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RESEARCH PAPERS

J. Basic Eng. 1966;88(2):287-294. doi:10.1115/1.3645849.

The concept of hydraulic capacitance is applied to pneumatic bellows assemblies. This concept is then used in the analysis of a number of different basic types of pneumatic controllers in common use. Its application allows the effect of the bellows system used for delaying the derivative action, and the effect of bellows movement generally, to be conveniently included in the analyses. Numerical values determined for a particular controller show that the effect of bellows movement can change the transfer function constants by as much as 20 percent.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1966;88(2):295-305. doi:10.1115/1.3645850.

The gain characteristics of a subsonic, pressure-controlled, proportional, fluid-jet amplifier are developed in relation to the control-port and receiver-design parameters. The analysis uses an assumed submerged-jet representation. Control ports are added and the control flows associated with various control-port positions are calculated. The control-port characteristics are presented in a series of design curves. The receiver characteristics are obtained also as a function of their inlet geometry. Optimum dimensions are found to maximize gain for pressure, flow, and power amplifiers. Linearity is discussed. The receiver characteristics are presented in a series of design curves. Complete amplifier characteristics are obtained by combining the control-port and receiver characteristics. An example calculation is shown.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1966;88(2):306-310. doi:10.1115/1.3645851.

The paper discusses optimal controls of processes with unknown constant parameters, where the processes are such that no measurements on the parameters are available during control periods. The general formulation of this optimal control problem is given for such systems, and it is shown that the formulation becomes quite simple when the equation for the observed-state vector is invertible, and that the problems of estimation and optimal controls cannot be separated for the class of problems discussed in the paper even when the systems are linear with quadratic criterion functions.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1966;88(2):311-315. doi:10.1115/1.3645852.

A graphical procedure for constructing the time response of a system from its phase-plane trajectory is described. From this, a graphical solution of the equation ẍ + θ(ẋ)g(x) + φ(ẋ) + h(x) = F(t) is presented. The construction procedure for the equation ẍ + φ(ẋ) + c(t)X(x) = F(t) is detailed as a special case of the more general form

ẍ + a(t)M(x)N(ẋ) + θ(ẋ)g(x) + b(t)Q(ẋ)
   + φ(ẋ) + c(t)X(x) + h(x) = F(t)
which can also be solved in a similar fashion. The extension of these techniques to higher-order systems is presented.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1966;88(2):316-321. doi:10.1115/1.3645853.

Theoretical and experimental investigations have been made on fluid-power transmission in hydraulic systems by pulsating flow. In particular, the system efficiency and the viscosity effect on the dynamic response of pulsating flow in the fluid line have been studied. Test results on the fluid-line dynamic response and on the system efficiency that obtained from the line-loss test setup and the miniaturized P-F hydraulic system setup, respectively, are presented.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1966;88(2):323-328. doi:10.1115/1.3645857.

Linear, stationary systems with multiple inputs when subject to performance indices quadratic in the state variables, but explicitly independent of the control variables, may be optimally governed by singular control. Necessary and sufficient conditions for the optimality of both partially singular control and totally singular control are obtained. Moreover, explicit formulae are presented for both open-loop and linear-feedback implementations of the optimal singular control.

Topics: Feedback
Commentary by Dr. Valentin Fuster
J. Basic Eng. 1966;88(2):329-336. doi:10.1115/1.3645858.

This paper describes an algebraic method of control system design using the system characteristic equation and the Routh array. This technique does not require finding roots of the characteristic equation and therefore is suitable for pencil and paper analysis. In addition, since no graphical steps are required, large system trade-off studies can be easily programmed for digital computer optimization. The method is based on the idea of validating a simple response approximation formed by truncating the characteristic equation. This validation is performed by placing a constraint on the ratio between the integrated square of the system impulse response and the corresponding integral of the approximation. The values of these integrals are obtained quite easily from the Routh array. Additional ratios and a generalized damping ratio are also defined. The design of a hydraulic control system is presented as an example problem illustrating the use of this method.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1966;88(2):337-342. doi:10.1115/1.3645859.

