# Under damp ing spring, eigen, 2-nd order eqn as 2 1-st order in matrix form.

• Yura
In summary, the problem is to find the general solution for a mass-spring-damper system with a mass of 4, a spring constant of 1, and a damping constant of sqrt(7). There are two different methods proposed, but neither is certain to be correct. The issue is mainly with expressing the solution in the correct form.

#### Yura

M a s s - s p r i n g - d a m p e r - s y s t e m.
(sorry about the random apostrophe's in some of the writing, i don't want someone to just google and take my working ><. i don't know if the apostrophe thing works though...)

i have two different methods but i don't think either is right, but close. i don't think there is a anything wrong with my calculations, but my problem is more of the form i need to express my answer in.

i'll try to upload a copy of my handwritten version, to make it easier to read, later tonight

## Homework Statement

mx" = -kx - cx'
mass = m = 4
Spring constant = k = 1
damp ing constant = c

find the gen'eral solu'tion for the case where the damp'ing cons'tant c = sqrt(7), (under damp ing)

y(t) = Matrix( [y1(t)]
[y2(t)])

in REAL FORM. ( i think the previous line means i have to put it in mtrix form, as two first order e quations)

## The Attempt at a Solution

after subbing in the constants into the equation i had:
4x" = -x - sqrt(7)x'

in a previous section i found the matrix to be
A = ( [0 1]
[-1/4 -sqrt(7)/4] )

solving the equation by using the "auxillary equation" method
( 4*lambda^2 + sqrt(7)*lambda + 1 = 0 )
i found lambda (the eigen values) to be:
lambda1 = -sqrt(7)/8 +3i/8
lambda2 = -sqrt(7)/8 -3i/8

here's where my methods split.
->METHOD I

using the underamping equation:
calling little omega "w", and alhpa "a")
lambda1 = -a + wi
lambda2 = -a - wi
(im not sure what happens to the negative when i sub)
A and B are constants
y = A*cos(wt)*e^(-at) + (B*cos(wt)*e^(-at)

y = A*cos((3/8)t)*e^(-(sqrt(7)/8)t) + (B*cos((3/8)t)*e^(-(sqrt(7)/8)t)

this being my gene'ral equation but it needed to be in m'atrix form
i think i need to take that negative i mentioned before into account and use it in my equation so i have the matrix with one positive and one negative

so it would give me:
y = A*cos((3/8)t)*e^(-(sqrt(7)/8)t) + (B*cos((3/8)t)*e^(-(sqrt(7)/8)t)

method II

there's too much to write for the second method but:
instead of subbing into the underdamping equation, i sub into the general
y = (C1)*(X1)*e^(lambda1*t) + (C2)*(X2)*e^(lambda2*t)

where matrix X is the eigen vector, i found by using the "A" matrix that i stated at the top of my solution attempt: (A - lambda*I) where "I" is matrix identity for a 2x2 matrix.

***([ ] [ ]) matrix, each set of square brackets represents a new column, each comma separates a row****

X1 = , [(sqrt(7)/8) - 3i/8])
X2 = , [(sqrt(7)/8) + 3i/8])

lambda1 = -sqrt(7)/8 +3i/8
lambda2 = -sqrt(7)/8 -3i/8 (from previous info given)

i subbed all my new found values into the gener'al solution and got:
y = (C1)*(, [(sqrt(7)/8) - 3i/8])*e^(((sqrt(7)/8) - 3i/8)*t)
+ (C2)*(, [(sqrt(7)/8) + 3i/8])*e^((-(sqrt(7)/8) - 3i/8)*t)

with the " e's " and their indexes, i then used the rule:
e^(a+ib) = (e^a)(cos(b) + i sin(b))
and then multiplied them into the matrixes beside them, (as i am following a very similar example) making my equation very very messy and complicated.

after this is where i stopped because i became really confused. the first method is my own i worked out using the help of a textbook. the second is a method of a given example which i think needs to be solved the same to this one.

in the example I am using for method two, they only take the first term of the equation to use the "e" rule on, ... after this part, everything falles apart for me and i can't tell left from right...

*im so sorry there's so much to read, i shortened it as much as i could without making it too confusing (i hope)

any help would be really appreciated.
thankyou for taking the time to read this.

