Equations of motion of damped oscillations due to kinetic friction

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The discussion focuses on the equations of motion for a damped oscillating system influenced by kinetic friction, represented as m\ddot{x} = -kx ± μmg. The equations vary based on the direction of velocity, with a suggestion to use the sign function to handle changes in motion direction. The participants question whether the system exhibits simple harmonic motion (SHM), concluding that it does not due to the presence of friction. Solutions to the second-order differential equation are explored, noting the complexity of finding a general solution. The overall consensus is that the system's behavior deviates from SHM due to the non-linear frictional force.
phantomvommand
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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.
 
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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.
 
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anuttarasammyak said:
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?
 
I observe the equation is a non linear one and do not expect to find general solution easily.
 
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anuttarasammyak said:
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?
 
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.
 
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phantomvommand said:
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.
 
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