Finite Element Modeling for Materials Engineers Using MATLAB®

Finite Element Modeling for Materials Engineers Using MATLAB®

Finite Element Modeling for Materials Engineers Using MATLAB®
... Finite Element Modeling for Materials Engineers Using MATLABÒ Oluleke Oluwole Finite Element Modeling for Materials Engineers Ò Using MATLAB 123 Dr Oluleke Oluwole ... Oluwole, Finite Element Modeling for Materials Engineers Using MATLABÒ, DOI: 10.1007/978-0-85729-661-0_1, Ó Springer-Verlag London Limited 2011 Chapter The Weak Formulation 2.1 Nodal Finite Elements ... shape of mesh elements and its construction over the whole domain; numbering of the nodes and elements and the coordinates O Oluwole, Finite Element Modeling for Materials Engineers Using MATLABÒ,...
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crc press - elem. math. and comp. tools for engineers using matlab - j. manassah

crc press - elem. math. and comp. tools for engineers using matlab - j. manassah
... MATHEMATICAL and COMPUTATIONAL TOOLS for ELECTRICAL and COMPUTER ENGINEERS USING MATLAB ® © 2001 by CRC Press LLC ELEMENTARY MATHEMATICAL and COMPUTATIONAL TOOLS for ELECTRICAL and COMPUTER ENGINEERS USING ... Elementary mathematical and computational tools for electrical and computer engineers using MATLAB/ Jamal T Manassah p cm Includes bibliographical references and index ISBN 0-8 49 3-1 08 0-6 Electrical engineering—Mathematics ... the if-else-end structure takes the form: if expression Commands evaluated if expression is True © 2001 by CRC Press LLC elseif expression Commands evaluated if expression is True elseif expression...
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Elementary mathematical and computational tools for electrical and computer engineers using Matlab - Chapter 2 pot

Elementary mathematical and computational tools for electrical and computer engineers using Matlab - Chapter 2 pot
... (x1(n+1) ^2) ); x2(n+1)=b2*y2(n)+a*x2(n) +2* (1-a)*(x2(n)) ^2/ (1+ (x2(n) ^2) ); y2(n+1)=-x2(n)+a*x2(n+1) +2* (1-a)*(x2(n+1) ^2) /(1+ (x2(n+1) ^2) ); end subplot (2, 1,1); plot(x1,y1,'.') title('a =-0 .99 b=1') ... y1(n+1)=b*x1(n)+a*(y1(n )-( x1(n)) ^2) ; x2(n+1)=a*x2(n)-b*(y2(n )-( x2(n)) ^2) ; y2(n+1)=b*x2(n)+a*(y2(n )-( x2(n)) ^2) ; end plot(x1,y1,'ro',x2,y2,'bx') 2. 8 .2. 1 Demonstration Different orbits for Hénon’s model ... p (2. 4) Since C is the original capital borrowed; At k = 2, using Eq (2. 2) and Eq (2. 4), we obtain: y (2) = (1 + r)y(1) – p = (1 + r)2C – p(1 + r) – p (2. 5) At k = 3, using Eq (2. 2), (2. 4), and...
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Elementary mathematical and computational tools for electrical and computer engineers using Matlab - Chapter 3 docx

Elementary mathematical and computational tools for electrical and computer engineers using Matlab - Chapter 3 docx
... theory © 2001 by CRC Press LLC FIGURE 3. 4 Profile of the signal of Pb 3. 31 FIGURE 3. 5 Profile of the first input to Pb 3. 32 Pb 3. 33 In DSB-AM (double-sideband amplitude modulation), the amplitude ... following sub-units: mF = 10 3 F (milli-Farad); nF = 10 −9 F (nano-Farad); fF = 10 −15 F (femto-Farad); µF = 10 –6 F (micro-Farad); pF = 10 −12 F (pico-Farad); aF = 10 −18 F (atto-Farad); Pb 3. 29 Assume ... Mfile for this polynomial using array operations Let p = [1 3] : Solution: function y=polfct(x) p=[1 3] ; © 2001 by CRC Press LLC L=length(p); v=x.^[(L-1) :-1 :0]; y=sum(p.*v); In-Class Exercises Pb 3. 13...
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Elementary mathematical and computational tools for electrical and computer engineers using Matlab - Chapter 4 docx

