initial position and released to produce a transient vibration. A series of experiments was performed by varying the frequency ν of the stiffness change at the electromagnetic mount. Starting from an initial displacement of 20 mm at the tip position, the transient displacement of a point close to the tip was measured by a laser displacement transducer. The effect of the parametric anti-resonance (induced modal interaction between modes 1 and 2 at ν 12 in Eq. (1)) is observed clearly in the frequency interval between 50 to 83 1/s in Fig. 5(a). In this range the damping of the system is artificially increased, hence, amplified. The decay reaches its minimum value of 8.05 1/s
at the optimum frequency of 73 1/s. Compared to the system with a constant stiffness, k(t ) = k0 , this corresponds to an amplification of damping by a factor of 4.5. Two samples of the transient time series at this frequency are plotted in Fig. 1(b). The dashed line represents the transient motion of the beam when the additional mount is inactive (constant mount stiffness k(t ) = k0 ). The solid line corresponds to the transient motion for active open-loop variation at a stiffness change of 80% and a frequency of ν 12 = 73 1/s. By employing this time-periodicity, the equilibrium position of the beam is reached after four vibration cycles. 4.2. Details on System 7 [17] The reason why System 6 in Table 1 shows a low performance index is because the main parametric anti-resonance was achieved at ω3 − ω2 . This means that the vibration modes 2 and 3 of the underlying nominal system are exchanging energy, however, the lightly damped first vibration mode is not affected by this. Parametric anti-resonances related to the first mode were all shadowed by parametric resonances. This is mainly due to the fact that the active magnetic bearings (AMBs) in System 6 were located close to the nodes of the first vibration mode which became less controllable. A new rotor test rig was designed, optimised and built such that its performance measure is 4 times higher than the one for System 6. The built Jeffcott rotor (System 7, see [17] for more details) is depicted in Fig. 6: A slender, flexible rotor shaft holding a single, centred, unbalanced disk. The shaft is supported by two active magnetic bearings (AMB1, AMB2) and is rotating at a constant
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Fig. 7. Active magnetic bearing system: (top) control loop, (bottom) setup. Fig. 6. Configuration of System 7: Jeffcott rotor supported by two active magnetic bearings (taken from [17]): (top) sketch, (bottom) setup.
rotational speed = 60 1/s provided by a driving motor. The total length of the shaft is 680 mm. The rotor system was balanced and a known unbalance was placed at the disk D. The electromagnetic forces generated in the AMBs depend on the rotor deflection and the magnetic field. The magnetic field can be changed in a wide range by currents provided to the electromagnets. The actual position of the rotor shaft is measured by inductive sensors (two for each radial direction). These signals are processed by the real-time controller hardware dSpace, which implements decentralised PID controllers to regulate the currents provided by power amplifiers to each of the electromagnets and, hence, to levitate the rotor, see Fig. [7] and [17]. The time-periodicity is implemented via periodically varying the proportional action of the PID controller in an open-loop control,
kP (t ) = kP (1 + ε sin ν t ).
(4)
This realises a periodic change in the active bearing stiffness which is implemented in both AMBs. Discretising the continuous shaft by finite beam elements, the time histories are calculated numerically for a sample speed of 600 rpm at a control amplitude of ε = 10% in (5) and summarised in the contour plot in Fig. 8. The levels are chosen starting slightly above the unbalance amplitude of the underlying constant system. Light areas depict small deviations from the unbalance response of the disk deflection |rD| and dark areas large deviations. The main parametric anti-resonance is clearly indicated at the white region close to the frequency 170 1/s ≈ ω3 − ω1 . Introducing this timeperiodicity induced an energy transfer between the first and the third rotor mode which results in an increase of the effective system damping. The time histories at this strong parametric anti-resonance are shown in Fig. 9 an amplitude of ε = 20% in (5). Experimental results are presented to underpin the theoretical findings from above. The rotor is levitated at 600 rpm in the AMBs by the PID controllers. Additionally, a periodic open-loop control varies
Fig. 8. Contour plot of time histories of disk D with periodic open-loop control at ε = 10% in (5) (taken from [17]).
