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Theoretical completion of Coriolis' acceleration formula and its practical applications

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2.0. Theoretical completion of Coriolis' acceleration formula and its practical applications

   2.1. Derivation of completed formula of Coriolis' acceleration

   2.2. Acceleration of relative motion arising in rotating centre of frame of reference
  

   2.3. Phenomenon of shock change of acceleration and its effects on flying apparatuses into the atmosphere mass

   Supplement 2.1. Theoretical completion of Coriolis' formula and some atmospheric reason of airplanes catastrophes

   Supplement 2.2. Hydrological reasons of catastrophes of high-speed submarines

   Concealed Force


   Copyright - Vladimir Sukhanov 2000, 2002, 2003
   Author - Vladimir Sukhanov, April 2008

   Translated by Valentina Sukhanova from Russian
  

2.0. Theoretical completion of Coriolis' acceleration formula and its practical applications

2.1. Derivation of completed formula of Coriolis' acceleration

   Derivation of Coriolis' acceleration formula with use of Energy Conservation Law is suggested in the article. It is showed some results from the formula.

   Initial conditions:

X"1 = Omega [Владимир Суханов]X'(2X-dX) / (X-dX) ;

X"2 = Omega [Владимир Суханов]X'(2X+dX) / (X+dX) ,

  -- Omega [Владимир Суханов] - angular velocity of rotating of frame of reference,
  -- X' - linear relative velocity of a material point when the vector of its motion crosses centre of rotation of the frame of reference,
  -- X - distance between the centre of rotation of the frame of reference and the material point,
  -- dX - increase X with tendency to zero,
  -- X"1 - Coriolis' acceleration of the material point coming to the centre of rotation,
  -- X"2 - Coriolis' acceleration of the material point going away from the centre of rotation.
  
X"1 = - X"2 (e.g. X" changes its own sign on contrary) when the material point has just crossed the centre of rotation. For the formula X"2 at negative value X: X"2 = - X"1. The formula X"2 can be got by following manner. In accordance with the Energy Conservation Law, energy E of the material point, when X' = const and Omega [Владимир Суханов]= const, equal sum of energies and has its constant value:
  

E = Ex + Ev + EOmega [Владимир Суханов] + Ek = const

  
where
  -- Ex - kinetic energy of rectilinear relative motion of the material point,
  -- Ev - kinetic energy of motion of the material point in a circle,
  -- EOmega [Владимир Суханов] - work which is made by centrifugal and centripetal forces, when of the material point displaces,
  -- Ek - work which is made by Coriolis' force (arising as a result of Coriolis' acceleration) when the material point displaces.
   Here:

Ex = M(X')2 / 2

   where M - mass of the material point;

Ev = Mv2 / 2

   where v - speed of the material point motion in a circle;

EOmega [Владимир Суханов] = M X"cX

   where X"c - centrifugal (centripetal) acceleration;

Ek = M X"2l

   where l - displacement of the material point in a circle.

   When the material point displaces that redistribution of energies in the sum of the energy E takes place: ones increase or decrease at the expense of others. That is

dEv + dEOmega [Владимир Суханов] + dEk = 0

  
since

dEx = 0 if M(X')2 / 2 = const.

  
Here:

dEv = M(v+dv)2 / 2 - Mv2 / 2

  
or

dEv = M(v2 + 2vdv + dv2 - v2) / 2 = M(v + dv/2)dv

  
where

v = XOmega [Владимир Суханов] , but dv = Omega [Владимир Суханов]dX ,

dEv = M(X + dX / 2)Omega [Владимир Суханов]2dX ;

dEOmega [Владимир Суханов] = MOmega [Владимир Суханов]2(X + dX / 2)dX

  
where

(X + dX / 2)Omega [Владимир Суханов]2 = X"c

   where X"c - middle value of the acceleration;

dEk = M (l + dl)X"2 ,

  
where

l = vT or l = XOmega [Владимир Суханов]T

dl = Tdv or dl = TOmega [Владимир Суханов]dX ;

  
or

dEk = M X"2(XOmega [Владимир Суханов]T + TOmega [Владимир Суханов]dX) .

  
or

dEk = M X"2 TOmega [Владимир Суханов](X + dX) .

