• No results found

sm ch (8).pdf

N/A
N/A
Protected

Academic year: 2021

Share "sm ch (8).pdf"

Copied!
11
0
0

Loading.... (view fulltext now)

Full text

(1)

CHAPTER 8

CHAPTER 8

8.1 8.1 >

>> > AAug ug = = [[ A A eyeeye(( sisi ze(ze( AA)) )) ]] Here’s an example session of how

Here’s an example session of how it can be employed.it can be employed. > >> > A = A = rr and(and( 3)3) A A == 0. 0. 959501 01 0.0. 484860 60 0.0. 45456565 0. 0. 232311 11 0.0. 898913 13 0.0. 01018585 0. 0. 606068 68 0.0. 767621 21 0.0. 82821414 >

>> > AAug ug = = [[ A A eyeeye(( sisi ze(ze( AA)) )) ]] Aug = Aug = 0. 0. 959501 01 0.0. 484860 60 0.0. 454565 65 1.1. 000000 00 0 0 00 0. 0. 232311 11 0.0. 898913 13 0.0. 010185 85 0 0 1.1. 000000 00 00 0. 0. 606068 68 0.0. 767621 21 0.0. 828214 14 0 0 0 0 1.1. 00000000 8.2 (a)

8.2 (a) [ [ A A]: 3]: 3

 2  2 [[ B B]: 3]: 3

 3  3 {{C C }: 3}: 3

 1  1 [[ D D]: 2]: 2

 4 4 [[ E  E ]: 3]: 3

 3  3 [[F F ]: 2]: 2

 3 3



GG



: 1: 1

 3 3

(b)

(b)square: [ Bsquare: [ B], [], [ E  E ]; column: {]; column: {C C }, row:}, row:



GG



(c)

(c)aa1212 = 5, b = 5,b2323 = 6, = 6, d d 3232 = undefined, = undefined, ee2222 = 1, = 1, f  f 1212 = 0, = 0, gg1212 = 6 = 6

(d) (d)

(1)

(1) (2) (2) [[ A A] + [] + [F F ] = undefined] = undefined 5 5 88 1133 [[ ]] [[ ]] 88 33 99 6 6 00 1100  E  E BB





 









(3) (4) (3) (4) 3 3 22 11 [[ ]] [[ ]] 66 11 33 2 2 00 22  B  B E E 





 







2 288 2211 4499 7 7[ ][ ] 77 1144 4422 1 144 00 2288  B  B





  







((55) ) [[ ] C C ] T T 

 

 

2 62 6 11





((66)) 2 211 1133 6611 [[ ]][[ ]] 3355 2233 6677 2 288 1122 5522  E  E BB





  







(7) (8) (7) (8) 5 533 2233 7755 [ ] [ ][[ ]] 3399 77 4488 1 188 1100 3366  B  B E E 





 









5 5 22 4 4 11 [[ ]] 3 3 77 7 7 55 T  T   D  D













(9) [ (9) [GG][][C C ] ] = = 56 56 (10)(10) 4 4 33 77 [[ ]] 11 22 66 2 2 00 44  I  I BB





  







(11) (12) (11) (12) C C C C  6 666 1122 5511 [[ ]] 1122 2266 3333 5 511 3333 8811 T  T   E  E E E 





 









T T [[ ]]

4411

(2)

8.3 The terms can be collected to give 1 2 3 0 6 5 50 0 2 7 30 4 3 7 50  x  x  x

 

   

 

   

   

 

Here is the MATLAB session:

>> A = [ 0 - 6 5; 0 2 7; - 4 3 - 7] ; >> b = [ 50; - 30; 50] ; >> x = A\ b x = - 17. 0192 - 9. 6154 - 1. 5385 >> AT = A' 0 0 - 4 - 6 2 3 5 7 - 7 >> AI = i nv( A) AI = - 0. 1683 - 0. 1298 - 0. 2500 - 0. 1346 0. 0962 0 0. 0385 0. 1154 0

