Equations of motion of damped oscillations due to kinetic friction

[phantomvommand]

Take rightwards as positive.
There are 2 equations of motion, depending on whether $\frac {dx} {dt}$ is positive or not.
The 2 equations are:
$m\ddot x = -kx \pm \mu mg$

My questions about this system:
Is this SHM?

Possible method to solve for equation of motion:
- Solve the 2nd ODE, although “joining” the equations when $\dot x$ changes from positive to negative is not easy.

 

[anuttarasammyak]

We may write the equation of motion as
$$m\ddot{x}=-kx- sgn(\dot{x})\mu mg$$
where sgn(a)=1 for a>0, 0 for a=0, -1 for a<0.

 

[phantomvommand]

We may write the equation of motion as
$$m\ddot{x}=-kx- sgn(\dot{x})\mu mg$$
where sgn(a)=1 for a>0, 0 for a=0, -1 for a<0.

Thanks for this; do you then go on to solve the 2nd ODE?
Also, is this SHM?

 

[anuttarasammyak]

I observe the equation is a non linear one and do not expect to find general solution easily.

 

[phantomvommand]

I observe the equation is a non linear one and do not expect to find general solution easily.

Is it just the sum of a particular function and complementary function?

 

[anuttarasammyak]

Say k=0 and $\dot{x}_0>0$ we get familiar relation of
$$\dot{x}=-\mu g t + \dot{x}_0$$
$$x=-\frac{1}{2}\mu g t^2 + \dot{x}_0 t+x_0$$
for 0<t<$\frac{\dot{x}_0}{\mu g}$, x= $\frac{\dot{x}_0^2}{2\mu g}+x_0$ for t beyond.

For k$\neq$0 similarly you can solve the equation until when $\dot{x}=0$. Then for time beyond it change sign of friction term until next time of $\dot{x}=0$ and so on.

 

[Dale]

Summary:: A spring has 1 end fixed to the wall, and the other end is connected to a block. Find the equation of motion of the block, given that it experiences only the spring force and a friction force = $\mu mg$.

Is this SHM?

No. SHM does not have anything like the friction force.