In the design of controllers for heat transfer systems, one must often describe the plant dynamics by partial differential equations. The problem of optimizing a controller for a system described by partial differential equations is considered here using exact and approximate methods. Results equivalent to the Euler-Lagrange equations are derived for the minimization of an index of performance with integral equation constraints. These integral equation constraints represent the solution of the partial differential equations and the associated boundary conditions. The optimization of the control system using a product expansion as an approximation to the transcendental transfer function of the system is also considered. The results using the two methods are in good agreement. Two examples are given illustrating the application of both the exact and approximate methods. The approximate method requires less computation.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1966;88(2):343-354. doi:10.1115/1.3645860.

A combined digital-analog mathematical model for the dynamic analysis of vertical U-tube natural-circulation steam generators is presented. The application of this model to the optimal design of a water-level controller for a steam generating unit is demonstrated. It is shown that a control system consisting of standard proportional and reset controls on water-level deviation from a desired set point and the difference between the steam and feedwater mass flow rates can be successfully employed for the control of water level in such a plant. The optimum values, as well as the range of the controller parameter sellings for which the steam generator exhibits a desired stable response, are determined.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1966;88(2):355-361. doi:10.1115/1.3645861.

A technique is presented for the design of a continuous nonlinear multivariable control algorithm from a set of simple, linear control models. The coefficients of the linear control models are dependent on the steady state operating point. The technique makes use of line integration and the method of least squares to bridge the gap between the sets of linear coefficients and the continuous nonlinear functions of the nonlinear algorithm. In general, the nonlinear control algorithm is nonunique. The method permits the extension of linear design procedures to the design of nonlinear control systems. It can be used to design a control system for a nonlinear process to maintain the dynamics of the entire system invariant with respect to changes in the operating point.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1966;88(2):362-368. doi:10.1115/1.3645862.

The problem considered is the sequential estimation of states and parameters in noisy nonlinear systems. The class of systems considered is those in which the dynamical behavior is described by an ordinary differential equation. No statistical assumptions are required concerning the nature of the unknown inputs to the system or the measurement errors on the output. For estimation purposes, a least-squares criterion is used. The new feature of the approach presented is that a sequential least-squares estimator is obtained for the class of problems considered. This estimator could be implemented in real time. Experimental results from several examples indicate that the proposed estimation scheme is feasible.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1966;88(2):369-378. doi:10.1115/1.3645863.

The core problem in the theory of measurements is to assign a value to represent quantitatively our knowledge of a physical property, given a finite set of observations. A corollary problem is to provide a statement as to the quality of the measurement. It is asserted that measurement is a process of drawing plausible inferences from incomplete data. A procedure for treating measurement processes is developed following Jaynes’ formalism, wherein probability is interpreted subjectively as a state of knowledge. In addition to the set of possible outcomes, it is necessary to include prior knowledge. In this paper, only stationary measurement processes are considered, employing repetitive observations of the same quantity. The set of possible observations is exhaustive and mutually exclusive. A theory that predicts infinite deviations cannot be accepted. Three conclusions are presented: (a) When only the expectation value and the variance are assumed known, the least biased, probability distribution is Gaussian; (b) when consideration of a possible malfunction is included, the probability distribution is bounded at three or possibly four standard deviations; (c) when the estimate of the expectation value is studied for a single set of observations, the probability distribution reduces to a rectangular function between the specified bounds. Uncertainty is defined by Shannon’s theorem. Imprecision is defined as the bounds on the probability distribution.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1966;88(2):385-391. doi:10.1115/1.3645865.

An optimum control problem which can be solved completely is the linear, time-invariant free-terminal-time, regulator problem with unrestricted control function and a quadratic-type performance index. However, if the control function is constrained, the solution of the regulator problem becomes complicated. In the present work, when the control function |u(t)| ≤ 1, the requirements of the existence of an optimal solution and a method for obtaining this solution are outlined.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1966;88(2):392-398. doi:10.1115/1.3645868.

