USING CIRCUIT ANALOGIES* L. Lcnriing u. Ozgiillert Department of Electrical Engineering The Ohio State Uiiiversity Columbus, OH 43210 Downloaded by UNIVERSITY OF ADELAIDE on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2736 Abstract Circuit analogies of mechanical systems are used to establish a framework in which large flexible structures may be analyzed. Impedance and scattering parameters of flexible structures are derived directly from either partial differential equation or finite element models of the structures. A cantilevered beam example is used for illustration and comparison of the two derivations. Component mode synthesis and controlled component synthesis of structures are discussed and related t o circuits. Various design methods for shaping scattering parameters of flexible structures for desired scattering properties are formulated including H,-based designs. Introduction Large space structures (LSS) may be viewed as a n interconnection of several substructures or networks. Essentially, the vibration control problem of flexible structures can be approached as a power transfer problem-similar t o problems in transmission line theory or electrical circuits. Thus, each of the substructures may be described in terms of scattering parameters similar t o those of an electrical network. T h e scattering parameters can be derived from either a partial differential equation (PDE) model of the structure or from the finite element model (FEM) of the structure. Like transmission lines, LSS are distributed parameter systems; therefore, they exhibit wave modes similar t o those encountered in the P D E approach. FEM’s are merely approximations of P D E models. For lumped parameter models such as FEhI’s, lumped parameter circuit analogies of LSS may be obtained directly from the FEM equations. Therefore, the scattering parameters of the structures may be obtained using circuit analysis techniques. T h e P D E models have been used in various control approaches such as the traveling wave approach [I, 2, 3, 41. In the traveling wave approach t o flexible structure “This paper is based on work sponsored by the Air Force Office of Scientific Research (AFSC) under Contract F49620-S9C-0046. t Member AIAA Copyright 0 1 9 9 1 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. control, the disturbances t o a structure are viewed as waves traveling through a medium. These waves can be reflected, partially reflected, or dissipated as they travel across the structure. At interfaces within the structure, scattering properties may be examined-how much of an incoming wave is reflected and how much is absorbed. With this perspective, the flow of these disturbances through the structure may be more easily understood. T h e propagation of disturbances in a mechanical system is mathematically similar t o t h a t of electrical or microwave circuits. T h e motivation for this work is to reconsider circuit analogies of mechanical systems and use existing results from circuit theory and network theory in the analysis and design of LSS. Inlthis paper, circuit and network theory will be reviewed with emphasis on circuit analogies of mechanical systems. Modeling of flexible structures will be considered beginning with existing P D E modeling and finite element modeling. T h e impedance and scattering parameters of a flexible structure will then be determined both from the wave formulation from the P D E model and from a lumped circuit analogy of the FERI. Component mode synthesis (CMS) techniques for modeling LSS will be considered in the framework of impedance and scattering parameters. Scattering parameter design methods will be discussed including the formulation of an H , design problem and a discussion of controlled component synthesis (CCS) techniques. Review of Circuit and Network Theory For decades, circuit analysis and network synthesis has received much attention in the research community. T h e theory in these areas is very well developed and several excellent texts are available [5, GI. A general electrical network is comprised of several interconnected circuit elements. Terminal pairs or ports allow voltage and current signals to be measured or applied a t various points in the network. A network with n ports is commonly called an n-port network or simply an n-port. For a general n-port network as shoirn i n Figure 1, the port-voltage and port-current vectors are defined 1208 I Scattering parameters exist for near parameter or distributed parameter passi works. Several quantities of interest t o powe mission may be simply and concisely repres terms of the scattering parameters. Thus, scattering parameters provide a convenient way t o represent the wave reflecting or absorbing behavior of a network. Again, consider the n-port shown in Figure 1 and let the following vectors and matrices be defined: Downloaded by UNIVERSITY OF ADELAIDE on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2736 NETWORK - Incident voltage vector V, - Reflected voltage vector V’ - “Ordinary” voltage vector I’ - “Ordinary” current vector 2’ - “Ordinary” open circuit impedance matrix R, - Normalizing matrix V - Normalized voltage vector I - Normalized current vector V, - Source voltage - vector 2, - Input (source) impedance matrix (diagonal) S - Scattering matrix 2 - Normalized impedance matrix Figure 1: General Representation of an n-port Network with Attached Sources and Loads Furthermore, let as Often, one of these vectors is taken t o be the excitation signal and the other is taken t o be the response signal. In the frequency domain, V(s) and I(.) represent the transforms of w(t) and i ( t ) ,respectively. Standard definitions for linearity, time-invariance, passivity, and causality may be found in most texts on network theory. Several theorems and excellent thorough discussions of these topics may be found in [7, 6, 51. Impedance Parameters In general, sources with internal impedances may be attached to the n-port network as shown in Figure 1. The open circuit impedance matrix, Z ( s ) , is defined such that V ( s )= Z ( s ) I ( s ) (3) V’ = &(E +K) I’ = &-l(v,-Vr) *here V’ and I’ are the “ordinary” voltage and current vectors which would actually be measured at the Ports, and RO is the normalizing matrix. Commonly, the vectors are normalized either with respect to the input impedances such that the normalizing matrix is chosen to be 1 Ro = a[zs(s)+ zs(-s)], (7) or with respect to unity such that R, = U where U is the identity matrix. This is equivalent to assuming unit input resistors. and’ V, are fictitious voltages representing incident and reflected voltage vectors. The normalized port-voltage and port-current vectors, V and I , are defined as where v = &-lv/ I = 6 s . Therefore, the incident and reflected voltage vectors (4) satisfy v i = -1 v + I ) For LTI n-port networks, the impedance matrices have interesting properties relating passivity and positive realness [7, 61. Scatterine Parameters Scattering parameters have been used in transmission line theory and circuit theory since the early 1920’s. v, = 2 1 2 v -I) 4 where = v,. The scattering matrix, S, is defined such that v, 1209 = sv, Fezt = M X + P X + ICX Text = J e +P i +K 6 where 0 5 ISijl L: 1 (13) A scattering parameter with a magnitude of 1 indicates that all incident power is completely reflected. Conversely, a scattering parameter with a magnitude of 0 indicates that all incident power is completely absorbed by the network. If the normalized impedance matrix, Z,is defined such that =ZI (‘4) then v S = (2+ U)-’(Z - U ) = ( 2 - U ) ( Z + U)-’ Modeling of Flexible Structures The flexible beam is a typical example of a flexible structure and has been widely used for flexible structure research. Therefore, the flexible beam will be used throughout this discussion as an illustrative example. (15) X where U is the identity matrix. This is equivalent t o Downloaded by UNIVERSITY OF ADELAIDE on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2736 (18) (19) S = (Z’+Ro)-’(Z’-Ro) = (Z’-Ro)(Z’+Ro)-’ (16) where 2’is the “ordinary” open circuit impedance matrix. Circuit analogies of mechanical systems have existed for many years. From the beginning of the study of electricity, electrical quantities such as inductances and capacitances were described in terms of mechanical quantities such as springs and masses. When the history of mechanical and electrical systems is considered, the complementary progress of the two areas is evident. Typical examples of circuit analogies for mechanical systems are shown in Table 1. Using these circuit Electrical voltage, V current, I , q charge, q Mechanical (Linear) velocity, v , x force, F momentum, P Mechanical (Rotational) angular velocity, w , 0 torque, r angular The Euler-Bernoulli beam equation for a onedimensional beam is given by where E is Young’s modulus, I is the area moment of inertia, p is the density of the material, and A is the cross sectional area of the beam. In the EulerBernoulli model, v ( z , t ) is defined t o be the transverse displacement of the beam at position x along the beam and at time t . Also, a translational force, f(x,t), is defined as the total external force acting on the beam as a function of position and time. This model assumes that the beam is homogeneous and the deflections are small. The model also ignores rotary inertia, shear effects, and gravity. The natural response of the beam can be found by considering the unforced system where f(x,t ) = 0. Using separation of variables, the solution is assumed to be of the form v(z,t)= R ( z ) T ( t ) . (21) The solution t o the spatial ODE is R ( z ) = a sin lcz + b cos k z + csinh lcz + d cosh lcz (22) where a , b , e, and d are constants to be determined from the boundary conditions, and analogies, the analogous systems of equations for electrical and mechanical systems are: Ie,t = CV+ (‘7) (23) is real. 12 10 Applying appropriate boundary conditions, the spatial solution for the clamped-free beam is found t o be R ( x ) = d (cosh kx - cos k x - cr(sinh k x - sin kx)) (24) where d is the constant from Equation 22. Assuming zero shear force at the tip of the beam, the frequency generating equation is found t o be cosh knL cos k,L +1=0 (25) where n = 0 , 1 , 2 , 3 , .. .is the mode number. This equation can be solved for k , and the angular frequencies, w,, can be determined from Downloaded by UNIVERSITY OF ADELAIDE on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2736 w, =ki :/ The impedance for the beam is