Adonis Diaries

Article #52, (September 12, 2006)

“Mathematics: a unifying abstraction for Engineering and Physics Phenomena”

A few examples of mechanical and electrical problems will demonstrate that mathematical equations play a unifying abstraction to various physical phenomena of entirely different physical nature.

Many linear homogeneous differential equations with constant coefficients can be solved by algebraic methods and their solutions are elementary functions known from calculus such as the examples in article 51.  For the differential equations with variable coefficients, the functions are non elementary and they fall within two classes and play an important role in engineering mathematics.

The first class consists of linear differential equations such as Bessel, Legendre, and the hyper geometric equations; these equations can be solved by the power series method.

The second class consists of functions defined by integrals which cannot be evaluated in terms of finite many elementary functions such as the Gamma, Beta, and error functions (used in statistics for the normal distribution) and the sine, cosine, and Fresnel integrals (used in optics and antenna theory); these functions have asymptotic expansions in the sense that their series may not converge but numerical values could be computed for large values of the independent variable.

Entirely different physical systems may correspond to the same differential equations, not only qualitatively, but even quantitatively in the sense that, to a given mechanical system, we can construct an electric circuit whose current will give the exact values of the displacement in the mechanical system when suitable scale factors are introduced.

The practical importance of such an analogy between mechanical and electrical systems may be used for constructing an electrical model of a given mechanical system. In many cases the electrical model provides essential simplification because it is much easier to assemble and the values easily measured with accuracy while the construction of a mechanical model may be complicated, expensive, and time-consuming.

An RLC-circuit offers the following correspondence with a mechanical system such as: Inductance (L) to mass (m), resistance (R) to damping constant (c), reciprocal of capacitance (1/C) to spring modulus (k), derivative of electromotive force to the driving force or input force, and the current I(t) to the displacement y(t) or output.

Here are a few elementary examples:

5)      Ohm’s law: Experiments show that the voltage drop (E) in a close circuit when an electric current flows across a resistor (R) is proportional to the instantaneous current (I), or E = R* I.

Also, that the voltage drop across an inductor (L) is proportional to the instantaneous time rate of change of the current, or E = L*dI/dt.

Also, the voltage drop across a capacitor (C) is proportional to the instantaneous electric charge (Q) on the capacitor, or E = Q*1/C.  Note that I(t) = dQ/dt.

6)    Kirchhoff’s second law:  The algebraic sum of all the instantaneous voltage drops around any closed loop is zero, or the voltage impressed on a closed loop is equal to the sum of the voltage drops in the rest of the loop. Thus,

E(t) = R*I + L*dI/dt + Q*1/C.

For example, a capacitor (C = 0.1 farad) in series with a resistor (R = 200 ohms) is charged from a source (E = 12 volts).  Find the voltage V(t) on the capacitor, assuming that at t = 0 the capacitor is completely uncharged.

7)    Hooke’s Law: Experiments show that when a string is stretched then the force generated from the string is proportional to the displacement of the stretch,

or F = k*s.  If a mass (M) is attached to a string, then when the string is stretched further more (y) after the system is in a static equilibrium, then: F = -k*s(0) – k*y.

Newton’s second law for the resultant of all forces acting on a body says that:

Mass * Acceleration = Force, or My” = -k*y.

Furthermore, if we connect the mass to a dashpot, then an additional force come into play, which is proportional to the rate of change of the displacement due to the viscous substance with constant (c).  The equation is then a homogeneous second order differential equation: M*y” + c*y’ + k*y = 0.  Depending on the magnitude of (c) we have 3 different solutions: either 2 distinct real rots, 2 complex conjugate roots, or a real double root [c(2) = 4*M*k)} corresponding respectively to the conditions of  over damping, under dumping, or critical damping.

For example, determine the motions of the mechanical system described in the last equation, starting from y = 1, initial velocity equal zero, M = 1 kg, k =1 for the various damping constant: c = 0, c = 0.5, c = 1, c =1.5, and c = 2.

8)    Laplace’s equation is one of the most important partial differential equations because it occurs in connection with gravitational fields, electrostatic fields, steady-state heat conduction, and incompressible fluid flow.  The solutions of the Laplace equation fall within the potential theory.

For example, find the potential of the field between two parallel conducting plates extending to infinity which are kept at constant potentials; or the potential between two coaxial conducting cylinders; or the complex potential of a pair of opposite charged sources lines of the same strength at two points.