ME 406 Assignment #1 Solutions

Transcription

1 Assignmen#1Sol.nb 1 ME 406 Assignmen #1 Soluions PROBLEM 1 We define he funcion for Mahemaica. In[1]:= f@_d := Ep@D - 4 Sin@D (a) We use Plo o consruc he plo. In[2]:= Plo@f@D, 8, -5, 5<, AesLabel Ø 8"", "fhl"<d fhl Ou[2]= (b) The graph shows ha here are wo posiive roos, one near 0.4 and he oher near 1.5. We use FindRoo o find hem. We check each roo afer we find i. In[3]:= FindRoo@f@D == 0, 8, 0.4<D Ou[3]= 8 Ø < In[4]:= f@d ê. % Ou[4]= 0. In[5]:= FindRoo@f@D == 0, 8, 1.5<D Ou[5]= 8 Ø <

2 Assignmen#1Sol.nb 2 In[6]:= f@d ê. % Ou[6]= 0. The checks show ha he roos are accurae. PROBLEM 2 A sandard parameric represenaion is given by := 2 Cos@D; y@_d := 3 Sin@D We use ParamericPlo o plo his. In[8]:= ParamericPlo@8@D, y@d<, 8, 0, 2 p<, AspecRaio Ø 1, PloRange Ø , 3.1<, 8-3.1, 3.1<<, AesLabel Ø 8"", "y"<d 3 y 2 1 Ou[8]= PROBLEM 3 We define he equaion for Mahemaica. We call i equa3. In[9]:= Clear@, m, c, k, a, bd;

3 Assignmen#1Sol.nb 3 In[10]:= equa3 := 8m ''@D + c '@D + == == a, '@0D == b<; We will use equa3 as an argumen of DSolve and NDSolve. Now we specify he parameer values. In[11]:= m = 2; c = 0.8; k = 8; a = 1; b = 2; (a) We solve wih DSolve. In[12]:= Ou[12]= ans3 = D 99@D Ø -0.2 H1. Cos@ D Sin@ DL== We conver he replacemen rule o a funcion named osc[]. In[13]:= Ou[13]= osc@_d ê. Flaen@ans3D -0.2 H1. Cos@ D Sin@ DL (b) We check firs he iniial condiions and hen he equaion. In[14]:= osc@0d Ou[14]= 1. In[15]:= osc'@0d Ou[15]= 2. In[16]:= m osc''@d + c osc'@d + k osc@d Ou[16]= H1. Cos@ D Sin@ DL I -0.2 H2.2 Cos@ D Sin@ DL H1. Cos@ D Sin@ DLM + 2 I -0.2 H-3.96 Cos@ D Sin@ DL H2.2 Cos@ D Sin@ DL H1. Cos@ D Sin@ DLM In[17]:= Simplify@%D Ou[17]= 0. Iniial condiions and equaion boh check. (c) We apply he plo command o osc[]. We name he graph so ha we can refer o i laer when we combine i wih anoher graph.

4 Assignmen#1Sol.nb 4 In[18]:= graph = Plo@osc@D, 8, 0, 8<, AesLabel Ø 8"", ""<D Ou[18]= (d) We now apply Plo o osc'[], again naming he graph. In[19]:= graphv = 8, 0, 8<, AesLabel Ø 8"", "v"<d 2 v 1 Ou[19]= -1-2 (e) We combine he graphs wih a Show command.

5 Assignmen#1Sol.nb 5 In[20]:= Show@graph, graphv, PloRange -> 8-2.5, 2.5<D 2 1 Ou[20]= -1-2 We can inser a more appropriae label on he verical ais. In[21]:= Show@graph, graphv, PloRange -> 8-2.5, 2.5<, AesLabel Ø 8"", " and v"<d and v 2 1 Ou[21]= -1-2 (f) Now we will use NDSolve o ge he soluion. We name he soluion (i.e., he inerpolaing funcion oupu) oscnum. In[22]:= oscnum = 8, 0, 8<D Ou[22]= 88@D Ø InerpolaingFuncion@880., 8.<<, <>D@D<< We conver his inerpolaing funcion oupu o an ordinary funcion, oscou[].

6 Assignmen#1Sol.nb 6 In[23]:= Ou[23]= osou@_d ê. Flaen@oscnumD InerpolaingFuncion@880., 8.<<, <>D@D Now we plo his funcion. In[24]:= graphnum = Plo@osou@D, 8, 0, 8<, AesLabel Ø 8"", ""<, PloLabel Ø "Numerical Soluion"D Numerical Soluion 1.0 Ou[24]= This is he same as our earlier graph, ecep for he newly added plo label. (g) We conver his equaion o a sysem of wo firs order equaions, by inroducing v = d/d. The resul is We define his equaion for Mahemaica. d d = v, dv d = - c m v - k m, wih H0L = a, v H0L = b. In[25]:= sysequa3 = 8'@D == v@d, v'@d == -Hc ê ml v@d - Hk == a, v@0d == b<; Now we use his as one of he argumens of NDSolve. In[26]:= sysnum = NDSolve@sysequa3, 8@D, v@d<, 8, 0, 8<D Ou[26]= 88@D Ø InerpolaingFuncion@880., 8.<<, <>D@D, v@d Ø InerpolaingFuncion@880., 8.<<, <>D@D<< We conver he inerpolaing funcion oupu o an ordinary funcion. We call he wo componens of he ordinary funcion osc[] and vosc[].

