Active structural elements within a general vibration control framework

Transcription

1 Active structural eleents within a general vibration control fraewor Jan Holteran Theo J.A. de Vries presented at the st IFAC Conference on Mechatronic Systes Mechatronics Darstadt, Gerany, 8- Septeber published in R. Iserann (editor), (), Proc. st IFAC Conference on Mechatronic Systes Mechatronics, Darstadt, Gerany, pp. 997-

2 ACTIVE STRUCTURAL ELEMENTS WITHIN A GENERAL VIBRATION CONTROL FRAMEWORK Jan Holteran Theo J.A. de Vries Cornelis J. Drebbel Institute for Systes Engineering, EL-RT, University of Twente PO Box 7, 75 AE Enschede, The Netherlands Phone: , Fax: echatronics@rt.el.utwente.nl, Abstract: High-precision achines typically suffer fro sall but annoying vibrations. As the ost appropriate solution to a particular vibration proble is not always obvious, it ay be convenient to cast the proble in a ore general fraewor. This fraewor ay then be used for frequency response analysis, which, together with close exaination of the disturbance sources, leads to a solution in general structural ters, lie vibration isolation, stiffness enhanceent or daping augentation. In case it is not possible or unpractical to control the vibration passively, a solution based on active structural eleents ay be considered. Keywords: active control, active eleents, frequency response ethods, odal control, ode analysis, structural paraeters, systes engineering, vibration dapers. INTRODUCTION High-precision achines typically suffer fro sall but annoying vibrations. These ay either be induced by environental vibrations transitted by the floor or the surrounding air, or result as an unwanted side effect due to actuator activity inside the achine. A straightforward solution to any vibration probles is to increase the stiffness within a achine. There is however a practical liit in passively increasing the stiffness of a structure. Furtherore, due to dedicated structural design rules (Koster, 998), vibrations in high-precision achines are typically badly daped (Van Schothorst, 999). Introducing additional passive daping into high-precision structures however is coplicated, as the stresses and strains to be daped are very sall (Fanson, et al., 99). Because of the indicated passive restrictions, worldwide uch research effort is put into active vibration control strategies. Because of the relative ease of ipleentation and stability robustness, especially active structural eleents for vibration control with co-located sensing and actuation, have gained uch interest (Bronowici, et al., 994; Anderson, et al., 997; Preuont, 997). ACX for exaple has developed the SartPac TM (Spangler, et al., 997); JPL has developed an active strut (Anderson, et al., 99). At the Cornelis J. Drebbel Institute for Systes Engineering at the University of Twente, research is devoted to the developent of a Sart Disc, which is envisioned as a load-bearing active structural eleent, with integrated sensing, actuation and control (Holteran, et al., 998; Van Schothorst, 999). Active Control experts, Inc., 5 First Street, Cabridge, MA 4 Jet Propulsion Laboratory, California Institute of Technology, 48 Oa Grove Drive, Pasadena, CA 99-8

3 The past decades, uch research has been aied at the solution of several dedicated probles in the area of vibration control. The distinctive ai of the present paper is to provide a fraewor for classifying generalized vibration probles in highprecision achines, and to indicate the virtues of active structural eleents within this fraewor. As such, the central questions addressed here are: Given a certain generalized vibration proble.. How to control it in general structural ters?. How to control it using active structural eleents?. MODELLING A HIGH-PRECISION MACHINE In this paper a largely stylized odel of a highprecision achine is considered (see figure ). The odel is ept as general, and thus as siple, as possible. As a consequence, the odel represents only one single doinant vibration ode of the achine itself, caused by the fact that the connection ( ) between two iportant parts of the achine is not infinitely stiff. Relative oveent of these parts, referred to as the upper frae ( ) and the base frae ( ; with, in general, > ), is undesirable, as this is assued to preclude proper operation of the achine. In order to iniize the effect of inevitable floor vibrations (t), high-precision achines are often resiliently supported (indicated by a stiffness parallel to a daper d ; with, of course, << ). As a consequence, the odel incorporates a second vibration ode which is assued to correspond roughly to the joint oveent of the upper frae and the base frae with respect to the floor. F d( t) F d ( t ) x ( t) d xd ( t) x t ( ) : upper frae : large but finite stiffness : base frae and d : resilient isolation (t): floor vibrations Fig.. Stylized odel of a high-precision achine Besides floor vibrations, in practice actuators within the achine and acoustic waves guided by the surrounding air ay also excite vibrations. This iplies that in the odel, at least two other disturbance sources should be incorporated: disturbing forces on both asses, F d (t) and F d (t). It is iportant to note that the odel presented here is truly a drastic siplification of a high-precision achine in practice. The odel for instance displays only one diension, whereas reality is threediensional. Nevertheless, in order to coe up with a general as possible fraewor for high-precision achine vibration control the odel should be preferably as siple as possible, incorporating only the phenoena that are of ost iportance for understanding the nature of the vibration proble. Daping between the upper frae and the base frae for instance is not odelled; its agnitude, and therefore its influence, is assued to be rather sall. 3. FREQUENCY RESPONSE ANALYSIS For proper operation of the odelled achine, the positional difference x - x should be as sall as possible, despite the fact that two disturbing forces and a disturbing oveent act upon the structure. In order to coe up with eans to iniize the positional difference, its response to the three disturbance sources should be exained (for siplicity, the daper d is left out of the analysis): x with s x D s F s F s + - = - D s D s x + - ( ) ( ) ( ) d d d 4 D ( s) = s + ( + + ) s +. () () The frequency responses of the positional difference to the three disturbance sources (see figure ) are characterized by three frequencies, for << and > given by: resonances:, w e + ª, + w e ª antiresonance: =. w a (3) (4) The generalized odel of figure is easily seen to be characterized by six structural paraeters: two daping values, deterining the height of the