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ficient experimental configurations are discussed in more detail showing the operation principle and impact of an induced timeperiodicity - an artificial amplification of existing system damping. The additional artificial damping provided to the system becomes significant and most effective for vibration mitigation of the lower vibration modes during transient vibrations. The present overview shows that the proposed concept was validated by different mechatronic prototypes. A proper tuning of the parametric excitation is needed in order to achieve increasing the effective system damping by induced modal interaction. The concept was validated theoretically, numerically and experimentally and is ready for industrial implementation.
References
Fig. 9. Time histories of disk D after ping at AMBs without and with periodic open-loop control at parametric anti-resonance ν ≈ ω3 − ω1 = ν 13 and ε = 20% in (1) (taken from [17]): (top) numerical calculation, (bottom) experiment.
the proportional action according to (5) at the parametric antiresonance frequency of 170 1/s. An impulse is applied to the rotor system is realised by adding an impulse-like current to the controller current which leads to transient vibrations. Evaluating the logarithmic decrement of these time histories gives a good measure of the effective damping present in the rotor system. For the present system configuration the logarithmic decrement can be enhanced by a factor of 3.5 (numerical prediction and experiment). The beneficial effect of a parametric antiresonance is confirmed. Without the proposed periodic open-loop control (ε = 0), the system damping can be increased simply by the active damping in the AMBs introduced by the PID controllers, too. However, this increase is limited by the level of measurement noise in the control loop and its amplification by the derivative action. The control parameters for the PID controllers are optimised for lowest additive noise sensitivity in the closed loop according to the procedure outlined in [32]. Starting from these PID values, the derivative action can be amplified maximally by a factor of 1.6 for System 7. Introducing the proposed periodic open-loop control at the control settings in Fig. 9, allows an increase of the effective system damping above this physical limitation. The maximum amplification factor for the effective damping that was achievable for the present setup experimentally is 1.9. This value lies above the limit realisable by conventional active damping. Further investigations are needed on how to increase this factor. 5. Conclusions A physical interpretation on the working principle of a parametric anti-resonance and its experimental verifications were demonstrated. It was highlighted that introducing a specific timeperiodicity in a flexible structure can induce modal energy exchange which exhibits enhanced damping properties. The improved dissipation mechanism is owed to modal coupling which creates higher frequency components. The possibility of a cascaded energy transfer was shown. Tuning the time-periodicity to a sufficiently strong parametric anti-resonance leads to a coupling of only two of the many vibration modes which results in an induced energy transfer between mainly between these modes. All existing experimental verifications of this concept are benchmarked against a performance measure. The two most ef-
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Tuning transient dynamics by induced modal interaction in mechatronic systemsR Fadi Dohnal Department of Biomedical Computer Science and Mechatronics, Division for Mechatronics Lienz, Private University for Health Sciences, Medical Informatics and Technology (UMIT), Lienz, 9900, Austria
a r t i c l e
i n f o
Article history: Received 1 September 2015 Revised 17 March 2017 Accepted 27 May 2017 Available online xxx Keywords: Semi-active vibration control Modal energy transfer Parametric anti-resonance
a b s t r a c t Introducing time-periodicity in one or more system parameters may lead, in general, to a dangerous and well-known parametric resonance. In contrast to such a resonance, a properly tuned time-periodicity is capable of transferring energy between vibration modes. Time-periodicity in combination with system damping is capable of efficiently extracting vibrational energy from the system and of amplifying the existing damping affecting transient vibrations. Operating the system at such a specific time-periodicity, the system is tuned at a parametric anti-resonance. The present contribution outlines the basic physical interpretation of this concept and summarises the experimental validation for different mechatronic systems. Starting with a theoretical performance measure, all experiments related to this concept are compared qualitatively and the two most successful implementations are discussed in more detail. © 2017 Elsevier Ltd. All rights reserved.