  
Consequently
  

Omega [Владимир Суханов]M X"2T(X + dX) = MOmega [Владимир Суханов]2 (X + dX / 2)dX + MOmega [Владимир Суханов]2 (X + dX / 2)dX

  
where

dX / T = (X')

  
from this

X"2 = Omega [Владимир Суханов]V(2X + dX) / (X + dX)

   Derivation of the formula X"1 can be got by analogy but

dEv = M[(v - dv)2 - v2] / 2

dEOmega [Владимир Суханов] = MOmega [Владимир Суханов]2(X - dX / 2)dX

dEk = M X"1TOmega [Владимир Суханов] (X - dX)

   When the material point approximates to the centre of rotation on a distance X=dX (or X=-dX at removal from the centre) different effect, which has been unwritten yet, arises and the suggesting formula does not work. Objects will be deviated from their own original trajectory when they are coming to the centre of rotation on distance X=dX. It lows probably that the objects will come exactly in the distance X=dX.

   In number of cases (for other initial conditions) it can be obtained formulas. For example:
  -- when is initial change of (X'), e.g. that is the material point has itself acceleration (X'')
  -- when is initial change of Omega [Владимир Суханов], e.g. that is frame of reference has itself angular acceleration of equal-slowing rotation Epsilon [Владимир Суханов].
  
In first case, when X"Neravno [Владимир Суханов]0, new component of energy arises in the equation of energy E:

Ex = MX"X , here dEx = MX"dX

  
From equation

dEx + dEv + dEk + dEOmega [Владимир Суханов] = 0

  
it can be obtained next formula:

X"2 = [Omega [Владимир Суханов](X' + X"dX / X')(2X + dX) + X"X'/Omega [Владимир Суханов]] / (X + dX)

   The formula can be simplified to generally level for approximate calculations:

X"1;2 = 2Omega [Владимир Суханов](X') :

X"1;2 = 2Omega [Владимир Суханов](X') + X"(X') / Omega [Владимир Суханов]X

  
or

X"1;2 = 2Omega [Владимир Суханов](X') + k X"

   where Omega [Владимир Суханов]X = v , and k = (X') / v is ratio of the linear relative speed to velocity of motion of the material point in a circle.

   The suggested formula has consequential:

if X" / X = 2Omega [Владимир Суханов]2 , that X"1;2 = 0

   That is Coriolis' acceleration is not always: namely, when equal-slowed (equal-accelerated) motion of a material point to a centre (from a centre) of frame of reference takes place during of all its way with above condition if.

   In the second case, when Epsilon [Владимир Суханов]Neravno [Владимир Суханов]0, new component of energy arises in the equation of energy E:
  

EEpsilon [Владимир Суханов] = MEpsilon [Владимир Суханов]Xl

  
Then

dEEpsilon [Владимир Суханов] = MEpsilon [Владимир Суханов](X + dX/2)(l+dl)

  
or

dEEpsilon [Владимир Суханов] = MOmega [Владимир Суханов]Epsilon [Владимир Суханов]T(X + dX/2)(X+dX).

  
From equation

dEk + dEv + dEOmega [Владимир Суханов] + dEEpsilon [Владимир Суханов] = 0

  
it can be obtained next formula:

X"2 = (Omega [Владимир Суханов]X' + XEpsilon [Владимир Суханов]/2)(2X+dX) / (X+dX) .

   The formula can have a look for approximate calculations:

X"1;2 = 2Omega [Владимир Суханов](X') + Epsilon [Владимир Суханов]X

   But for the case when X"Neravno [Владимир Суханов]0 and Epsilon [Владимир Суханов]Neravno [Владимир Суханов]0 the formula take its next view:

X"1;2 = 2Omega [Владимир Суханов](X') + Epsilon [Владимир Суханов]X + kX" .