8.4 (a) Here are all the possible multiplications: >> A=[ 6 - 1; 12 7; - 5 3] ; >> B=[ 4 0; 0. 6 8] ; >> C=[ 1 - 2; - 6 1] ; >> A* B ans = 23. 4000 - 8. 0000 52. 2000 56. 0000 - 18. 2000 24. 0000 >> A* C ans = 12 - 13 - 30 - 17 - 23 13 >> B* C ans = 4. 0000 - 8. 0000 - 47. 4000 6. 8000 >> C* B ans = 2. 8000 - 16. 0000 - 23. 4000 8. 0000

(b) [ B][ A] and [C ][ A] are impossible because the inner dimensions do not match: (2

2) * (3

2)

(3)

8.5 >> A=[ 3+2*i 4; - i 1] >> b=[ 2+i ; 3] >> z=A\ b z = - 0. 5333 + 1. 4000i 1. 6000 - 0. 5333i 8.6 f unct i on X=mmul t ( Y, Z)

% mmul t : mat r i x mul t i pl i cat i on % X=mmul t ( Y, Z)

% mul t i pl i es t wo mat r i ces % i nput :

% Y = f i rst mat r i x % Z = second mat r i x % out put :

% X = pr oduct

i f nar gi n<2, er r or ( ' at l east 2 i nput ar gument s r equi r ed' ) , end [ m, n] =si ze( Y) ; [ n2, p] =si ze( Z) ;

i f n~=n2, err or( ' I nner matr i x di mensi ons must agr ee. ' ) , end f or i =1: m f or j =1: p s=0. ; f or k=1: n s =s +Y( i , k) * Z( k, j ) ; end X( i , j ) =s ; end end

Test of function for cases from Prob. 8.4: >> A=[ 6 - 1; 12 7; - 5 3] ; >> B=[ 4 0; 0. 6 8] ; >> C=[ 1 - 2; - 6 1] ; >> mmul t ( A, B) ans = 23. 4000 - 8. 0000 52. 2000 56. 0000 - 18. 2000 24. 0000 >> mmul t ( A, C) ans = 12 - 13 - 30 - 17 - 23 13 >> mmul t ( B, C) ans = 4. 0000 - 8. 0000 - 47. 4000 6. 8000 >> mmul t ( C, B) ans = 2. 8000 - 16. 0000 - 23. 4000 8. 0000 >> mmul t ( B, A)

(4)

?? ? Er r or usi ng ==> mmul t

I nner mat r i x di mensi ons must agr ee. >> mmul t ( C, A)

?? ? Er r or usi ng ==> mmul t

I nner mat r i x di mensi ons must agr ee. 8.7

f unct i on AT=mat r an( A)

% mat r an: mat r i x t r anspose % AT=mt r an( A)

% generat es t he t r anspose of a mat r i x % i nput : % A = or i gi nal mat r i x % out put : % AT = t r anspose [ m, n] =si ze( A) ; f or i = 1: m f or j = 1: n AT( j , i ) = A( i , j ) ; end end

Test of function for cases from Prob. 8.4: >> mat r an( A) ans = 6 12 - 5 - 1 7 3 >> mat r an( B) 4. 0000 0. 6000 0 8. 0000 >> mat r an( C) ans = 1 - 6 - 2 1 8.8 f unct i on B = permut ( A, r 1, r 2)

% permut : Swi t ch r ows of mat r i x A wi t h a permut at i on mat r i x % B = permut ( A, r 1, r 2)

% i nput :

% A = or i gi nal mat r i x

% r 1, r 2 = r ows t o be swi t ched % out put :

% B = mat r i x wi t h r ows s wi t ched [ m, n] = si ze( A) ;

i f m ~= n, er r or ( ' mat r i x not squar e' ) , end i f r 1 == r 2 | r 1>m | r 2>m

err or( ' r ow numbers ar e equal or exceed mat r i x di mensi ons' ) end P = zeros( n) ; P( r 1, r2)=1; P( r 2, r 1) =1; f or i = 1: m i f i ~=r 1 & i ~=r 2 P( i , i ) =1; end

