УДК 517.977.56


© 2015 г. H. Aschemann, G. Kostin, A. Rauh, V. Saurin

Germany, Rostock, Chair of Mechatronics, University of Rostock Russia, Moscow, Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences

Received July 11, 2014

In this paper, control-oriented models are derived for an experimental setup representing the structure of a typical high bay rack feeder. To develop a real-time applicable control algorithm, a frequency analysis is performed for the original viscoelastic double-beam structure. This leads to a simplified Bernoulli beam model with specific boundary conditions. On the basis of the proposed model, a feedforward control strategy is designed. The control objective under consideration is to move the flexible structure to a desired position in a given time interval and to minimize the relative mean energy stored in the beams during the process. A modification of the method of integrodifferential relations, which is based on a projection approach and a suitable finite element technique, is employed to optimize the controlled motions. Results of numerical simulations are presented and compared with experimentally measured data from the experimental setup.

DOI: 10.7868/S0002338815020031

Introduction. The design of control strategies for dynamic systems with distributed parameters has been actively studied in recent years. Processes such as oscillations, heat transfer, diffusion, and convection are part of a large variety of applications in science and engineering. The theoretical foundation for optimal control problems with linear partial differential equations (PDEs) and convex functionals was established by Lions [1]. Linear hyperbolic equations are treated, e.g., in [2, 3]. An introduction to the control of vibrations can be found in [4]. Oscillating elastic networks were investigated, e.g., in [5].

Different approaches to the discretization of dynamic models with distributed parameters are developed to reduce the original initial-boundary value problem to a system of ordinary differential equations (ODEs). In this context, variational and projection methods are powerful tools to solve control problems for structures with flexible elements. The method of integrodifferential relations (MIDR) was proposed in [6] for the design of optimal control laws for elastic beam motions. A projection approach was developed as a modification of the Galerkin method in the frame of the MIDR for dynamical systems described by linear parabolic PDEs in [7]. In the current paper, this approach is combined with the finite element method developed in [8, 9]. It is extended to modeling and optimal control of rack feeder systems with distributed elastic and inertial parameters, which have already been considered by using an alternative system representation in [10].

In Section 2, a typical structure of a flexible rack feeder system is considered. A Bernoulli beam model describing its structure is presented in Section 3. After that, a Fourier analysis is performed in Section 4 to reduce the model to the necessary degrees of freedom. As differ from the previous study, the initial-boundary value problem accounting not only lateral but also longitudinal viscoelastic displacements is formulated for rack feeder dynamics in Section 5. In the next section, an optimal feedforward control problem for a damping of beam vibrations excited during structure locomotion is stated. A modification of the finite element method (FEM) is described in Section 7. The proposed control strategy is demonstrated, numerically verified, and experimentally validated in Section 8. Finally, the paper is concluded with an outlook on future research in Section 9.

1. A typical structure of flexible rack feeders. The experimental setup that has been built up at the Chair of Mechatronics of the University of Rostock represents the structure of a typical high bay rack feeder as shown in Fig. 1 (left). The flexible structure consists of two identical beams clamped vertically to a horizontally movable carriage. The plane motions of the system are described in the noninertial Cartesian ref-

*This work was supported by the Russian Foundation for Basic Research, project nos. 12-01-00789, 13-01-00108, 14-0100282, the Leading Scientific Schools Grants NSh-2710.2014.1, NSh-2954.2014.1.

Fig. 1. Beam structures: original (left) and reduced (right)

erence frame with the vertical axis x, the horizontal axis z and the origin O located equidistantly from the points where the beams are attached to the carriage [11].

Both beams are rigidly connected at their tips by a pulley block, which is necessary for the vertical positioning of a cage. The cage represents a payload sliding along the left beam. In addition, the beams are coupled by means of a rigid rod. The following parameters of the structure are used in the subsequent modeling: the length of the beams l, their cross section area A, the moment of area I, the distance 2b between the beams, Young's modulus E and the volume density p of the beam material, the masses ml and ma of the pulley block and the cage, their moments of inertia Il and Ia, the vertical position a(t) of the cage, and the height l/2 of the hinging. The velocity v(t) of the carriage is supposed to be the control input of this dynamical system with distributed parameters.