   Full formula of Corioli' acceleration (with all components) is more interest:

X"2 = [(Omega [Владимир Суханов]X'+ X"Omega [Владимир Суханов]dX/ X'+Epsilon [Владимир Суханов]X/2 +Epsilon [Владимир Суханов]dX/2 +1)(2X+dX) + X" X'/Omega [Владимир Суханов])] / (X+dX)

  
or

X"2 = Epsilon [Владимир Суханов](X + dX/2) + [Omega [Владимир Суханов](X'+X"dX/X')(2X+dX) + X" X'/Omega [Владимир Суханов])] / (X+dX)

  
or

X"2=(Omega [Владимир Суханов]X'+X"Omega [Владимир Суханов]dX/X'+Epsilon [Владимир Суханов]X/2+Epsilon [Владимир Суханов]dX/2+2X"X'X/Omega [Владимир Суханов]+X"X'dX/Omega [Владимир Суханов]) (2X+dX)/(X+dX)

  



2.2. Acceleration of relative motion arising in rotating centre of frame of reference

   Suggested completed of Coriolis' acceleration formula has theoretical useful and may be used for calculations of hinge mechanisms where trajectories of movement of axles of ones hinges can cross trajectories or placement of others ones with creating of relative movement. Described additional acceleration can be found in movement above the Poles of Earth and in streams of circulation atmosphere.

   Coriolis' acceleration formula of a material point X" is known:

X" = 2Omega [Владимир Суханов]V

   where
  -- Omega [Владимир Суханов] - angular velocity of rotation of frame of reference,
  -- V - is linear relative velocity of a material point, when the vector of its motion crosses centre of rotation of frame of reference,
  
If a moving material point crosses centre of rotation of frame of reference that
  

X" = Omega [Владимир Суханов]V

   That is Coriolis' acceleration decreases from usual one in two times.

   Full formula of Coriolis' acceleration (see supplement 2.1. "Derivation full formula of Coriolis' acceleration"), when the material point approximates to the centre of rotation, can be presented in next view:

X" = Omega [Владимир Суханов]V(2X - dX) / (X - dX)

  
where
  -- X - distance between centre of rotation of frame of reference and the material point,
  -- dX - value of increase of the distance X which has tendency to zero.
  
In majority of cases X does not aspire to zero, consequently:
  

(2X - dX) / (X - dX) = 2

   It follows to note that when material point approaches to the centre of rotation on distance of a few dX, that X" begins increase and for XStrelka [ Владимир Суханов]dX acceleration X" aspires to infinity and exchanges its own sign in opposite. Coriolis' acceleration begins to act in opposite direction in comparison with usual. Farther, X" decreases to zero (for XStrelka [ Владимир Суханов]0,5dX) and increases again. At motion through the centre of rotation (X=0) X"=Omega [Владимир Суханов]V. When the material point going away from the centre of rotation, X'' aspires to 2Omega [Владимир Суханов]V and if X reaches of a few dX that X" will be equal 2Omega [Владимир Суханов]V already.

   Full note of the formula when the material point going away from the centre of rotation is:

X" = Omega [Владимир Суханов]V(2X+dX) / (X+dX) (see graph).

Graphyc [Владимир Суханов]

The graphic 1. Acceleration of relative motion which arises in centre of rotation of frame of reference

  
Motion to the centre of rotation presents danger for mechanism (and for all) because of shock change of Coriolis' acceleration from normal to zero trough infinity with change its sign. Here destruction does not expel.

   Suggested completion can be explained the surprises which arise during a flight of flying apparatuses in turbulent atmosphere and near to the geographic poles.
  


2.3. Phenomenon of shock change of acceleration and its effects on flying apparatuses into the atmosphere mass

   It is presented description of the atmosphere phenomenon, which creates disturbance for safety of motion of flying apparatuses. The nature and character of this phenomenon is given there.

   Flying apparatus for its motion (conditionally rectilinear relative to the Earth surface) is exposed by removing because of motion of atmospheric mass (relative to the Earth). Here components of the relative motion arise. Thus for arc or turbulent or concentric motion of the atmospheric mass Coriolis' acceleration X" and the Coriolis acceleration (X" ), which arises from interaction of meridian component velocity Vm of the flying apparatus and angular velocity of the Earth rotation ZOmega [Владимир Суханов] , start to act upon the flying apparatus:

(X" ) = 2ZOmega [Владимир Суханов]Vm

   Each arc motion of the atmosphere is the moving of frame of reference and has itself metacentre and itself reduced angular velocity of rotation Omega [Владимир Суханов]
, and each relative motion in a moving of frame of reference has component of velocity V , which cross the metacentre. Coriolis' acceleration X" during motion in the moving atmosphere, which has atmospheric metacentre, is:

X" = 2ZOmega [Владимир Суханов]V

   Atmospheric metacentre is named that area above which perpendiculars to directions of motion of the atmospheric mass crosses.

