Object oriented modeling of a smart structure

Transcription

1 Object oriented modeling of a smart structure D.S. Necsulescu Department of Mechanical Engineering, University of Ottawa, Canada. Abstract The Object Oriented Modelling approach permits modular development of a smart structure simulator. Textual programming of smart structures simulators is time consuming and often requires reprogramming to account for design changes. Object Oriented Modelling facilitates interchangeabi 1 ity of modules, permits the development of a library of reusable modules and can be easily formulated as a graphical programming approach. This approach permits rapid prototyping and results in an increase in the number of design alternatives that can be evaluated in a short period of time. This paper presents the Object Oriented Modelling approach using acrossthrough cut variables and multipole models of the flexible structures and actuators. The across-through cut variables allow the analysis of power transfer from one module to another. Modular modelling using matrix and Simulink representations will be illustrated for a flexible shaft driven by a motor. 1 Introduction Modular modelling of physical systems using a library of modules permits to maintain the one-to-one mapping of physical and simulated modules and to emulate in model development the assembly process of physical counterpart Cellier [1], Object Oriented programming is based on concepts and procedures that enhance modular modelling features, Rumbach [2], Fishwick [3], TenEyck [4]. Modelling physical systems using Object Oriented approach requires acrossthrough cut variables for system decomposition and integration, Cellier [1], Mattson [5], Paynter [6]. Paynter [6] called this across-through modeling a noncausal description because the direction of the power flow in the junction is

2 54 Computational Methods for Smart Structures and Materials II bilateral as opposed to the block diagram description in which signals have unidirectional flow. Paynter [6] developed Bond Graph modeling based on two port descriptions. The same description, using across-through two pole ports, is used in Object Oriented Modeling of mixed systems. The theoretical background of this description can be found in Hamiltonian dynamics for obtaining power transfer equations., Necsulescu [11]. Multi-body mechanical systems and multi-loop electrical networks modelled using Object Oriented approach are represented by a system a differentialalgebraic equations solvable numerically only for lower index systems, Mattson [5], Simeon [7]. Object Oriented models can be programmed as computer codes for execution using Object Oriented programming languages (for example C++)' TenEyck [4]. While the rigor of Object Oriented programming languages is a plus, often the convenience of already known non-object Oriented language, like C or Fortran, justifies their use even if they can not reproduce all the features of Object Oriented programming languages Rumbach[2], Fish wick [3]. MATLAB, after version 4, and Simulink were developed using Object Oriented programming languages. Simulink permits graphical programming for simulating systems based on signal transmission and it uses block diagram modeling. Signal flow and power transmission modeling of mechatronic systems can be illustrated using Simulink. Examples of Object Oriented modeling simulators for mixed engineering systems are Dymola and Omola, Cellier [1], Mattson [5], Kasper [8]. Object Oriented modelling of physical systems has been developed mostly for lumped parameters systems, often for mechatronic systems, Kasper [8], Ferretti [9], Mann [10]. Analytical dynamics can be used as common formalism for mechanical part of the system and the electrical part of the actuators. For this purpose, Lagrangian dynamics was proposed by Mann [10], while Hamiltonian dynamics can be considered as well due to the explicit power transfer equation, Necsulescu [11]. Distributed parameters modelling, developed lately for flexible structures as part of flexible arms robots and smart structures development, use second order differential equations and finite elements models, Stanway [12], Torby [13], Necsulescu [14]. Flexible muitibody systems modelling, using multiport modules linked by generalised velocity and its dual force for interconnections, was proposed by Yoshimura [15]. In this paper is investigated Object Oriented modelling of smart structures in the simple form of a series system containing flexible components and actuation. The illustrations are given for an electric motor with a flexible shaft. 2 Two port models of aflexiblecomponent Two port models were introduced for representing components of electric networks using two terminals for each port. Alternative names for two port components of a network are four terminal network or quadripole [6]. The two

