Applications of numerical calculation in engineering design.
 presented that computational thought, experimental and theoretical thoughts were three human scientific ways of thinking, that further enriched the three major scientific ways in natural sciences: theory, experiment ways and calculation ways , they all emphasized the calculation was one of the three equipotent science means. With the popularization of computer applications, numerical calculations have been widely used in the production practice. Numerical calculation in engineering design is using not only its possessing highly abstract and rigorous scientific features of mathematics, but also its emphasizing the practical methods of engineering practice problems; furthermore it can fully meet the accuracy and effectiveness of specific methods .
2. Drawing moving orbit with no Analytical orbit equations
2.1 The trajectory of a charged particle in a non-uniform magnetic field
Some a non-uniform magnetic field with the field intensity B, which increases proportional to z, i.e. B = [alpha]z ([alpha] for scale factor), B vector directed to positive z-axis orientation constantly. A charged particle with its mass m, electric quantity q, initial velocity [v.sub.o], from the original point into this magnetic field, the initial moving direction of [v.sub.o] deviated from z- axis positive direction [theta], how is the trajectory of the particle?
Particle acceleration is due to the Lorentz force, the differential equation of particle motion vector as F = ma = qv x b, its 3 component equations in Cartesian coordinate system as:
m [dv.sub.x]/dt = [alpha][qzv.sub.y],
m [dv.sub.y]/dt = [alpha][qzy.sub.x], (1)
m [dv.sub.z]/dt = 0
Integrating the third one of above equations and with the initial condition, we can get z = [v.sub.o]t cos[theta]. Then of the equations (1), the first equation divided by the second equation, the differential equation was obtained as
[v.sub.x] [dv.sub.x] + [v.sub.y] [dv.sub.y] = 0 (2-1)
Supposing [v.sub.o] direction in the xoz plane, the initial condition can also be expressed as [v.sub.xo] = [v.sub.o] sin[theta], [v.sub.yo] = 0, using this condition to integrate the differential equation (2-1), got
[v.sup.x.sup.2] + [v.sub.y.sup.2] = [v.sub.o.sup.2] sin[theta] (2-2)
[v.sub.x] = [v.sub.o] sin[theta] cos[phi], [v.sub.y] = [v.sub.o] sin[theta] sin[phi] is a solution of the eqution (2-2), [phi] being the angle variable.
Using the solution [v.sub.x] = [v.sub.o] sin[theta] cos[phi], [v.sub.y] = [v.sub.o] sin[theta] sin[phi] and formula z = [v.sub.o] t cos[theta] to replace those in the first equation of (1), so obtained a new differential equation as the following, integrating the new equation by using the condition t = 0, [phi] = 0, the [phi] angle function is got.
-m [dj/dt] = [alpha] [qv.sub.o] t cos[theta] [??]
[phi] = - [alpha] [qv.sub.o] cos[theta]/2m [t.sup.2] (3)
Substituting j = -[alpha] [qv.sub.o] cos[theta]/2m [t.sup.2] (3) in the solution [v.sub.x] = [v.sub.o] sin[theta] cos[theta], so obtained
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)
Setting [omega] = [square root of ([alpha] [qv.sub.o]/m)], u = [square root of (cos[theta]/2)][omega]t, the expression of x rewritten as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)
The integral is called Fresnel cosine integral.
Similarly, Fresnel sine integral can be obtained as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)
Formula (5) and (6) only existing numerical solution, and when the integral upper limit tends to infinity, the two Fresnel integrals all tend to [square root of (2[pi]/4)], i.e.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (7)
For simplicity, [v.sub.o], R all take values of a unit, coefficient a is assigned the reciprocal of the nuclear-mass ratio; when the incidence angle [theta] of the charged particles is of 80[degrees], by using formula (5) and (6), the x and y components of a particle trajectory vary with time as shown in Figure 1-a; then combined with z = [v.sub.o] t cos[theta], the actual orbit of a particle is obtained, which as depicted in Figure 1-b.
Simulation results show that the coordinates x and y changing over time is cyclical, but the amplitude value's decreasing; z coordinate increases linearly over time; the trajectory of a charged particle is spiral, the longer the time, the higher the particle moves along z direction, the smaller the track circle radius. The extreme position of the track is at the point (2.0944, -2.0944), i.e.
[xu.sub.[right arrow][infinity]] = 2.0944 = - [y.sub.u[right arrow][infinity]] (8)
[FIGURE 1 OMITTED]
2.2 Drawing the motion orbits of spacecrafts
The equation of a spacecraft motion trajectory is not difficult to establish, but the process of equation derivation is too cumbersome and complex. If introduction of numerical calculation for this kind of problem, drawing the orbits has another way, so it needn't to seek their analytic forms of orbit equations.
Taking into account the centripetal force of a spacecraft motion, according to Newton's motion laws, it is easy to set up the following dynamical equations of the spacecraft motion .
m[d [sup.2]r/[dt.sup.2] - r(d[theta]/dt)] = -G [M.sub.E]m/[r.sup.2] = -g[R.sup.2.sub.E]m/[r.sup.2] (9)
-r d[sup.2][theta]/[dt.sup.2] + 2 [dr/dt] [d[theta]/dt] = 0 (10)
When the above equations are combined with the angular momentum conservation equations, spacecraft orbits equations can be got theoretically. If the design is simply to depict their orbits, the numerical solutions of the differential equations can be used to do so directly, and the transformation of polar coordinates to rectangular coordinates is fairly simple, so to avoid the derivation of the orbit equation.