defined usin velocit and force as the mechanical equivalents of vo tag current, respectively: ff (34) Evaluating at the tip of the beam, x = L , the impedance is Z(L) = EjIw k3 [ sin kL cosh kL - cos kL sinh kL 1 +coskLcoshkL Recall that k is a function of frequency as shown in Equation 23. Thus, this expression for the impedance is an irrational function in w . -&=, The impedance found in the previous section can be used to find the scattering parameter. The scattering parameter evaluated at the tip of the beam is given by S(L) = 1 R n ( z )= - (coshk,z - cosk,z - gn(sinhknz - sink,x)) Jz (27) + + cosh k , L cosk, L (28) sinh k , L sin k , L ' These spatial solutions are orthonormal such that u, 1' = R i ( z ) R j ( z ) d x= 6ij s=-j w - c j w 3- c A? The impedance of a beam can be used to model the system. Using an impedance model, several circuit and network synthesis methods may be applied in the design of structures and controllers. Using the circuit analogies of mechanical systems from Table 1, the impedance parameters for the mechanical systems [8] will be defined in terms of the electrical analogies. A lateral shear force f applied to the beam imposes the boundary condition = -EI- a3v ax3 & Using Equations 21 and 24 where d = and taking the Fourier transform with respect to time, JI; [sin kx - sinh kz + u(cosh kz + cos kz)] n ( w ) (31) where V , F , and 0 are the Fourier transforms of v, f, and T , respectively. The lateral velocity of the beam in the frequency domain is found by taking the Fourier transform: .T it --+ j w v = jwR(z)n(w) (32) j w = -[cosh kz - cos k z - u(sinh k z - sin kz)]n ( w ) ( 3 3 ) dz + (37) where V i , j = 0 , 1 , 2 , 3 , .. . ImDedance Parameters F=- Z(L) - R Z(L) R where R is the real para-hermitian part of the source impedance. Using Equation 35, the scattering parameter can be written as C = REIk3 (29) f (35) Scat terine: Parameters - By using these values of k and w and letting d = the normalized spatial solutions are found from Equation 24 t o be where ]. + 1 cos kL cosh kL sinkLcoshkL - coskLsinhkL is real. Thus, the scattering parameter is all-pass with unit magnitude. Like the expression for the impedance parameter, this expression for the scattering parameter is an irrational function of w . The scattering parameter characterizes the scattering properties of the beam at the tip. Cantilevered Beam Example Consider the cantilevered beam shown in Figure 2. As an example, the parameters given in Table 2 for an aluminum beam have been used. Assuming a unit source impedance ( R = l), frequency responses of the open circuit impedance and the scattering parameter have been generated numerically using a FORTRAN program. The phase variation of the impedance is shown in Figure 4. The phase of the scattering parameter is shown in Figure 5. Modal frequencies for the first ten modes of the beam are shown in Table 3. As expected, the scattering parameter has unit amplitude over the entire frequency range. This indicates that all energy applied to the structure over all frequency ranges is completely reflected-as expected for a cantilevered beam. Note that the scattering paranieter has zero phase at the resonant frequencies. This indicates that the reflection is in phase with the driving force at the tip which results in a standing wave. 12 11 inductance matrix, but is the matrix of the reciprocals of the inductances; that is, Table 2: Parameters for Aluminum Beam The upper left element in the impedance matrix, 211, corresponds t o the impedance observed at the tip of the beam. The impedance matrix can be used t o find the scattering matrix from Downloaded by UNIVERSITY OF ADELAIDE on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2736 S(s) = Frequency (Hz) P D E I FEM 0.0988 0.0988 0.6192 0.6193 1.7339 1.7350 3.3977 3.4054 5.6167 5.6492 + where R is the source impedance. Like the impedance matrix, the scattering matrix is composed of rational functions of s. The upper left element of the scattering matrix, S11, characterizes the scattering properties of the beam at the tip. Table 3: Modal Frequencies of Beam from EulerBernoulli and Finite Element Models Mode Number 1 2 3 4 5 Z(S) - R Z(s) R J Circuit Eauivalents 4 Finite Element Model As mentioned previously, finite element modeling is another common method for modeling flexible structures [9, 101. The equations of motion for the beam can be represented as (39) where M and I< are the mass and stiffness matrices of the structure, respectively. Fezt is the external force applied t o the nodes. The vectors $c and qc are the nodal acceleration and displacement vectors, respectively. The impedance parameters can be derived from the finite element model as they were from the PDE model. Using the mass and stiffness analogies of capacitance and inverse of inductance, the impedance matrix is defined as Z(s) = (sM + 4). (40) where s is the Laplace variable of complex frequency. Thus, the impedance matrix is composed of rational polynomials in s. Note that in the matrix case, the analogy for the stiffness matrix is not the inverse of the Figure 3: Circuit Equivalent of Single Element Usin.g the consistent mass matrix [9], the circuit equivalent of a single finite element is shown in Figure 3. The values for each