Equation for Damping in an Ideal Mass-Spring System

[Auror]

In an ideal mass–spring system with mass m, spring constant k, and damping coefficient b,
restoring force,$\vec F_r=-kx\hat{x}$
and damping force,$\vec F_d=-bv\hat{v}$
When the oscillating object goes towards amplitude position from equilibrium position,direction of velocity and position($\displaystyle \vec x$) is same. For that , net force,$\vec F=\vec F_r+\vec F_d$ can be written as(just putting values)- $F=m\ddot x=-kx-bv...(1)$.
But when the oscillating object moves back from that amplitude position to equilibrium position,direction of velocity changes,but direction of position doesn't. So direction of damping force changes. Hence direction of restoring force and direction of damping force are opposite. So net force should be, $F=m\ddot x=-kx+bv...(2)$
While solving differential equation for damping- we used (1) only which gave the differential equation,
$m\ddot x+kx+b\dot x=0$
But what about the equation 2 which we didn't deduce but is valid simultaneously? Why doesn't change of direction of damping force count here?

 

[Ken G]

When the velocity changes direction, the sign of the damping term automatically switches. You don't need to change the form of the equation manually, the same form is needed in all cases, and the sign takes care of itself. So equation (2) is wrong, there is no need to change the form of equation (1), or the differential equation it results in, when the velocity changes sign. Put another way, when you switched your notation from vector notation to the one-dimensional form, you dropped the unit vectors, which is fine, but v is no longer a magnitude-- it now has a sign too. That's an important difference between the v-hat unit vector and the x-hat unit vector-- the latter always points in the same direction, but not the former. So when you write x times x-hat, the x has a sign, but when you write v times v-hat, the v does not have a sign, it is always just a magnitude. That difference means that in the one-dimensional form, both x and v have a sign, whereas only x did in the vector notation.

ETA: I see Orodruin has answered similarly, and note that in his interpretation, your x-hat always points in the direction of the displacement, it is not an x coordinate vector. That is indeed more consistent with what you wrote, though the notation can be ambiguous so it may be better to use r-hat than x-hat for that purpose. Anyway, if x-hat means the unit vector in the direction of the displacement, and not the x-direction unit vector, then both x and v are magnitudes, and neither has a sign in the vector notation, but both acquire signs in the one-dimensional version.

 

[Orodruin]

Put in another way, when you write

restoring force,$\vec F_r=-kx\hat{x}$
and damping force,$\vec F_d=-bv\hat{v}$

your $x$ and $v$ are the magnitudes of the displacement and velocity, respectively. You can write it like this and yes, if you do you need to account for the sign change manually. However, doing so is just introducing complication as you can just use the velocity and actual displacement instead which will lead you to an equation where you (a) do not need to worry about the sign change and (b) get more information out of.