Supposing that [v.sub.II] is the spacecraft's second cosmic speed, it also known as the escape velocity; with it, the spaceship can get rid of the Earth's gravity to infinitely far away, its value [v.sub.II] = [square root of (2G[M.sub.E]/[R.sub.E])] . Not considering the effect of the Earth's rotation, spacecraft's gravitation on the Earth's surface is taken for gravity, and [GM.sub.E] = g[R.sub.E.sup.2] . So its speed of escaping out of the Earth [v.sub.II] = [square root of (2g[R.sub.E])] = 11180 m/s.
Calculation results as shown in Figure 2, [v.sub.0] is the lauching velocity of a spacecraft, when the value of [v.sub.0]/[v.sub.II] less than 1, the spacecraft orbit is an ellipse; when [v.sub.0]/[v.sub.II] greater than 1, the spacecraft trajectory is hyperbolic. The whole drawing process takes a simple procedure preparation, various spacecrafts's orbits can be got corresponding to their different flying speeds, the result shows a vivid trajectory distribution.
[FIGURE 2 OMITTED]
3. The electrode size design of an electrostatic precipitator-solving transcendental equation
As an important application of static electric field, the electrostatic precipitators are essential equipments in many highly polluted enterprises such as coal-fired power plants . The device is a coaxial metal cylinder (see Figure 3), it has a external radius [r.sub.e], and a coaxial inner cylinder radius [r.sub.i], the positive and negative endpoints of an external electrode are linked to the center fine line and ektexine of the metal cylinder. When coupled with a high voltage at the external electrode, a strong electric field is generated in the cylinder, particularly close to the centerline, there is the maximum electric field; the inner air will be ionized and becoming air ions. When the outer air flowing in and through the cylinder, its dust particles will collide with the air ions and charged , the positively and negatively charged particles head for the outer wall and center solid cylinder of the barrel under the effect of the strong electric field, then precipitated to the bottom of the cylinder and cleared out. After several rounds of this cycle, the air becomes clean up .
[FIGURE 3 OMITTED]
Now such an electrostatic precipitator in a power plant, its cylindrical outer radius [r.sub.e] = 1.84m, the applied high voltage was adjusted to U = 60KV, center metal shaft radius [r.sub.i] designed to make the nearby high-field to 8 MV/m (the designed value greater than the usual breakdown field strength). Seeking the answer needs solution of transcendental equation, which can be done with numerical analysis by using computer.
Setting the charged density of the center line X, then from the electric field intensity in the cylindrical cavity, the potential difference between the inner and outer walls of the cylinder was deduced ,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)
Ordering r = [r.sub.i], then equation (11) combined with the electric field strength in the cylindrical cavity E = [lambda]/2[pi][[epsilon].sub.o]r, the owing formula can be got
E = U/[r.sub.i]ln [r.sub.e]/[r.sub.i] (12)
The equation is a transcendental one, and its solution is the intersection of the straight line (E = 8 Mv/m) and the curve of transcendental equation (12).
The first intersection is relatively easy to find, numerical calculation results as shown in Figure 4-a, at the dimension of [r.sub.i] = 1.8241m. Of the left side of Figure 4-a there should be another point of intersection; the intersection radius is relatively small, finding it needs numerical approximation in a higher sensitivity . After several attempts, finding the intersection point location as shown in Figure 4-b, this intersection corresponds to the cylinder centerline radius [r.sub.i] = 0.0010m. The first point of intersection corresponding to a big radius of the cylinder centerline, not only a waste of material, but also occupying too much space, the dust removal efficiency is lower . So the latter smaller radius is the ideal result.
[FIGURE 4 OMITTED]
The numerical calculation used in engineering design, brings great convenience to the design, and its accuracy can be controlled, fully meeting a variety of needs. With this, not only some seemingly complex, tedious theoretical calculation becomes easy and simple, and because the visualization of results, the design process is no longer so boring. Successful introduction of the numerical calculation in engineering design, also is of a a further evidence of important position and role for it in research activities.
Received: 2 April 2012, Revised 12 June 2012, Accepted 18 June 2012
This research was supported by Shaanxi Province Research Foundation of Natural Science (2011JM8021); it is also supported by Weinan Normal University key research project (12YKF019).
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School of Physics and Electrical Engineering
Weinan Normal University
Corresponding author: Lei, Q. Z. is an associate professor in School of Physics and Electrical Engineering, Weinan Normal University, China. He received his master degree in radio physics and is working for PhD in Xidian University. His main research fields include electromagnetic wave propagation in random medium and signal processing. For the last 3 years, he has been author of more than 20 articles, of which about 10 articles were published in international journals, others in chinese core journals.
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|Publication:||Journal of Digital Information Management|
|Date:||Oct 1, 2012|
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