CHAPTER 5a. SIMULTANEOUS LINEAR EQUATIONS

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1 CHAPTER 5. SIMULTANEOUS LINEAR EQUATIONS A. J. Clrk School of Engineering Deprtment of Civil nd Environmentl Engineering by Dr. Ibrhim A. Asskkf Spring 00 ENCE 0 - Computtion Methods in Civil Engineering II Deprtment of Civil nd Environmentl Engineering University of Mrylnd, College Prk Systems of simultneous equtions cn be found in mny engineering pplictions nd problems. Systems tht consist of smll number of equtions cn be solved nlyticlly using stndrd methods from lgebr. Systems of lrge number of equtions require the use of numericl methods nd computers. ENCE 0 CHAPTER 5. SIMULTANEOUS LINEAR EQUATIONS Slide No.

2 Simultneous Liner Equtions nd Engineering The system of simultneous equtions is probbly one of the most importnt topics in modern engineering computtions. This is not n exggertion if one considers tht recent technologicl dvnces were mde possible by the bility of solving lrger nd lrger systems of equtions. ENCE 0 CHAPTER 5. SIMULTANEOUS LINEAR EQUATIONS Slide No. Simultneous Liner Equtions nd Engineering Smll Systems = 5 = = 9 0.Y 0.Z = Y 0.Z = Y + 0Z = Lrger System M + L+ 0 + L+ + L 8 + L = 0. = = 6.5 = ENCE 0 CHAPTER 5. SIMULTANEOUS LINEAR EQUATIONS Slide No.

3 Simultneous Liner Equtions nd Engineering Technologicl dvnces in engineering tht involve lrge number of simultneous equtions include: The finite element method The finite difference method The nlysis of structurl, mechnicl, nd electricl systems. ENCE 0 CHAPTER 5. SIMULTANEOUS LINEAR EQUATIONS Slide No. 4 Simultneous Liner Equtions nd Engineering In the pst, the numericl methods (such s the finite element nd finite difference methods) tht were used to solve systems of simultneous equtions were not ttrctive owing to the tremendous mount of clcultions involved. However, computers hve chnged tht nd ltered our pproch to engineering problem solving. ENCE 0 CHAPTER 5. SIMULTANEOUS LINEAR EQUATIONS Slide No. 5

4 Electricl Circuit V V I = current V = voltge R = resistnce I R I R R I ENCE 0 CHAPTER 5. SIMULTANEOUS LINEAR EQUATIONS Slide No. 6 Electricl Circuit V V R I R I R I Kirchhoff s first lw sttes tht the lgebric sum of current flowing into junction of circuit must equl zero Kirchhoff s second lw sttes tht the lgebric sum of the electromotive forces round closed circuit must equl the sum of voltge drops round the circuit. ENCE 0 CHAPTER 5. SIMULTANEOUS LINEAR EQUATIONS Slide No. 7 4

5 Electricl Circuit V V c e I R I R R I b d f ENCE 0 CHAPTER 5. SIMULTANEOUS LINEAR EQUATIONS Slide No. 8 Electricl Circuit Applying Kirchhoff s first lw t junction c, yields the following liner eqution: I + I I = 0 Applying Kirchhoff s second lw to loop cdb yields the following liner eqution: V + = RI RI ENCE 0 CHAPTER 5. SIMULTANEOUS LINEAR EQUATIONS Slide No. 9 5

6 Electricl Circuit Applying Kirchhoff s second lw to loop efb yields the following liner eqution: V V = RI RI V V R I R I R I ENCE 0 CHAPTER 5. SIMULTANEOUS LINEAR EQUATIONS Slide No. 0 R Electricl Circuit V V I I R R If R =, R = 4, R = 5, V = 6, nd V =, then I I I I + 4I + 5I = 0 = 6 = 4 The solution to these three equtions produces the current flows in the network. ENCE 0 CHAPTER 5. SIMULTANEOUS LINEAR EQUATIONS Slide No. I I 6