3 Computational Methods for Smart Structures and Materials II 55 pole port has associated a current I variable and a voltage V that permit the calculation of the power P= VI transferred through the port. For obtaining a two port lumped parameters model for a mechanical system, a flexible horizontal shaft is assumed. The shaft can be cut from the system using two pairs of across-through variables {T,,co,} and {T], 0)2}, as shown in Fig. 1. T, (Dl <JL)2 Figure 1: A flexible shaft For illustrating Object Oriented issues of smart structures, five models of the flexible shaft will be considered, Necsulescu [14]: (a) lumped parameter model with torsional spring coefficient k; (b) lumped parameter model with torsional spring coefficient k and lumped inertia J; (c) single finite element model; (d) three finite element model. These models are obtained as follows: (a) assuming the flexible shaft represented by a lumped parameters model with torsional spring coefficient k, the following equations can be obtained: T,(t)=k(8,(t)-&2(t)) Given that m = do/dt and using Laplace transform, this system can be solved to obtain the relationship between the pairs of across-through variables {T,,coi} and {T?, co?}: or in matrix form This model is suitable only for shafts with low moment of inertia and ignores non-minimum phase property of flexible shafts, Necsulescu [14]. (b) assuming the flexible shaft represented by a torsional spring coefficient k and lumped inertia J, the following matrix equation can be written for the case of splitting j into two J/2 at the two ends of a spring k: T,(s)=(J/2) s'8i(s)+ k(&,(s)-&2 (s)) T2 (t)=(j/2) 8^2(8)+ k(&2(s)-&i (s)) or, in matrix form: T,(s) "(J/2) 0 0 (J/2) i K K 1-k k

4 56 Computational Methods for Smart Structures and Materials II This equation shows that the flexible shaft is represented by an inertia matrix with no cross-coupling and a compliance matrix with cross-coupling. This is a minimum phase model, while the flexible shaft is a non-minimum phase system. This model was called inconsistent; a finite element model can be used to obtain a consistent model. The relationship between the pairs of across-through variables {Ti,001} and {T2, 0)2} in matrix form is: T,(s) -s/k +1) (k/s)(((j/2k)g This matrix equation gives the relationship between the pairs of cut variables T and 8 function only of internal variables J and k. (c) assuming theflexibleshaft represented by a single finite element model the following equations can be obtained, Necsulescu [14]: T,(s)=(J/3) s'8,(s) )+(J/6) s%(s)+ k(8,(s)-&2 (s)) T, (t)= )=(J/6) s'8,(s)) +(J/3) s^2(s)+ k(&2(s)-&, (s)) or in matrix form: F(J/3) (J/6) - s\ [(J/6) (J/3) k -k 0)2(5) This equation shows that the flexible shaft is represented by an inertia matrix and a compliance matrix with cross-coupling and gives one positive zero. This is a non-minimum phase model, i.e. a consistent model of theflexibleshaft. The relationship between the pairs of across-through variables (T,,001} and {T2, 0)2} in matrix form is: T,(s) (Js/3 -+- k/s)(js/6 - k/s)"* (Js/6 - k/s) - (Js/3 + k/s)^ (Js/6 - k/s)" (Js/6 - k/s)"' - (Js/3 + k/s)(js/6 - k/s)"* (d) assuming the flexible shaft represented by three finite elements, a matrix equation, with inter-elements angular position variables 8% and &b can be obtained, Necsulescu [14]: ~T,(s)" / "J/9 J/ " 3k -3k 0 0 \ J/18 2J/9 J/ = s + (l/s) 0 J/18 2J/9 J/18-3k 6k -3k 0 0-3k 6k -3k J2(S)_ V 0 0 J/18 J/ k 3k J This is a consistent model of the flexible shaft and is more accurate than the single finite element model and can use the inter-elements links for introducing extra actuators to build a smart structure. The two middle scalar equations can be used to calculate o)%and coy function of Oi and 0)2. After replacing this result in the first and last scalar equation the relationship between the pairs of acrossthrough variables (Tj,0),} and {T2, 0)2} is obtained. 0) i a Wb 0)2

