1. ## Mass-spring system

Vibration Free

$\displaystyle m \frac{d^2x}{dt^2} + kx = 0$

Where frequency is

$\displaystyle w = \sqrt{\frac{k}{m}}$

$\displaystyle \frac{d^2x}{dt^2} + \frac{k}{m}x = 0$

The characteristic equation is:

$\displaystyle r^2 + w^2 = 0$
$\displaystyle r = +or- iw$ where $\displaystyle i^2 = -1$

Then:
$\displaystyle x(t) = C_1e^{iwt} + C_2e^{-iwt}$

Calculating I can get
$\displaystyle x(t) = a_1 \cos(wt) + a_2 \sin(wt)$

Now, I need to do to get the following equation ?

$\displaystyle x(t) = Acos(wt - \delta)$ (I think this is the equation we need to get the free vibration)

2. Originally Posted by Apprentice123
Vibration Free

$\displaystyle m \frac{d^2x}{dt^2} + kx = 0$

Where frequency is

$\displaystyle w = \sqrt{\frac{k}{m}}$

$\displaystyle \frac{d^2x}{dt^2} + \frac{k}{m}x = 0$

The characteristic equation is:

$\displaystyle r^2 + w^2 = 0$
$\displaystyle r = +or- iw$ where $\displaystyle i^2 = -1$

Then:
$\displaystyle x(t) = C_1e^{iwt} + C_2e^{-iwt}$

Calculating I can get
$\displaystyle x(t) = a_1 \cos(wt) + a_2 \sin(wt)$

Now, I need to do to get the following equation ?

$\displaystyle x(t) = Acos(wt - \delta)$ (I think this is the equation we need to get the free vibration)
There are several threads in the trigonometry subforum that show how to go from $\displaystyle a_1 \cos(wt) + a_2 \sin(wt)$ to $\displaystyle Acos(wt - \delta)$. You will need to spend the time finding them. I'm certain that you will also be able to find the required calculation using Google.

3. Originally Posted by mr fantastic
There are several threads in the trigonometry subforum that show how to go from $\displaystyle a_1 \cos(wt) + a_2 \sin(wt)$ to $\displaystyle Acos(wt - \delta)$. You will need to spend the time finding them. I'm certain that you will also be able to find the required calculation using Google.

4. Originally Posted by Apprentice123
That is research for you to do, not me.

5. Originally Posted by Apprentice123
Vibration Free

$\displaystyle m \frac{d^2x}{dt^2} + kx = 0$

Where frequency is

$\displaystyle w = \sqrt{\frac{k}{m}}$

$\displaystyle \frac{d^2x}{dt^2} + \frac{k}{m}x = 0$

The characteristic equation is:

$\displaystyle r^2 + w^2 = 0$
$\displaystyle r = +or- iw$ where $\displaystyle i^2 = -1$

Then:
$\displaystyle x(t) = C_1e^{iwt} + C_2e^{-iwt}$

Calculating I can get
$\displaystyle x(t) = a_1 \cos(wt) + a_2 \sin(wt)$

Now, I need to do to get the following equation ?

$\displaystyle x(t) = Acos(wt - \delta)$ (I think this is the equation we need to get the free vibration)
There is nothing to do, just observe that these are just two different ways of writing an arbitary sinusoid of angular frequency $\displaystyle \omega$

(and by the way the angular frequency is usually represented by $\displaystyle \omega$ (omega) and the phase by $\displaystyle \varphi$ (phi))

CB

6. Originally Posted by CaptainBlack
There is nothing to do, just observe that these are just two different ways of writing an arbitary sinusoid of angular frequency $\displaystyle \omega$

(and by the way the angular frequency is usually represented by $\displaystyle \omega$ (omega) and the phase by $\displaystyle \varphi$ (phi))

CB

Thank you. A is a constant? What is $\displaystyle \delta$ ?

7. Originally Posted by Apprentice123
Thank you. A is a constant? What is $\displaystyle \delta$ ?
It is also a constant (it is the phase) Both $\displaystyle A$ and $\displaystyle \varphi$ (or $\displaystyle \delta$ if you prefer) are determined by the initial conditions.

What you have is a general solution to a second order ODE, and there should be two constants, these are $\displaystyle A$ and $\displaystyle \varphi$ in one form of the solution and $\displaystyle C_1$ and $\displaystyle C_2$ in the other form for the solution.

CB

8. Originally Posted by CaptainBlack
It is also a constant (it is the phase) Both $\displaystyle A$ and $\displaystyle \varphi$ (or $\displaystyle \delta$ if you prefer) are determined by the initial conditions.

What you have is a general solution to a second order ODE, and there should be two constants, these are $\displaystyle A$ and $\displaystyle \varphi$ in one form of the solution and $\displaystyle C_1$ and $\displaystyle C_2$ in the other form for the solution.

CB
The initial conditions

$\displaystyle x(0) = Acos(- \delta) = x_o$
$\displaystyle x'(0) = -Awsin(- \delta) = v_o$

I find $\displaystyle \delta = tan^{-1} \frac{v_o}{x_o w}$

How I find A ?

9. Look at $\displaystyle A^2cos^2(\delta)+ A^2sin^2(\delta)= A^2$.

10. Originally Posted by HallsofIvy
Look at $\displaystyle A^2cos^2(\delta)+ A^2sin^2(\delta)= A^2$.
Thanks