An approach to control system design is described, based on an extension of the multilevel control concept. In essence, the problem is divided into simpler subproblems: (a) The controlled process is decomposed into a number of subprocesses, each with its own control system operating on a local suboptimal performance criterion; (b) each subprocess controller is decomposed into a hierarchy of control functions which distribute the load and responsibility for satisfying the control objective. The primary effort of the paper is in the development of the control hierarchy as a useful concept in control system design and implementation. The hierarchy is described in the context of the multilevel approach; however, an important distinguishing feature is the association of levels with disturbance sets roughly classified according to the frequency dependency of their effects on the overall performance. Also associated with the hierarchy is an ordering with respect to time scale, complexity of computation, degree of uncertainty, and so on, which relate to important design considerations for the overall system.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1966;88(2):399-406. doi:10.1115/1.3645869.

In this paper the problem of the stability of motion of the equilibrium solution x1 = x2 [[ellipsis]] = xn = 0 is studied, in the sense of Lyapunov, for a class of systems represented by a system of differential equations dxi /dt = Fi (x1 , x2 [[ellipsis]]xn , t), i = 1, 2[[ellipsis]]n or ẋ = A (x,t)x . Various x1 are known as state variables and Fi (0, 0[[ellipsis]]0, ∞) = 0. The various elements of square matrix A (x , t) are functions of time as well as functions of state variables x . Two different methods for generating Lyapunov functions are developed. In the first method the differential equation is multiplied by various state variables and integrated by parts to generate a proper Lyapunov function and a number of matrices α, α1 [[ellipsis]]αn , S 1 , S 2 [[ellipsis]]S n . The second method assumes a quadratic Lyapunov function V = [x ′ S (x ,t)x ], x ′ being the transpose of x . The elements of S (x ,t) may be functions of time and the state variables or constants. The time derivative V̇ is given by V̇ = x ′ [B ′ A + Ṡ ]x = x ′ T (t,x )x where B x gives the gradient ∇V, and Ṡ is defined as ∂S /∂t. For the equilibrium solution x1 = x2 [[ellipsis]] = xn = 0 to be stable it is required that V̇ should be negative definite or negative semidefinite and V should be positive definite. These considerations determine the sufficient conditions of stability.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1966;88(2):408-414. doi:10.1115/1.3645872.

Inlet and outlet temperature fluctuations of the fluid passing through the inner tube of a concentric-tube counterflow heat exchanger are related by a temperature transfer function. This function is affected by the velocity distribution of the stream, a distribution which is usually in the shape of a paraboloid. Such a distribution has a center-line velocity that is higher than the bulk velocity. The center-line velocity is responsible for longitudinal temperature diffusion effects which cannot be ignored. Slug flow at bulk velocity, a convenient assumption made by many authors, does not account for the shape of experimental fluid-temperature transients discussed in this paper. Rather, the center-line velocity and the bulk velocity, both convenient quantities, are recommended as effective aids in the successful prediction of temperature transients.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1966;88(2):415-418. doi:10.1115/1.3645873.

For a class of linear systems the conditions are given under which drift compensation is possible by adjustments of the control parameters. In such a case the system is called “completely adaptable.” The equations for the adjusting mechanism are derived by use of a functional derivative technique, admitting a remarkably simple implementation. Finally, the stability conditions for the total system are given.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1966;88(2):419-428. doi:10.1115/1.3645874.

