Foreword Preface to the Second Edition Preface 1. Maximization, Minimization, and Motivation 2. Vectors and Matrices 3. Diagonalization and Canonical Forms for Symmetric Matrices 4. Reduction of General Symmetric Matrices to Diagonal Form 5. Constrained Maxima 6. Functions of Matrices 7. Variational Description of Characteristic Roots 8. Inequalities 9. Dynamic Programming 10. Matrices and Differential Equations 11. Explicit Solutions and Canonical Forms 12. Symmetric Function, Kronecker Products and Circulants 13. Stability Theory 14. Markoff Matrices and Probability Theory 15. Stochastic Matrices 16. Positive Matrices, Perron's Theorem, and Mathematical Economics 17. Control Processes 18. Invariant Imbedding 19. Numerical Inversion of the Laplace Transform and Tychonov Regularization Appendix A. Linear Equations and Rank Appendix B. The Quadratic Form of Selberg Appendix C. A Method of Hermite Appendix D. Moments and Quadratic Forms Index.

Introduction to Matrix Analysis. By Richard Bellman. 2nd Edition. Pp. xxiii, 403. £7·50. (McGraw-Hill.)

Systematic variations of the endplate geometry for two endplate propellers are made in order to assess the impact of the blade tip shape on propeller performance Several types of modifications are analyzed including variations in plate width, contraction angle, plate sweep and flap angle The effect of endplate geometry modifications in the radial distribution of circulation is illustrated The magnitude of the circulation is derived from the analysis of the tangential velocities behind the propeller The contribution of pressure and friction components to the total endplate forces is also shown Comparisons are made at full scale A special procedure for the generation of the computational grids is implemented in order to minimize computational errors in the comparison of the alternative geometries RANS code FINFLO is used in the evaluation

Evaluation of endplate impact on tip loaded propeller performance using a RANSE solver

The effect of tip tabs on the flow field of a three bladed rotor/propeller is investigated experimentally. The experiments are run at chord Reynolds numbers of R e c = 0.4 – 2.9 × 10 5. The tab angles of attack of 0 ° , ∓ 3 ° , and ∓ 5 ° with respect to the rotation of the rotor are used to vary the tab loading. The rotor wakes and thrust characteristics at positive angles of attack, when the tip loading is outward, are qualitatively similar to those with no-tabs. In contrast, when the tip loading is inward at zero and negative angles of attack, the vortex wake is radically altered; the thrust nearly vanishes, even reverses with increasing inward loading. The key factors influencing the behavior of the wake are the vortex systems of the tabs and their associated downwash. The downwash is inward for the outward tab loading and causes increased volume and momentum flux in the rotor wake, and it is outward for the inward tab loading and causes expansion of the wake and nearly complete loss of thrust. When the tab loading is inward, a quasi-steady bound ring vortex system forms around at the rim of the rotor disk. At tab angle of − 5 °, the flow direction on the pressure side of the rotor disk reverses.

Vortex wakes of tip loaded rotors at low Reynolds numbers

Sea trials of a 45 ft seine boat, fitted first with a conventional tip, and subsequently with a ducted tip, 36 in. diameter propeller have been conducted. The trials consisted of careful propeller efficiency measurements at a number of advance ratios, and underwater video imaging of the different propellers under Bollard pull conditions. The trials have shown that flow-through ducts installed at the blade tips suppress the tip vortex roll-up, thus resulting in a substantial delay in the onset of tip vortex cavitation. The cavitation inception index for the ducted tip propeller is 50 percent lower than for the conventional propeller. This cavitation improvement comes without any efficiency penalty. In fact, the efficiency of the ducted tip propeller is up to 6 percent higher (at high advance ratios) than the efficiency of the same propeller fitted with conventional tips. The ducted tip propeller was originally thought to be useful only for boats with heavily loaded propellers, such as tug boats and trawlers. The current research has shown that the ducted tips may have much broader application.

