Engineering Analysis - Assignment Example

Engineering Analysis By Presented to September 9, Engineering Analysis Question at a) I) UsingCharacteristic equation Finding the characteristic equation of the equation Thus the characteristic equation of the above given equation is; Finding the roots of the characteristic equation using the formula, we have Thus the roots are either Or The roots of the differential equation are –3 and –2 this implies that he roots of the equation are distinct and real thus the general solution of the equation is ii) Using Laplace transform is transformed into Decomposing the above into partial fractions we have Thus by correspondence, the it implies that the inverse Laplace transform of the above given differential equation is c) Advantages of using Laplace Transform This method of solving differential equations has many advantages over the old classical methods. This advantages includes the following, a Able to solve higher order differential equations- with this method, you can solve equations of more than second degree equations as compared with classical methods which allow us to solve only first and second degree order equations. b The values of the constant are not necessary need us to insert them since they are automatically put in the equation. c Through application of this method, one is able to change ordinary differential equations into more convenient and easily solvable algebraic equations d When dealing with Laplace Transform, the external force is either off or on for some irregular or regular period of time or the external force can either be an impulse (example is that the force lasts for a short time) e You do not need to find the arbitrary constants as this method yields a particular solution for a certain differential problem without need to first solving the general solution. f This method is shorter and when you have constants coefficients it is easy to solve the linear differential equations. g Work is systematic as the boundary and initial condition are included. h Continues inputs can be easily treated. Laplace Transform method is therefore best applicable in engineering for solving such problems. Disadvantages a These methods yield erroneous results especially if they are used or applied mechanically without the considerations of the actual theory involved. Its highly encouraged to use different methods apart from the above mentioned since they are able to yield precisely better results, a way to counter this problem. b Many Laplace Transform software have been developed to solve the differential equations, this software need someone with skills in computers and knowledge in the Laplace so as to use the software efficiently and to be able to guide it to produce error free results. Its encouraged to use the right software that has been tested for to reduce these errors. c Numerically inverting the Laplace requires special type of method for solving them. Hence this method may be inappropriate in solving such a problem. d The method is tiresome and one mistake in the process can produce wrong results. This can problem is easily countered by looking carefully at the equations and also making sure each variable is corresponding to the value its assigned. Therefore, it’s clear that, Laplace Transform has more advantages than the other method and its best applicable in solving the ordinal differential equations. Application of the method The Laplace transform is a method that is applied in solving linear ordinary differential equations. It finds its way in different fields such as in Physics, Control Engineering, electrical Engineering, Signal processing and also Optics. Here we look at different areas were its applied a Application in Physics (It is part of engineering study)- it can be used to solve the harmonics vibrations of the beams at their both end. The formula is correctly inserted to calculate these harmonics b Application in Electric Circuit theory- in parallel or series circuits, the Laplace Transform can be applied to solve the switching transient phenomena. c In power system load frequency control- power systems are comprised of transmissions, generations and distributions of systems.There must be turbines which drives a generator that serves the load through distributing and transmission lines. Question 2 a) Taking the condition, and substituting the condition in the above equation, Thus the solution to the differential equation is b) (i) Euler’s Method Initial condition (2, 0.4) and Using the general equation below, Then (ii) Improved Euler’s Method (2, 0.4) and Runge Kutta x y=f(x) 0.4 2 0.46 2.04743 0.52 2.10121 0.58 2.16114 0.64 2.22692 0.7 2.29817 0.76 2.37438 0.82 2.45488 0.88 2.53887 0.94 2.62534 1 2.7131 C) The analytic solution does not give the exact solution of the y value. Furthermore, analytic method cannot be used to solve complex differential equations. Analytic solutions can only be used to solve ordinary differential equations of systems where approximations are allowed hence they cannot be used in the design of medical systems and other sensitive engineering systems. Analytic methods are used to solve system equations where complex processes are involved such as the control systems in linear systems. The drawback of analytic method is that it is time consuming and cannot be applied in cases of emergency designs and implementations. Numerical methods give accurate solutions hence they are the best methods of solving differential equations, these methods do not consume much time since they involve the use of computers which are provided with instructions. Apart from the above advantages, numerical methods have several flows which include the lack of accuracy as the solution goes dipper and their lack of computation power. Example when using Euler method in graphing, the solution gets worse as you go further in the solution which gives a poor estimate of the initial solution. Numerical methods find most of the applications in the design of engineering systems such control systems where evaluation of inputs, outputs and control feedback is necessary for proper functioning of the system. These methods are also used in the design of buildings specifically when comparing the weight and the materials needed in construction; this is because these methods are more accurate as compared to analytical methods and hence they provide reliable solutions to problems faced by civil engineers. In communications and electronic engineering, numerical methods are used in finding the Fourier and other polynomials needed in solving problems in communication systems such as digital/electronic cameras; this is because these systems make use of the first Fourier technology hence numerical methods are used in their design. In the design of cars, numerical methods are used when finding the characteristic equations for the systems which specifically involves mass (car and passengers) the damper (shock absorbers) and the friction factor experienced by the car when moving. Question 3 a) Characteristic Polynomial: Real Eigenvalues: Eigenvectors: For Eigenvalue -4: For Eigenvalue -3: For Eigenvalue 5: b) Creating a matrix Thus the solution is the general solution for the equations are; Thus the characteristic equation is C) Eigen values are used in solving problems of multivariable systems. These systems are used to force a given group of processes to behave in some desired and prescribed way by either fulfilling some requirements of the time or frequency domain or achieving the best performances as expressed by an optimization index. Study of eigenvectors and Eigen values forms a very crucial part in science and engineering. These values find their applications mainly in physical science that is physics. Eigen vectors are used in analysis of rotation of rigid physical bodies since these rigid bodies have irregular shapes which make them to move in the direction of rotation of preference. The direction preferred by the rigid bodies are referred to as the inertia axes and are obtained by evaluating a vectors Eigen values called inertia tensor or can also be referred as moments of inertia. Eigen values are also applied in evaluating the systems theory in differential equations especially equations that relate to two variables or components x (t), y (t). That is x’(t)=Bx, where B= this implies that the system has an eigenvector, v which has a specific meaning such that process variables in a system have specific outputs and inputs. Substances such as oil and dirt when analyzed give an output of linear systems which have various eigenvalues hence analysis of these values are applied by oil companies in oil exploration since these values can provide information on the location of oil reserves. Eigen values and Eigenvectors can be used in the design of cars this is because most of the car designers used the analysis of these values so as to remove the noise produced in the car such that the occupants enjoy a quite ride. The analysis of these values is used in the design of stereo systems in cars so sound produced is directed in the right way to the passengers. Read More