This paper presents a new technique for constructing Lyapunov functions for estimating the domain of asymptotic stability of nonlinear control systems. A machine program assigns figures of merit (based on the quality of the stability-boundary estimates) to the members of a set of Lyapunov functions and then searches for the one which maximizes this measure. The method is applied to relay-control systems, which are not tractable by other systematic techniques such as the method of Zubov. The paper is divided into two parts. The first part contains a discussion of the motions of relay-control systems and the technique of investigating their stability domains with Lyapunov functions. The second part contains a discussion of existing methods of generating Lyapunov functions and a description of the new method, along with a number of second and third-order examples.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1966;88(2):429-436. doi:10.1115/1.3645875.

The paper consists principally of three parts. In the first, an original analytic representation is introduced for systems where differential equations are available. In the second, the structure of the functional is analyzed with nonzero initial conditions. The third introduces functional representations for systems described by past measured input-output records.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1966;88(2):437-443. doi:10.1115/1.3645876.

To find the optimum control law u = u(x) for the process ẋ = f(x, u), the Hamiltonian H = p′ f is formed. The optimum control law can be expressed as u = u* = σ(p, x), where u* maximizes H. The transformation from the state x to the “costate” p entails the analytic solution of the nonlinear system: ẋ = f(x, σ(p, x)); ṗ = −fxp with boundary conditions at two points. Since such a solution generally can not be found, we seek a quasi-optimum control law of the form u = σ(P + Mξ, x), where x = X + ξ with ‖ξ‖ small, and X, P are the solutions of a simplified problem, obtained by setting ξ = 0 in the above two-point boundary-value problem. We assume that P(X) is known. It is shown that the matrix M satisfies a Riccati equation, −Ṁ = MHXP + HPX M + MHPP M + HXX , and can be computed by solving a linear system of equations. A simple example illustrates the application of the technique to a problem with a bounded control variable.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1966;88(2):444-450. doi:10.1115/1.3645877.

The considerations necessary for minimization of solar radiation pressure effects for gravity-gradient stabilized vehicles are presented here. Owing to the rather weak restoring forces available for gravity-gradient stabilized vehicles, solar pressure torques represent a prime source of attitude errors unless steps are taken to minimize their effects. The solar torque minimization procedure generally consists of four distinct steps for a given vehicle configuration: (a) Derivation of the solar torque expressions for the characteristic vehicle configuration, including such effects as diffuse reflection, multiple reflections, and so on; (b) identification of the relative contribution of the solar torques on the various surfaces, and facilitation of solar torque minimization by balancing torque contributions of similar time variation and opposite sign against one another; (c) minimization of the torque about the vehicle axis with the weakest restoring torque (usually the local vertical) via optimization of reflectance characteristics and other physical parameters (using a steepest descent or similar approach); and (d) determination of the vehicle attitude response for the nominal configuration and reflectances, suggesting any configurational changes which might reduce peak attitude errors if necessary. The minimization procedure is performed in this paper using the NASA / Hughes Applications Technology Satellite (ATS) as a prime example of a gravity-gradient-stabilized satellite in an environment where solar pressure is the predominant external disturbance. The application of the solar balancing techniques to the ATS configuration resulted in peak yaw torques of less than 1 dyne-cm for the synchronous altitude satellite, and corresponding peak attitude errors of less than 1 deg in all axes due to solar pressure torques. Although the torque minimization procedures presented here are applicable in the general sense, the application of the techniques to a specific configuration requires derivation of the solar torque expressions for that particular configuration; therefore, the torque minimization example for the NASA/Hughes ATS vehicle can serve as a guide for other configuration applications.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1966;88(2):451-456. doi:10.1115/1.3645878.

The fluid squeeze-film produced by relative axial or tilting motion of two closely spaced plates provides viscous damping action over certain ranges of operation. When gas is the working fluid, a damper can be realized which is operable over a wide frequency range in the presence of extreme temperatures and intense radiation. A linearized analysis and approximate design equations, verified by a limited experimental program, are presented for several useful damper configurations.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1966;88(2):457-462. doi:10.1115/1.3645879.