1. Introduction Transient vibrations in mechanical systems are a common problem in engineering, see e.g. [1–3]. All concepts to tackle this problem can be classified as passive, semi-active, active or hybrid means. For passive concepts, the system parameters are designed once and do not change during operation of the vibrating structure. Design parameters for handling transient vibrations are damping as well as mass and stiffness parameters. One class of passive concepts is re-designing the mass and stiffness distribution in order to shift the natural frequencies outside from the frequency range of operation or to tune corresponding vibration modes to acceptable deformation shapes. Once the main system design is set, the last passive possibility to affect the transient vibration is the choice of a proper damping. Semi-active concepts summarise concepts that change system parameters actively. For example, utilising electro- or magnetorheological fluids, the force transferred between the fluid and the system depends mainly on the effective damping which is set by an external electric or magnetic field. Active concepts incorporate real-time control of the force acting on the system directly. Finally, hybrid concepts combine features of all concepts. This contribution discusses a semi-active concept, the parametric vibration absorption or parametric anti-resonance.
R This paper is dedicated to Aleš Tondl, the pioneer of parametric anti-resonance. The content of this paper was presented as a keynote during the 3rd International ´ Conference MECHATRONICS – Ideas for Industrial Applications in Gdansk, Poland, May 11–13, 2015. E-mail address: [email protected]
In contrast to the classic passive dynamic vibration absorber, the parametric vibration absorption does not need an additional subsystem consisting of mass, stiffness and damping elements. Knowing only the natural frequencies of the system of focus, a parametric vibration absorption is implemented by varying one or more system parameters periodically in time. Tuned properly, a parametric anti -resonance is created which couples the vibration modes of the underlying Hamiltonian system and enables an energy transfer between a lightly and a strongly damped vibration mode. This targeted energy transfer has been observed only for strongly nonlinear vibration absorbers outlined in [4] so far. However, parametric vibration absorption does not need nonlinearity and works for purely linear systems, too. Systems of differential equations with time-periodic coefficients, also termed parametrically excited systems, have been the focus of research since many decades. Parametrically excited vibrations occur if one or more parameters of the equations of motion are not constant but are described explicitly by a function of time; periodic and independent of the system motion. Classic examples are the pendulum with periodically varying length or periodically moving pivot point leading to the famous Mathieu, Hill or Meissner equation. Parametrically excited systems and structures have been studied extensively in the past because of the interesting phenomena that occur in such systems. A parametrically excited system may exhibit a destabilising parametric resonance if at least one system parameter is varied close to a parametric excitation frequency (see e.g. [5–7])
ν kl,n =
|ωk ∓ ωl | n
,
k, l, n = 1, 2, . . .
(1)
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Please cite this article as: F. Dohnal, Tuning transient dynamics by induced modal interaction in mechatronic systems, Mechatronics (2017), http://dx.doi.org/10.1016/j.mechatronics.2017.05.010
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Herein ωk and ωl denote the k-th and l-th angular frequency of the underlying undamped system with constant coefficients (Hamiltonian system). The denominator n represents the order of the parametric resonance. This frequency is called a principal parametric resonance for k = l, and a parametric combination resonance for k = l. The systems analysed so far showed that the first order resonances, n = 1, are most significant, see e.g. [5–7]. Fig. 1. Linear example system possessing four degrees of freedom xj .
