What speed v(t) enables constant power by solenoid pull?

[cairoliu]

Assuming the position x = 0 if no air gap, max stroke of plunger is L.
Bordery conditions: x(t=0) = L, x(t=T)= 0.
Given magnetic force F(x) = kx(t)^-2, here k is constant.
If uncontrolled, there is big crash noise afer solenoid actuated, so I wish it pull plunger at constant power.
Help me find the dynamic position x(t), then I can get velocity.
I know Wolfram software may resolve it, but I have no money to buy it.

 

[jack action]

Bordery conditions: x(t=0) = L, x(t=T)= 0.
Given magnetic force F(x) = kx(t)^-2, here k is constant.
If uncontrolled, there is big crash noise afer solenoid actuated, so I wish it pull plunger at constant power.

If x(t=T)= 0, then F(x(t=T)) = kx(t=T)^-2 = k(0)^-2 = $\infty$. How do you expect not having a big crash noise at t=T ?

 

[cairoliu]

If x(t=T)= 0, then F(x(t=T)) = kx(t=T)^-2 = k(0)^-2 = $\infty$. How do you expect not having a big crash noise at t=T ?

Now that noise not good, I can reserve a minimal air gap 𝛿, then x(t=T)= 𝛿, and still ask for the solution of constant power.

 

[jack action]

How is constant power helping you? If you have power $P$ at the end of the stroke, then if $F = 0$, $v$ must be infinite, or if $v = 0$, then $F$ must be infinite. Any way you choose, you will have a lot of energy to disperse rapidly when the plunger suddenly stops.

To avoid the big crash noise, you would normally remove the energy - with a damping force - before it reaches the end. Which means power would gradually go to zero.

 

[hutchphd]

Or a spring, depending upon duration.