   If flying apparatus (for its motion) comes into metacentre of the atmospheric motion that the Coriolis' acceleration X" increases striving for infinite and changes its sign on opposite, farther X" fall down to zero, restores to its usual half value and at going away from the metacentre it restores to its usual value. It means that change of Coriolis' acceleration X" has shock character.

   If flying apparatus comes into metacentre of rotation of atmosphere above the geographical poles of the Earth, that (X" ) change analogy X".

   The atmosphere in area of geographical poles moves by itself laws and makes displacement of atmospheric metacentres above the poles. Complicated motion of the atmospheric mass in area of the poles can bring to arise (formation) of a few metacentres of motion of the atmosphere.

   When apparatus is flying near an atmospheric metacentre, the atmospheric mass cannot make irreversible deformation to the apparatus' body but can quickly exchange regime of its flying and disturb stability of its motion: stripping reaction into "spin" or flying with slip.

   Shock change of X" has perpendicular direction to course of the apparatus motion (lateral X" gives special risk for the flying apparatus). "Spin" into the atmospheric metacentre can be lengthy or can be watched consecutive come in and go away into the series of "spins", also can be wrong (broken) "spin". In such case it will be very danger, after regeneration of normal flying regime with partial losing of high, to take back course.

   Motion of flying apparatus near of metacentre on distance which is commensurable with size of the flying apparatus is very risk.

   Apparatus is flying through the atmospheric metacentre can be exposed by lateral blow, which is more than safety factor of the flying apparatus by its value, and as a result it will be destroyed. Size of flying apparatus, undoubtedly, is much more than value dX, but in a number of cases dX can considerably increase relatively of the flying apparatus, and the flying apparatus itself can become equivalent of material point.

   Conception of the atmospheric metacentre is conditioned/relative. In the nature metacentre can have a view of a curve line because of variable curvature of directions of motion of the atmospheric masses, and metacentre can have a view of whole area because of mutual displacement of the atmospheric masses by all volume of their motion

   The local atmospheric metacentres, which are results of cyclones and anticyclones, are the most risk for flights.

   The suggested formulas are theoretical and simplified but with their help we can determine more risk atmospheric space.

   When flying apparatus is coming into a metacentre, it will be drifted from itself course and it will be returned. By values of the drift and return we can calculate where the metacentre is or, as a last resort, its how near distance it is.

   Prognosis of arise of the atmospheric metacentres, their finding and deviation from them of flying apparatus are solutions of safe of flights.

   The author - Суханов Владимир Николаевич (Suhanov Vladimir Nikolaevich)
   It was registered in VNTIC the 01 of december 2000 by the number 72200000039.
   The article was published in the book "Inventive Creation" in Russian language in 2003.
   Copyright - Vladimir Sukhanov 2000, 2001, 2003
  
  


   Copyright - Vladimir Sukhanov 2000, 2002, 2003

Russian

  
Author - Vladimir Sukhanov
Translation - Valentina Sukhanov

Supplement 2. Coriolis Formula

Supplement 2.1. Theoretical completion of Coriolis' formula and some atmospheric reason of airplanes catastrophes

   Formulas, which can explain reasons of some (it would be seems inexplicable) airplanes catastrophes, are given here.
Atmospheric cyclones and anticyclones, and also concentric atmospheric turbulences present large obstacles for safe flights. In those cases flying apparatus can get
large lateral overload (accelerations), which can throw down it into "spin".