capacitor and inductor are shown in the figure where I<, I<l = pAh 420 ’ h3 = EI ’ (43) (44) and the other parameters are as defined in Table 2. From left to right, the node voltages of this circuit are analogous to the generalized nodal velocities of the element: 6,6, ir’, and respectively. Note that the two ports on the left correspond the the left nodal variables and the two ports on the right correspond t o the right nodal variables. In the circuit shown, some capacitances and inductances are negative. This indicates that active elements 1212 e’, Downloaded by UNIVERSITY OF ADELAIDE on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2736 Fmqucncy (Hz) Figure 4: Phase of Impedance Parameter of Cantilevered Beam at Tip (PDE) Figure 6: Phase of Impedance Parameter of Cantilevered Beam at Tip (FEM) must be employed to implement this realization. The impedance matrix of the element, however, is positive real. Thus, the circuit is passive and a circuit realization exists which uses only passive elements. Imposing the boundary conditions for coupling elements together is analogous to connecting the rightmost ports of one element to the leftmost ports of the next element. phase of the impedance is shown in Figure 6. The phase of the scattering parameter is shown in Figure 7. Modal frequencies for the first ten modes from the finite element model are shown in Table 3. These frequencies are very close to the frequencies obtained from the Euler-Bernoulli beam model as shown in the same c table. The lower frequencies are nearly identical. Since the FEM is a finite dimensional approximation of the PDE model, the models are expected to match Cantilevered Beam Example very closely. Indeed, the frequency responses shown in Consider again the cantilevered beam example. Unit Figures 6 and 7 closely match the responses shown in source impedances were assumed such that R = 7.7, Figures 4 and 5 . where U is the identity matrix. Frequency responses The amplitude of the scattering parameter from the Of zll(s) and sll(s), corresponding to the FEM is also unity over all frequency ranges as expected parameter and the scattering parameter at the tip of and discussed previous~y. the beam, have been generated using MATLAB. The Figure 5: Phase of Scattering Parameter of Cantilevered Beam at Tip (PDE) 1213 Figure 7: Phase of Scattering Parameter of Cantile'vered Beam at Tip (FEM) Downloaded by UNIVERSITY OF ADELAIDE on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2736 Component Mode Circuit analogies of large flexible structures also have interesting applications to the well known component mode synthesis (CMS) methods [11, 12, 13, 14, 15, 16, 171. In CMS, a large flexible structure is broken into components or substructures. Each component or substructure is typically modeled using finite element modeling techniques. The order of each of these component models is reduced using the assumed-modes method or the Rayleigh-Ritz method with constraint equations. The reduced order components are then mathematically coupled together with interface compatibility equations t o form a reduced order model of the entire structure. The basic idea in CMS is to find a set of component modes for each component that make the greatest contribution t o the modes of the composite structure. Several methods exist t o select these modes. Perhaps the most widely known methods are the Craig-Bampton method, the MacNeal-Rubin method, and the BenfieldHruda method. The Craig-Bampton method [12], a refinement of the original CMS method proposed by Hurty [ll], uses fixed-interface normal modes and constraiiit modes. Fixed-interface normal modes are the natural modes of vibration which result when all of the component interface nodes are fixed. These are the modes which result when a single component is allowed to vibrate freely but all of the attached components are held rigid. Note that there are an infinite number of normal modes. Thus, in practice, the set of normal modes is truncated and some low frequency subset selected. The constraint modes are the displacement or shape functions of the component resulting from a statically imposed displacement at a single component interface node while keeping all other interface nodes fixed (rigid with zero displacement). Thus, the fixed-interface normal modes and the constraint modes are linearly independent and the mode set comprised of these modes can be used t o describe the motion of the component. The MacNeal-Rubin method [13, 141 uses freeinterface normal modes and residual modes. Freeinterface normal modes are the natural modes of vibration which result when all of the component interface nodes are allowed to move freely. Again, note that there are an infinite number of normal modes. Thus, in practice, the set of normal modes is truncated and some low frequency subset selected. Attachment modes are the static displacement or shape functions which result when a unit force is exerted on a single node and all other nodes are free. The residual modes are basically attachment modes which are linear combinations of the truncated normal modes representing the contributions of the truncated modes to the component motion. The Benfield-Hruda method [15] uses loadedinterface normal