DEPARTMENT OF MECHANICAL ENGINEERING
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Item Energy audit of commercial buildings (a case study of global fleet oil ltd. and national mirror houses(2012) Elusakin, J. E.; Ajide, O. O.; Salau, T. A. O.; Idowu, O. G."The dearth and cost of energy supply in Nigeria calls for planning and management through energy audit. The object of this paper is to carry out a comprehensive energy audit of the two commercial buildings commonly referred to as Energy House and Mirror House which are the head offices of the Global Fleet Oil & Gas Ltd. Company and National Mirror Newspaper respectively, using their 2011 energy consumption data. The power supplies to both offices are from the same generators which provide two third of the power supply to the building in 2011 but with different transformers of 1000 kV A each for the PHCN power supply. The facilities load were assessed by direct inspection and conversion were done where necessary. The capacities of the available three generators were 400, 640 & 1000 kW respectively (using 0.8 power factor) and the maximum load of the facility at any time in a day is 398 kW. This study has revealed the monumental financial waste on the un-utilized energy and facilities that could be replaced with energy saver equipment. Therefore, it is imperative that a holistic energy planning and regular assessment of energy requirements are considered as key components of building projects. "Item Development and investigation of Runge-Kutta coefficients depenedent stability polynomial(2013-07) Salau, T. A. O.; Ajide, O. O."Simulation of reliable solutions of nonlinear engineering problems by means of stable numerical algorithms is a frequent and acceptable practice. This study focuses the development and investigation of Runge-Kutta coefficients dependent stability polynomial for the second, third and fourth orders Runge-Kutta schemes. The development utilized matrix inversion operation procedure that involves determinant and cofactors computation of relevant matrix. The validation was made referencing the standard result of [1] and extended to several cases. The resulting polynom ials obtained consist of combination of the scheme coefficients with increasing power of time step that follows a rhyme pattern.The validation test case result agreed perfectly with test standard result. Selected studied version of different schemes shows wide variation in the shape of stability curve and region bounded. It is interesting to note that the popular second, third and•fourth order schemes have stability curve that bounded larger region than their respective counterpart. It is concluded that the study results can be utilized as reliable platform for stability analysis for different versions of the second, third and fourth order schemes. "Item Correlation and distribution analyses of estimated fractal dimensions and husrt's exponent from waveforms of excited non linear pendulum(Global Journals Inc (USA), 2013) Salau, T. A. O.; Ajide, O. O."This study utilised correlation and distribution analyses to investigate the acceptability of Parameters. selection sensitive simulation of the excited nonlinear pendulum waveforms was performed with the constant time step fourth order Runge-Kutta algorithm with codes developed in FORTRAN90. However, the waveforms validated by Gregory and Jerry (1990) and treated as time series were characterized using developed codes of Carlos (1998) and Hurst fractal dimension estimation procedures. The validation results compare. qualitatively well and the correlation coefficients between Carlos (1998)-based and Hurst's exponent based dimension estimate for the angular displacement and velocity are respectively R2 = 0.68 and R2 = 0.66. A higher correlation coefficient (R2 = 0.84) existed between the estimated Hurst's exponent of the angular displacement and velocity. The Hurst distribution exhibited both full spectrum and peak values range 0.04 to 1.00 and percentage probability range 2 to 12. The sum of this study results is the interchange possibility and utility of the two fractal dimension estimators as waveforms characterising tool. "Item Comparative analysis of simulation time in non linear and harmonically excited pendulum and duffing oscillators(Council for Innovative Research, 2013-08) Salau, T. A. O.; Ajide, O. O."The motivation for the present study is derived from the fact that time rnanagernent is an integral part of good engineering practice. The present study investigated the quantification of the required computation time using two nonlinear and harmonically excited oscillators (Pendulum and Duffing) as case studies. Simulations with personal computer were effected for Runge-Kutta schemes (RK2, RK3, RK4, RK5, RK5M) and one blend (RKB) over thirty five thousand and ten excitation periods consisting the unsteady and steady solutions. The need for validation of the develope,d FORTRAN90. codes by comparing Poincare results with their conterpart from the literature informed the choice of simulation parameters However, the simulation time was monitored at three lengths of excitation period (15000, 25000 and 35000) using the current time subroutine call command. The validation Poincare results obtained for all the schemes including RKB compare well with the counterpart available in the literature for both Pendulum and Duffing. The actual computation time increases with increasing order of scheme. But suffered a decrease for the blended scheme. The diffencerence in