Solving for time in damped fall equation with exponential and logarithm

[SystemTheory]

This equation specifies displacement x in terms of motion time t (starting from rest).

$$x = \tau g\left(t + \tau e^{-t/\tau} -\tau\right)$$

where tau = m/b is the system time constant of a mass m suspended from a mechanical braking device with rectilinear damping constant b and g is standard gravity.

Can anyone help me find t as a function of displacement x? I'm having trouble with the logarithm algebra. I'm new to the forum, so any tips are appreciated.

 

[arildno]

This is a trancendental equation for "t", you won't be able to find an exact solution.

 

[SystemTheory]

Thanks for that help.

I'm trying to predict the time down a ramp of known length xL as shown in the diagram. I'd like a closed form approximation to write in a code statement. This will set the transient analysis time based on the input data inside a computer simulation.

The differential equation solved for acceleration looks like this:

$$\frac{dv}{dt} = \frac{mg sin\theta - bv}{m}$$

Any ideas appreciated.

 

[SystemTheory]

Bump. Cross-reference to related thread on the differential equation (DE) section.

https://www.physicsforums.com/showthread.php?t=359199