2. Parametric anti-resonance Almost all investigations on the dynamics of a single or of coupled differential equations having time-periodic coefficients are focusing on the resonant behaviour of parametric excitation; see e.g. [5–9]. The main focus there was to investigate the destabilising effect of parametric excitation, i.e. the instability boundary curves in the domain of system parameters. The non-resonant cases were not considered since they do not compromise the operation of a machine or do not amplify the system vibration in sensor applications. The benefit of introducing a non-resonant parametric excitation in an unstable coupled system was first highlighted in the pioneering work by Tondl in [10]. He showed that an unstable self-excited system can be stabilised by introducing a timeharmonic stiffness coefficient, a parametric excitation, at a specific parametric combination resonance frequency. Since this occurs at the frequency of a parametric resonance, the mechanism was named parametric anti-resonance. Tondl’s work initiated theoretical studies of several self-excited two-mass systems with time-harmonic stiffness using numerical simulation in [11,12] and analytical studies in [13,14]. The concept was enhanced to systems with time-harmonic damping [15] or inertia coefficients [16]. Almost all these investigations considered the variation of a single system parameter. Comprehensive analytical and numerical investigations for time-harmonic variations of a single as well as multiple physical parameters - a simultaneous variation of stiffness, damping and inertia coefficients - were carried out in [12,17]. This is especially relevant for implementations because in general it is rather difficult to control a single physical property without changing other properties in a specific system. For a long time it had been believed that parametric antiresonance can occur only in combination with self-excitation, see [11] and the review article [19]. However, the stabilising mechanism of a parametric anti-resonance was identified as the coupling of eigenvalues (vibration modes) of the underlying system with constant coefficients, see [20]. This qualitative interpretation drastically enlarged the applicability of the concept of damping by parametric excitation, since a properly chosen parametric antiresonance not only stabilises an already unstable system (stabilisation by parametric excitation) but is also capable of enhancing the already existing damping; independent of self-excitation being present or not. Furthermore, introducing a parametric antiresonance in a system with multiple degrees of freedom offers the unique possibility of coupling only two of the many vibration modes of the original system and induces an energy transfer between these selected vibration modes while the remaining vibration modes stay decoupled, at least in first order approximation [17,20]. The main theoretical contributions with respect to parametric anti-resonances in this context can be found in [10–12,17]. Recent reviews on the parametric anti-resonance can be found in [17,18] from which some findings are summarised in the following.
3. Basic physical interpretation The physical interpretation of a parametric anti-resonance as an induced modal interaction, which incorporates a modal energy
Fig. 2. Time series of modal displacements for the undamped system with cascaded time-periodicity (taken from [21]).
transfer, was proposed in [19] and the energy flow has been considered in [12,21]. This physical interpretation was rediscovered in [22]. In the following, the basic working principle of a parametric anti-resonance is outlined on a simple chain mass system with four degrees of freedom for which different parametric antiresonances are realised by a harmonic stiffness variation, see Fig. 1. The physical interpretation proposed in [19] of coupling two of the many vibration modes of the underlying constant system is discussed in means of physical and modal time histories according to [17]. This interpretation leads intuitively to the calculation of the energy flow of each vibration mode and leads to clear physical insight of how parametric anti-resonances work. Employing a properly tuned, time-periodic variation of at least one system parameter enables two things: 1. Inducing an energy transfer between only two of the many vibration modes of the underlying system with constant coefficients. 2. Increasing the energy dissipation due to improved action of the already existing damping element. It has to be highlighted that vibration mitigation (energy dissipation) is only achieved due to the interaction of a parametric anti-resonance and the existing system damping. Both components are needed in order to realise a desired vibration mitigation. It was shown in [17,20,21] that choosing specific parametric anti-resonance frequencies induces an energy transfer between different modes. This knowledge is exploited by introducing a cascaded time-periodic change of the stiffness element k1 (t) in Fig. 1 as shown in Fig. 2 on the top. For the present study only n = 1 is considered which in general induces the strongest modal interaction. Starting with the undamped system with deactivated time-periodicity and initial conditions corresponding to the first mode, the stiffness coefficient is kept constant during the first 0.5 s. Then, the parametric anti-resonance frequency ν 12 in (1) is
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Fig. 3. Time series of physical displacements for the undamped system with cascaded time-periodicity (taken from [21]).
activated with a 30% amplitude variation of the stiffness parameter. This induces an energy transfer between the first and the second vibration mode. At the time instant of 2.2 s most of the vibration energy of the first mode has been transferred to the second mode. Now, the parametric anti-resonance is instantly switched to another frequency ν 23 in (1). This induces an energy transfer between the second and the third mode as clearly visualised in the modal displacements in Fig. 2. At the time instant of 2.5 s most of the vibration energy has been transferred to the third mode and the time-periodicity is switched off again. Such a cascade of parametric anti-resonances enables the transfer of low-frequency vibration energy into high frequency vibration energy. The corresponding physical displacements are shown in Fig. 3. Evaluating the time history of the vibrational energy for this example shows that the system gains additional energy through the time-dependent stiffness element. The maximum damping that is achievable by induced parametric anti-resonance can be approximated by perturbation theory as, see [17] for more details, 0 0 DPA max = Dk + Dl
ωl , ωk
ωk < ωl .