5 Computational Methods for Smart Structures and Materials II 57 3 Modeling systems using Object Oriented representation The examples of lumped parameters models the components are resistance, inductance, capacitance and transformers for electrical systems and mass, spring damper, gears for mechanical systems. In an Object Oriented representation each object incorporates the function or method (differential and / or algebraic equations) and communicates (can be connected) with other objects using only by message passing (input and output variables). Object Oriented modeling of electromechanical systems has specific features: -the objects are connectable to each other at ports with effort-flow variables (with the product effort*flow giving the power transfer in the port) -the model of the system is build from objects by imposing constraints on message passing such that power balance is respected at the connections of objects. In bond graph models, power dissipation is modeled at the point where it occurs, while in objects only the power at ports is apparent. Power balance equations take specific forms in particular connections. For electric components connected at the same node, given that all links to that node have the same voltage, power balance equation is reduced to current balance equation, i.e. Kirchhoff first law. For mechanical components, in a free body diagram, at a cut all links have the same velocity (and position) and force balance equation replaces power balance equation, Cellier [1]. 4 Modular model of DC motor with aflexibleshaft A load with a moment of inertia Jf, and an damping coefficient Bf is assumed to be linked by a flexible shaft with stiffness coefficient k to a PM (Permanent Magnet) - DC motor (Fig. 2). The DC motor model, for the cut variables T(s) and co(s), in matrix form, is given by: )/%m % o)(s) I l/k,n -(Ls + R)/K_ II i(s) 1 ^ ' 1 1_ '" ^ 'in _JL_ \ / _ where J,^, B,^ and K,^ are motor inertia, damping coefficient and constant, respectively. For the flexible shaft, the cut variables are T(s), co(s) toward the motor rotor and Tf(s), Of (s) toward the fan. The relationship of the between the pairs of cut variables the model (a) for the flexible shaft is given by: T,(s)l f -1 OjT(s)' -s/kf

6 58 Computational Methods for Smart Structures and Materials II Figure 2: Through-across cuts for a DC motor with a loaded flexible shaft For the shaft load, given that it is a passive load, the torque applied to the load is zero and the relationship between the pairs of cut variables is In order to obtain a relationship between C0f{s), U(s) and i(s), the two pairs of cut variables T(s), oo(s) and Tf (s), (Of (s) have to be eliminated using the matrix equation of the DC motor. A matrix equation that maintains the mapping between the model components and the physical system components is obtained: [ shaft load model ][ flexible shaft model ][ PM - DC motor model ][U(s) i(s)] t where the subscript t denotes vector transpose. For the output of the system (the angular velocity Of of the fan), the right and side of the of the equation contain the 2 by 2 matrices corresponding to: the shaft load model (with parameters Jf and Bf), the flexible shaft model the DC motor model (with parameters!,, b^, K^,L, and R) and the inputs vector [U(s) i(s)]t to the DC motor. The conservation of the mapping represents a modularity feature of the model and can be used for changing various components of the model while conserving the same cut variables. A modular model permits the replacement of components having the across-through cuts for the power transmission components. For example, flexible shaft model can be any of the (a), (b), (c) or

7 Computational Methods for Smart Structures and Materials II 59 (d) models presented in section 2. Only series systems, (as, the example of a motor drive) were investigated here. Electric networks and multi-body dynamic systems often result in models described by differential algebraic equations. 5 Simulink modular modeling using encapsulation Simulink is a graphical programming language that is based on data-flow approach, i.e. each block is executed the moment all input values are available. For this purpose, in Simulink models, the assignment of the input and output variables is required and this makes it a non-object Oriented programming language. Simulink procedure for "Encapsulating a Subsystem" facilitates, however, modular modeling of electro-mechanical systems. For this reason, Simulink, similar to other non- Object Oriented programming languages (for example C or Fortran), can be used for implementing Object Oriented design in a hybrid form, Rumbach [2]. ' scope fc-tr ju 11 1 i Current balanceequation i=(au)u - (Ao)Omega H Scope Bi of rvr~^ d(c mega /dt BT Newton seed" hd law d(omega)/dt = (B )i-(bt)t,»a "scope" T Figure 3: Simulink model of the PM-DC motor

8 60 Computational Methods for Smart Structures and Materials II For the PM-DC motor, after assigning the input as U and the output i, toward DC supply, and the input T and the output co, towards the flexible shaft, the Simulink model shown in Fig. 3 can be obtained. In this case, the assignment for the load (flexible shaft and shaft load), is the dual, i.e. input co and the output T, while for the DC supply is the input i and the output U. By encapsulating the three subsystems: [Load] =[ shaft load model ][ flexible shaft model ] [ PM - DC motor] [DCsupply]=[U(s) i(s)] the Simulink model is reduced to the three blocks shown in Fig. 4.,» om ega LoacJ T,» T ' omega *,» i U rjlksup Sy \w ^ U i m PM-DC motor " Figure 4: Encapsulated Simulink model of the PM DC motor, DC supply and load.

