On Optimum Propulsion by Means of Small Periodic Motions of a Rigid Profile. I. Properties of Optimum Motions

The problem of optimum thrust generation by means of a rigid profile performing small arbitrarily periodic motions in an inviscid incompressible fluid is studied. The motions considered have to generate a prescribed mean value of thrust and must be such that the contribution to this mean thrust by the suction at the leading edge does not exceed a certain given value. Furthermore, the motions are in general subjected to a maximum type constraint on their amplitude. For this infinite dimensional, nonconvex and nonsmooth optimization problem, a generalized Lagrange multiplier rule is derived. In case the constraint on the amplitude is omitted, the optimum motions are calculated analytically; for the general case a number of properties of the solutions are derived from the Lagrange multiplier rule.     

On Optimum Propulsion by Means of Small Periodic Motions of a Rigid Profile II. Optimization of the Period and Numerical Results

We present some numerically calculated optimum thrust generating small amplitude periodic motions of a rigid profile in an inviscid imcompressible fluid. The motions considered consist in general of both a heaving and a pitching part and have common period. Apart from the prescribed thrust, the motions have to satisfy the demand that the contribution to the total thrust of the suction at the leading edge not exceed a given number and are furthermore subjected to a constraint on their amplitude. Solutions of an analogous optimization problem for pure heaving motions are also discussed. Furthermore, the problem of optimizing the period of the motions is considered.     

流体流动中波状游动平板最优运动的数值方法

提出了一种求解波状游动平板最优运动方式的优化方法．最优化问题表述为固定推力的条件下,使得输入功率最小．由于存在不可见模态,使得该问题具有奇性,用通常的Lagrange乘子法计算得到的可能不是最优解,而是一个鞍点值．为了消除这一奇性,增加了一个关于幅值的不等式约束,并利用逐步二次规划的优化方法求解该问题．将该方法运用到二维和三维的波动板的几个例子上,获得了最优解．     

On the existence of small-amplitude optimum hydrofoil propulsion

The trust production by means of a periodic heaving motion of a rigid profile is discussed. It is proved by using Bauer's maximum principle, that when a constraint is put on the magnitude of the amplitude, optimum motions do exist.     

Optimum control of the forced motions of systems with continuously distributed parameters : PMM vol. 38, n 6, 1974, pp. 1056–1062

We propose one of the possible versions of the optimum control of the forced motions of elastic systems of the type of rods, plates, and shells. We apply the procedure developed to elementary problems on the transition of a freely-supported rod or plate from an initial state φ, ψ to the rest state in the least possible time T in the presence of a constraint on the forcing load. We use the elementary results of theory of the l-problem of moments of Krein [1–3].     

On the existence of optimum large amplitude hydrofoil propulsion

The possibility of optimum thrust-production by means of periodic motions of a flexible wing is discussed. The disturbance velocities are assumed to be small so that a linearized theory is valid. It is proved that when adequate constraints are put upon the amplitude of the path of the wing and on the amplitude of its slopes and curvature, an optimum path exists.     

Maximum Divert for Planetary Landing Using Convex Optimization

This paper presents a real-time solution method of the maximum divert trajectory optimization problem for planetary landing. In mid-course, the vehicle is to abort and retarget to a landing site as far from the nominal as physically possible. The divert trajectory must satisfy velocity constraints in the range and cross range directions and a total speed constraint. The thrust magnitude is bounded above and below so that once on, the engine cannot be turned off. Because this constraint is not convex, it is relaxed to a convex constraint and lossless convexification is proved. A transformation of variables is introduced in the nonlinear dynamics and an approximation is made so that the problem becomes a second-order cone problem, which can be solved to global optimality in polynomial time whenever a feasible solution exists. A number of examples are solved to illustrate the effectiveness and efficiency of the solution method.     

On the relativistic theory of rocket flight

It is shown by macroscopic analysis that, when the entire mass of a rocket is consumed for creating thrust, objects may be obtained as a result having energy but zero mass, moving the velocity of light. It is shown that the boost process of such massless objects can be realized in finite time from the observer's point of view. The vast stellar luminosity of quasars and certain jet motions observed in remote space can be explained by the production of massless radiation with internal motions connected with the separation of large energies inside the stars.     