7 Anlysis of Stticlly Determinnt Truss 000 lb 90 o H 0 o 60 o V V ENCE 0 CHAPTER 5. SIMULTANEOUS LINEAR EQUATIONS Slide No. Anlysis of Stticlly Determinnt Truss 000 lb H V = H = V = 0 + V 000 = 0 H 0 o 90 o 60 o V V ENCE 0 CHAPTER 5. SIMULTANEOUS LINEAR EQUATIONS Slide No. 7

8 Anlysis of Stticlly Determinnt Truss 000 lb H V 0 60 = cos0 + cos60 = 0 = 000 sin 0 H sin 60 = 0 90 o 0 o 60 o 000 lb V V () ENCE 0 CHAPTER 5. SIMULTANEOUS LINEAR EQUATIONS Slide No lb H 0 90 o H V = H = V V 0 + cos0 + + sin 0 = 0 = 0 H () 0 o 60 o V V ENCE 0 CHAPTER 5. SIMULTANEOUS LINEAR EQUATIONS Slide No. 5 8

9 Anlysis of Stticlly Determinnt Truss 000 lb o H V V = = V + cos60 = 0 sin 60 = 0 () H 0 o 60 o V V ENCE 0 CHAPTER 5. SIMULTANEOUS LINEAR EQUATIONS Slide No. 6 H V Anlysis of Stticlly Determinnt Truss Combining the three systems of equtions (systems,, nd ), we obtin one system of simultneous eqution s shown in the next viewgrph: = cos0 + cos60 = 0 H = cos60 = 0 = 000 sin 0 sin 60 = 0 V = V + sin 60 = 0 = H + cos0 + = 0 H V = V + sin 0 = 0 ENCE 0 CHAPTER 5. SIMULTANEOUS LINEAR EQUATIONS Slide No. 7 9

10 H 0 o Anlysis of Stticlly Determinnt Truss 90 o 000 lb 60 o V V H + V + V = 0 = 000 = 0 = 0 = 0 = 0 ENCE 0 CHAPTER 5. SIMULTANEOUS LINEAR EQUATIONS Slide No. 8 H Anlysis of Stticlly Determinnt Truss The solution to the previous system of equtions provides the following force vlues for the truss: = 500 lb 0 o 90 o 000 lb ENCE 0 CHAPTER 5. SIMULTANEOUS LINEAR EQUATIONS 60 o V V = 4 lb = 866 lb H = 0 (s expected) V = 50 lb V = 750 lb Slide No. 9 0

11 Engineering Dynmics Exmple: Suppose tht tem of three prchutists is connected by weightless cord while freeflling t velocity of 5 m/s s shown in the figure. Compute the tension in ech section of the cord nd the ccelertion of the tem, given the msses of ech prchutist nd the dreg coefficients s provided in the tble. ENCE 0 CHAPTER 5. SIMULTANEOUS LINEAR EQUATIONS Slide No. 0 Engineering Dynmics Exmple R Prchutist Mss (kg) Drg Coefficient (kg/s) 5 8 T ENCE 0 CHAPTER 5. SIMULTANEOUS LINEAR EQUATIONS Slide No.

12 Engineering Dynmics Exmple ree-body digrms re needed for ech of the prchutists s shown in the next viewgrph. Summing forces in the verticl direction nd using Newton s second lw of motion, gives: m g T c v m g + T c v R = m m g = m c v + R = m (4) ENCE 0 CHAPTER 5. SIMULTANEOUS LINEAR EQUATIONS Slide No. Engineering Dynmics Exmple c v c v R c v T m g R m g T m g ENCE 0 CHAPTER 5. SIMULTANEOUS LINEAR EQUATIONS Slide No.