   In this book (part 1, supplement 2) was represented derivation of completion of Coriolis' formula. Result of that derivation is:

X2"=Omega [Владимир Суханов]X'(2X+dX)/(X+dX)+ Epsilon [Владимир Суханов] (X+dX/2)+X"[X'/Omega [Владимир Суханов](X+dX)+Omega [Владимир Суханов](2X+dX)dX/X'(X+dX)],

   where
  -- X2 " - Coriolis acceleration;
  -- Omega [Владимир Суханов] - angular velocity of rotating cyclone or anticyclone;
  -- X' - linear speed of flying apparatus,
  -- X - linear displacement of flying vehicle from the metacentre;
  -- dX - increase of linear displacement with tendency to zero;
  -- Epsilon [Владимир Суханов] = Omega [Владимир Суханов]' - angular acceleration of the rotation of the atmospheric section;
  -- X" - linear acceleration of motion of flying apparatus.
   or

X2"= Omega [Владимир Суханов]X'(2X+dX)/(X+dX)+Epsilon [Владимир Суханов](X+dX/2)+X"X'/Omega [Владимир Суханов](X+dX)+Omega [Владимир Суханов]X"(2X+dX)dX/X'(X+dX) .

   Formula contains four components:
1)

Omega [Владимир Суханов]X'(2X+dX) / (X+dX)

  
2)

Epsilon [Владимир Суханов](X+dX/2)

  
3)

X"X' /Omega [Владимир Суханов](X+dX)

  
4)

Omega [Владимир Суханов]X"(2X+dX)dX / X'(X+dX)

   In the components: 1), 3) and 4) accelerations make analogous risk for flying apparatus. That risk and also some recommendations how avoid it were described in the part 1 of this book.

   The component 2) of the formula presents description of acceleration which influences on regime of piloting, but it cannot make risk of overloads.

   The component 3) of the formula describes risk of lateral acceleration of flight apparatus for motion through all section of the atmospheric concentric motion. This risk quickly increases if angular speed Omega [Владимир Суханов] of the concentric motion has tendency to decrease to zero. In this case acceleration can arise not less strong and risk than from the acceleration 1). This means that flight apparatus should not change its linear speed (should not allow linear acceleration X ") during crossing of the section of the atmosphere with the concentric motion, whose circulation has tendency to slowing down.

   Change of the linear speed of the flight apparatus is risk action for crossing of all section of the atmosphere circulation. In this case the most risk phases of flight are:
  -- ascend after fly up,
  -- lowering before landing.

   In these cases height of flight apparatus is low and fall down into the "spin" is very risk. If above the airfield and in its surroundings is the atmosphere circulation (even insignificant), then ascends and landings of flight apparatuses must be forbidden, special for civilian airliners. Ignoring of this rule led already to series of catastrophe, including passenger airliners.

   The atmosphere concentric motion can catch flight apparatus in any phase of its flight. Knowledge of the described phenomenon and the production of adequate recommendations for pilots will increase flights safety.

   Later safety ways for flights in circulating atmosphere, must be foresee in system of flight apparatus itself, and also in safety rules of air navigation and their provide round-based services.
  

Supplement 2.2. Hydrological reasons of catastrophes of high-speed submarines

   Suggested formulas can explain some reasons catastrophes of submarines (and flying apparatuses).

   Moving submarine (by analogy with flight apparatus) after falling into whirlpool (circulation of the atmospheric) has effect of Coriolis overloads:

X2" = Omega [Владимир Суханов]X'(2X+dX) / (X+dX)+ Epsilon [Владимир Суханов](X+dX/2)+X"[X ' /Omega [Владимир Суханов](X+dX)+Omega [Владимир Суханов](2X+dX)dX / X'(X+dX) ],

  
(see symbols meanings in supplement 2.1)

   Those overloads are analogous to those, which act on airliner, which is crossing the atmosphere circulation.

   Circulations of the water or air atmosphere can have not only vertical rotational axis but also horizontal ones. In this case motion and manoeuvrings near to bottom (earth's surface) is very risk since Coriolis' overloads can throw down the submarine (or flying apparatus) and it will be struck on the bottom that can destroy or break it.

   In this case direction of rotation of flow of the water (air) is important. From that will be depending next: first the submarine (flying apparatus) will be thrown down and then thrown up, or vice versa - first will be thrown up and then will be thrown down. When a submarine crossing symmetrically circulating medium, that such thrusts, which were described above, will not change the course of the submarine and it different, but in reality approximation to rotation axis of medium can make different, and going from the axis increase of its different. Different on it prow can cast down it on the ground or cast down it into risk depth of its immersion.