modes. Loaded-interface normal modes are the natural modes of vibration which result when the free component interface nodes are loaded with representative mass and stiffness contributions from the attached components. This method generates component modes which more closely resemble the system modes by including approximate dynamic effects of the remaining components. Isolated Boundary Loadinn Isolated boundary loading developed by Young [18] is a component modeling method which is based on the boundary loading ideas of the Benfield-Hruda method. Again, dynamic contributions of other components are included in the model of a given component. In isolated boundary loading, the interface loading matrices are merely the stiffness and mass matrices corresponding t o the boundary nodes of the adjoining component. The modified boundary submatrices are the same for each of the components involved and may be found either by loading the first component onto the second or by loading the second onto the first. The coupling of substructures using isolated boundary loading is slightly different than the coupling using CMS techniques. Rather than using boundary compatibility conditions, isolated boundary loading employs superposition to couple the components at the boundary. Bssentially, the equation of motion for the boundary of the coupled structure is found from superposition of the forces and motions at the boundaries of the individual components A and B such that AB Qi = fi”” = QA + Q ? fd” +fi”. (45) (46) Using isolated boundary loading for component modeling and this superposition-based approach for substructure coupling, the coupled model is the exact model for the composite system. Circuit Aiialorries The CMS methods which have been presented and the issues which are addressed may be analyzed using circuit analogies. The components or substructures of the system can be represented as subcircuits or subnetworks within a larger circuit or network. Coupling substructures is analogous to connecting circuit networks as shown in Figure 8. The ports of an n-port circuit are equivalent to the interface nodes of a component. Internal nodes of a circuit are equivalent to the interior or internal coordinates. Ports which are electrically shorted to ground represent fixed-interface nodes. Similarly, ports which are electrically open represent free-interface nodes. Thus, fixed-interface normal modes of the component are equivalent to the circuit modes which exist when 1214 Downloaded by UNIVERSITY OF ADELAIDE on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2736 Figure 8: Structure/Circuit Coupling Analogy all ports are shorted t o ground. Free-interface modes are equivalent t o circuit modes when all ports are open. A constraint mode is basically the DC voltage profile of the circuit when a unit voltage is applied to a single port while all other ports remain grounded. An attachment mode is basically the DC voltage profile of the circuit when a unit current is applied t o a single port while all other ports remain open. Rigid body modes are equivalent t o the DC steady-state response of the circuit when dealing with rigidly connected substructures. Models for interconnected multibody structures such as a multiple-link manipulator are nonlinear. These can be handled with some nonlinear circuit elements. CMS from the circuits perspective reduces to finding a set of circuit modes for each subnetwork that make the greatest contribution t o the modes of the composite network. The number of internal nodes used in the circuit is reduced and the capacitances and inductances of the subnetwork are modified t o retain some of the properties of the truncated modes. The previously discussed CMS techniques can obviously be applied t o the circuit equivalents derived from the finite element model by merely replacing the mass and stiffness matrices with the capacitance and inverse of inductance matrices, etc. A more interesting issue is that of boundary loading as addressed with the Benfield-Hruda method and isolated boundary loading in controlled component synthesis. From the Benfield-Hruda method, the concept of loaded-interface normal modes is developed where a component is loaded with approximate mass and stiffness properties of the adjoining components. This loading is equivalent to loading the corresponding subnetwork with capacitance and inductance properties of the adjoining subnetworks. The Benfield-Hruda method does not provide intuitive physical insight into what is happening other than the fact that these properties are being projected onto the component. From the circuits perspective, isolated boundary loading attaches all adjoining subnetworks to the principal subnetwork. Essentially, the ports or nodes of the adjoining networks are connected to the ports of the principal subnetwork. All parallel and series circuit element combinations are reduced to their equivalent elements. Thus, only the values of the circuit elements attached t o the ports ar ements attached only t o interi After these element values are changed, t responding t o the adjoining components so that only the “original” no principal subnetwork are retained. An alternative approach t o these loading methods might be to reduce the adjoining