(5)

end B=P* A; Test script: cl c A=[ 1 2 3 4; 5 6 7 8; 9 10 11 12; 13 14 15 16] B = per mut ( A, 3, 1) B = per mut ( A, 3, 5) A = 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 B = 9 10 11 12 5 6 7 8 1 2 3 4 13 14 15 16

?? ? Er r or usi ng ==> per mut

r ow number s ar e equal or exceed mat r i x di mensi ons Err or i n ==> per mut Scr i pt at 4

B = per mut ( A, 3, 5)

8.9 The mass balances can be written as

15 12 1 31 3 01 01 12 1 23 24 25 2 ( ) ( ) Q Q c Q c Q c Q c Q Q Q c

  

23 2 31 34 3 03 03 24 2 34 3 44 4 54 5 0 ( ) Q c Q Q c Q c Q c Q c Q c Q c

15 1 25 2 54 55 5 0 ( ) 0 Q c Q c Q Q c

The parameters can be substituted and the result written in matrix form as

1 2 3 4 5 9 0 3 0 0 120 4 4 0 0 0 0 0 2 9 0 0 350 0 1 6 9 2 0 5 1 0 0 6 0 c c c c c

 

 

 

    

   

    

    

   

  

The following MATLAB script can then be used to solve for the concentrations cl c Q = [ 9 0 - 3 0 0; - 4 4 0 0 0; 0 - 2 9 0 0; 0 - 1 - 6 9 - 2; - 5 - 1 0 0 6] ; Qc = [ 120; 0; 350; 0; 0] ; c = Q\ Qc c =

(6)

28. 4000 28. 4000 45. 2000 39. 6000 28. 4000

8.10The problem can be written in matrix form as

1 2 3 2 2 3 0.866025 0 0.5 0 0 0 0 0.5 0 0.866025 0 0 0 2000 0.866025 1 0 1 0 0 0 0.5 0 0 0 1 0 0 0 1 0.5 0 0 0 0 0 0 0.866025 0 0 1 0 F  F  F   H  V  V 

  

   

   

   

   

 

 

MATLAB can then be used to solve for the forces and reactions, cl c; f or mat shor t g A = [ 0. 866025 0 - 0. 5 0 0 0; 0. 5 0 0. 866025 0 0 0; - 0. 866025 - 1 0 - 1 0 0; - 0. 5 0 0 0 - 1 0; 0 1 0. 5 0 0 0; 0 0 - 0. 866025 0 0 - 1] ; b = [ 0 - 2000 0 0 0 0] ' ; F = A\ b F = - 1000 866. 03 - 1732. 1 0 500 1500 Therefore, F 1 = –1000 F 2 = 866.025 F 3 = –1732.1  H 2 = 0 V 2 = 500 V 3 = 1500 8.11 cl c; f or mat shor t g k1=10; k2=40; k3=40; k4=10; m1=1; m2=1; m3=1; km=[ ( 1/ m1) *( k2+k1) , - ( k2/ m1) , 0 - ( k2/ m2) , ( 1/ m2) *( k2+k3) , - ( k3/ m2) 0, - ( k3/ m3) , ( 1/ m3) *( k3+k4) ] x=[ 0. 05; 0. 04; 0. 03] ; kmx=- km*x km = 50 - 40 0 - 40 80 - 40 0 - 40 50 kmx = - 0. 9 - 2. 2204e- 016 0. 1

(7)

Therefore,

 x

1 = –0.9,

 x

2= 0 , and

 x

3= 0.1 m/s2.