(2)
Herein, D0i are the modal damping coefficient and k, l the indices of the angular frequencies of the modes k and l of the underlying system with constant coefficients.
3
System 2, 2006: Artificial two-mass system: Air track with two gliding rigid bodies connected by helical springs. One body is selfexcited electromagnetically while the second body is attached to a string with time-periodic tension [23]. System 3, 2006: Artificial two-mass system: two rigid bodies connected by helical springs and hanging on long inextensible wires. One body is connected to a long-stroke (up to 40 mm) electromagnetic actuator driven by a variable current. The electromagnetic device was designed by the author in [24]. Successful tests were reported in [25]. System 4, 2008: Flexible cantilever under transversal force realised by a semi-active electromagnetic mount [26]. System 5, 2009: Flexible cantilever under mainly axial force realised by a string which tension is controlled by a piezoelectric actuator [27]. This device was built for testing an active control concept [30] and was reused to resemble the system in [31]. System 6, 2010: Flexible rotor with multiple disks supported by active magnetic bearings [28]. System 7, 2012: Jeffcott rotor supported by two active magnetic bearings [17]. All these experiments are benchmarked in [17] (extended compared to [18]) and listed in Table 1 that is based on an analytically derived performance index
αkl = max(ε )
Qkl Qlk
4ωk ωl kk ll
(3)
Herein, max(ε ) expresses the maximum achievable time-periodic variation of a stiffness parameter in terms of amplitude/constant value, ii are related to the modal damping coefficients in (2) and Qkl are the stiffness-related cross-couplings between different modes. This performance measure is derived from an analytical approximation using a perturbation technique, see [18] for more details, and enables an objective comparison between the effectiveness of different experimental configurations. This comparison helps to identify potentials and limitations in the application field of this concept and identifies clearly which of the presently existing system configurations need to be redesigned. The most successful configurations (System 4 and System 7) are well designed since they are close to the optimally tuned System 1. These configurations were realised by electromagnetic variablestiffness actuators driven by a periodic open-loop control at a single frequency, the parametric anti-resonance frequency in (1). These are the most promising configurations for potential applications. The remaining system configurations could perform better and a redesign should be taken into consideration. In the following, more details are given for the successful configurations System 4 [26] and System 7 [17].
4.1. Details on System 4 [26] 4. Experimental verification In recent years several attempts were undertaken to verify the existence of parametric anti-resonances experimentally. Starting with analogue computer calculations [10], followed by discrete two-mass systems of an artificial nature [23–25], the concept of damping by parametric excitation was confirmed for simple continuous flexible cantilevers [26,27] or on flexible rotor supported by two active magnetic bearings [17,28]. Recently, a piezoactuated journal bearing was built to test the influence on the onset of bearing instability [29]. A chronological order of experimental verification of a parametric anti-resonance is listed here [17]: System 1, 1998: Analogue computer calculation by Tondl [10] of a nonlinear self-excited two-mass system exhibiting parametric anti-resonance.
