# Mass-spring-damper system solve for x(t) using power series

• maciejj
In summary, the student attempted to solve a differential equation of motion involving power series, but did not get the coefficients a1, a2, a3, ... correct.
maciejj

## Homework Statement

A mass of 10kg is suspended from vertical spring of stiffens 100N/m and is provided with dashpot damper having damping coefficient of 1000Ns/m.
The mass is pulled down the distance of 4cm from its equilibrium position and than released.
Establish the differential equation of motion and solve using power series for variation in displacement with time.

## The Attempt at a Solution

Started with
mx''(t)=-kx(t)-cx'(t)
and initial conditions
t0=0,x(t0)=4cm=0.04m, x'(t0)=0,x''(t0)=0 here assume that velocity and acceleration=0
at t=0
after solving using power series got:
x(t)=a0+a1x+a2x2+a3x3+... a0=0.04 but all the rest of coefficients =0

its not 0 for the rest for coefficient , u just have to treat the equation as some function and take its derivative, and since u know x' and x u know x'' by the equation , taking the derivative will get u the equation consist of x''' , x'' and x' and x'', x' have already been known

mx^''=-kx-cx'
x^''+c/m x^'+k/m x=0
Putting known values into equation:
x^''+100/10 x^'+1000/10 x=0
x^''+10x^'+100x=0
Now solving using power series:
Let assume that:
x(t)=a_0+a_1 t+a_2 t^2+a_3 t^3+..
t_0=0,x_0=0.04 so a_0=0.04 From initial conditions

x^' (t)=a_1+〖2a〗_2 t+〖3a〗_3 t^2+〖4a〗_4 t^3+..
t_0=0,〖x'〗_0=0 so a_1=0 From initial conditions

x^'' (t)=〖2a〗_2+〖6a〗_3 t+〖12a〗_4 t^2+20a_5 t^3+..
t_0=0,〖x''〗_0=0 so a_2=0 From initial conditions
Putting these into differential equation
x^''+10x^'+100x=0
〖(2a〗_2+〖6a〗_3 t+〖12a〗_4 t^2+20a_5 t^3+..)+10(a_1+〖2a〗_2 t+〖3a〗_3 t^2+〖4a〗_4 t^3+..)+
+100(a_0+a_1 t+a_2 t^2+a_3 t^3+..)=0

Rearranging
100a_0+10a_1+2a_2+(100a_1+20a_2+6a_3 )t+(100a_2+30a_3+12a_4 ) t^2+
+(100a_3+40a_4+20a_5 ) t^3=0
And a_0=0.04 ,a_1=0,a_2=0 -From initial conditions
0.04+(100a_1+20a_2+6a_3 )t+(100a_2+30a_3+12a_4 ) t^2+(100a_3+40a_4+20a_5 ) t^3=0

this is where I m now
but don't konw how to get rest of coefficents

Are you trying to solve this by power series for some reason? Although that should work, it is certainly the hard way to do a constant coefficient DE.

hi ,yes its one of task for my math assignment and its specified to solve it using power series

actually x''(0) is not zero since the object is at the amplitude it should have maximum accelaration toward equilibrium point

maciejj said:

## Homework Statement

A mass of 10kg is suspended from vertical spring of stiffens 100N/m and is provided with dashpot damper having damping coefficient of 1000Ns/m.
The mass is pulled down the distance of 4cm from its equilibrium position and than released.
Establish the differential equation of motion and solve using power series for variation in displacement with time.

## The Attempt at a Solution

Started with
mx''(t)=-kx(t)-cx'(t)
and initial conditions
t0=0,x(t0)=4cm=0.04m, x'(t0)=0,x''(t0)=0 here assume that velocity and acceleration=0
at t=0
after solving using power series got:
x(t)=a0+a1x+a2x2+a3x3+... a0=0.04 but all the rest of coefficients =0

Show your work. Without some indication of where you went wrong, it would be impossible to assist you.

RGV

Hi ,have uploaded a pdf file showing what I have done.
So what initial conditions would you use?

#### Attachments

• q5.pdf
136.3 KB · Views: 301
maciejj said:
Hi ,have uploaded a pdf file showing what I have done.
So what initial conditions would you use?

People keep telling you that x''(0) is nonzero, but you don't listen. The DE x'' = -10 x' - 100x gives x''(0) = -100*4/100 = -4.

That will give you an equation of the form C1*t + C2*t^2 + C3*t^3 + ... = 0, where the C1, C2, C3,... are linear combinations of your a1, a2, a3, ... . The equation is supposed to be an identity in t, so that means that all the coefficients must vanish; that is, we must have C1 = 0, C2 = 0, C3 = 0, ... . Solving those will give you the coefficients a1, a2, a3, ... in the expansion of x(t).

RGV

have changed x''(0)
is that correct now?

#### Attachments

• q5.pdf
182.8 KB · Views: 278

## 1. What is a mass-spring-damper system?

A mass-spring-damper system is a mechanical system that consists of a mass connected to a spring and a damper (or shock absorber). The mass is free to move in one dimension, and the spring and damper provide resistance to this movement. This system is commonly used in engineering and physics to model and analyze various physical phenomena.

## 2. How do you solve for x(t) in a mass-spring-damper system?

To solve for x(t) in a mass-spring-damper system, we use the equation of motion, which is a second-order linear differential equation. This equation can be solved using various methods, such as the power series method, Laplace transform, or numerical methods.

## 3. What is the power series method?

The power series method is a mathematical technique used to solve differential equations by expressing the solution as a series of polynomial terms. In the context of a mass-spring-damper system, we can use this method to find an approximate solution for x(t) by expanding the solution in terms of powers of time.

## 4. What are the advantages of using the power series method to solve for x(t)?

One advantage of using the power series method is that it allows us to find an approximate solution for x(t) without having to solve the differential equation directly. This can save time and effort, especially when the equation is complex. Additionally, the power series method can provide a more accurate solution compared to other approximation methods.

## 5. Are there any limitations to using the power series method?

Yes, there are some limitations to using the power series method. This method is only applicable to linear differential equations, and the accuracy of the solution depends on the number of terms included in the series. Additionally, the convergence of the series may be affected by the behavior of the solution at certain points, such as singularities or discontinuities.

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