8.12 Vertical force balances can be written to give the following system of equations,

1 2 2 1 1 1 2 3 3 2 2 2 1 3 4 4 3 3 3 2 4 5 5 4 4 4 3 5 5 5 4 ( ) 0 ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 m g k x x k x m g k x x k x x m g k x x k x x m g k x x k x x m g k x x

  

 

 

 

 

 

 

0 0 0

   

   

   

Collecting terms, 1 2 2 1 1 2 2 3 3 2 2 3 3 4 4 3 3 4 4 5 5 4 4 5 5 5 5 k k k x mg k k k k x m g k k k k x m g k k k k x m g k k x m g

    

   

    

    

   

 

After substituting the parameters, the equations can be expressed as ( g = 9.81),

1 2 3 4 5 120 40 637.65 40 110 70 735.75 70 170 100 588.60 100 120 20 735.75 20 20 882.90  x  x  x  x  x

 

 

 

   

  

   

   

  

  

The solution can then be obtained with the following MATLAB script: cl c; f or mat shor t g g=9. 81; m1=65; m2=75; m3=60; m4=75; m5=90; k1=80; k2=40; k3=70; k4=100; k5=20; A=[ k1+k2 - k2 0 0 0 - k2 k2+k3 - k3 0 0 0 - k3 k3+k4 - k4 0 0 0 - k4 k4+k5 - k5 0 0 0 - k5 k5] b=[ m1*g m2*g m3*g m4*g m5*g] ' x=A\ b A = 120 - 40 0 0 0 - 40 110 - 70 0 0 0 - 70 170 - 100 0 0 0 - 100 120 - 20 0 0 0 - 20 20 b = 637. 65 735. 75 588. 6

(8)

735. 75 882. 9 x = 44. 758 118. 33 149. 87 166. 05 210. 2

8.13 The position of the three masses can be modeled by the following steady-state force balances

2 1 1 1 3 2 2 2 1 3 3 2 0 ( ) 0 ( ) ( 0 ( ) k x x m g kx k x x m g k x x m g k x x

  

  

)

Terms can be combined to yield

1 2 1 1 2 3 2 2 3 3 2 2 kx kx m g kx kx kx m g kx kx m g

 

 

Substituting the parameter values

1 2 3 30 15 0 19.62 15 30 15 24.525 0 15 15 29.43  x  x  x

   

  

  

  

   

A MATLAB script can be used to obtain the solution for the displacements cl c; f or mat shor t g g=9. 81; k=15; K=[ 2*k - k 0; - k 2*k - k; 0 - k k] m=[ 2; 2. 5; 3] ; mg=m*g x=K\ mg K = 30 - 15 0 - 15 30 - 15 0 - 15 15 mg = 19. 62 24. 525 29. 43 x = 4. 905 8. 502 10. 464

(9)

12 52 32 65 52 32 54 43 54 12 52 65 43 1 6 1 1 1 0 0 0 0 0 1 0 1 1 0 0 0 0 1 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 i i i i  R R R R i  R R R i V V 

0

  

  

  

  

  

   

   

   

   

  

 

or substituting the resistances

12 52 32 65 54 43 1 1 1 0 0 0 0 0 1 0 1 1 0 0 0 0 1 0 0 1 0 0 0 0 0 1 1 0 0 10 10 0 15 5 0 5 10 0 20 0 0 200 i i i i i i

 

 

 

 

   

    

    

    

    

 

   

This system can be solved with MATLAB, cl c; f or mat shor t g R12=5; R52=10; R32=10; R65=20; R54=15; R43=5; V1=200; V6=0; A=[ 1 1 1 0 0 0; 0 - 1 0 1 - 1 0; 0 0 - 1 0 0 1; 0 0 0 0 1 - 1; 0 R52 - R32 0 - R54 - R43; R12 - R52 0 - R65 0 0] B=[ 0 0 0 0 0 V1- V6] ' I =A\ B A = 1 1 1 0 0 0 0 - 1 0 1 - 1 0 0 0 - 1 0 0 1 0 0 0 0 1 - 1 0 10 - 10 0 - 15 - 5 5 - 10 0 - 20 0 0 B = 0 0 0 0 0 200 I = 6. 1538 - 4. 6154 - 1. 5385 - 6. 1538 - 1. 5385 - 1. 5385 i21 = 6.1538 i52 =