9 Computational Methods for Smart Structures and Materials II The encapsulation, as illustrated in Fig. 4, permits modular replacement of blocks as long as the designations of inputs and outputs are respected. The encapsulation in Simuiink model has the advantage of retaining the identity of the subsystem blocks not only during graphical programming of the model, but also during the execution of the simulation. Data flow approach leads to the output computation for each block as soon as all its inputs are available. While this data flow approach slows down the simulation, it has advantages during program debugging. In order to speed up the execution, the Simuiink model has to be compiled in a high level language program that can be executed faster. The results presented in section 2, regarding the models of the flexible shaft, can be reused also in this Simuiink encapsulation example, by using for the [ flexible shaft model ] any of the (a), (b), (c) or (d) models. Moreover, as shaft load can be added a beam model with Tf (s), Of (s) cut variables. Conclusions Object Oriented modelling of smart structures permits modular modelling using a library of component modules. Series systems can be modeled modularly by using the property of matrix multiplication of two pole module models. Simuiink programming can also achieve modular modelling, using encapsulation, and facilitates module replacement. References [1] Cellier, F., Continuous System Modeling, Springer Verlag, [2] Rumbach, J. et al. Object Oriented Modeling and Design, Prentice Hall, [3] Fishwick, P.A., Simulation Model Design and Execution, Prentice Hall, [4] TenEyck, J., Object-Oriented Programming, in The Handbook of Software for Engineers and Scientists, editor-in-chief P. Ross, CRC Press, pp , [5] Mattson, S., Andersson, M., Astrom, K., Object-Oriented Modeling and Simulation, in CAS for Control Systems, edited by D. Linkens, Marcel Dekker, pp , 1993, [6] Paynter, H.M., Analysis and Design of Engineering Systems, MIT Press, pp ,1960. [7] Simeon, B., Fuhrer, C., Rentrop, P., Differential-algebraic equations in vehicle system dynamics, Surveys on Mathematics for Industry, I, pp [8] Kasper, R., Koch, W., Object-Oriented Behavioral Modelling of Mechatronic systems, Proc. Third Conference on Mechatronics and Robotics, editor J. Luckel, 4-6 Oct., pp , [9] Ferretti, G. et al., Modular Dynamic Virtual-Reality Modeling of Robotic systems, IEEE Robotics and Automation Magazine, pp , Dec., 1999.

10 62 Computational Methods for Smart Structures and Materials II [10] Mann, H., Modular Approach to Lagrange's Formalism for Mechatronic Systems, Proc. Int. Symp. Mechatronics and Advanced Motion Control, pp , May, [HJNecsulescu, D.S., Skowronski, J.M., Shaban-Zanjani, H., Low Speed Motion Control of a Hamiltonian System, in Mechanics and Control, R.S. Gutalu-editor, Plenum Press, pp , [12] Stanway, J. et al., Comparison and Validation of Dynamics Simulation Models for a Structurally Flexible Manipulator, J. Dyn. Syst. Measurement and Control, pp , Sept., [13]Torby, B.J., Kimura, I., Dynamic Modeling of a Flexible Manipulator with Prismatic Links, J. Dyn. Syst. Measurement and Control, pp , Dec., [14]Necsulescu, D.S., de Carufel, J., Finite Element Model Based Predictive Control of Smart Structures, Dynamics and Control of Structures in Space, editors C.L. Kirk, D. J. Inman, Computational Mechanics Publications, [15] Yoshimura, H., Kawasa, T., A Network-Theoretical Formalism for Flexible Multibody Dynamics, Proc. 4^ Int. Conf. Dynamics and Control of Structures in Space, May, 1999.