   The author - Sukhanov Vladimir Nikolaevich
   It was registered in VNTIC the 16 December 2002 by number 72200200011.
It is published in the bulletin VNTIC "Ideas. Hypothesises. Solutions" N 2, 2002 years.
   The article was published in the book "Inventive Creation" in Russian language in 2003.
Copyright -
Vladimir Sukhanov 2001, 2002, 2003


   Copyright - Vladimir Sukhanov 2004
   Author - Vladimir Sukhanov

   Translated by Valentina Sukhanova from Russian

Concealed Force

The surrounding us world has lots of free energy and for it description and using it is not needed nay exotic and anomaly theories because we have still enough our knowledge for opening of large resources of the universe nature.

In the beginning of the 19-th century Gaspard-Gustave Coriolis, French mathematician, mechanical engineer and scientist, using kinematical principles and researching relative movement objects revealed and calculated acceleration and force, which arise when object moves into rotating frame of reference. Later the acceleration and the force were named after his name. Much time passed from that moment and science mechanics contains except static and cinematic else and dynamic but we, as kind latterly time, continue to calculate the Coriolis' force by the principle of cinematic.

I we observe motion of an object into rotating frame of reference using Energy Conservation Law, we got not exactly the same as Coriolis got. If we take

- X" - Coriolis' acceleration
- X' - linear relative velocity of an object when the vector of its motion crosses centre of rotation of the frame of reference
- Omega [Владимир Суханов] - angular velocity of rotation of frame of reference,

That we can write the well know Coriolis' formula in next view:

X" = 2Omega [Владимир Суханов]X'

But if we take
- Ex = M(X')2 / 2 - kinetic energy of rectilinear relative motion of an object
- Ev = Mv2 / 2 - kinetic energy of motion of the object in a circle
- EOmega [Владимир Суханов] = M X"cX - work which is made by centrifugal and centripetal forces, when of the object displaces
- Ek = M X"l - work which is made by Coriolis' force (arising as a result of Coriolis' acceleration) when the object displaces
- M - mass of the object
- V - speed of the object motion in a circle
- X"c - centrifugal (centripetal) acceleration
- I - displacement of the object in a circle and the equation of Energy Conservation Law takes next view for that case

E = Ex + Ev + EOmega [Владимир Суханов] + Ek = const,

And make its solution relating to the Coriolis' acceleration, that we can obtain

X" = Omega [Владимир Суханов]X'(2 X + d X)/(X + d X)
where
- X - a distance from the object to the frame of reference
- dX - an increasing of X

On a graphic it looks such:

Graphyc [Владимир Суханов]

Fig.

We can see the acceleration and, consequently, the force which exposes on the object have two segments, where the force aspires to infinity but on each of them to different directions so total energy of the object on observing segments does not change. But if we examine each segment of the object trajectory in separate, we reveal huge force which deflects the object from straight trajectory.

It is a surprise. Why not anybody has taken into consideration those huge forces? And why those forces are concealed? Conditions can arise in which overcoming of the concealed force can be as overcoming of thin armour. Its thickness aspires to zero. It is needed other armour, but thicker than one, for overcoming it. For example, not all flight apparatus have some armour. So space apparatuses turning back from orbit on the Earth with the first space speed can be destroyed absolutely because huge linear relative velocity X' when they meet with concentric movement the atmosphere. Similar catastrophes with airplanes and speed submarines we have known enough but full Coriolis' force is concealed for the majority specialists until now.

Examining numerous ways mechanical using of concealed energy, we can reveal that in some cases some inventors still use, intuitively, practically the Coriolis' force in it extremity. We can use the concealed force in its fully and with realization having this full Coriolis' formula.

Using principles of dynamic for determine of Coriolis' force open for us, as we could see, earlier unknown possibilities. What happen if we will use for calculation of Coriolis' force principle of relativity or quantum mechanic? We will be able to open much interesting things....

Literature


1. Book "Inventive Creation" in Russian in 2003, ISBN: 5-94990-002-2.
2. Magazine "New Energy Technologies" No 5-6, 2003. ISSN 1684-7288.
3. Patents of the USA http://www.rexresearch.com/intertial/intrtial.htm


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