Analysis and optimization of the motion of a body controlled by means of a movable internal mass

The controlled horizontal motion of a body in the presence of dry friction forces is investigated. Control is accomplished by means of a movable mass that can move within the body in a bounded range. Some simple modes of periodic relative motions of the movable mass, under which the entire system moves as a whole, are investigated. Constraints are imposed on the relative displacement, velocity and acceleration of the movable mass. The optimum parameters of this relative motion, under which the maximum mean velocity of the body is reached, are determined.     

Using Surrogate Constraints in a Lagrangian Relaxation Approach to Set-Covering Problems

Lagrangian relaxations have been used in a variety of IP problem settings. The main thrust of such efforts is to obtain bounding information for use in a branch-and-bound procedure. This paper examines the effect of adding a single surrogate constraint to Lagrangian subproblems in an attempt to improve upon the bounds produced by conventional Lagrangian relaxation. Computational results on some randomly generated set-covering problems are reported.     

A special case of the motion of a point mass

When investigating the motion of a point mass in a plane. Zhukovskii [1] pointed out a case when, without finding the general integrals of the equations of motion, one can specify particular integrals of these motions. To obtain the particular integrals in explicit form, a certain constraint was imposed on the force function. The case of motion without this constraint is investigated.     

Multiple-Subarc Gradient-Restoration Algorithm, Part 2: Application to a Multistage Launch Vehicle Design

In Part 1 (see Ref. 2), a multiple-subarc gradient-restoration algorithm (MSGRA) was developed with the intent of enhancing the robustness of gradient-restoration algorithms and also enlarging the field of applications. Indeed, MSGRA can be applied to optimal control problems involving multiple subsystems as well as discontinuities in the state and control variables at the interface between contiguous subsystems.In Part 2 (this paper), MSGRA is applied to compute the optimal trajectory for a multistage launch vehicle design, specifically, a rocket-powered spacecraft ascending from the Earth surface to a low Earth orbit (LEO). Single-stage, double-stage, and triple-stage configurations are considered. For multistage configurations, discontinuities in the mass occur at the interfaces between consecutive stages.The numerical results show that, given the current levels of the engine specific impulse and spacecraft structural factor, the single-stage version is not feasible at this time, while the double-stage and triple-stage versions are feasible. Further increases in the specific impulse and decreases in the structural factor are needed if the single-stage configuration has to become feasible.Also, the numerical results show that the optimal trajectory requires initially maximum thrust, followed by modulated thrust so as to satisfy the maximum acceleration constraint, followed by nearly zero thrust for coasting flight, followed by a final burst with moderate thrust so as to increase the spacecraft velocity to the circular velocity needed for LEO insertion. The above properties of the optimal thrust time history are useful for developing the guidance scheme approximating in real time the optimal trajectory for a launch vehicle design.     

Steepest ascent can be exponential in bounded treewidth problems

We investigate the complexity of local search based on steepest ascent. We show that even when all variables have domains of size two and the underlying constraint graph of variable interactions has bounded treewidth (in our construction, treewidth 7), there are fitness landscapes for which an exponential number of steps may be required to reach a local optimum. This is an improvement on prior recursive constructions of long steepest ascents, which we prove to need constraint graphs of unbounded treewidth.     

Variational problems in space flight mechanics

The effect of weight limitations on the parameters of the optimum motion of a variable-mass body in a gravitational field is considered. An analysis is made of the optimum motion for three possible propulsion systems: limited exit velocity, limited power, and limited thrust. Solutions are presented for some typical cases. The problems of selecting the optimum weight parameters, finding the optimum propulsion system controls, and determining the optimum trajectory are solved simultaneously.     

Dynamo Action of Isotropically Driven Motions of a Rotating Fluid

Fluid motions are driven by a random isotropic body force in a rotating system. The fluid is incompressible and infinite in extent. The dynamo action of the resulting fluid motions is investigated by Fourier decomposition. The global effects of the turbulent motions are expressed in terms of gradients of the local mean magnetic field, Bo . Two aspects of the problem are novel. The spectral approach is used to solve for the turbulent quantities while gradients of Bo are retained, and secondly a non-zero mean emf. is obtained from an entirely isotropic forcing.     

