Mathematics in Construction and Engineering - Term Paper Example

CONSTRUCTION & ENGINEERING: ENGINEERING MATHEMATICS REPORT Name Institution Course Tutor Date Contents Executive Summary 2 Introduction 3 Method and Strategies 3 Calculations, Results & Analysis 5 Task 1 5 Current and phase angle 5 Power factor of the system 6 Structural framework 7 Task 2 8 Logic Circuit 8 De – Moivre’s Theorem 9 Discussion 11 Conclusion 24 Executive Summary In construction and engineering, mathematics is the integral aspect of knowing the type of variables required for a system, whether electrical or mechanical. It is through mathematical concepts and processes that the exact values of requirements of a system is obtained for its effectiveness and efficiency in line with the primary intentions. The paper demonstrates that different variables of mechanical force and electrical systems have different values and levels for their effectiveness and efficiency of the system, some of which entails complex number that only certain approaches would solve. Mathematical approaches and concepts such as De Moivre’s Theorem, amongst others have been used to analyse and evaluate engineering systems. Construction & Engineering: Engineering Mathematics Report Introduction As the paper intends to demonstrate, different variables of mechanical force and electrical systems have different values and levels for their effectiveness and efficiency of the system, some of which entails complex number that only certain approaches would solve. The report explores different mathematical concepts to analyse and evaluate various engineering systems, which for this case include electrical and mechanical force systems. Method and Strategies Electrical systems entail the use or provision of alternating current (AC) facilitations from or to other devices that are essential for the objective of the system set up. The idea of power factor is an integral concept in all of electric system. Various electric systems are balanced in such a way that the factor is always at its preferable values. In Task 1, an analysis of an electrical system consisting of: a capacitor connected in series for the rationale of improving the power factor; load of a simple inductive coil of a resistance of 500Ω and inductance of 10mH, and a capacitor of 10µF, connected in series with the load. For this analysis, arithmetic operations, as well as application of complex numbers, involving the above mentioned values was used to determine the amount of current which would flow in the circuit and its phase angle. In all electrical systems, understanding the idea of power factor is essential to engineers, considering that power factor determines the ability of a circuit to result in the intentions of the system. However, it is imperative to note that there are various methods by which power factor of an electrical system can be improved. The idea of improving the power factor of the circuit utilised for this task has been explored and discussed in the Discussion section of this paper. Additionally, arithmetic operations and the idea of complex numbers in assessment and evaluation have been employed to analyse a structural framework supporting a roof. The framework consists of three members with three forces of A=60kN at 45o from +X-axis, B=30kN at 95o from the +X-axis, and C=90kN at 145o from +X-axis. The information above was used to determine the necessary adjustments upon which equilibrium and stability of the structure can be realised. For the results presented in the sections that follow, discussions and conclusions concerning the positioning of the forces to obtain stability are presented. In most thermodynamics departments of many companies, the idea and equations used for their set up and up keep are majorly cubic. It is these types of equations that most thermodynamic experiments employ for effectiveness and efficiency. For Task 2, Newton-Raphson procedure was used to determine solution to experimental data and the final equation, which commenced with the following. In the same perspective, logic system was devised in the following sections using nand- gates with a target of meeting the requirement of Z = A + B – C. For further analysis and provision of solutions to thermodynamics problems presented by Task 1, de-Moivre’s theorem was used, alongside Argand Diagram and Gant Chart. Calculations, Results & Analysis Task 1 Current and phase angle Phase angle is associated with the idea of angular component that is present in a circuit due to voltage effects on flow of current. For this task, an electrical system connected to a 250V, causing a potential difference (pd.) at a frequency of 50Hz produced the following values. The circuit can be represented by the following diagram. The current flowing and angular component of phase shift can be determined by the following criterion. Current Power factor of the system For efficiency, the above factor is within the accepted values. With a minimum value of 0.8, all value above and close to it can be accepted to imply efficient electric system. However, it is imperative to note that power factor can take any value within a scale of 0 to 1, which factor 1 is normally rare in most electrical systems due to the idea that real world systems have other factors which hinder the value being one. However, it can be achieved theoretically by re-arranging the inductive and capacitive reactance formulas. In this case, the capacitive reactance would be the variable whilst inductance remains constant. Therefore, the impedance power factor of the electrical system with the fore-mentioned values would be determined as follows. Structural framework The framework consisting of three members with three forces of A=60kN at 45o from +X-axis, B=30kN at 95o from the +X-axis, and C=90kN at 145o from +X-axis, pin jointed together to support a roof with a single concurrent co-planar force system would result in values obtained by the following procedure concerning the idea of equilibrium. Based on the above idea of angular effect of direction and magnitude, the following values are the impacts of the three members acting for the