How Is the Solution to the Vibration Equation Derived?

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In summary: Not sure whether it will talk about that integral, but certainly about equations of this form. A standard textbook about differential equations may cover this.
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How to solve this vibration equation?
[tex] \ddot{x} + \omega_{0}^2 x = {e \over m} E_{x} [/tex]
Hello,everyone:
I got an equation:
[tex]
\ddot{x} + \omega_{0}^2 x = {e \over m} E_{x}

[/tex]
I know the solution is:
[tex]
x(t) = {e \over {\omega_{0} m}} \int_{0}^{t} E_{x}(\xi) \sin{ \omega_{0} } (t - \xi) d \xi \\
[/tex]
[tex] x(0) = \dot{x} (0) = 0[/tex]

My attempt to verify this solution is by using formula:
[tex]
{d \over dx } \int_{\alpha (x)}^{\beta (x)} f(x,y)dy
= \int_{\alpha (x)}^{\beta (x)} { {\partial f(x,y)} \over {\partial x}} dy
+ f[x, \beta (x)] \beta^{\prime}(x)
- f[x, \alpha (x)] \alpha^{\prime}(x)

[/tex]
But I don't know how the physicist obtained it. Can anyone give an answer? Thanks.
 
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  • #2
We don't give answers. You need to show what attempt you made to solve it and I don't see that in your post. We in turn will give you hints or tell you where you went wrong. This is how our site works.

We also have a template to fill out for homework type questions when posted into one of our homework forums.

1) Homework statement

2) Relevant equations

3) Attempt at a solution
 
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  • #3
jedishrfu said:
We don't give answers. You need to show what attempt you made to solve it and I don't see that in your post. This is how our site works.
Thanks. Is there any book talking about equations of this form?
 
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thaiqi said:
Summary:: How to solve this vibration equation?
[tex] \ddot{x} + \omega_{0}^2 x = {e \over m} E_{x} [/tex]

ut I don't know how the physicist obtained it. Can anyone give an answer? Thanks.
You basically have an equation of the form ##x''(t)+cx(t)=d## which is a simple second order differential equation. There are a lot of techniques to solve differential equations in general, but this one is especially simple and occurs very often in physics. You can e.g. set ##x(t))=a \sin(\alpha t)+b\cos(\beta t)## and solve the equation system which results from differentiating. ##x'' \sim x ## as given, already looks like sine and cosine, since those functions "reproduce" themselves by differentiation twice.
 
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fresh_42 said:
You basically have an equation of the form ##x''(t)+cx(t)=d## which is a simple second order differential equation. There are a lot of techniques to solve differential equations in general, but this one is especially simple and occurs very often in physics. You can e.g. set ##x(t))=a \sin(\alpha t)+b\cos(\beta t)## and solve the equation system which results from differentiating. ##x'' \sim x ## as given, already looks like sine and cosine, since those functions "reproduce" themselves by differentiation twice.
Thanks first. It is rather of ##x''(t)+cx(t)=d(t)## type.
Second, I learned from some articles that this can be solved using what is called "Duhamel integral", will a common book on ordinary differential equation talk about that? If not, what book is suitable?
 
  • #9
thaiqi said:
... will a common book on ordinary differential equation talk about that? If not, what book is suitable?
Not sure whether it will talk about that integral, but certainly about equations of this form. A standard textbook about differential equations is a good recommendation for any physicist. However, I cannot name a special one.
 
  • #10
Some folks use Arfken and Weber Mathematical Methods for Physicists or Boaz.

Theres also Nearings book available free online.

http://www.physics.miami.edu/~nearing/mathmethods/

not sure if duhamel is mentioned though. You could search on diff equation books duhamel to find one since google indexed so many so far.
 
  • #11
You are starting with a second order differential equation.
That rule you wrote involving the integral below is called Leibniz's rule. It is in the mathematical handbook by Schaums. The physicist (probably) attained it when (s)he was taking Calculus or perhaps advanced calculus classes. The rule (and even the method of solution) is also found in many books on modern control theory in engineering. Electrical and Aerospace engineers are well acquainted with these types of problems; but allow me to be of even more help.
Your starting point is a second order differential equation, and Leibniz rule involves an (i.e. one) integral. In order to use it you should express your second order differential equation as a first order differential equation involving matrices. Then you would obtain the state transition matrix. etc.
Your best bet would be to read relevant sections in Modern Control theory such as Modern Control Theory by Brogan. If you are a physicist, this book (and others in this area), are likely unfamiliar to you, and I do not know of the references to help.
 
  • #12
Thanks for you all.
 
  • #13
Sorry for the kinda stupid qustion, but is [tex] E_x [/tex] a function E differentiated with respect to x, or is [tex] E_x [/tex] just a constant?
 
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Looking at the form of the equation, it involves time t, so Laplace transformation with respect to time t would be helpful. Looking at the form of the solution, it is the Laplace transformation of a convolution. So Laplace transformation is the solution.
 
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  • #15
Laplace transform is surely one way to go. Also (maybe not sure) Fourier transform. But i think it can be solved also with the Green's function method because the ODE is of the form $$Lf(t)=s(t)$$ where L a linear differential operator and s(t) a source function.
 

Related to How Is the Solution to the Vibration Equation Derived?

1. What is a vibration equation?

A vibration equation is a mathematical representation of the motion of a vibrating system, such as a spring or a pendulum. It describes the relationship between the forces acting on the system and the resulting displacement or acceleration.

2. How do I solve a vibration equation?

The first step in solving a vibration equation is to identify the type of vibration, such as free or forced, and the type of system, such as single or multi-degree of freedom. Then, you can use various techniques such as the method of undetermined coefficients or the Laplace transform to solve the equation and determine the system's response.

3. What are the common assumptions made when solving a vibration equation?

Some common assumptions made when solving a vibration equation include small displacements, linear behavior of the system, and negligible damping. These assumptions simplify the equation and make it easier to solve, but they may not always accurately represent real-world systems.

4. How do I determine the natural frequency of a vibrating system?

The natural frequency of a vibrating system can be determined by solving the vibration equation for the system's eigenvalues, which represent the frequencies at which the system will naturally vibrate. The natural frequency is a characteristic of the system and is dependent on its mass, stiffness, and damping.

5. What are some practical applications of solving vibration equations?

Solving vibration equations is essential in many engineering fields, such as mechanical, civil, and aerospace engineering. It is used to analyze and design structures and machines, such as buildings, bridges, and vehicles, to ensure they can withstand vibrations and operate efficiently and safely.

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