Elementary mathematical and computational tools for electrical and computer engineers using Matlab - Chapter 4 docx
... +a*D2(k-1)+u(k)); D(k)=(2/dt)*(y(k)-y(k-1))-D(k-1); D2(k)= (4/ dt^2)*(y(k)-y(k-1) )-( 4/ dt)*D(k-1)-D2 (k-1); end plot(t,y,t,u,' ') The dashed curve is the temporal profile of the source term In-Class ... D2(1)=(1/a(1))*(-b(1)*D(1)-c(1)*y(1)+u(1)); for k=2:N y(k)=( (4* a(k)/dt^2+2*b(k)/dt+c(k))^ (-1 ))* (y(k-1)* (4* a(k)/dt^2+2*b(k)/dt)+D(k-1) * (4* a(k)/dt+b(k))+a(k)*D2(k-1)+u(k)); D(k)=(2/dt)*(y(k)-y(k-1))-D(k-1); ... dt=(tfin-tin)/(N-1); u=sin(w*t); y(1)=0; D(1)=0; D2(1)=(1/a)*(-b*D(1)-c*y(1)+u(1)); for k=2:N © 2001 by CRC Press LLC y(k)=( (4* a/dt^2+2*b/dt+c)^ (-1 ))* (y(k-1)* (4* a/dt^2+2*b/dt)+D(k-1)* (4* a/dt+b)+...
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Elementary mathematical and computational tools for electrical and computer engineers using Matlab - Chapter 5 potx

Elementary mathematical and computational tools for electrical and computer engineers using Matlab - Chapter 5 potx
... a(1)) (5. 25) a(2) = c(1) = a(1) + r(b(1) − a(1)) (5. 26) b(2) = b(1) (5. 27) Now, replacing the values of a(2) and b(2) from Eqs (5. 26) and (5. 27) into Eq (5. 25) , we are led to a second-degree ... The command is poly poly(r) In-Class Exercise Pb 5. 14 Find the roots of the polynomial p = [1 pute their sum and product 3], and com- Pb 5. 15 Consider the two polynomials: p1 = [1 3] and p2 = ... function M-file, the following command: © 2001 by CRC Press LLC [xmin,ymin]=goldensection('cos',3,3 .5, 10^ (-4 )) 5. 3.3 MATLAB fmin and fmins Built-in Function Following methodically the same steps using...
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Elementary mathematical and computational tools for electrical and computer engineers using Matlab - Chapter 6 pdf

Elementary mathematical and computational tools for electrical and computer engineers using Matlab - Chapter 6 pdf
... LLC (6. 67) and where ˜ ˜ ˜ Vtot = Atot e jφtot = V1 + V2 (6. 68) Preparatory Exercise Pb 6. 35 Write the analytical expression for Atot and φtot in Eq (6. 68) as functions of the amplitudes and phases ... signals? Pb 6. 42 Two- and three-phase power can be extended to N-phase power In such a scheme, the N-110-V /60 -Hz signals are given by: © 2001 by CRC Press LLC πn   Vn = 110 cos 120t +   N  and ... b=(cos(w1)-cos(w2))^2; p=[1 a-b -8 *a 14*a-2*b-2 -8 *a a-b 1]; rr=roots(p); r=rr(find(rr>0 & rr...
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Elementary mathematical and computational tools for electrical and computer engineers using Matlab - Chapter 7 pot