4.6154 i32 =

1.5385 i65 =



i54 =

1.5385 i43 =

1.5385

(10)

Here are the resulting currents superimposed on the circuit: 6.1538 1.5385 6.1538 1.5385 1.5385        1 .        5        3        8        5 0 200  –6.1538    –        4  .        6        1        5        4

8.15 The current equations can be written as

0

52 23 21

i

i

i

0

43 35 23

i

i

i

0

54 43

i

i

0

54 65 52 35

i

i

i

i

Voltage equations: 2 21 20 35 V  i

5 4 54 15 V V  i

2 3 23 30 V V  i

3 5 35 7 V V  i

4 3 43 8 V V  i

5 2 52 10 V V  i

5 65 140 5 V  i

21 23 52 35 43 54 65 2 3 4 5 1 1 1 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 1 0 1 1 0 0 0 0 35 0 0 0 0 0 0 1 0 0 0 0 30 0 0 0 0 0 1 1 0 0 0 0 0 0 8 0 0 0 1 1 0 0 0 0 0 0 15 0 0 0 1 1 0 0 0 7 0 0 0 0 1 0 1 0 0 10 0 0 0 0 1 0 0 1 0 0 0 0 0 0 5 0 0 0 1 i i i i i i i V  V  V  V 

0 0 0 0 20 0 0 0 0 0 140

  

  

  

  

  

  

  

  

   

   

   

   

   

   

   

   

 

A MATLAB script can be developed to generate the solution, cl c; f or mat shor t g

R12=35; R52=10; R32=30; R34=8; R45=15; R35=7; R25=10; R65=5; V1=20; V6=140;

A=[ - 1 - 1 1 0 0 0 0 0 0 0 0; 0 1 0 - 1 1 0 0 0 0 0 0;

(11)

0 0 0 0 - 1 1 0 0 0 0 0; 0 0 - 1 1 0 - 1 1 0 0 0 0; R12 0 0 0 0 0 0 - 1 0 0 0; 0 R32 0 0 0 0 0 - 1 1 0 0; 0 0 0 0 R34 0 0 0 1 - 1 0; 0 0 0 0 0 R45 0 0 0 1 - 1; 0 0 0 R35 0 0 0 0 - 1 0 1; 0 0 R25 0 0 0 0 1 0 0 - 1; 0 0 0 0 0 0 R65 0 0 0 1] B=[ 0 0 0 0 - V1 0 0 0 0 0 V6] ' I =A\ B I = 2. 5107 - 0. 55342 1. 9573 - 0. 42429 0. 12913 0. 12913 2. 5107 107. 87 124. 48 125. 51 127. 45 Thus, the solution is

i21 = 2.5107 i23 =

0.55342 i52 = 1.9573 i35 =

0.42429 i43 = 0.12913

i54 = 0.12913 i65 = 2.5107 V 2 = 107.87 V 3 = 124.48 V 4 = 125.51

References

Related documents

Based on the international migration data from 35 LDCs and 15 high-income OECD countries in 1990 and 2000, we find that inward FDI deter the out-migration of individuals with

The figure below depicts the main elements of the search form in this mode and the corresponding search process behavior in the CQC Catalogue (quality reports not in the scope of

Economic impact assessments are the most commonly used form of measuring the economic benefits of cultural sector organisations, so a second case study that used this method has

What if the project team were able to provide a complete set of the business requirements previously captured, review them with the users who have the changes in

The qualitative data collected based on the interview of this study were analysed to look for patterns and finally the researcher identified four challenges faced by

The effect on the spouse. The individual’s decision to start receiving Social Security benefits before reaching his FRA may also affect a spouse’s benefits. Unless the spouse has

We take cyber security seriously and have invested substantial resources into our efforts to promote and improve the ability of our company, our peers and others to provide

Order of the Minister of Foreign Affairs and the Minister for Foreign Trade and Development Cooperation of 12 July 2013, MinBuZa-2013.209411, laying down administrative rules and