support of the roof. Magnitude and direction calculations: As the x value is positive and the y value is negative, we know the angle lies in the second quadrant. Therefore, to work out the angle referenced from 0o. Task 2 Using Newton- Raphson procedure with the following equation, evaluating the experimental data would result in: For the solution of R&D project problems in an electrical department in an organization with respect to the above would result in values presented by the following table. The table entails conversion of 0.5937510 to a binary number, via octal. Logic Circuit The logic circuit representing the three members with their magnitudes would be as represented below, based on a truth table that has follows. De – Moivre’s Theorem Using de – Moivre’s theorem to determine all the complex roots for higher power, in Polar and Cartesian forms, and showing the roots in an Argand diagram would result in the following calculations and figures. For, the theorem would result in the following procedure first root second root And for , the theorem results in the following. a) therefore, it lies in the second quadrant. using demtvres: Discussion In an electrical system with all the integral variables for the achievement of energy output purposes, the concept of current flow and phase angle addresses some of the important aspects of balancing the variables. The idea of causing current to flow in a circuit to a level and phase angle that best meet the rationales or objectives is an essential concept in engineering. In this case, a source of practical voltage for the electrical system was of 250V at 50Hz, which can be replaced by a current source and vice versa (Bird 2013). With the equations such as or , voltage source can be used with variables such as resistance and capacitance, amongst others, to determine the amount of current that the voltage would be able to impact on the circuit. The idea of phase angle can be argued to be the angular or directional component of complex numbers represented in a function. This concept is obtained from the fact that a flowing current in a circuit would most likely face resistance. According to (…), electrical impedance describes the measure of opposition or resistance that a current faces when a specific measure of potential difference is impacted on the through voltage application. Unlike resistance, which has only the magnitude, impedance has both the magnitude and phase, hence an AC circuit is realised. Therefore, for Task 1, a voltage of 250V at 50Hz would result in an impedance of I = From the results above, it is demonstrated that inductors do not behave as resistors, hence the phase difference. Whilst resistors determines the flow of current through them by opposing their flow, inductors tend maintain the flow of current to a stable state by resisting changes that occur to the flow, hence the concept of phase difference represented by an angle and obtained from , illustrating the change of current with voltage drop in a circuit (Sandoval n.d). Therefore, the phase angle between the current flowing in a circuit of a capacitor connected in series for the rationale of improving the power factor; load of a simple inductive coil of a resistance of 500Ω and inductance of 10mH, and a capacitor of 10µF, is Using the above values of phase angle and current, a diagram representation of the two are as follows. Impedance Current About power factor in an electrical system, it is essential that engineers comprehend the implications that it has on true, reactive and apparent power, especially in power distribution electrical equipment. Power in an electrical system is described as the measure of energy delivery rate, which cannot be realised in an AC circuit despite the flow of current in and out of a load. Mathematically, the idea of energy delivery in a system can be represented by Power (P) = Volts (V) x Amps (A). With the flow of current in and out of the load, an oscillatory movement is obtained, hence reactive or harmonic current, which results in apparent power. According to (Kreyszig & Norminton 1993), apparent power, which is a product of Volt and Amps, is quite higher than the true or actual power. It is this difference that result in a power factor. Therefore, it can be argued that power factor can vary depending on the arrangement of the load in the circuit; that is, linear and non-linear or sinusoidal voltage have different or varying power factor. In this perspective, is can be argued that true power is always associated with linear load circuits whilst apparent power is associated with the other non-linear voltages. In this case, the load was connected in series with the other variables; however, the idea of inductance results in alternating current that impacts a sinusoidal voltage. Therefore, the power factor for the circuit is 0.846 considering the non-linearity of voltage producing an angle phase of with the current of flowing in the circuit. With the phase angle, it is evident that the energy delivered by the circuit would not be as exact as actual power, implying that power factor can never be practically 1. Further, it implies that, despite the electrical system being efficient in distributing energy at a power factor of 0.846, the system cannot be practicably 100% efficient, but only in theory. To achieve the ideal power factor of 1, inductive and capacitive reactance formula can be re-arranged and the idea of complex numbers applied as shown below, which the method of improving the energy distribution efficiently in the system. About the mechanical structure framework with various forces acting at a point to form a concurrent planar support system, the magnitude and angular positions of the members forces result in stability hence support. A single concurrent planar system can be described as a system of two or more forces of action converging at a point from different points. This case presents a force of support made by three different members that converge their magnitudes from different angular positions to provide support to the roof. The convergence of the members can be represented as follows. The above