auto-translated version of patent automatisch vertaalde versie van patent CS86069   |   Addendum to “Instruments for solving cubic equations” Addendum bij “Instrumenten voor het oplossen van derdegraads vergelijkingen”

CZECHOSLOVAK REPUBLIC

OFFICE FOR INVENTIONS AND STANDARDIZATION

Class 42 m, 15   Issued January 15, 1957.

PATENT DOCUMENT 86069

Dr. Antonin Svoboda, Prague and Ing. Vaclav Vurcfeld, Prague.

Analog mathematical method for calculating the roots of algebraic equations.

Patent dependent on patent 85,204.
  Filed November 30, 1954 (PV 2683-54).       The patent is valid from November 30, 1954.

The subject of the invention is the improvement of an analog mathematical machine for calculating the coefficients of algebraic equations according to the basic patent No. 85,204, which allows the calculation not only of real coefficients, but also of complex coefficients.
  According to the basic patent, the calculation was achieved by finding the balance of only that component of the moments of gravity — eccentrically loaded wheels — acting on the rocker, which acts by the tilting moment of the wheels of the suspension axis of the rocker, while the balance of the second component of the tilting moments, corresponding to the tilting moments of the axis perpendicular to the suspension axis, is not monitored. The tilting moments of this second component are automatically compensated by the tilting moment of the reactions of the supports of the suspension.
  According to this patent, it was also not possible to solve complex equations using a single rocker system.
  The essence of the present invention lies in the fact that the machine according to the basic patent is designed in such a way that the rocker is suspended in a "Cardan" hinge (see Fig. 2). This enables monitoring of the balance of the tilting moments of the weight of individual discs, pairs of discs, groups of pairs of discs to all the corresponding axes in the horizontal plane and can be solved with the help of a single rocker system not only real, but also complex equations.
  The numerical basis of the device function follows from the following: we will write out the variable x having the general complex value ξ + jη in the polar form x = ξ + jη = ρ ejφ after substituting into the original equation:
A0 + A1 X + A2 X2 + … An Xn = 0 1
  we get
A0 + A1 ρ ejφ + A2 ρ2 ej2φ + … An ρn ejnφ = 0 1a
  Let's make the substitution: ρ = cos β.
  After substituting into equation 1) we will get:
  (done for example for a third degree equation)
A0 + A1 cos β ejφ + A2 cos2β ej2φ + A3 cos3β ej3φ = 0 1b
  After decomposing the higher powers of cos β to the first powers, the multiple cos angle β will be:
A0 + A1 cos β ejφ + ½ A2 cos 2β ej2φ + ¼ A3 cos 3β ej3φ + ¾ A3 cos β ej3φ = 0 1c
  Let us further denote: B0 = A0,   B1 = A1,   B2 = ½ A2,   B3 = ¼ A3,   B’0 = ¾ A3
  We obtain: B0 + B1 cos β ejφ + B2 ej2φ + B2 cos 2β ej2φ + B3 cos 3β ej3φ + B’3 cos β ej3φ = 0 1d
  The left side of the equation can be graphically expressed in the Gaussian plane as the sum of vectors whose rotations from the fundamental direction are, as can be seen, in the ratio forming the natural series of numbers:
φ0 = 0,   φ2 = φ3,   φ2 = 2φ,   φ3 = 3φ,   …   φn = n φ;
  The magnitudes of the individual vectors can then be created as follows: Divide the entire equation by the absolute value of the largest coefficient B (which we will denote by BN), the individual coefficients will then be:
b0 = B0
(BN)		 ,   b1 = B1
(BN)		 ,   …   etc.
  - b0 - and - b2 - (in the third term) are formed as the sum of two unit vectors symmetric to the unit vectors e0, ej2φ, and deviated from these by o ∢ α0, α2 about which the following holds: cos α0 = b0,   cos α2 = b2 ,
 
  Furthermore, the values: b1 cos β,   b2 cos 2β,   b3 cos 3β,   b’3 cos β , as the sum of individual vectors formed in a way from two pairs of unit vectors (see Fig. 1). In Fig. 1a there is an example of the creation of the value by b0 cos α0 and in Fig. 1b an example of the creation of the value R1 = b1 cos β.
  The unit vectors in the first and second pair belonging to the same quadrilateral are always rotated by the same angle:
o ∢ α1 in the square creating value  b1 cos β
o ∢ α’2 — ,, — b2 cos 2β
o ∢ α3 — ,, — b3 cos 3β
o ∢ α’3 — ,, — b’3 cos β
  Therefore, the following holds about the angle of their rotation:
cos α1 = ½ b1,   cos α’2 = ½ b2,   cos α3 = ½ b3,   cos α’3 = ½ b’3;
  so the equation will take the final form:
cos α0 + 2 cos α1 cos β ejφ + cos α2 ej2φ + 2 cos α’2 cos 2β ej2φ + 2 cos α3 cos 3β ej3φ + 2 cos α’3 cos β ej3φ = 0 1e
  We will then carry out the addition of vectors in the world of physical analogies as vector addition of the tilting moment of the weight of the eccentrically weighted balancing disc on the vertical pin "Č", on which the discs are threaded, and by finding such a mutual position of the disc (or double and square disc) according to equation 1e, in which the sum of the tilting moments is zero.
  Each individual tilting moment vector is represented by one eccentrically loaded disk of the same execution.
  The disappearance of the resulting tilting moment acting on the pin will be observed by attaching the unit (i.e. the pin Č with the discs K) to the cardanic suspension Z so that the center of gravity of the unit is below the plane of the cardanic suspension, whereupon the vertical position of the pin indicates that the condition of zero eccentric tilting moment has been satisfied, and that the mechanism with the value of the angle β and the angle φ gives the right-hand solution of the root x of the original equation.
  Each individual disc has its own drive (designed suitably as a latch with a ratchet wheel, electrically controlled).
  The procedure for working with the device is as follows:
  1. All discs are set to the "basic" position, i.e. to the position, below all. The values: α, β, φ are all = 0.
  2. The individual disks forming the pair are rotated by a multiple of the corresponding angle α so that their resulting tilting moment vector remains in the reference direction. In this way, we have brought the device to the "initial" state, i.e. we have set the constants = b.
  3. Finally, we will change the value ∢ β and ∢ φ for individual disks or pairs of disks in accordance with equation 1e systematically (for example, by a suitably chosen proportional relation between ∢ β and ∢ φ) until the resulting balance of all tilting moments on the vertical pin is reached.
  In this position, the drives are stopped by a suitable device. The value of ∢ β and ∢ φ corresponding to this position indicates the original x solution of equation 1 according to relation 2.

Subject of the patent:

  Analog mathematical machine for calculating the roots of algebraic equations according to basic patent No. 85,204, designed as a rocker (V), on which eccentrically loaded wheels formed by groups of two or more balancing discs (K) are rotatably mounted, in which the central arm of the centroid of the base-wheel created by the relative rotation of two pairs of disks gives the magnitude of the vector of the sought-after tilt in the Gaussian plane in the equilibrium position of the pendulum, and the rotation of the arm of the center of gravity of the basic wheel gives the rotation of the tilt vector from the basic direction in the Gaussian plane, while the rotations of the mentioned wheels are maintained in the ratio given by the natural forces and their rotation balances the constant mechanical moment of the pendulum created by the additional load, characterized by the fact that the pendulum is suspended in a cardanic suspension (Z), so it is possible to monitor the equilibrium of the tilting moments of individual disks, pairs of disks or groups of pairs of disks to all parallel axes in the horizontal plane, and so in addition to real roots, complex roots are also obtained.
Fig.1 Fig.2