computation time required between RK5 and RK5M is negligible for all studied cases. The actual computational time for Duffing (5-33seconds) remain consistently higher for corresponding Pendulum (3-23seconds) with difference (2-10seconds). Interestingly, the quantitative difference betweenthe corresponding normalised computation time for systems and schemes is insignificant. It is insensitive to systems and schemes and fonmed a simple average ratio{ (1.0) :(1.5) : (2.0) : (3.1) :(3.1) : (2.4)} for RK2, RK3, RK4, RK5, RK5M and RKB respectively. It is concluded that the end justified the means provided that computation accuracy is assured using the higher order scheme (with higher computational time ratio). "Item Fractal characterization and iso-mapping of parameter plane of harmonically excited pendulum(International Journal of Scienctific and Engineering Reasearch, 2013-07) Salau, T. A. O.; Olabode, A. A."The motivation for the present study is derived from the fact that time rnanagernent is an integral part of good engineering practice. The present study investigated the quantification of the required computation time using two nonlinear and harmonically excited oscillators (Pendulum and Duffing) as case studies. Simulations with personal computer were effected for Runge-Kutta schemes (RK2, RK3, RK4, RK5, RK5M) and one blend (RKB) over thirty five thousand and ten excitation periods consisting the unsteady and steady solutions. The need for validation of the develope,d FORTRAN90. codes by comparing Poincare results with their conterpart from the literature informed the choice of simulation parameters However, the simulation time was monitored at three lengths of excitation period (15000, 25000 and 35000) using the current time subroutine call command. The validation Poincare results obtained for all the schemes including RKB compare well with the counterpart available in the literature for both Pendulum and Duffing. The actual computation time increases with increasing order of scheme. But suffered a decrease for the blended scheme. The diffencerence in computation time required between RK5 and RK5M is negligible for all studied cases. The actual computational time for Duffing (5-33seconds) remain consistently higher for corresponding Pendulum (3-23seconds) with difference (2-10seconds). Interestingly, the quantitative difference betweenthe corresponding normalised computation time for systems and schemes is insignificant. It is insensitive to systems and schemes and fonmed a simple average ratio{ (1.0) :(1.5) : (2.0) : (3.1) :(3.1) : (2.4)} for RK2, RK3, RK4, RK5, RK5M and RKB respectively. It is concluded that the end justified the means provided that computation accuracy is assured using the higher order scheme (with higher computational time ratio). "Item Application of average positive lyapunov in estimation of chaotic response peak excitation frequency of harmonically excited pendulum(International Journal of Advances in Engineering and Technology, 2013-07) Salau, T. A. O.; Ajide, O. O."The fact that the drive parameters space of harmonically excited pendulum consist of mix parameters combination leading to different dynamics phenomena including chaotic and periodic responses is a strong motivation for this study aim at estimating the peak frequency that favour chaotic response. Simulation of pendulum and estimation of the average Lyapunov exponents by Grahm Schmidt Orthogonal rules at parameter nodal points selected from damp quality (2.0≤q≤ 4.0). excitation amplitude (0.9≤g ≤1.5) and drive frequency (0.5 ≤ ωD≤1.0) were effected using popular constant time step Runge-Kutta schemes (RK4, RK5 and RK5B) from two initial conditions through transient and steady periods. The impact of resolution on the measure of percentage of parameters combination leading to chaotic response (PPCLCR) was examined at resolution levels (RI to R5) for increasing drive frequency. The validation cases were from those reported by Gregory and Jerry (1990) for (ώᶹ,q,g≡ 2/3,4,1.5) and (ωυq,g≡ 2/3,4,1.5) simulated from (0. 0) initial conditions. Corresponding validation results compare well with reported results of Gregory and Jerry (1990). The estimated peak frequency (0.6 radian /s) is the same across studied resolutions initial conditions and Runge-Kutta schemes. The peak value of PPCLCR is 69.5. 69.4 and 69.4 respectively for RK4. RK5 and RK5B at initial conditions (0. 0). When initial conditions is (I. 0) the corresponding PPCLR value changes in significantly to 69.6. 69.7 and 69.6 for RK4, RK5 and RK5B. Therefore affirms the utility and reliability of Lyapunov exponent as chaotic response identification tool. "Item A novel graphic presentation and fractal characterisation of poincare solutions of harmonaically excited pendulum(International Journal of Advances in Engineering and Technology, 2013-07) Salau, T. A. O.; Ajide, O. O."The extensive completed research and continuous study of pendulum is due to its scientific and engineering importance. The present study simulate the Poincare solutions of damped, nonlinear and harmonically driven pendulum using FORTRAN 90 coded form of the popular fourth and fifth order Runge-Kutta schemes with constant time step. Validation case studies were those reported by Gregory and Jerry (I 990) for two damping qualities q1,q2= 2,4), fixed drive amplitude and frequency (g = 1.5,ώD= 2/3). A novel graphic presentation of the displacement and velocity components of the Poincare solutions for 101-eases each drawn from the parameters