Pressure amplitude distribution: Horn equation.

The pressure amplitude distribution p(x) of harmonic oscillation, p(x,t) = p(x) cos(wt), in the horn is governed by the equation:

where A(x) is the cross-section are, k is the wavenumber, w=2 pi f is de angular frequency, c = 340 m/s is the sound speed in air, f is the frequency.

Boundary conditions of horn opened at the both sides are: p(0) = p(L)=0.

It shows the first three modes of the second order differential equation given above. It shows clearly the pressure null at ends.

It shows the first 10 modes.

Click on BUY NOW, we have solved the horn equation, step-by-step. We have used MATLAB to solve this equation, we included the code.

RLC series circuit

For the circuit shown find an equation which shows how the current varies with time. Assume zero initial conditions when the switch is closed. Use Laplace transform and Kirchhoff's voltage law:

Click on BUY NOW to see the step-by-step solution, we have plotted the solution with MATLAB.

Signal and System Analysis

It shows the continuous signal as blue line, whereas the red line is the low-pass filter signal with a cutoff frequency of 75 Hz.

It shows the continuous signal as blue line, whereas the red line is when the low-pass filter has been applied. It has a cutoff frequency of 30 Hz.

We have the step-by-step solution of Frequency Modulate and Amplitude Modulate problems... click on BUY  NOW to get the final solution.

Cosmic Microwave Background and Point sources at mm wavelength.

We show a 10x10 sq deg map of the CMB, beam size = 10 arcmin.

A map 2x2 deg^2  CMB + point sources at 220 GHz, beam size = 1.5 arcmin.

This is my own WORK. Prohibited its reproduction.

Optimisation Methods and Simulation

Consider the minimization problem

There is not any constraint on x1 and x2.

We have solve the above problem using Simulated Annealing algorithm, with my own code and MATLAB in-built function, click on BUY NOW to see the step-by-step solution and many more.

Pressure distribution around an airfoil

Theoretical calculations for NACA 23012

Thin Airfoil Theory

Introduction

An airfoil is defined by first drawing a “mean” camber line. The straight line that joins the leading and trailing ends of the mean camber line is called the chord line. The length of the chord line is called chord, and given the symbol ‘c’.To the mean camber line, a thickness distribution is added in a direction normal to the camber line to produce the final airfoil shape. Equal amounts of thickness are added above the camber line, and below the camber line.

                                                                                (a)

                                                                                (b)

The solid blue line represents thin airfoil theory for Cp vs x/c, a) The solid red line is at AOA = 5 deg, b) The solid red line is at AOA = 10 deg.


We have an extensive analysis of pressure distribution around an airfoil, click on BUY NOW... MATLAB code is included.

Control Systems

What is the importance of Pole-zero diagrams in control systems?


What is the importance of Laplace Transformation in control systems?

Solve the differential equation system using MATLAB and analytically using Laplace Transform.

at t=0, x=4, y = 2 and x' = 0 and y' = 0.

We have the step by step solution and many more... 
click on BUY NOW to see the procedure.

Block Reduction Method

Reduce the following block diagram using Block Reduction Method and obtain the overall transfer function

We have obtained step by step the equivalent transfer function....
click on BUY NOW to see the procedure.

Hard-disk read/write head controller and speed control of a DC motor

We have controlled a simple model for the read/write head with differential equation,

We have the analytical and Matlab code for design control, for example:

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Spring-mass damping system

The spring-mass damping system as shown in the graph:

Where k is the spring constant, m is the mass  and c is the damping factor.

Solve its dynamic equation for x(t) and compare it with experimental data.

We have the analytical and Matlab code for numerical solution....

We have used the 4th order Runge-Kutta method for numerical exercise.

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System Identification using Frequency Response

Determine the transfer function of a simple RC low pass filter given in Figure 1 (a) and a high pass filter in Figure 1 (b). The resistance is 50KOhms and the capacitance is 100nF.

Figure 1: (a) A RC Low Pass Filter

Figure 1: (b) A RC High Pass Filter

We have the analytical and Matlab code for numerical solution....

In addition, we have used Laplace Transform and plot Bode diagrams and more...

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Partial Differential Equation

Consider a thin rod of length L. Fourier's law, relating heat flux q'' and temperature T is then q(x;T)=-kdT/dx, where 0<=x<=L is the position along rod,  t is time, and k  is the thermal conductivity. Using this expression, and applying conservation of energy to a small element of the rod between x and x+ \delta x  yields the heat equation

where: alpha=k/cc is the thermal diffusivity, in which \rho is the mass density and cc is the specific heat capacity;  Ta is the ambient temperature; and c is the heat transfer coefficient between the rod and the ambient air. The last term in Eq. (1) represents the distributed heat flux due to the temperature difference between the rod and air. In general, the parameters \alpha and c can depend on position, and Ta can depend on position and time.

We have the analytical and Matlab code for numerical solution.... click on BUY NOW.

Normal Modes

This diagram below shows three masses connected by three springs resting in equilibrium on a smooth horizontal table. Thus the force of the weight of each mass in, say, the y direction is matched by the reaction of the table. The only result forces on the masses come from the springs. The diagram shows the masses at rest, but the masses are constrained to move only in the x direction.

Three masses are connected by springs on a smooth table and can move only in a straight line. Assume the motions are sufficiently small so that the springs remain linear.

* Derive the system of equations of motion:

* Determine the frequency of the normal modes of the system and the normal nodes of motion.

We have the analytical and Matlab code for numerical solution.... click on BUY NOW.

Plot the displacement for all three masses vs time

1D Heat equation: Numerical solution

A stainless steel body of conical section (see Figure 1) is immersed in a fluid at a temperature Ta. The body is of circular cross-section that varies along its length, L. The large end is located at x=0  and is held at a temperature Ta=5. The small end is located at x=L=2 and is held at Tb=4.

A heat balance equation can be developed at any cross-section of the body using the principles of conservation of energy. When the body is not insulated along its length and the system is at a steady-state, its temperature satisfies the following o.d.e. (ordinary differential equation).

d^2T/dx^2 + a(x)dT/dx+b(x)T=f(x)

Where a(x), b(x) and c(x)  are functions of the cross-section area, heat transfer coefficients and heat inside the body. In the current example they are given by:

We have the analytical and numerical solution.... click on BUY NOW.

1.- Solve the equation using the shooting method as follows.

(a) Convert the second-order o.d.e. to a system of two first-order o.d.e.

(b) Use the shooting method to numerically solve the system of equations with a step size of 0.5

(c) Plot the temperature distribution along the body.