结构件间隙伸缩系统的建模与振动抑制

1、DOI: 10.2195/LJ_Ref_Arnold_E_0320061Modeling and Vibration Suppression for Telescopic Systems of Structural Members with ClearancePROF. DR. DIETER ARNOLDDIPL.-ING. MARTIN MITTWOLLEN DIPL.-ING. FRANK SCHNUNG,UNIVERSITT KARLSRUHE, INSTITUT FR FRDERTECHNIK UND LOGISTIKSYSTEME)PROF. DR.-ING. JRG WAUER D

2、IPL.-ING. PIERRE BARTHELSUNIVERSITT KARLSRUHE, INSTITUT FR TECHNISCHE MECHANIKAbstractTelescopic systems of structural members with clearance are found in many applications, e.g., mobile cranes, rack feeders, fork lifters, stacker cranes (see Figure 1). Operating these machines, undesirable vibratio

3、ns may reduce the performance and increase safety problems. Therefore, this contribution has the aim to reduce these harmful vibrations. For a better understanding, the dynamic behaviour of these constructions is analysed. The main interest is the overlapping area of each two sections of the above d

4、escribed systems (see markings in Figure 1) which is investigated by measurements and by computations. A test rig is constructed to determine the dynamic behaviour by measuring fundamental vibrations and higher frequent oscillations, damping coefficients, special appearances and more. For an appropr

5、iate physical model, the governing boundary value problem is derived by applying Hamiltons principle and a classical discretisation procedure is used to generate a coupled system of nonlinear ordinary differential equations as the corresponding truncated mathematical model. On the basis of this mode

6、l, a controller concept for preventing harmful vibrations is developed.Fig.1: Graduated multi-section systems with clearance 2006 Logistics Journal. Reviewed publications - ISSN 1860-7977Page 141. IntroductionGraduated multi-section systems of structural components extending and retracting inside ea

7、ch other are interesting technical systems. They are found, e.g., in mobile cranes, rack feeders, fork lifters, stacker cranes (see Figure 1). As the main duty of these machines is not driving around but loading and unloading goods to and from racks, trucks etc., a great number of acceleration and d

8、eceleration operations occur. Combined with the extending and retracting motion of the sections, bending vibrations of the system perpendicular to the telescopic axis occur. These vibrations lead to a significant reduction of the performance due to the required waiting time for relaxation and to saf

9、ety problems so that controlled vibration suppression seems to be necessary.The objective of the present paper is to develop a controller concept for preventing harmful vibrations. First, a system without clearance and with a fixed telescopic length which can be characterized by a time-invariant sys

10、tem of linear differential equations, is reduced to its dominating modes. Using this reduced model, a concept of state control via pole placement is designed which exhibits the desired effects. Introducing a so-called Luenberger observer, straightforward measurements of the motion of the telescope b

11、ase and of the control variable of the actuator are sufficient to operate the controller. For real telescopic operations an adaptive controller and observer are introduced. The controller, developed for the reduced linear system model, is applied to the significantly more complicated system with cle

12、arance for studying the influence of clearance on the vibration suppression during telescopic motions.2. Test RigAs seen in Figure 1, the appearance of graduated telescopic multi-section systems with clearance can be very different. Nevertheless the main problem can be concentrated to the overlappin

13、g area of each two sections (see markings in Figure 1). Only the orientation in space is different as shown in the functional sketch of the system (Figure 2). This sketch leads on the one hand to an appropriate test rig and on the other hand to an appropriate physical model.Main interest: overlappin

14、g area of two sections. Difference: orientation in space.Fig. 2: Sketch of the main item the overlapping areaThe test rig, as shown in Figure 3 consists of two sections, made of slender steel-beams. The lower section is fixed to a rigid block on the ground; the upper section is connected to the lowe

15、r section with screws. For choosing an assembly without clearance or with a well defined clearance, a special system of sliding bolts and bushes is used to reduce friction to a minimum and to enable free translation and rotation movement. This test rig is made of elements with a very simple geometry