spaces 2≤q≤4 and 0.9≤ g≤ 1.5 at 100-equal steps were characterised using the fractal disk dimension analysis. Corresponding validation results compare well with reported results of Gregory and Jerry (1990). There is observed quantitative variations in the corresponding consecutive Poincare solutions prescribed by Runge-Kutta schemes with increasing number of excitation period however the quality of the overall Poincare section is hard to discern. Non uniform variation of scatter plots per area of solutions space characterised chaotic and periodic responses as against average uniform variation for a random data set The plots of periodic response distribute restrictedly on the solutions space diagonal while probabilities of chaotic responses on the studied parameters space is between 21.5% and 70.6%. Estimated fractal disk dimension variation is in the range 0.00≤ Df≤ 1.8 I for studied cases. The study therefore has demonstrated the utility of the novel graphic plots as a dynamic systems characterising tool. "Item Runge-Kutta schemes coefficients simulation for comparison and visual effects(SciRes, 2013-05) Salau, T. A. O.; Ajide, O. O."Runge-Kutta scheme is one of the versatile numerical tools for the simulation of engineering systems. Despite its wide and acceptable engineering use, there is dearth of relevant literature bordering on visual impression possibility among different schemes coefficients which is the strong motivation for the present investigation of the third and fourth order schemes. The present study capitalise on results of tedious computation involving Taylor series expansion equivalent supplemented with Butcher assumptions and constraint equations of well-known works which captures the essential relationship between the coefficients. The simulation proceeds from random but valid specification of two out of the total coefficients possible per scheme. However the remaining coefficients are evaluated with application of appropriate function relationship. Eight and thirteen unknown coefficients were simulated respectively for third and fourth schemes over a total of five thousand cases each for relevant distribution statistics and scatter plots analysis for the purpose of scheme comparison and visual import. The respective three and four coefficients of the slope estimate for the third and fourth schemes have mix sign for large number of simulated cases. However, none of the two schemes have above three of these coefficients lesser than zero. The percentages of simulation results with two coefficients lesser than zero dominate and are respectively 56.88 and 77.10 for third and fourth schemes. It was observed that both popular third and fourth schemes belong to none of the coefficients being zero classification with respective percentage of 0.72 and 3.28 in total simulated cases. The comparisons of corresponding scatter plots are visually exciting. The overall difference 'between corresponding scatter plots and distribution results can be used to justify the accuracy of fourth scheme over its counterpart third scheme. "Item Chaos diagram of harmonically excited vibration absorber control duffing's oscillator(International Journal of Scienctific and Engineering Reasearch, 2013-02) Salau, T. A. O.; Ajide, O. O."This study utilised positive Lyapunov exponents' criteria to develop chaos diagram on the parameters space of 4-dimensionalharmonically excited vibration absorber control Duffing's Oscillator. Relevant simulations were effected by choice combination of constant step Runge-Kutta methods and Grahm Schmidt Orthogonal rules. Simulations of 4-dimensional hyper-chaotic models of modified Lorenz and RÖsier were used for validation purposes. Lyapunov's spectrums were obtained at (197 x 301) mesh points of parameters space (µ,αa). Lyapunov's spectrum of modified Lorenz system by constant time step (NRK1) fourth order Runge-Kutta method (04208.01650. - 0.0807, -26.4603) compare correspondingly well with (0.4254, 0.1286, 0.0000, -26.5493) reported by Yuxia et et. Similarly, Lyapunovs spectrum of modified Rosier system by constant time step (NRK1) fourth order Runge-Kutta method (0.1424, 0.0051, -0.0041, -24.0831) compare correspondingly and qualitatively with (0.1287, 0.0149, -0.0056, -22.8617) reported by Marco (1996). The sum of Lyapunov exponents (-22.7237, -31.3107, -27.8797) in Rosier compare correspondingly and qualitatively with variation matrix measure -AVERT (- 24.0181, -30.9462, -28.1991) respectively for fourth, fifth and modified fifth order Runge-Kutta methods. The chaos diagram results suggested preferentially higher mass ratio for effective chaos control of Duffing's Oscillator main mass. The parameters space in the region of relative lower mass ratio suffered irregular boundaries. The practical applications of this chaos diagram plot include, by instance, walking in the parameters-space of vibration absorber control Duffing's Oscillator along suitable engineering paths. "Item Parametric investigation and classification of quadratic equation(International Journal of Scienctific and Engineering Reasearch, 2013-01) Salau, T. A. O.; Ajide, O. O.This study investigated the papramters space of simple and familiar quadratic equation often encounter in science based disciplines and in particular engineering.The aim is strange images creation to excite and launch novices to quest for general understanding of fractal taking parallel clue from Julia and Mandelbrot sets creation using the iterative mechanisms of complex quadratic mapping