33-231 Physical Analysis Fall 2003 Problems: Set 11 (due Wednesday, November 12, 2003) Note: In Problems 43 through 45, all matrix calculations should be done two ways. First do them by hand, then check them using Maple. In each problem, some intermediate results are given as guideposts, to help you stay on track. 43. This problem concerns the same mechanical system as Problem 38. Use the same coordinates as in Problem 38. a) Obtain the M and K matrices, as discussed in class, and write the equations of motion in matrix form, i.e., in the form MxKx&&.=− M K=FHGIKJ=−−FHGIKJ3 00 22mmk kk k, b) Form the matrix K M−λ, obtain the secular determinant K M− λ , set it equal to zero, and solve the resulting equation to find the possible values of λ, i.e., the eigenvalues of the problem. You may want to number them consistently with your work in Problem 38. (I recommend putting them in ascending order, with the smallest one first.) Compare your values of λ with your results from Problem 38. λ ω λ ω1 122 226= = = =kmkm, . c) For each eigenvalue find the corresponding eigenvector. Let a be the eigenvector corresponding to λ1, and let b be the eigenvector corresponding to λ2. Thus show that a and b satisfy the equations (K − λ1M)a = 0 and (K −λ2M)b = 0. a b=FHGIKJ=−FHGIKJ1 1132, . d) Form a matrix A with the eigenvectors a and b as columns and express the normal-coordinate transformation as a matrix equation, in the form x = Aq. Obtain expressions for x1 and x2 in terms of q1 and q2, and compare with your results from Problem 38. Also obtain the inverse transformation q = A−1x. A =−FHGIKJ=+= −1 1132321 1 22 1 2,,.x q qx q q (continued)Problems: Set 11 (page 2) Fall 2003 43. (continued) e) Express the equations of motion (in matrix form) in terms of the normal coordinates, and show that the differential equations for the q's can be separated into an equation containing only q1 and its derivatives, and an equation containing only q2 and its derivatives. M xKxMAqKAq&&,&&,=−=− 3 3233 2221 2 1 21 2 1 2mq mqkq kqmq mqkq kq&& && ,&& &&.+ = − −− = − + (Add and subtract, with appropriate multipliers, to obtain uncoupled equations.) f) Determine the normal-mode frequencies from the separated equations for q1 and q2, and compare your results with those from part (b). 44. Repeat the calculations of Problem 43, but using a different coordinate system. Let x1 be the coordinate of mass 3m in an inertial frame of reference, as before, and let x2 be the displacement of mass 2m relative to mass 3m, i.e., the elongation of the right-hand spring. Following are guideposts corresponding to those in Problem 43. a) M K=FHGIKJ=−FHGIKJ3 02 2 0mm mk kk, . b) λ λ1 26= =kmkm, . c) a b=FHGIKJ=−FHGIKJ2112, . d) A =−FHGIKJ2 11 2, x1 = 2q1 + q2, x2 = q1 – 2q2 e) M xKxMAqKAq&&,&&,=−=− 6 3 36 2 21 2 1 21 2 1 2mq mq kq kqmq mq kq kq&& &&,&& && .+ = − −− = − +Problems: Set 11 (page 3) Fall 2003 45. Carry out the same calculations as for Problems 43 and 44, for the system of Problem 39. Following are guideposts as before. a) M K=FHGGIKJJ=FHGGIKJJ=−− −−FHGGIKJJ=−− −−FHGGIKJJmmmmk kk k kk kk0 00 00 01 0 00 1 00 0 10201 1 01 2 10 1 1, . b) K M− =− −− − −− −= − − − −λλλλλ λ λk m kk k m kk k mk m k m k k m0202 222b g b g b g. λ λ λ1 2 303= = =, , .kmkm c) For λ1: ( ) , .K M a a− =−− −−FHGGIKJJFHGGIKJJ= =FHGGIKJJλk kk k kk kaaa0200111123 For λ2, ( ) , .K M b b− =−− −−FHGGIKJJFHGGIKJJ= =−FHGGIKJJλ0 00 00101123kk k kkbbb For λ3, ( ) , .K M c c− =− −− − −− −FHGGIKJJFHGGIKJJ= = −FHGGIKJJλ2 00 20121123k kk k kk kccc d), ,.,A = −−FHGGIKJJ= + += −= − +1 1 11 0 21 1 121 1 2 32 1 33 1 2 3x q q qx q qx q q q e) M xKxMAqKAq&&,&&,=−=− Equations for q1, q2, and q3 are the same as for Problem 39(c). By adding and subtracting multiples of these equations, separated equations for the q's can be obtained, just as in Problem