Yura said:
M a s s - s p r i n g - d a m p e r - s y s t e m.
(sorry about the random apostrophe's in some of the writing, i don't want someone to just google and take my working ><. i don't know if the apostrophe thing works though...)
You don't know if it is right or wrong but you are someone might "take it"?
How thoughtful of you. Wouldn't want someone else to turn in a wrong solution!

i have two different methods but i don't think either is right, but close. i don't think there is a anything wrong with my calculations, but my problem is more of the form i need to express my answer in.

i'll try to upload a copy of my handwritten version, to make it easier to read, later tonight

## Homework Statement

mx" = -kx - cx'
mass = m = 4
Spring constant = k = 1
damp ing constant = c

find the gen'eral solu'tion for the case where the damp'ing cons'tant c = sqrt(7), (under damp ing)

y(t) = Matrix( [y1(t)]
[y2(t)])

in REAL FORM. ( i think the previous line means i have to put it in mtrix form, as two first order e quations)

## The Attempt at a Solution

after subbing in the constants into the equation i had:
4x" = -x - sqrt(7)x'

in a previous section i found the matrix to be
A = ( [0 1]
[-1/4 -sqrt(7)/4] )

solving the equation by using the "auxillary equation" method
( 4*lambda^2 + sqrt(7)*lambda + 1 = 0 )
i found lambda (the eigen values) to be:
lambda1 = -sqrt(7)/8 +3i/8
lambda2 = -sqrt(7)/8 -3i/8

here's where my methods split.
->METHOD I

using the underamping equation:
calling little omega "w", and alhpa "a")
lambda1 = -a + wi
lambda2 = -a - wi
(im not sure what happens to the negative when i sub)
A and B are constants
y = A*cos(wt)*e^(-at) + (B*cos(wt)*e^(-at)

y = A*cos((3/8)t)*e^(-(sqrt(7)/8)t) + (B*cos((3/8)t)*e^(-(sqrt(7)/8)t)
?? You are aware the this is the same as y= Ccos((3/8)t)*e^(-(sqrt(7)/8)t) where C= A+ B aren't you? You need a "sin((3/8)t) also!
(Oh, and why is this "y" when your original equation was for "t"?)

this being my gene'ral equation but it needed to be in m'atrix form
i think i need to take that negative i mentioned before into account and use it in my equation so i have the matrix with one positive and one negative
One positive and one negative what?

so it would give me:
y = A*cos((3/8)t)*e^(-(sqrt(7)/8)t) + (B*cos((3/8)t)*e^(-(sqrt(7)/8)t)
Again, that is exactly the same as y= C*cos((3/8)t)*e^(-(sqrt(7)/8)t) with C= A+ B. You do NOT have two "independent" solutions.

method II

there's too much to write for the second method but:
instead of subbing into the underdamping equation, i sub into the general
y = (C1)*(X1)*e^(lambda1*t) + (C2)*(X2)*e^(lambda2*t)

where matrix X is the eigen vector, i found by using the "A" matrix that i stated at the top of my solution attempt: (A - lambda*I) where "I" is matrix identity for a 2x2 matrix.

***([ ] [ ]) matrix, each set of square brackets represents a new column, each comma separates a row****

X1 = , [(sqrt(7)/8) - 3i/8])
X2 = , [(sqrt(7)/8) + 3i/8])

lambda1 = -sqrt(7)/8 +3i/8
lambda2 = -sqrt(7)/8 -3i/8 (from previous info given)

i subbed all my new found values into the gener'al solution and got:
y = (C1)*(, [(sqrt(7)/8) - 3i/8])*e^(((sqrt(7)/8) - 3i/8)*t)
+ (C2)*(, [(sqrt(7)/8) + 3i/8])*e^((-(sqrt(7)/8) - 3i/8)*t)

with the " e's " and their indexes, i then used the rule:
e^(a+ib) = (e^a)(cos(b) + i sin(b))
and then multiplied them into the matrixes beside them, (as i am following a very similar example) making my equation very very messy and complicated.

after this is where i stopped because i became really confused. the first method is my own i worked out using the help of a textbook. the second is a method of a given example which i think needs to be solved the same to this one.

in the example I am using for method two, they only take the first term of the equation to use the "e" rule on, ... after this part, everything falles apart for me and i can't tell left from right...

*im so sorry there's so much to read, i shortened it as much as i could without making it too confusing (i hope)

any help would be really appreciated.
thankyou for taking the time to read this.