16、, because the elastic properties of these elements are well known and the measurement results are comprehensible.acc.tip of upper sectiontest rig with most simply geometry for:Stimulstimulationacc.tip of lower section analysing dynamic behaviour analysing controller behaviouroverlapping area with/wi

17、thout clearanceFig. 3: Test rigIn the following, two typical assemblies are shown in two movies. Movie 1 shows the dynamic behaviour of a two-sectional graduated system without clearance. On the left side, you can see the whole test rig, in the upper window, the overlapping area can be seen and in t

18、he lower window, the development of the vibrations can be observed, presented by the acceleration signals of the sections tips. The movie is shown in slow-motion; the fundamental frequency is 0.53Hz. As expected, the vibration development is absolutely harmonic. Mov. 1: System without clearanceMovie

19、 2 shows the dynamic behaviour of a two-sectional graduated system with clearance (2mm). On the left side, you can see the whole test rig, in the upper window, the overlapping area can be seen and in the lower window, the development of the vibrations can be observed, presented by the acceleration s

20、ignals of the sections tips. The movie is shown in slow-motion; the fundamental frequency is 0.53Hz. As expected, the vibration-development is not longer harmonic. Only the characteristic vibration of upper and lower section is in phase balance. Each time the lower end of the upper section hits the

21、lower section, high frequent vibrations (e.g. 6.35Hz) are stimulated due to the impact. As seen in Movie 2, the acceleration signal of the high frequent vibrations shows an antiphase development.Mov. 2: System with clearance3. Physical ModellFrom the viewpoint of mechanics, a non-linear multi-field

22、problem of vibrating structural members with variable geometry has to be considered. Material surface areas of particular components move along surface areas of other components and define complicated boundary and transition conditions. The clearance produces non-linear effects. In many applications

23、 the different segments are slender and can be modelled as Bernoulli/Euler beams mounted on a rigid vehicle unit and carrying at some location, e.g., at the end of the last section, a load unit assumed to be rigid. The vehicle unit together with the first deformable segment and all the other segment

24、s (one of them together with the load) perform transverse motions and the extending or retracting motion of the sections is supplemented. The contact regions between two sections are modelled as discrete point contacts. A special feature of the modelling is to introduce the reaction forces at the co

25、ntact points in the form of distributed line loads (by using Dirac impulse functions), so that for the contacting sections elementary boundary conditions remain. The contact formulation itself takes place via one-sided spring-damper elements.The procedure is illustrated in Fig. 4a for a two-section

26、telescopic beam system mounted on a rigid traverse, performing a translational motion accompanied by an extending motion of the two beam segments with defined clearance between them. Beam 1 is fixed at a rigid vehicle unit; beam 2 carries a point load at its end. The vehicle is driven by a horizonta

27、l force F as excitation. The deformation of the beams (including vehicle mass and load) is represented by the absolutedisplacementsw(x1 , t)and v(x2 , t) . The model is defined by the following parameters: beamlengths l1,2 , constant cross-sectional areas A1,2 , constant cross-sectional moments of i

28、nertia I1,2 ,density r and Youngs modulus E of the two flexible components, masses of load and vehiclemL andmT , respectively, and telescopic lengthl A (t) . The contact between the beams isrealised (see Figure 4b) via discrete spring-damper systems in the form of a so-called displacementcondition (

29、not a force condition) 3, the given number n of contact points, the clearancelS ,spring stiffness c , and damping coefficient d . c can be estimated from the geometry and the material of the contact partners whereas the estimation of d is more complicated. As the purpose of the model is the creation

30、 of a control concept for vibration suppression, it is important that the equations of motion stay as simple as possible. In the controlled system the clearance plays the role of an external disturbance and as the controller has to work for every kind of contact, a very accurate estimation of d is n

31、ot necessary. In the axial direction it is assumed that there is no friction. This assumption is justifiable as the bearing between the different segments is realised as roller bearing in many applications. It is assumed here that the force flow leads from the upper part into the lower part.mLx2E, r