Elementary mathematical and computational tools for electrical and computer engineers using Matlab - Chapter 7 pot
... Replacing l by l – in Eq (7. 94) and eliminating Pl′ ( x) from Eq − (7. 95), we find that: (1 − x ) dPl ( x) = lPl−1 ( x) − lxPl ( x) dx (7. 97) Differentiating Eq (7. 97) and using Eq (7. 95), we obtain: ... unique For example, in 4-D space, the canonical four-unit orthonormal basis vectors are given, respectively, by: ê1 = [1 0 0] (7. 15) ê2 = [0 0] (7. 16) ê3 = [0 0] (7. 17) ê4 = [0 0 1] (7. 18) r and ... Product and Scalar Triple Product* In this section and in Sections 7. 6 and 7. 7, we restrict our discussions to vectors in a 3-D space, and use the more familiar conventional vector notation 7. 5.1...
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Elementary mathematical and computational tools for electrical and computer engineers using Matlab - Chapter 8 pdf

Elementary mathematical and computational tools for electrical and computer engineers using Matlab - Chapter 8 pdf
... n v m = λ n v m (8. 83) On the other hand, if we dotted Eq (8. 78) on the left with the bra-vector , we obtain: H v m = λ m v m = λ m v m (8. 84) Now compare Eqs (8. 83) and (8. 84) They are equal, ... ilamp=circuit872(RL) M=[1 0 0 0;1 -1 -5 0 0;0 -1 -1 00 0; 0 -3 00;0 -RL 0;0 0 -1 -1 ]; Vs=[5;0;0;0;0;0]; VI=M\Vs; ilamp=VI(5); Then, from the command window, we proceed by calling this function and plotting ... following script M-file provides the solution to the above circuit: NOTE M=[1 0 0 0;1 -1 -5 0 0;0 -1 -1 00 0; 0 -3 00;0 -2 0;0 0 -1 -1 ]; Vs=[5;0;0;0;0;0]; VI=M\Vs In-Class Exercise Pb 8. 4 Use the same...
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Elementary mathematical and computational tools for electrical and computer engineers using Matlab - Chapter 9 ppt

Elementary mathematical and computational tools for electrical and computer engineers using Matlab - Chapter 9 ppt
... transformations, in 2-D, are respectively: s Sx =  x 0 0 1  (9. 8) 1 Sy =  0 0 sy   (9. 9) In-Class Exercises Pb 9. 12 Find the transformation matrix for simultaneously compressing the x-coordinate ... 2, while expanding the y-coordinate by a factor of Apply this transformation to the trapezoid of Example 9. 1 and plot the result Pb 9. 13 Find the inverse matrices for Sx and Sy 9. 1.5 Translation ... −1  (9. 3) −1 Py =  0 0 1  (9. 4) In-Class Exercise Pb 9. 1 Using the trapezoid of Example 9. 1, obtain all the transformed G’s as a result of the action of each of the three transformations...
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Elementary mathematical and computational tools for electrical and computer engineers using Matlab - Chapter 10 docx

Elementary mathematical and computational tools for electrical and computer engineers using Matlab - Chapter 10 docx
... Solution: The number of ways of selecting 10 items from a batch of 100 items is: N= 100 ! 100 ! 100 = = C10 10! (100 10) ! 10! 90! n where Ck is the binomial coefficient and represents the number of combinations ... (10. 43) Again, using Eqs (10. 37) and (10. 43), we have: P( Ai B) = P(B Ai )P( Ai ) (10. 44) P(B) Now, substituting Eq (10. 41) in the denominator of Eq (10. 44), we obtain Eq (10. 42) Example 10. 10 ... N ( A) = 95! 95 = C10 10! 85! and the probability for the event A is P( A) = 95 C10 86 × 87 × 88 × 89 × 90 = = 0.5837 100 96 × 97 × 98 × 99 × 100 C10 In-Class Exercises Pb 10. 1 A cube whose faces...
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Discrete element modeling for flows of granular materials

Discrete element modeling for flows of granular materials
... DISCRETE ELEMENT MODELING FOR FLOWS OF GRANULAR MATERIALS BY LIM WEE CHUAN ELDIN (M.Eng., B.Eng (Hons.), NUS) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF CHEMICAL ... University for helpful discussions via video-conferencing on the subject of granular attrition which subsequently led to the formulation of a theoretical approach for modeling bulk granular attrition ... contact force fcn,ij normal component of contact force fct,ij tangential component of contact force fd,ij viscous contact damping force fdn,ij normal component of viscous contact damping force...
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