Compensation of control systems by means of linear filters is a basic and well-known art. However, the limitations of linear compensation are also well known. In particular, the interdependence of the phase and gain characteristics of linear filters is always in conflict with the desires of the control-system designer. For example, it is impossible to introduce attenuation into a system without introducing undesirable phase lag as well. The development of a split-path nonlinear filter (SPAN filter) concept promises to yield a practical device which has independent phase and gain characteristics. These characteristics are independent of input signal amplitude. This filter can, therefore, have desirable characteristics which are unattainable with conventional linear filters. The design of the nonlinear filter to meet a desired gain-phase, describing-function specification is simplified by results given in the paper. Analysis and simulation of three sample problems has shown that a great deal of flexibility is available with this nonlinear filter. It is possible to obtain phase lead and amplitude attenuation by a proper choice of linear filtering in the two channels. Analog-computer simulation has verified the analytical work with respect to stability margins, and provided transient-response data, in the absence of analytical methods.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1966;88(2):463-468. doi:10.1115/1.3645880.

A method is given for the identification of distributed parameter systems. Normal operating records or experimental data may be used. The method involves the determination of arbitrary parameters in an assumed partial differential-equation model of the system. The method applies equally well to linear and nonlinear equations, and equations with varying coefficients. The accuracy of the results depends upon the exactness of the model, the amount of data used, the error in numerical integration, and the amount of noise which is present in the data. Examples are given which illustrate the application of the method. Results using the method for the identification of a physical system are given.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1966;88(2):469-474. doi:10.1115/1.3645881.

The use of the describing-function method has been generally limited to systems with nonlinearities which can be combined in a block diagram of a feedback control system which is represented by a scalar variable. This paper deals with an extension of the method to some autonomous systems with more than one nonlinearity. The paper also shows that nonlinear systems can have hypersurfaces or manifolds in state space which divide regions of different stability characteristics. Each of these manifolds can be regarded as a boundary element of a family of surfaces related to a Lyapunov function. The describing-function method is applied in such a way that the hypersurfaces are approximated by a quadratic form. Two cases in an example, for which data obtained by analog computer and describing function approximations are compared, show that the amount of error caused by the approximation can be acceptably small, particularly in the subspace of the state space in which a system exhibits nonlinearities.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1966;88(2):475-479. doi:10.1115/1.3645882.

The purpose of this paper is to obtain stability conditions for a class of nonlinear distributed-parameter systems by using a generalization of Liapunov’s direct method. Sufficient conditions for local stability and instability of the equilibrium state are derived. An application is given in which conditions are obtained for stability of a chemical-reactor process.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1966;88(2):480-488. doi:10.1115/1.3645883.

The object of this work was to evaluate quantitatively the Bauschinger effect in a 4330 modified steel as a function of strength level and structure as derived from variations in heat-treatment. Material having martensitic, pearlitic, and bainitic structures was studied utilizing a uniaxial tension-compression specimen. Various ways of defining the magnitude of the Bauschinger effect are explained. One is a conventional approach as suggested by Welter, the other a technique which takes strain-hardening into account. The results show the Bauschinger effect to be independent of yield strength for three different strength levels of the martensitic material. It is only mildly influenced by material structure and independent of the direction of overstrain. The Bauschinger effect increases with increasing permanent strain up to approximately 2 percent and thereafter remains essentially constant.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1966;88(2):489-499. doi:10.1115/1.3645884.

Lead bars of various height-to-width ratio have been rolled down in a model rolling mill to reductions of as great as 60 percent between rolls designated as smooth, partially rough, and rough. The results of experiments to determine the amount of spread which each bar undergoes are presented nondimensionally, and these are compared with the spread formulas of Wusatowski, Hill, and Sparling.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1966;88(2):500-506. doi:10.1115/1.3645885.