4 F d s F d s + + e e 4 db/dec -4 db/dec e e a s s s db/dec + + e e 4 db/dec -4 db/dec Fig.. Frequency responses fro disturbances to positional difference resonance peas, and two asses and two stiffness values which together deterine the asyptotes of the responses. With respect to the latter four structural paraeters, their influence on the frequency responses is suarized in figure 3. It is iportant to note that the responses shown in figure and 3 only represent a view on half of the proble. The other half of the proble relates to the doinant disturbance sources. Vibration control is in general only possible after all aspects of the vibration proble at hand have been exained thoroughly, i.e., aspects concerning the structure suffering fro vibration as well as aspects concerning the disturbance sources (Mead, 998). 4. VIBRATION CONTROL IN GENERAL STRUCTURAL TERMS In general, the first attept to control the vibration should be in reduction of the disturbance fro the source (Mead, 998). If this is not possible or unpractical, the only solution left is in reshaping the responses, that is, in lowering the resonance peas and/or anipulating the response asyptotes. Once the desired changes in the responses have been established, the solution to the proble can easily be forulated in ters of the structural paraeters appearing in the odel. Addition of daping, which corresponds to lowering a resonance pea, for instance is easily seen to be a successful approach only in case the vibration proble is caused predoinantly by an initially badly daped structural resonance. The overall response shape however is hardly affected by daping augentation. This iplies that, in case the vibration proble anifests itself in a rather broad frequency band, sole daping augentation is in general not useful. With respect to figure, in that case the responses should be lowered in a certain non-resonant frequency range, which can only be achieved by a change of the general structural paraeters ass and/or stiffness, not daping (Mead, 998). In finding the ost appropriate eans to reshape the frequency responses, figure 3 can be used in a very convenient way. In case the ain disturbance source is the floor vibration, vibration isolation (response sets c and d) is usually considered. Coparison of both response sets now indicates that in general an increase of the base frae ass is preferred above a decrease of the support stiffness, the ain difference being in the responses to F d (t). Liewise, response sets a and b indicate that an increase of the achine stiffness (stiffness enhanceent) is preferable above a decrease of the ass of the upper frae, the ain difference here being in the responses to F d (t). Note that, intuitively, an increase of the stiffness can also be seen to be the overall best solution 5. ACTIVE VIBRATION CONTROL Once the solution to the vibration proble has been forulated in general ters, it should be deterined how to ipleent the solution in practice. It was already entioned in the introduction that for highprecision achine vibration probles, ipleentation by passive eans ight be ipossible or unpractical. In that case a solution based on active eleents ay be considered (Preuont, 997; Mead, 998). In this paper we consider two slightly distinct active vibration control approaches, both based on colocated sensing and actuation. The virtues of both approaches are discussed for the siplest possible odel, describing only a single ode of a structure. We consider a ass-spring syste at a vibrating floor, (t) (figure 4). A position actuator u(t) and a sensor, both appropriately co-located, together with a

5 controller constitute the active structural eleent (indicated by the shaded boxes in figure 4) the Drebbel Institute ais for. F d ' < F d ' > ' e ' e F d F d ' e ' e ' e ' e a b F d ' > F d ' < ' e F d F d ' e ' a ' a ' e c b d b Fig. 3. Overview of effects of paraeter changes on frequency responses

