Formula for calcutaing Impact force when using a compression spring?

Question

I need a formula to calculate impact force.

I am Using a compression spring, therefore the spring will be pushing. I have the distance of push (yes the spring stops short of impact) i have the spring rate in lbs. And i have the mass of the object it is pushing in grams.

Answer 1

Force on a spring while compressing it is F = k dx; where k is the spring constant in N/m and dx is the change in the spring's length measured from its uncompressed length.  For example, if k = 10 N/m and I push a 1 meter spring to compress it dx = .5 meters, then the force I am holding that spring down with is F = k dx = 10*.5 = 5 Newtons.

As an extra added bonus to this answer, the work I do in pushing that spring down .5 m is W = f dx, where f is the average force over the dx compression.  As the force is linearly proportional to the compression dx, the average force occurs mid way between dx > 0 and dx = 0.  Thus the average force f happens when dx/2 occurs.  That is, f = k dx/2 = F/2; so that W = f dx = k dx/2 dx = 1/2 k dx^2 = the work I did to compress the spring dx from its neutral position.

Now to your question...just as your source of impact (e.g., a lead weight) touches the end of the uncompressed spring, we have a force of impact of F = k dx = k*0 = 0.  There is no force of impact.  And the weight's total energy TE = 1/2 mV^2 all kinetic energy.  

But, when the weight compresses the spring to, say, dx > 0, some of the kinetic energy of the weight will have been converted to the spring's potential energy pe = 1/2 kdx^2.  So the weight's total energy TE = 1/2 mv^2 + 1/2 kdx^2; ke = 1/2 mv^2 and v < V is the reduced velocity as the weight slows down.  When the spring is fully compressed dX, the weight's total energy TE = PE = 1/2 kdX^2.  At this point ke = 0 because there is no velocity in the weight.

Recall the force on/by a spring is F = k dx.  Thus the impact force of the weight while compressing the srping is just F = 0 before engaging the spring.  And F = k dx while the spring is still compressing.  Finally F = k dX when the spring is fully compressed.

Using the conservation of energy, we can solve for dX, the maximum compression if we know the kinetic energy of the weight as it just hits the spring.  That is KE = 1/2 mV^2 = 1/2 k dX^2 = PE when the spring is fully compressed and the force of impact is F = k dX.   Solve for dX = sqrt(mV^2/k).  You can also note that V^2 = 2gh; where h is the height your weight (m) was dropped from and g = 9.81 m/sec^2.  So that dX = sqrt(2mgh/k) which is rather straightforward to solve.