Fig. 4 shows the experimental realisation of the system, a highly flexible aluminium cantilever (390 × 10 × 1 mm) with an additional electromagnetic mount, see Fig. 4 for more details. The cantilever performs lateral vibrations in the horizontal plane such that gravity is not effective. The parametric anti-resonance is implemented by an electromagnetic actuator that consists of two pairs of permanent magnets and electromagnets. Different to System 3, the electromagnetic actuator consists of permanent magnets moving between two fixed electromagnets and no ferro-magnetic housing is used. The distance between the electromagnets can be adjusted in order to achieve an almost linear force characteristic within the deflection range of the beam. The permanent magnets are attached to the beam while the electromagnets were fixed. The magnetisation was chosen such that repulsive forces, i.e. forces that simulate a spring, were generated between the permanent
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F. Dohnal / Mechatronics 000 (2017) 1–7 Table 1 Performance measure (3) of all existing experiments on parametric anti-resonance (taken from [17], extended from [18]). System configuration
Performance index
System 1, 1998: Tondl [10]: Analog computer calculation of self-excited two-mass system System 2, 2006: Dohnal et al. [23] System 3, 2006: Dohnal et al. [24,25] System 4, 2008: Dohnal and Mace [26] System 5, 2009: Ecker and Pumhössel [27] System 6, 2010: Dohnal and Markert [28] System 7, 2012: Dohnal [17]
α lk
αlk /αlkTondl
40.0 1.6 1.0 35.0 5.3 6.7 25.3
(100%)% (4%) (2%) (88%) (13%) (17%) (63%)
Fig. 5. Transient responses of System 4 (taken from [18]): (top) effective damping at a parametric excitation amplitude of ε = 80%, (bottom) comparison of the nominal (unactuated) system and actuated mount at 73 1/s.
Fig. 4. Configuration of System 4: Cantilever with additional electromagnetic mount (taken from [18]): (top) sketch, (bottom) setup, view from top.
magnets and electromagnets driving the beam to its equilibrium position. These forces depend on the beam deflection and the magnetic field. The magnetic field can be changed in an arbitrary manner by the current provided to the electromagnets. A parametric anti-resonance is realised by driving the electromagnets by a predefined periodic current in an open-loop manner. Both coils are fed by a single amplifier such that exactly the same current flows through both coils. An electrically operated latch was used to hold the tip of the cantilever at same initial position and released to produce a transient vibration. A series of experiments was performed by varying the frequency ν of the stiffness change at the electromagnetic mount. Starting from an initial displacement of 20 mm at the tip position, the transient displacement of a point close to the tip was measured by a laser displacement transducer. The effect of the parametric anti-resonance (induced modal interaction between modes 1 and 2 at ν 12 in Eq. (1)) is observed clearly in the frequency interval between 50 to 83 1/s in Fig. 5(a). In this range the damping of the system is artificially increased, hence, amplified. The decay reaches its minimum value of 8.05 1/s
at the optimum frequency of 73 1/s. Compared to the system with a constant stiffness, k(t ) = k0 , this corresponds to an amplification of damping by a factor of 4.5. Two samples of the transient time series at this frequency are plotted in Fig. 1(b). The dashed line represents the transient motion of the beam when the additional mount is inactive (constant mount stiffness k(t ) = k0 ). The solid line corresponds to the transient motion for active open-loop variation at a stiffness change of 80% and a frequency of ν 12 = 73 1/s. By employing this time-periodicity, the equilibrium position of the beam is reached after four vibration cycles. 4.2. Details on System 7 [17] The reason why System 6 in Table 1 shows a low performance index is because the main parametric anti-resonance was achieved at ω3 − ω2 . This means that the vibration modes 2 and 3 of the underlying nominal system are exchanging energy, however, the lightly damped first vibration mode is not affected by this. Parametric anti-resonances related to the first mode were all shadowed by parametric resonances. This is mainly due to the fact that the active magnetic bearings (AMBs) in System 6 were located close to the nodes of the first vibration mode which became less controllable. A new rotor test rig was designed, optimised and built such that its performance measure is 4 times higher than the one for System 6. The built Jeffcott rotor (System 7, see [17] for more details) is depicted in Fig. 6: A slender, flexible rotor shaft holding a single, centred, unbalanced disk. The shaft is supported by two active magnetic bearings (AMB1, AMB2) and is rotating at a constant
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Fig. 7. Active magnetic bearing system: (top) control loop, (bottom) setup. Fig. 6. Configuration of System 7: Jeffcott rotor supported by two active magnetic bearings (taken from [17]): (top) sketch, (bottom) setup.
rotational speed = 60 1/s provided by a driving motor. The total length of the shaft is 680 mm. The rotor system was balanced and a known unbalance was placed at the disk D. The electromagnetic forces generated in the AMBs depend on the rotor deflection and the magnetic field. The magnetic field can be changed in a wide range by currents provided to the electromagnets. The actual position of the rotor shaft is measured by inductive sensors (two for each radial direction). These signals are processed by the real-time controller hardware dSpace, which implements decentralised PID controllers to regulate the currents provided by power amplifiers to each of the electromagnets and, hence, to levitate the rotor, see Fig. [7] and [17]. The time-periodicity is implemented via periodically varying the proportional action of the PID controller in an open-loop control,
kP (t ) = kP (1 + ε sin ν t ).