32、, I2 , A2clearancev(x2 , t)x1w(x11 , t)E, r, I1 , A1extensionmTF body 1body 2spring- damper system c, d Fig. 4: a) System modelb) Contact formulation4. FormulationBoundary value problemApplying Hamiltons principlett1d t (T -U ) dt + Wdt = 0,1tvirt(2.1)00the governing boundary value problem can be de

33、rived. T is the kinetic energy, U the potential energy and Wvirt the virtual work of forces without potential of the considered system. The kinetic energy reads1l1* 21l2* 2,2 02 0T = r A1 wt dx1 +r A2 vt dx2(2.2)*wherer A1= r A1 + mT d ( x1 )andr A* = r A + m d ( x - l )and the symbold (.)22L22repre

34、sents Diracs delta-function. If the action of the spring-damper systems is completely included into the virtual work, for the remaining potential energy one obtainsU = 1l1 EI w2- r g (11lA*dx +l2A*dx)w2 dx2 0 1 x1x1 x11 022x1 1l2(2.3)+ 1l2 EI v2- r gA*dx v2 dx .2 0 2 x2 x2 x22x2 22Since no internal

35、damping of the beam segments will be taken into consideration, as the worst case for control, the virtual work contains all the contact forces between the beams and the locallyconcentrated driving force of the vehicle as distributed loads couple the resulting field equations:f1 ( x1, t )andf2 ( x2 ,

36、 t )whichl1Wvirt = 0l2f1d wdx1 + 0f2d vdx2 .(2.4)Due to the formulation of all these locally concentrated forces by distributed loads using Dirac impulses, the boundary conditions will be homogeneous. Evaluating Hamiltons principle (2.1)introducing T , U and Wvirtgoverning field equationsaccording t

37、o equations (2.2), (2.3) and (2.4), respectively, yields ther A1 wtt + EI1wx x x x + r A1 g (l1 - x1 ) wx+ g (mL + r A2l2 ) wx x *1 1 1 11 x11 1(2.5)1=f1 ( x1 , t ) + d ( x1 - l1 ) g (mL + r A2l2 ) wx ,r A2 vtt + EI2vx x x x + r A2 g (l2 - x2 ) vx+ gmLvx x 2*2 2 2 22 x2 2(2.6)=f2 ( x2 , t ) - d ( x2

38、 ) g (mL + r A2l2 ) vx + d ( x2 - l2 ) gmLvx22and the corresponding boundary conditionsw (0, t ) = 0, w(0, t ) = 0, w(l , t ) = 0, w(l , t ) = 0,(2.7)x1x1x1x1x1x11x1x1x11v(0, t ) = 0, v(0, t ) = 0, v(l , t ) = 0, v(l , t ) = 0(2.8)x2 x2 x2 x2 x2 x2 x22x2 x2 x22for the two bodies.For the special case

39、 in which the beam segments contact each other at the two pointsx2 = 0 only, the distributed forces are specified asx1 = l1 andf = dx F + dx - lt Fx t+ d xt Dx t1( 1 )( 1A ( )K ( 1 ( )1 ( )dtK ( 1 ( )(2.9)+ d x - l Fxt+ d xt Dxt( 11 ) K ( 2 ( )2 ( )dtK ( 2 ( ) ,f = -d x Fx t+ d xt Dx t2( 2 ) K ( 1 (

40、 )1 ( )dtK ( 1 ( )(2.10)- d x- l - lt Fxt+ d xt Dxt( 2( 1A ( )K ( 2 ( )2 ( )dtK ( 2 ( ).The non-linear characteristic of the spring force FK (x (t ) takes into account the fact that in therange of backlash no forces can be transferred. The same is valid for the assumed damping coefficient DK (x (t )

41、 :F (x (t ) = c x (t ) - 1 x (t ) + lS sign x (t ) + lSK2 2 2 + 1 x (t ) - lS sign x (t ) - lS ,(2.11)2 2 2 D (x (t ) = d 1- 1 sign x (t ) + lS+ 1 sign x (t ) - lS,(2.12)K22 22 x1 (t ) = v (0, t ) - w(lA (t ) , t ),x2 (t ) = v (l1 - lA (t ), t ) - w(l1, t ).(2.13)DiscretisationThe discretisation of