On the basis of the finiteness of the flow strength of structural materials, the pressures permitted by current methods of pressure vessel design are limited. In this paper the analysis of a new method of design of commercial large volume pressure vessels is presented. This new design, which is a controlled fluid fill, may be used for many excursions to very high pressures without failure. The pressure that may be attained seems limited only by material property changes at extreme hydrostatic pressures. Large volume commercial vessels to 500,000 psi with reasonable outer to inner diameters may be built. The controlled fluid-fill pressure vessels in addition to piercing the current upper pressure barrier is also competitive with the shrink-fit method at pressures as low as 15,000 psi.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1966;88(2):509-517. doi:10.1115/1.3645889.

Dynamic tensile tests were conducted on uranium-10 weight percent molybdenum alloy in as-cast and wrought conditions. Carbon content was varied from 71 to 800 ppm. Heat-treatment was also varied. Tests were conducted on a universal testing machine at strain rates of 4.8 × 10−5 and 0.057 in/in/sec and on a modified Dynapak at strain rates up to 100 in/in sec. Tests were run from 75 to 600 F. Relatively low strain rates were used to determine the susceptibility to stress corrosion while the higher strain rates demonstrated the sensitivity of various mechanical properties to strain rate effects.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1966;88(2):518-524. doi:10.1115/1.3645890.

In the low-temperature range, the engineering yield strength of polycrystalline bcc metals can change by a factor of 10 or more with serious consequences appearing in the form of catastrophic brittle fracture. Engineering variables known to have an important effect on the yield behavior are state of stress, temperature, loading or strain rate, composition, and microstructure. For iron, chromium, molybdenum, and tungsten, it is shown that yield behavior can be represented by a single-valued relation between two dimensionless parameters.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1966;88(2):525-531. doi:10.1115/1.3645891.

Observations of inception and desinent cavitation numbers in small submerged jets, and observations of the flashing of small, highly superheated, free jets, are presented. The observations, made over a range of jet sizes and water temperatures, reveal a strong influence of geometric scale in both systems. The water temperature has little effect upon the cavitation number for the submerged jet except at high temperatures. Flashing occurs in the free jet after a delay time which is shown to vary as the (−7/2) power of the liquid superheat, and inversely as the jet area.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1966;88(2):533-537. doi:10.1115/1.3645894.

This paper presents the theory of secondary flow generation for an incompressible fluid of spatially varying density due to rotation of the frame of reference. It is pointed out that, while Coriolis forces may deflect the relative streamlines, the generation of secondary vorticity and secondary circulation generally requires an appropriately oriented momentum gradient. In particular, secondary vorticity will not be generated by rotation of the frame of reference in the absence of momentum gradients perpendicular to the flow direction when the streamlines are plane curves. Consideration is given to the rotating dishpan experiment and to the apparently analogous physical examples of the vortex formation when water drains very slowly from a large tank, and the generation of circulation in a hurricane.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1966;88(2):539-547. doi:10.1115/1.3645897.

A review is made of Ringleb’s theory on snow cornice-like flows and of the published, unsuccessful attempts at experimental verification. Arguments are presented which render the theory an unlikely representation of real fluid flows. The flow over a downstream-facing step in a channel, with either of two types of sinks located immediately downstream of the step, was investigated experimentally. A vortex and associated smooth expansion of the exterior flow behind the step were established for sufficiently high sink rates. A model of the step-sink flow is suggested and found consistent with results presented here and elsewhere.

Commentary by Dr. Valentin Fuster

DISCUSSIONS

Commentary by Dr. Valentin Fuster
Commentary by Dr. Valentin Fuster
Commentary by Dr. Valentin Fuster

TECHNICAL BRIEFS

J. Basic Eng. 1966;88(2):550-552. doi:10.1115/1.3645900.
Abstract
Commentary by Dr. Valentin Fuster
J. Basic Eng. 1966;88(2):552-554. doi:10.1115/1.3645901.

Three sets of strain-cycled fatigue tests have been conducted at three laboratories on wrought 70-30 Cu-Ni. The data reported herein provide an unnotched fatigue-failure curve of this material suitable for design purposes.

Commentary by Dr. Valentin Fuster

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