(4)
This realises a periodic change in the active bearing stiffness which is implemented in both AMBs. Discretising the continuous shaft by finite beam elements, the time histories are calculated numerically for a sample speed of 600 rpm at a control amplitude of ε = 10% in (5) and summarised in the contour plot in Fig. 8. The levels are chosen starting slightly above the unbalance amplitude of the underlying constant system. Light areas depict small deviations from the unbalance response of the disk deflection |rD| and dark areas large deviations. The main parametric anti-resonance is clearly indicated at the white region close to the frequency 170 1/s ≈ ω3 − ω1 . Introducing this timeperiodicity induced an energy transfer between the first and the third rotor mode which results in an increase of the effective system damping. The time histories at this strong parametric anti-resonance are shown in Fig. 9 an amplitude of ε = 20% in (5). Experimental results are presented to underpin the theoretical findings from above. The rotor is levitated at 600 rpm in the AMBs by the PID controllers. Additionally, a periodic open-loop control varies
Fig. 8. Contour plot of time histories of disk D with periodic open-loop control at ε = 10% in (5) (taken from [17]).
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ficient experimental configurations are discussed in more detail showing the operation principle and impact of an induced timeperiodicity - an artificial amplification of existing system damping. The additional artificial damping provided to the system becomes significant and most effective for vibration mitigation of the lower vibration modes during transient vibrations. The present overview shows that the proposed concept was validated by different mechatronic prototypes. A proper tuning of the parametric excitation is needed in order to achieve increasing the effective system damping by induced modal interaction. The concept was validated theoretically, numerically and experimentally and is ready for industrial implementation.
References
Fig. 9. Time histories of disk D after ping at AMBs without and with periodic open-loop control at parametric anti-resonance ν ≈ ω3 − ω1 = ν 13 and ε = 20% in (1) (taken from [17]): (top) numerical calculation, (bottom) experiment.
the proportional action according to (5) at the parametric antiresonance frequency of 170 1/s. An impulse is applied to the rotor system is realised by adding an impulse-like current to the controller current which leads to transient vibrations. Evaluating the logarithmic decrement of these time histories gives a good measure of the effective damping present in the rotor system. For the present system configuration the logarithmic decrement can be enhanced by a factor of 3.5 (numerical prediction and experiment). The beneficial effect of a parametric antiresonance is confirmed. Without the proposed periodic open-loop control (ε = 0), the system damping can be increased simply by the active damping in the AMBs introduced by the PID controllers, too. However, this increase is limited by the level of measurement noise in the control loop and its amplification by the derivative action. The control parameters for the PID controllers are optimised for lowest additive noise sensitivity in the closed loop according to the procedure outlined in [32]. Starting from these PID values, the derivative action can be amplified maximally by a factor of 1.6 for System 7. Introducing the proposed periodic open-loop control at the control settings in Fig. 9, allows an increase of the effective system damping above this physical limitation. The maximum amplification factor for the effective damping that was achievable for the present setup experimentally is 1.9. This value lies above the limit realisable by conventional active damping. Further investigations are needed on how to increase this factor. 5. Conclusions A physical interpretation on the working principle of a parametric anti-resonance and its experimental verifications were demonstrated. It was highlighted that introducing a specific timeperiodicity in a flexible structure can induce modal energy exchange which exhibits enhanced damping properties. The improved dissipation mechanism is owed to modal coupling which creates higher frequency components. The possibility of a cascaded energy transfer was shown. Tuning the time-periodicity to a sufficiently strong parametric anti-resonance leads to a coupling of only two of the many vibration modes which results in an induced energy transfer between mainly between these modes. All existing experimental verifications of this concept are benchmarked against a performance measure. The two most ef-
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