the picture looks like this, carbon dioxide,system of 3 masses and 2 springs M k
ID: 2938847 • Letter: T
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the picture looks like this, carbon dioxide,system of 3 masses and 2 springs M k m k M 0-------o--------0 l--> l--> l---> x y z The given equations: Mx''= k(y-x) my"= -k(y-x)-k(y-z) Mz"= -k(z-y) x1= x-y x2= y-z x3= x1' x4= x2' Write the system of equations as a first ordersystem x'(vector)=Ax(vector) =k/m =k(1/M+1/m) so that the coefficient matrix A will be in termsof and Then find the fundamental frequencies and normalmodes of the system Hint: the characteristic equation is quartic,but it can be solved as a quadratic for the square of theeigenvalue. I will give out as many karma points as I can for help. thanks in advance! (I do not have a text bookfor this class..though it would be nice!) What i've got so far Mx''=-kx1 Mz''=kx2 my''=kx1-kx2 x2=y-z x2''=y''-z'' x1=x-y x1''=x''-y'' x1''=(-kx1/M)-(k/m)(x1-x2) x2''=(k/m)(x1-x2)-(k/M)x2 x1'=x3 x2'=x4 x3'=x1''=(-k/M)x1-(k/m)(x1-x2) x4'=x2''=(k/m)(x1-x2)-(k/M)x2 x'(vector)=AX(vector) I substitute and where applicable to get my Amatrix ( I apologize I do not know how to make a matrix onhere) 0 0 1 0 0 0 0 1 - 0 0 - 0 0 Then for the solution C(vector)et=AC(vector)et (A-I)C(vector)=0 Then I need to find the determinant of (A-I) = 0 Solve for , then find the modes of the system and thefundamental frequencies. What do I do to find the modes? the picture looks like this, carbon dioxide,system of 3 masses and 2 springs M k m k M 0-------o--------0 l--> l--> l---> x y z The given equations: Mx''= k(y-x) my"= -k(y-x)-k(y-z) Mz"= -k(z-y) x1= x-y x2= y-z x3= x1' x4= x2' Write the system of equations as a first ordersystem x'(vector)=Ax(vector) =k/m =k(1/M+1/m) so that the coefficient matrix A will be in termsof and Then find the fundamental frequencies and normalmodes of the system Hint: the characteristic equation is quartic,but it can be solved as a quadratic for the square of theeigenvalue. I will give out as many karma points as I can for help. thanks in advance! (I do not have a text bookfor this class..though it would be nice!) What i've got so far Mx''=-kx1 Mz''=kx2 my''=kx1-kx2 x2=y-z x2''=y''-z'' x1=x-y x1''=x''-y'' x1''=(-kx1/M)-(k/m)(x1-x2) x2''=(k/m)(x1-x2)-(k/M)x2 x1'=x3 x2'=x4 x3'=x1''=(-k/M)x1-(k/m)(x1-x2) x4'=x2''=(k/m)(x1-x2)-(k/M)x2 x'(vector)=AX(vector) I substitute and where applicable to get my Amatrix ( I apologize I do not know how to make a matrix onhere) 0 0 1 0 0 0 0 1 - 0 0 - 0 0 Then for the solution C(vector)et=AC(vector)et (A-I)C(vector)=0 Then I need to find the determinant of (A-I) = 0 Solve for , then find the modes of the system and thefundamental frequencies. What do I do to find the modes? x'(vector)=AX(vector) I substitute and where applicable to get my Amatrix ( I apologize I do not know how to make a matrix onhere) 0 0 1 0 0 0 0 1 - 0 0 - 0 0 Then for the solution C(vector)et=AC(vector)et (A-I)C(vector)=0 Then I need to find the determinant of (A-I) = 0 Solve for , then find the modes of the system and thefundamental frequencies. What do I do to find the modes?Explanation / Answer
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