6 F ( t) d x( t) F d ( t ) x( t) Table Effect (in ters of figure 3) of active eleent insertion in the achine odel (figure ) u( t) xd ( t).. x( t) -H( s) acceleration sensor controller constitute the active structural eleent (indicated by the shaded boxes in figure 4) the Drebbel Institute ais for. 5. Acceleration feedbac First we consider the actuator to be steered upon a easureent of the acceleration of the ass. Application of a static feedbac law: u = - H( s) s x = -Hs x boils down to a virtual ass change, according to = + H In case the feedbac law is dynaic rather than static, it is also possible to add daping to the syste: Direct Velocity Feedbac (Preuont, 997) is nown to correspond to a daper between x and the ass. 5. Force feedbac Next we consider the actuator to be steered upon force easureent in the supporting structure. Application of a static feedbac law: u = H( s) F = HF then boils down to a virtual stiffness change, according to: = / ( + H ) In case the feedbac law is dynaic rather than static, it is also possible to add daping to the syste: Integral Force Feedbac (Preuont, 997) is nown to correspond to a daper between the ass and the vibrating floor ( ). u( t) xd( t) x x F ( t) force sensor H( s) Fig. 4. Acceleration feedbac (left) and force feedbac (right) (5) (6) (7) (8) Active structural eleent location Acceleration feedbac Force feedbac between and a b between and - b between and c d between and - d The effect of the proposed active structural eleents in ters of the response sets shown in figure 3, (tered a, b, c, or d) is suarized in table. An iportant rear that should be ade here with respect to active vibration control is the fact that it is only effective within a liited bandwidth. In general a vibration control syste would consist of an active part for low frequencies, and a passive part for higher frequencies (Hansen, and Snyder, 997). 6. EXAMPLE: SMART DISC PROOF-OF- CONCEPT EXPERIMENT As an exaple of the use of the fraewor presented in this paper, we consider the Sart Disc proof-ofconcept experient, in which the deflection of a siple bea was copensated for actively (Holteran, et al., 998). The experient was set up such that a disturbing force could be applied at the top of the bea. This force, being far ore doinant than any other disturbance source, was intended to act in a liited frequency range (. Hz), far below the lowest natural frequency of the syste. The experient as described above, can easily be cast in the general fraewor presented in this paper. It is obvious that we should only be concerned about the response to F d. The frequency range of interest is below w e, iplying that a resonance is not the proble; daping augentation thus is not useful. Fro figure 3 it is furtherore easily seen that the structural paraeter to be changed is. We thus conclude to consider solution b. As our intention has been to evaluate the concept of vibration control using an active structural eleent, we refer table, and decide upon force feedbac and insertion between and (due to the experient setup insertion between and was not possible). The experient perfored showed that the prototype Sart Disc was able to suppress vibration in the intended frequency range to 3 db, corresponding to an active stiffness enhanceent factor of 3 to 3 (figure 5).

7 gain [db] -, gain = deflection (active control). deflection (original). -4,.. frequency [Hz] deflection [ µ ] ; dist. force [N] - - F (t) d deflection (active control) deflection (original).5 tie [s] Fig. 5. Experient frequency response Fig. 6. Response to bloc-shaped disturbance Subsequent experients with a bloc-shaped disturbing force revealed that a further increase of the control bandwidth should be used for an increase of the stiffness in a broader frequency range, rather than the addition of daping at the resonance frequency of the syste (figure 6). 7. CONCLUSION With respect to high-precision achine vibration probles, the following general ethod for arriving at a practical solution can be used. () Forulate a copetent odel, incorporating the doinant harful vibration odes of the syste at hand and the doinant disturbance sources, and characterize the disturbance sources in the frequency doain. () If the proble is doinated by a resonance, then try to increase the daping (by passive eans); otherwise, based on figure 3, reshape the frequency responses by changing the general structural paraeters ass and/or stiffness (passively). (3) In case it turns out ipossible or unpractical to ipleent the proposed solution in general ters by passive eans, consider a solution based on active structural eleents, based on figure 3 and table. REFERENCES Anderson, E.H., D.M. Moore, J.L. Fanson, and M.A. Ealey (99). Developent of an active truss eleent for control of active structures. In: Optical Engineering, Vol. 9, No., pp Anderson, E.H., M.D. Holcob, A.X. Bogue, and F.R. Russo (997). Integrated Electro-echanical Devices for Active Control of Vibration and Sound. Presented at Adaptive Structures and Materials Systes Syposiu, International Mechanical Engineering Congress and Exposition, Dallas, TX, Noveber 997. Bronowici, A.J. J. Innis, S. Casteel, G. Dvorsy, O. Alvarez, and E. Rohleen (994). Active Vibration Suppression using Modular Eleents. In: Proc. SPIE Sart Structures and Intelligent Syetes, Vol. 9, pp Fanson, J.L., E.H. Anderson, and D. Rapp (99). Active structures for use in precision control of large optical systes. In: Optical Engineering, Vol. 9, No., pp Hansen, C.H., and S.D. Snyder (997). Active Control of Noise and Vibration, Chapan & Hall, London, Great Britain. Holteran, J., T.J.A. de Vries, and M.P. Koster (998). Experient to evaluate the feasibility of the Sart Disc concept. In: Proc. 6 th UK Mechatronics Foru International Conference Mechatronics 98, Sövde, Sweden, pp. 7-. Koster, M.P. (998). Constuctieprincipes (in Dutch), Twente University Press, Enschede, The Netherlands. Mead, D.J. (998). Passive Vibration Control, Wiley & Sons, Chichester, Great Britain. Preuont, A. (997). Vibration Control of Active Structures, An Introduction, Kluwer Acadeic Publishers, Dordrecht, The Netherlands. Spangler, R.L., F.M. Russo, and D.A. Palobo (997). A Copact Integrated Piezoelectric Vibration Control Pacage. In Sart Structures and Integrated Systes, Proc. SPIE Sart Structures and Materials, Vol. 34, pp Van Schothorst, G. (999). Active Vibration Control using Piezoelectric Sart Discs. In: Matheatics and Control in Sart Structures, Proc. SPIE Sart Structures and Materials, Vol. 3667, pp