representation depicts that a force, support magnitude and direction of action from the three members are: The above values imply that, even though support to the roof is obtained, the stability is yet to be at its optimum level considering that equilibrium is not achieved by the member force acting towards support. Therefore, to obtain equilibrium and determine the positions that the three member forces would be positioned, the arbitrary point of reference would be 0o, considering that X values are positive and Y values are negative (Wyatt & Hough 2013). The angle above lies in the second quadrant, hence: To achieve equilibrium there must be an exact and opposite force applied to the calculated force. By adding 180o to the current phase and keeping the same magnitude the load will be become balanced. The opposing force can be noted by – F for calculation and discussion purposes to imply a force acting on the opposite direction to the magnitude of the support provided by the three member forces. Final value for – F The Argand diagram of the member forces acting at equilibrium for optimum stability of the roof would be as follows. The diagram shows angular positions of the three forces to provide equilibrium support to the roof in a single concurrent-planar system. The idea of using cubic of equations to represent and evaluate thermodynamic problems implies determination of the properties of a given thermodynamic system from a cubic equation of state, which entails obtaining roots of numbers. In many organisations, the idea of equation of state and the concept of thermodynamic are very essential, especially in modelling a vast variety of industrial processes. An accurate and firm thermodynamic framework in a company would be identified or recognised by its higher level modelling and simulation properties. Data that has different variables in thermodynamics can be represented by a function that can then be solved using cubic equations of state to determine the possible roots representing the values of the variables in question (Kreyszig & Norminton 2011). In this case, an equation can be solved to obtain the roots with which would represents values of the variables of the function. For this equation, the other root is, implying the possible values determining the state of the functions are and. The concept and process of converting numbers into binary via octal numbering system entail dividing the intended number into groups of three bits, which are also similar to the hexadecimal numbering system. In this perspective, the octal number would have eight digits of 0, 1, 2, 3, 4, 5, 6, and 7. It is from this concept that the octal numbering system denoted by q is given the base value 8. Therefore, . In this case, the conversion of 0.59375 to binary number via an octal one would result in number represented by the table below. The idea of logic in mathematics entails values referred to as truth value or logical values. It is these values that are considered to determine whether a device has a logical aspect. In this case, logical circuit would have true values that satisfy the equation as represented in the figure below. The logical values that meet the requirement are as presented in the table below. Therefore, to achieve a NAND only, the following rule can be employed. In the rule, the AND and OR are replaced with NAND configurations above to obtain a NAND-only circuit represented by the diagram below. Further, the above logic NAND circuit can be simplified by removing the two consecutive NAND gates shown above, since they can be argued to cancel each other out. Therefore, the final simplified logic circuit can be represented as shown below. Using de-Moivre’s theorem to solve for complex roots for higher powers entails considering functions in polar form, which can be obtain as follows: If z = r(cos θ+i sin θ) and w = t(cos φ+i sin φ), then the product zw is: zw = rt(cos(θ + φ) + i sin(θ + φ)) Particularly, if r = 1, t = 1 and θ = φ (i.e. z = w = cos θ + i sin θ), then (cos θ + i sin θ)2 = cos 2θ + i sin 2θ, which multiplying each side by cos θ + i sin θ results in (cos θ + i sin θ)3 = (cos 2θ + i sin 2θ)(cos θ + i sin θ) = (cos 3θ + i sin 3θ) upon adding the arguments of the terms in the product. From the above expression and equations as the baseline, De Moivre’s Theorem can be represented as follows for higher power values. (eiθ)p = eipθ (Bird 2013: James 2007), which according the task in this case would result in and as the first roots, and and as the second roots. Representing the resultant roots in Argand diagrams gives: However, getting roots of complex numbers in their higher power entails considering the quadrant in which the angles are found (Bird 2013: Kuldeep 2003). As in this case, the Y is negative whilst X is positive, implying that the value of the angle obtained is found in the second quadrant, which then produces, and using demtvres , the second beomes Conclusion In construction and engineering, mathematics is the integral aspect of knowing the type of variables required for a system, whether electrical or mechanical. It is through mathematical concepts and processes that the exact values of requirements of a system is obtained for its effectiveness and efficiency in line with the primary intentions. As demonstrated in this case, different variables of mechanical force and electrical systems have different values and levels for their effectiveness and efficiency of the system, some of which entails complex number that only certain approaches would solve. Reference List Bird, J 2013, ‘Higher engineering mathematics’, Oxford: Routledge James, G 2007, ‘Modern engineering mathematics’, Pearson Education. Kreyszig, E & Norminton E 2011, ‘Advanced engineering mathematics (9th Ed.)’, New York: John Wiley & Sons, Inc. Kuldeep, S 2003, ‘Engineering mathematics through application’, Macmillan Sandoval, G n.d., ‘The factor in electrical power systems with non-linear loads’. Available from: https://pdfs.semanticscholar.org/0d3b/a6010bcb726a477362fa90ceb026f81055e9.pdf Wyatt, K & Hough, R 2013, ‘Principles of structure (5th Ed.)’, New York: Taylor & Francis Group. Read More