Browsing by Author "Duncan, W. J."
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Item Open Access Assessment of errors in approximate solutions of differential equations(College of Aeronautics, Cranfield, 1947-12) Duncan, W. J.The term assessment is applied to any process which enables us to set rigid bounds to the error or to estimate its value. It is shown that upper and lower bounds can be assigned whenever the Green's function of the problem is one- signed; this is true in many important problems. Another method is applicable to step by step -solutions of ordinary differential equations, linear or non-linear, and depends on use of the "index" of the process of integration. Lastly, the error in a linear problem can be estimated when an approximation to the Green's function is known.Item Open Access The characteristics of systems which are nearly in a state of neutral static stability(College of Aeronautics, Cranfield, 1950-01) Duncan, W. J.It is shown that the rate of subsidence or divergence ﮑ of a system which is near a state of neutral static stability can easily be calculated for knowledge of the mode of displacement in the neutral state and this model is found by solving a set of linear algebraic equations. The first order correction to the mode can also be found and in important cases this can be made the basis for calculating a second approximation to ﮑ; if necessary a further correction to the mode can now be found from this a still more accurate root can be calculated. The method can be extended to continuous systems having infinitely many degrees of freedom.Item Open Access Flutter of systems with many freedoms(College of Aeronautics, Cranfield, 1948-08) Duncan, W. J.Experience has shown that it is often necessary to retain many degrees of freedom in order to calculate critical flutter speeds reliably, but this entails much labour. Part 1 discusses the choice of a minimum set of freedoms and suggests that this should be based on the equation of energy and the use of the Lagrangian dynamical equation corresponding to any proposed additional freedom. The method for conducting flutter calculations so as to minimise labour are treated in Part 2.Item Open Access Ignoration of distortional co-ordinates in the theory of stability and control(College of Aeronautics, 1946-12) Duncan, W. J.Gates and Lyon have proposed to treat theoretically the stability and control of deformable aircraft by a method in which the distortion co-ordinates are ignored and the influence of distortion is allowed for by suitable modifications of the derivatives and other coefficients. in the 'present paper an exact method for eliminating the distortion co-ordinates is given and the conditions in which the true eliminant conforms with the simplification of Gates and Lyon are examined. In general the simplification is not justified mathematically, but in certain circumstances it provides an acceptable approximation. it will not be practically valid unless the structural distortions occur so relatively slowly that the associated inertia forces are negligible, i.e. the distortions must be quasi-static.Item Open Access Some related oscillation problems(College of Aeronautics, Cranfield, 1949-04) Duncan, W. J.Two simple means for establishing a relation between a pair of oscillation problems are briefly discussed. In the first, the displacements are connected by use of a differential operator. The set of natural frequencies is identical for the two problems and results of interest are obtained when the transformed boundary conditions can be physically interpreted. In this manner it is shown, for example, that flywheel on a uniform shaft can be transformed into a flexible coupling and a mass carried on a uniform beam into a flexible hinge. In the second, the connection is established by use of the concept of mechanical admittance. Here the frequency equations are simply related but the frequencies are not.Item Open Access Technique of the step-by-step integration of ordinary differential equations(College of Aeronautics, Cranfield, 1947-02) Duncan, W. J.In Part 1 step-by-step methods are examined critically and emphasis is placed on the dependence of the error on the number n of steps used for a given range of the independent variable. The index of a process is defined and it is shown that the errors can be assessed and partially corrected when the index is known and results obtained for two or more values of n. Attention is drawn to the advantages in certain cases of a part-analytical process. In Part 2 methods of numerical integration in general are classified and briefly reviewed. The chart, Table 2.2.1, summarises the classification.