Механика математическое моделирование (стр. 6 )

+ Mgl sinj0) /IO + kj’’’ -(d1-d2)]/IO,

where j’’’ -(d1-d2) is known from the preceding segment [d2 , d1) of the motion with hj’’ -(t-d1) in (55) not yet in action. Now, for t ³ d1 the motion is defined by the third order differential equation, and with the approximation d2 @ 0, this equation can be written as ordinary DDE: lkj’’’(t) =- IO j’’(t) - al - Mg l sin j (t) – l hj’’(t-d1), t ³ d1 with j(d1), j’(d1) defined as end-point values in the previous segment of j(t) over [0, d1], and j’’(d1) given by the consistency condition.

3.2. If d1 @ 10-8 sec, small, but d2 is relatively large, then in (55) we have, in fact, the second order differential equation with discontinuity in the right-hand side. Indeed, until after t* > d2 the third derivative at right of (55) is not in action, thus, setting d1 @ 0, we get from (55) the equation (IO + lh)j’’(t) = - al - Mglj (t), different from the equations in (50), due to seemingly heavier disc and additional term - al, but with the same initial conditions. This equation exists until t* = d2 at which moment the third derivative in (55) comes into play, changing the right-hand side for t > d2 as follows:

IO j’’ = - al - Mg l sin j (t) – l[ hj’’(t- d1) +

+ kj’’’(t- d 2)], t > d2 . (56)

This is the same equation as (55) with all right derivatives. However, the third derivative at right does not project the motion as it did in Case 3.1, due to a greater delay d2 > d1 . It adds an additional force Df(t) = - lkj’’’ -(t-d2) depending on the rate of change of the actually realized values of past acceleration j’’(t-d2) for t > d2 assuring softer rate of change in acceleration which is good for a vehicle and for the people in the vehicle, if we consider in place of the pendulum a swing with people at entertainment centers.

N. B. In the theory of DDEs, the functions with delays in the right-hand sides must be defined prior to the start of the motion. For example, to define a unique solution in (55) for t ³ 0, cf. (50), the theory requires to define Q*(.) over the prior segment [-d , 0] where d = max(d1 , d2). With time delays due to information transmittal [16], delayed terms in forces Q*(.) cannot be "defined" on prior intervals because they physically do not exist in those time intervals. Setting them at zero may bring contradictions. Indeed, if d1 < d2 and we set j’’ - = j’’’ -º 0 over [0, d1 ) with j (0) = j0 > 0, then by continuity (Lemma 5.1) we have also j’’º 0 at left in (55), so that at t = 0 we get in (55): 0 =- al- Mgl sinj0 <0, an absurdity. For these reasons, we do not mention prior segments of definition for delayed terms which can be dealt with as they come into action.

3.3. The absence of time delays in mathematical descriptions of motion may lead to substantial errors, especially for small particles at high velocities. In deterministic consideration, this can be seen on example of a linear harmonic oscillator by comparison of the magnitude of its period with the order of natural time delays. Suppose that gravitation acts on the electron in the same way as on a metal pendulum and that it is added to other forces according to the parallelogram rule. Then we can imagine that small oscillations are superimposed on the rotational motion of an electron around the nucleus which would distort its uniform rotation. In the oscillatory part of the motion along the bottom arc 2j0 , we can consider the electron as a point-wise mass, so that the second equation in (50) with r = 0, for small j0 takes the form j’’ + gj /l = 0, irrespective of the mass of the electron, and the solution is j = j0 sin w t, where w 2 = g / l, with the period T = 2p(l/g)0.5. If we take l = a0 = 0.529 ´ 10-8 cm which is the radius of the first (innermost) Bohr orbit in the hydrogen atom (Bohr radius), then we have T = 1.460 ´sec. This is just at the middle of the time uncertainty segment for delays d1 , d 2 within [10-10, 1] sec considered above, so that model (50) is inapplicable to the study of harmonic oscillations of the electron in the hydrogen atom. In deterministic studies, time delays should be taken into account, when possible, especially if computations are involved in experiments, or particles move in a field of controlled forces, in which cases time delays due to information transmittal really take place.

Remark 10.1. It is clear that differential equations with the left time derivatives in the right hand sides present the limiting case of functional differential equations (FDEs) with delayed arguments [17–21] if delays tend to zero. However, the limit as d ® 0 cannot be attained if those delayed derivatives are the Newton – Leibniz right time derivatives. In this case, one can consider the connection by reverting to the left time derivatives in the limit, or replacing right derivatives by the left ones when approaching the limit.

Conclusions

This paper presents the causal approach to theoretical mechanics and engineering, different from the current textbook considerations based exclusively on the classical right time derivatives which ignore the natural orientation of increasing time t thus do not physically exist, although are actually used in the mathematical constructs to investigate prospective trajectories as approximations to real motions over some intervals of time.

In contrast to the usual Newton – Leibniz right time derivatives at the left hand side which project the motion into the future, the left and possibly delayed time derivatives are considered in the right hand sides of the equations of motions and processes that take into account the external influences and controls depending on the measured parameters and the immediately preceding exterior actions which are then transmitted into the power train of the motion. The results allow us to introduce the causal corrections into the current representations of the general laws of dynamics, into the Lagrange and Hamilton equations in independent coordinates, and to improve the design of autopilot systems in aviation which is currently based on the use of outboard Pitot tubes that, on occurrence of random wind gusts, give distorted evaluation of the average relative velocity of the airplane, and may get frozen and fail altogether in bad weather which has already happened in the Air France flight 447, Rio de Janeiro – Paris, on May 31st, 2009.

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Generalized equations of motion for mechanical systems with variable masses and forces depending

on higher order derivatives

E. A. Galperin

Departement de mathematiques Universite du Quebec a Montreal

C. P. 8888, Succ. Centre Ville, Montreal, Quebec H3C 3P8, Canada

galperin. *****@***ca

The Buquoy generalization of Newton’s second law of motion for systems of bodies with variable masses driven by reactive forces produced by ejected burnt fuel (Mestschersky) is considered, with its extension for motions subject to external forces depending on accelerations and higher order derivatives of velocities. Such forces are exhibited in Weber’s electro-dynamic law of attraction; they are produced by the Kirchhoff-Thomson adjoint fluid acceleration resistance acting on a body moving in a fluid and are also involved in manual control of aircrafts and spacecrafts that depends on acceleration of the craft itself. The causality of systems driven by such forces is assured by consideration of the left higher order derivatives in the right-hand sides of the equations of motion. The consistency condition and a new solution method are presented, and the existence and uniqueness of solutions for equations of motion driven by such forces is proved. The notion of effective forces is discussed, and the parallelogram law is verified for the effective forces in mechanical systems with left higher order derivatives in controls. On this basis, the new autopilot design is proposed for added security in civil aviation, independent of the currently used Pitot tubes which may fail or render the local measurements of wind gusts instead of the correct estimates for the average relative velocity of the aircraft with respect to the wind in flight or to the airstrip at landing.

Key words: Motion of bodies with variable masses; Forces with the left higher order derivatives of velocity; Generalized equations of minimum order; Autopilot design in aviation.

© Galperin E. A., 2014