Sea Trials of the Ducted Tip Propeller

Steady flow over rounded and ducted tip hydrofoils has been studied computationally using the CFDRC-ACE(U) flow solver and a k-e turbulence model. The flow domains were gridded with a combination of C-H, H-H, tetrahedral and prism grid blocks and mesh sizes ranged from 350,000–550,000 cells. A good agreement in flow pattern was achieved between the numerical solutions and available experimental data. The computations show that the ducted tip hydrofoil sheds less bound circulation over the majority of the wing span than does a rounded tip hydrofoil with the same cross section and aspect ratio. Observation of the streamwise component of vorticity immediately downstream of the different hydrofoils shows that the vorticity from the ducted tip hydrofoil is shed in the shape of a duct instead of the concentrated circular vortex shed by the rounded tip hydrofoil.© 2002 ASME

Computational Study of the Flow Around a Ducted Tip Hydrofoil

A new type propeller derived from “winglets” was investigated. This propeller is also fitted small blades at the blade tips likely to “winglets”, so we named it “bladelets”. These are for the purpose of reducing induced drag from tip vorteces. This paper, as the first report, mainly presents the results of the series tests which are concerned with the arrangement of these small blades. Propeller open test, flow visualization and flow velocity measurements were performed, and one of the best arrangement of “bladelets” were found out. The results looks 1 or 4% better compared the efficiency of “bladelet propeller” with that of the original one within the bound of real working condition.

Study of the propeller with small blades on the blade tips (1st Report)

In the first report, the authors carried out the various experiments with 1.3m ship model on manoeuvrability and response to various disturbances, in the circulating water channel. With these results, optimal control method which minimized the criterion function of cost or damage, was designed and was examined to be realizable by experiments. But criterion function depend on the type of ships, so optimal control method with various criterion function must be examined. In this report, the authors investigate how to change the optimal control with the changes of criterion-function-coefficients upsilon sub 1, upsilon sub 2 which were defined in the first report, Next, where the ship navigates on the settled route in uniform wind, how to make optimal control method realizable is examined.

Model experiments on manoeuvrability and automatic navigation on settled route in the circulating water channel (the third report)

At first, in order to obtain the manoeuvrability and response to wind, waves and oblique current, various experiments on a 1.3m self-propelled model of passenger ship were carried out in the circulating water channel. Utilizing these values, the optimal control system of this model, which reduces the following integral to a minimum, was analysed.I=∫tt0 {γ1 (Δη) 2+γ2 (Δθ) 2+δ2} dtwhereΔη : trajectory errorΔθ : azimuth errorδ : helm angleγ1, γ2 : coefficientst : timeMoreover, the experiments of optimal automatic navigation on settled route with this ship model were performed by use of above calculated value in calm water, wind, waves and oblique current, in circulating water channel.

Model Experiments on Manoeuvrability and Automatic Navigation on Settled Route in the Circulating Water Channel (First Report)

To investigate the relation between manoeuvering tests using a free running ship model and ones using a forced oscillator, the authors, at first, carried out the fundamental experiments : sinusoidal steering tests (ω' =0.042.2) and forced oscillation tests (ω' =0.040.15, h0/ Lpp= 0.174.7, Lpp=1.3m).Utilizing these results, they discuss convenient forced ocillation tests, which are not only available as the above forced oscillation tests, but also effective to study how to simulate free running conditions. Practically, according as the external force and moment delivered to the ship by the apparatus became small, the ship conditions on the convenient test became similar to those of the free running model. And finally, when the external force and moment were negligible small, a well agreement between both conditions was recognized.

Tetsuo Tagori

Papers

Study of the propeller with small blades on the blade tips (1st Report)

Model experiments on manoeuvrability and automatic navigation on settled route in the circulating water channel (the third report)

Model Experiments on Manoeuvrability and Automatic Navigation on Settled Route in the Circulating Water Channel (First Report)

Study on the Relation Between Manoeuvering Tests Using a Free Running Model and Forced Yawing Ones