2. Natural vibrations of coupled beams. The motion of the rack feeder is studied by applying the Bernoulli beam model. Let us first analyse the natural vibrations of the structure. For this purpose, the vertical position of the cage is fixed to a = l/2 in the undeformed state and the carriage velocity v(t) = 0 is chosen. In this section, the viscous friction in the beam material is ignored and only lateral and longitudinal elastic motions in the Oxz plane are taken into account. The eigenforms w-(x) and u-(x) of the coupled horizontal and vertical vibrations for the first and second beam (j = 1,2) are described by the following system of ODEs

sj - Apw2w,- = 0, Elw'j -s, = 0,

1 У 1 1 1 x e (0,a) U(a,l). (2.1)

f'j + Apw2Uy = 0, EAu'j - fj = 0,

In this system, the vertical coordinate x is an independent variable, ю denotes the unknown eigenfre-quency, s- (x) is the bending moment in the beam cross section and f-(x) is the stretching force along the j th beam. The primed symbols stand for the derivatives with respect to the spatial variable x . The eigen-forms of the elastic vibration for the lower (x < a) and upper (x > a) elements of both beams are obtained as a general solution of the system (2.1) according to

Wj = cд sin Xx + j cos Xx + Сд sinh Xx + Сд cosh Xx, u- = d-k sin ^x;

j = 1,2, k = 1 if x e (0, a) or k = 2 if x e (a, l), (2.2)

X = 4Ap®2E '/_1, ц = -\/pE_1ю.

The boundary conditions at the bottom and top of the beams have the form

x = 0: wj (0) = 0, wj (0) = 0, uj (0) = 0; x = l: w1(l) = w2(l), 2bw1(l) = 2bw2(l) = u1(l) - u2(l),

m^2 (ui(l) + U2(l)) = 2fi(l) + 2f2(l), (2.3)

ml®2Wi(l) = -s'i(l) - s2(l),

Ww[(l) = si(l) + S2(l) + b (fi(l) - f2(l)).

In the proposed model, the lateral and longitudinal motions are further constrained by the interelement conditions at the fixed position of the cage:

x = a : Wj (a - 0) = Wj (a + 0), Uj (a - 0) = Uj (a + 0), wj (a - 0) = wj(a + 0), w1(a) = w2(a),

maw w1(a) = sj(a + 0) + s2(a + 0) - s1(a - 0) - s2(a - 0), (2.4)

mam\(l) = f1(a - 0) - fi(a + 0), f2(a + 0) = f2(a - 0),


Iaw w1(a) = s1(a - 0) - s1(a + 0), s2(a + 0) = s2(a - 0).

The terms with the coefficients ma and Ia appear in (2.4) due to the rigid coupling of the first beam with

the cage. The coefficients cд, djk in (2.2) and the eigenfrequency ю are defined from the degeneracy condition for the system of boundary and interelement constraints (2.3), (2.4).

3. Reduced double beam model. To derive a control-oriented model for the experimental setup, the three lowest eigenmodes of the rack feeder structure, which are excited noticeably during typical motions, are estimated. The following known geometrical and mechanical parameters are used in the vibration analysis:

l = 1.07 m, b = 0.0245 m, p = 2700 kg • m-3,

mi = 0.906 kg, ma = 0.95 kg, E = 70 GPa, (3.1)

A = 3.10 • 10-4 m2, I = 2.138 • 10-9m4.

It can be shown numerically that the influence of the moments of inertia I l and I a on the values of the first eigenfrequencies юи, n = 1,2,3, is negligibly small. Therefore, they are set to zero in (2.3) and (2.4).

The eigenvalues for the system (2.1), (2.3), (2.4) obtained by the Fourier method (see [12]) are summarized in Table 1. It can be seen that the longitudinal wave numbers цn are much smaller than the corresponding lateral ones X n for n < 3. This means that the trigonometric functions of vertical extension and compression Uj(x), j = 1,2, from (2.2) can be replaced rather accurately by linear approximations with respect to the coordinate x. Moreover, the lateral displacements w1(x) and w2(x) do not differ noticeably from each other for these modes. These facts lead to a reduced model of the rack feeder (see Fig. 1, right) as a single Bernoulli beam with the doubled geometrical parameters of the cross section area 2A and the moment of area 2I.

For this new beam model, the differential equations of natural motions with respect to the spatial variable x are

s2Ap®2w = 0, 2EIw"- s = 0, x e (0,a)U (a,l), (3.2)

where w(x) denotes the lateral displacement of the introduced beam and s(x) is its bending moment. Under the above-mentioned assumption of linear longitudinal displacements, the simplified boundary and interelement conditions are directly derived from equations (2.1), (2.3), (2.4) . These conditions have the form

Table 1. Eigenvalues for elastic and viscoelastic models

Mode 1st 2nd 3rd

Two coupled elastic

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