Regular and Chaotic Dynamics of a Chaplygin Sleigh due to Periodic Switch of the Nonholonomic Constraint

The main goal of the article is to suggest a two-dimensional map that could play the role of a generalized model similar to the standard Chirikov–Taylor mapping, but appropriate for energy-conserving nonholonomic dynamics. In this connection, we consider a Chaplygin sleigh on a plane, supposing that the nonholonomic constraint switches periodically in such a way that it is located alternately at each of three legs supporting the sleigh. We assume that at the initiation of the constraint the respective element (“knife edge”) is directed along the local velocity vector and becomes instantly fixed relative to the sleigh till the next switch. Differential equations of the mathematical model are formulated and an analytical derivation of mapping for the state evolution on the switching period is provided. The dynamics take place with conservation of the mechanical energy, which plays the role of one of the parameters responsible for the type of the dynamic behavior. At the same time, the Liouville theorem does not hold, and the phase volume can undergo compression or expansion in certain state space domains. Numerical simulations reveal phenomena characteristic of nonholonomic systems with complex dynamics (like the rattleback or the Chaplygin top). In particular, on the energy surface attractors associated with regular sustained motions can occur, settling in domains of prevalent phase volume compression together with repellers in domains of the phase volume expansion. In addition, chaotic and quasi-periodic regimes take place similar to those observed in conservative nonlinear dynamics.     

Optimization of functionals subject to integral constraints

The classical method for optimizing a functional subject to an integral constraint is to introduce the Lagrange multiplier and apply the Euler-Lagrange equations to the augmented integrand. The Lagrange multiplier is a constant whose value is selected such that the integral constraint is satisfied. This value is frequently an eigenvalue of the boundary-value problem and is determined by a trial-and-error procedure. A new approach for solving this isoperimetric problem is presented. The Lagrange multiplier is introduced as a state variable and evaluated simultaneously with the optimum solution. A numerical example is given and is shown to have a large region of convergence.     

Minimum-Time Control of a Crane with Simultaneous Traverse and Hoisting Motions

The problem of the minimum-time control of a crane having simultaneous traverse and hoisting motions is considered. We propose an approach that converts this problem into a finite-dimensional optimization problem via control parametrization with an appropriate basis function. Such an approach simplifies the treatment of the constraints and allows for the easy satisfaction of the endpoint constraints. This optimization problem is solved using a novel two-stage optimization process. Under additional conditions, the solution obtained from this process can be shown to be the optimum. When these conditions are not met, a near-optimal solution is obtained. Several numerical examples are provided, including the case where there is unequal cable length at the endpoints. The validity of the solution is verified experimentally on a test rig.     

On the Dynamics of Gear Shifting

Continuously or discretely variable transmissions are used in a variety of machines to match prime mover characteristics to varying loads. In most cases, the transmissions provide a fairly rigid constraint on the motions of inertial elements associated with the prime mover and the load although compliance will allow some high frequency oscillatory behavior. In this paper, compliance effects are neglected and the emphasis is on correct representations of impulsive forces associated with rapid gear ratio changes. The aim is to represent jumps in speeds, momenta and energy at the shifting times. This allows an efficient computer model of the overall system dynamics to be made without the necessity of representing the high frequency dynamics of the transmission itself. Bond graph models of automotive systems will be used to illustrate some of the pitfalls associated with models of various types of shiftable transmissions but the concepts apply to a wide variety of machines.     

A multi-product continuous review inventory system in stochastic environment with budget constraint

Common characteristics of inventory systems include uncertain demand and restrictions such as budgetary and storage space constraints. Several authors have examined budget constrained multi-item stochastic inventory systems controlled by continuous review policies without considering marginal review shortage costs. Existing models assume that purchasing costs are paid at the time an order is placed, which is not always the case since in some systems purchasing costs are paid when order arrive. In the latter case the maximum investment in inventory is random since the inventory level when an order arrives is a random variable. Hence payment of purchasing costs on delivery yields a stochastic budget constraint for inventory. In this paper with mixture of back orders and lost sales, we assume that mean and variance of lead time demand are known but their probability distributions are unknown. After that, we apply the minimax distribution free procedure to find the minimum expected value of the random objective function with budget constraint. The random budget constraint is transformed to crisp budget constraint by chance-constraint technique. Finally, the model is illustrated by a numerical example.