13 Engineering Dynmics Exmple c v c v R c v T m g R m g T m g m g + R cv = m m g + T cv R = m m g cv T = m ENCE 0 CHAPTER 5. SIMULTANEOUS LINEAR EQUATIONS Slide No. 4 Engineering Dynmics Exmple Substituting the vlues for prchutists msses nd drg coefficients, the system of equtions provided by Eq. 4, gives 80(9.8) T (5) = 80 70(9.8) + T 5(5) R = 70 50(9.8) 8(5) + R = 50 ENCE 0 CHAPTER 5. SIMULTANEOUS LINEAR EQUATIONS Slide No. 5

14 Engineering Dynmics Exmple After rerrnging nd simplifying, 80 + T = T + R = 6 50 R = 400 The solution to the bove system of equtions gives the following vlues: = 8.7 m/s, T = N, nd R = 5 N ENCE 0 CHAPTER 5. SIMULTANEOUS LINEAR EQUATIONS Slide No. 6 Generl orm for System of Equtions Definition A liner eqution is one in which vrible only ppers to the first power in every term of given eqution. Thus, system of m liner equtions in n unknowns j, j =,,,n, cn be represented s n j= ij j = C, i =,, L, m i ENCE 0 CHAPTER 5. SIMULTANEOUS LINEAR EQUATIONS Slide No. 7 4

15 Generl orm for System of Equtions Expnded orm m m + L+ + L+ M + L+ n n mn n n n = C = C = C m (5) ij = known coefficients of the equtions j = unknown vribles C i = known constnts ENCE 0 CHAPTER 5. SIMULTANEOUS LINEAR EQUATIONS Slide No. 8 Generl orm for System of Equtions Expnded orm (m = n) n n + L+ + L+ M + L+ n n nn n n n = C = C = C n (5b) ij = known coefficients of the equtions j = unknown vribles C i = known constnts ENCE 0 CHAPTER 5. SIMULTANEOUS LINEAR EQUATIONS Slide No. 9 5

16 Generl orm for System of Equtions Clssifiction of Systems of Equtions. A set of equtions in which the number of unknowns is equl to the number of equtions (n = m). A set of equtions in which the number of unknowns is less thn the number of equtions (n < m). A set of equtions in which the number of unknowns is greter thn the number of equtions (n > m) ENCE 0 CHAPTER 5. SIMULTANEOUS LINEAR EQUATIONS Slide No. 0 Solution of System of Two Equtions The solution of two eqution gives insight to understnd the clssifiction of systems of equtions bsed on grphicl interprettion in twodimensionl spce. If n equls, then Eq. 5 reduces to + = C (6) + = C (6b) ENCE 0 CHAPTER 5. SIMULTANEOUS LINEAR EQUATIONS Slide No. 6

17 Solution of System of Two Equtions A solution for simple system cn be obtined by substitution. Solving Eq. 6 for gives C = The expression for in the bove eqution cn be substituted into Eq. 6b: C = + C (7) (7b) ENCE 0 CHAPTER 5. SIMULTANEOUS LINEAR EQUATIONS Slide No. Solution of System of Two Equtions Eqution 7b is single eqution with one unknown,. This eqution cn be solved for to give C C = (8) Eq. 8 cn be substituted bck into Eq. 7 to give C C = (8b) ENCE 0 CHAPTER 5. SIMULTANEOUS LINEAR EQUATIONS Slide No. 7

18 Solution of System of Two Equtions It seems tht the solution procedure for this set of equtions is simple to pply. One should imgine the effort tht would be required to solve 5 or 0 simultneous equtions using the substitution procedure. Mny complex engineering problems involve hundreds or even thousnds of simultneous equtions. Hence, the need for lterntive solution procedures is justified. ENCE 0 CHAPTER 5. SIMULTANEOUS LINEAR EQUATIONS Slide No. 4 Types of Numericl Procedures Becuse of the wide-spred use of computers nowdys, numericl solution methods re widely used. There re three generl types:. Elimintion methods,. Itertion methods, nd. Method of determinnts. ENCE 0 CHAPTER 5. SIMULTANEOUS LINEAR EQUATIONS Slide No. 5 8

19 Clssifiction of Systems of Equtions Bsed on Grphicl Interprettion Systems of equtions cn be clssified bsed on their solutions to the following types:. Systems tht hve solutions,. Systems without solution, nd. Systems with n infinite number of solutions. ENCE 0 CHAPTER 5. SIMULTANEOUS LINEAR EQUATIONS Slide No. 6 9