subnetworks to their Thevenin or Norton equivalents or some circuit approximation of these equivalents. To generate these approximations, any of the several circuit approximation techniques could be used. The equivalents could then be connected to the principal subnetwork to determine a mode set which closely resembles the modes of the composite system. This method of loading provides insight regarding how the loading is accomplished. Cantilevered Beam Example The cantilevered beam example which has been considered previously is now assumed t o be composed of two identical components with the same material properties as before. Thus, the parameters for the components are as specified in Table 2 with the exception that the length and mass are halved. Figure 9: Input Impedance at Tip of the Coupled Cantilevered Beam Figures 9 and 10 show the impedance and scattering parameters for the coupled structure using the CraigBampton mode set. These impedance and scattering parameters are very close to the impedance and scattering parameters from the PDE model and the full FEM as shown in Figures 4, 5, 6, and 7. Modal frequencies for the first eight modes of the coupled structure using the Craig-Bampton method are shown in Table 4. These frequencies are very close t o the frequencies obtained from the Euler-Bernoulli beam model and the FEM a s shown in Table 3. The lower frequencies are identical to those from the FEM. 1215 Figure 11: Cascaded Circuits/Structures Downloaded by UNIVERSITY OF ADELAIDE on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2736 defined such that Figure 10: Scattering Parameter at Tip of the Coupled Cantilevered Beam Table 4: Modal Frequencies of Beam Using CraigBampton Method 4 5 6 0.0988 0.6193 1.7353 3.4195 5.6818 8.6777 where the superscripts i and r denote incident and reflected voltages, respectively. SA and SB are the scattering matrices for subnetworks A and B , respectively. V2 and Vb represent the port voltage vectors for the m ports of A and B , respectively, that are connected together. VI represents the port voltage vector for the n ports of A that are not connected to B but may be connected to some other network or structure. The scattering matrix for A can be partitioned as = [ s 1 1 s12 SZl s 2 2 1 (49) wherg the submatrices have the appropriate dimensions. If the two subnetworks are cascaded together, the scattering parameter, S, for the resulting n-port network can be found such that vy = sv;. By cascading the two networks and imposing the appropriate boundary conditions, the scattering matrix can be written as Scattering Parameter Control Design s = S11 + sIZsB(I - S22sB)-1S21 As mentioned previously, scattering parameters are closely related to power transfer properties of a network or structure. Often, it may be important to control the scattering of energy through the structure. Therefore, scattering parameter design is of great interest. This type of design objective differs fundamentally from the approach of adding damping to a structure or minimizing motion of a structure. The actual scattering of energy throughout the structure may be considered and controlled. Disturbances may be reflected, trapped, redirected, or absorbed to some extent using scattering parameter design. Consider the cascaded networks shown in Figure 11. Assume that subnetwork A is a n m-port network and subnetwork B is a m-port network. In general, subnetworks A and B may be cascaded as shown where the m ports of B are connected to m of the ports of A . The scattering parameters for the subnetworks are + (51) assuming that SB # ST; such that ( I - S ~ ~ S Bex)-~ ists. If subnetwork B is taken to be a controller for a structure represented by subnetwork A, then SB may be designed for desired properties. For example, suppose no reflection is desired at the port such that S = 0. B would be designed such that SB = [STiSiiSF:Szz + I]-' [ST~SIIST:] (52) assuming that ST; and ST: exist. For a system with # nz, S 1 2 and S 2 1 are not square and the inverses do not exist. Even if n = m and these inverses do exist, causality and realizability problems still may arise. However, if some reflection is allowed such that the scattering parameter is S, B could be designed to have the scattering parameter n 12 16 SB = [SGI(S - S11)S,;~S22 + I] -I [S,l(S - Sll)S,-,l]. (53) Perhaps it is important t o minimize th.e reflections over some frequency range. It should be noted that the model of the structure is passive, linear, and timeinvariant. Thus, the scattering matrix is bounded-real and the scattering parameter for the cascaded network given in Equation 51 can be minimized over the desired frequency range using H2 or H , design techniques. The key idea is t o make IS(jw)l small over some desired frequency ranges. A weighting function, may be chosen such that IW(jw)l is large over the desired frequency ranges where the scattering parameter is to be minimized. The maximum of I W ( j w ) S ( j w ) l over all w can be minimized over SB. The maximum over all w can be expressed in terms of the H,-norm defined such that Controlled ComDonent Svnt hesis Controlled component synthesis (CCS) is a control design method in which controllers are designed for individual components based on component models [18, 191. This decentralized control strategy extends the component mode synthesis modeling concepts into control concepts by allowing component controllers based on component models to be developed independently. The interlocking control concept is one of the key ideas behind CCS. With interlocking control, collocated sensors are placed at the internal boundary coordinates of each component. The component controllers are designed to minimize motion at the internal boundary nodes. One or more controllers are inserted near the substructure interfaces. By minimizing the internal boundary motion, the transmission of disturbances between components is suppressed. Although any technique may be used for controller design, the linear quadratic optimal regulator (LQR) is perhaps the most convenient [18]. The weighted internal boundary states are t o be included in the cost function with the weighted control inputs. Assuming velocity measurements, the output for component A is defined to be yA = ifbecause the sensors are located at the internal boundary coordinates. The control inputs are uA = because the actuators are also lecated at the internal boundary coordinates. Using the LQR approach, the control law for component A is found t o minimize the quadratic cost function Downloaded by UNIVERSITY OF ADELAIDE on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2736 w(~), (54) where 8 denotes the largest singular value. Therefore, the objective is to find SB which, if possible, achieves where SB stabilizes the scattering properties of the structure in some sense and is the scattering matrix of a realizable network or structure. If achieving infimum is impossible, perhaps it is possible to come arbitrarily close t o infimum. This problem is similar to the four-block problem from H, theory. It should be noted that in the FEM case, the scattering parameters of the structure are rational matrices in s. In the PDE case, however, the scattering parameters are complex irrational functions of w . sensitive A disturbance sources Figure 12: Configuration for Disturbance Isolation Several other similar problems may be formulated. The network shown in Figure 12 illustrates how a controller might be used to isolate disturbances from sensitive areas of the structure. In this case, the scattering parameters representing the scattering properties from the disturbance ports t o the sensitive area ports should be minimized in some sense. Conversely, the scattering parameters representing the scattering properties from the sensitive area ports to other insensitive ports may be maximized t o allow disturbances to escape to the insensitive areas. Shaping the scattering parameters as desired can be a powerful tool for vibration control of large flexible structures. ft In general, a state feedback control law could be found to minimize this performance criterion. However, optimal output feedback can also be employed to achieve good results and is easier to implement than state feedback. CCS controller designs are inherently decentralized since only the component outputs or states are required for the component control law. The basis for the CCS method is closely related to the scattering parameters of the structure. The fundamental idea in CCS, as mentioned before, is to minimize the transmission of disturbances between components. In other words, certain component scattering properties are desired. With this concept in mind, scattering parameter design techniques such as those discussed previously may be used at the component level. Thus, the controllers for each component would be designed to achieve desired scattering properties directly. Conclusions In this paper, circuit analogies of mechanical systems have been used to establish a framework in which large 12 17 Downloaded by UNIVERSITY OF ADELAIDE on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2736 flexible structures may be analyzed. It has been established that impedance and scattering models of flexible structures can be derived directly from either partial differential equation models or finite element models of the structures. A cantilevered beam example has been used to parallel these discussions and illustrate the concepts. Component mode synthesis has also been discussed in terms of circuit analogies. It has been shown that issues related to component mode synthesis can be addressed in terms of circuit analogies. Using circuit analogies of flexible structures, the scattering properties of structures are characterized in terms of the scattering parameters. The circuit analogy method of modeling large flexible structures can provide additional insight into system characteristics such as scattering properties that conventional modeling techniques cannot usually provide. This knowledge can be exploited in controller design. The scattering parameters may be “shaped” as desired to achieve desired scattering properties. Controllers may be designed using H2 or H, design techniques to minimize scattering over certain frequency ranges of interest. This type of problem has been formulated and is very similar to the familiar four-block problem from H, theory. Controlled component synthesis was presented. 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