42、the coupled partial differential equations (2.5) and (2.6) (nonlinear and time- variant in general) together with the corresponding boundary conditions (2.7) and (2.8) is based onGalerkins method. For that, the approximate solutions by a series expansionw ( x1, t )andv ( x2 , t )are representedNw (

43、x , t ) = u (t ) cos(l x ) +cos (lil1 )cosh (l x ) ,(2.14)i 11ii=1cosh (lil1 )i 1 Nv ( x2 , t ) = uN +1 (t ) + uN +2 (t ) x2 + uN +i (t ) cosh (ki x2 ) + cos(ki x2 )i=3(2.15)(cosh (kil2 ) - cos (kil2 )- sinh (k x) + sin (k x )i 2sinh (kil2 ) - sin (kil2 )i 2fulfilling all boundary conditions (2.7) a

44、nd (2.8).Galerkins averaging leads to a system of ordinary differential equations of the typeMu& = F (u, u& , t ).(2.16)5. Vibration supression conceptTo suppress vibrations, a state space control concept is introduced. For a system without clearance and with a fixed telescopic length, equation (2.1

45、6) represents a time-invariant system of linear ordinary differential equations which can be reformulated asMu& + Cu = b* F (t )(3.1)where b* is a 2N -dimensional vector, M is the mass matrix and C is the stiffness matrix of the system.Reduction of orderEquation (3.1) represents a 2N -degree-of-free

46、dom system. The objective of an order reduction is to find, for a given model of high order, a model of significantly lower order whose dynamic behaviour approximates the original behaviour as well as possible. This means that theapproximate model has to contain the essential modes of the original s

47、ystem, since they dominate the dynamic behaviour of the original system Lunze01. For this purpose, the system equations (3.1) are transformed to principal coordinates y byu = MR y.(3.2)The columns of the (2N , 2N ) -matrix MRwill be composed of the right eigenvectorsmRi (i = 1, 2,., 2N )of the syste

48、m. ML is a matrix which contains the left eigenvectors of theLRsystem (MT = M-1 ) . There isT-1( 222 )ML M CMR = -diagw1 ,w2 ,.,w2 N= D(3.3)iand w 2 (i = 1, 2,., 2N ) are the eigenvalues of the system.From equation (3.1), using (3.2) and (3.3), it follows thatL&y& = -Dy + MT M-1b* F (t ).(3.4)If the

49、n the eigenvectors corresponding to the large eigenvalues are removed from MR and ML ,one obtains the reduced (2N , Nr ) -matrices model isMRrandMLr , and with that the approximate&y&= -D y + MT M-1b* F (t ).(3.5)rr rLrThis type of order reduction is justified since the state control should control

50、the rigid body motion and the lower modal vibrations of the system. The high-frequency oscillations possess a small magnitude and will diminish strongly by material damping effects. The number of higher-order modes to be included into the reduced model has to be determined depending on the applicati

51、on and the quality of control desired.Driving unitThe driving unit of the telescope will be represented by a scalar system of first orderTA F& + F = KAU(3.6)with time constant TAand amplification factorKA . U is the control voltage of the motor.Introducing the state variablesz1 = y 1,., z= y, z1 = y

52、& 1,., z2= y&, z21 = F ,(3.7)rNrrNrNr +rNrrNrNr +instead of (3.5) and (3.6) one obtainsz& = Az + bU .(3.8)Controllability and observabilityControllability and observability are checked by computing the so-called controllability matrixQS and observability matrix QB Fllinger78:QS = qS1, qS 2 ,., qSN *

53、 , qS1 = b, qSi+1 = AqSi ,TTQB = qB1 , qB 2 ,., qBN * , qB1 = cwF , qBi+1 = A qBi(3.9)(3.10)whererN * = 2N +1,TcwF = W1 (0) ,W2 (0) ,.,WN (0) , 0,., 0 MRr S(3.11)(3.12)and S comes fromyr = Sz.(3.13)The system is completely controllable if the determinant of QSdoes not vanish, and it iscompletely observable if the determinant of QBdoes not vanish. For real systems,