7 Assignmen#1Sol.nb 7 In[27]:= Ou[27]= 8osc@_D, vosc@_d< = 8@D, v@d< ê. Flaen@sysnumD 8InerpolaingFuncion@880., 8.<<, <>D@D, InerpolaingFuncion@880., 8.<<, <>D@D< We plo osc[]. In[28]:= graphsysosc = Plo@osc@D, 8, 0, 8<, AesLabel Ø 8"", ""<, PloLabel Ø "Numerical Soluion of Sysem"D Numerical Soluion of Sysem 1.0 Ou[28]= PROBLEM 4 We define he mari for Mahemaica, and display i using MariForm. In[29]:= A = 881, -2, 3, 0<, 8-2, 0, 3, 2<, 83, 3, 5, -6<, 80, 2, -6, 1<< Ou[29]= 881, -2, 3, 0<, 8-2, 0, 3, 2<, 83, 3, 5, -6<, 80, 2, -6, 1<< In[30]:= MariForm@AD Ou[30]//MariForm= (a) We calculae he inverse, calling i Ainv, and hen check i by muliplying A and Ainv. In[31]:= Ainv = Inverse@AD Ou[31]= :: , , , : , , 8 121, >, : , , , >, >, : , , 6 121, >>

8 Assignmen#1Sol.nb 8 In[32]:= MariForm@A.AinvD Ou[32]//MariForm= To solve he linear equaions AX = b, we firs define b for Mahemaica, and hen calculae he soluion as Ainv.b. In[33]:= b = 81, 0, 1, 0< Ou[33]= 81, 0, 1, 0< In[34]:= X = Ainv.b Ou[34]= : , , , > We can conver he eac raional numbers o numerical values wih he N command. In[35]:= N@XD Ou[35]= , , , < Now we check he soluion, sill using he eac raional values. In[36]:= A.X - b Ou[36]= 80, 0, 0, 0< Because he enries of A were all inegers, Mahemaica did eac calculaions. By using he N command o conver ineger o real resuls, we can do he calculaions in erms of reals raher han inegers. In[37]:= Ainv = Inverse@N@ADD Ou[37]= , , , <, , , , <, , , , <, , , , << In[38]:= X = Ainv.b Ou[38]= , , , < (b) Because all of he enries of A are eac inegers, Mahemaica will aemp an eac calculaion of he eigenvalues and eigenvecors. Le's look a jus he eigenvalues firs. In[39]:= Eigenvalues@AD Ou[39]= 9RooA Ò1-51 Ò1 2-7 Ò1 3 + Ò1 4 &, 4E, RooA Ò1-51 Ò1 2-7 Ò1 3 + Ò1 4 &, 1E, RooA Ò1-51 Ò1 2-7 Ò1 3 + Ò1 4 &, 3E, RooA Ò1-51 Ò1 2-7 Ò1 3 + Ò1 4 &, 2E= A small lesson here: "eac" isn' always useful. In principle eac soluions of a quaric equaion are possible. In pracice i is no worh our ime o unravel he above answer. We force a numerical approach by using he N funcion which convers eac values o numerical values.

9 Assignmen#1Sol.nb 9 In[40]:= Eigenvalues@N@ADD Ou[40]= , , , < Now he eigenvecors. In[41]:= eigvecs = Eigenvecors@N@ADD Ou[41]= , , , <, , , , <, , , , <, , , , << We check he pairwise orhogonaliy. Firs we erac he four vecors from he above lis. In[42]:= eig1 = eigvecs@@1dd Ou[42]= , , , < In[43]:= eig2 = eigvecs@@2dd Ou[43]= , , , < In[44]:= eig3 = eigvecs@@3dd Ou[44]= , , , < In[45]:= eig4 = eigvecs@@4dd Ou[45]= , , , < Now we check he orhogonaliy. In[46]:= eig1.eig2 Ou[46]= µ In[47]:= eig1.eig3 Ou[47]= µ In[48]:= eig1.eig4 Ou[48]= µ In[49]:= eig2.eig3 Ou[49]= µ In[50]:= eig2.eig4 Ou[50]= µ In[51]:= eig3.eig4 Ou[51]= µ Thus hey are orhogonal o wihin he numerical accuracy of he calculaion.