Is there any waste in energy by rectifying it as you've shown? I mean what happens to that other half of the wave?
I think a single diode can be better in some cases.
Basically you have a linear motor with a non-linear load which will result in poor efficiency with a prime mover being some external oscillating force and the load being some pulsed force and pulsed current
Energy is stored in the velocity of mass
or stored by compression in the spring
then stored into a charged capacitor
Series ESR during the pulsed current
*duty cycle in cap,diode and coil account for some losses.
So the question remains, how much stored energy can be easily generated?
Like an air pump action, you can get a lot more done with gravity and strong resistance in one direction only allowing potential energy to be created without the extra load.
Thus depending on the force required and direction one might choose a single diode to release in one direction and no diode to store energy in the spring with more velocity and longer swings rather than faster shorter strokes with resistance in both directions and higher frequency required to accumulate the same kinetic energy Em.
In order to maximize energy transfer from kinetic to spring to electromagnetic the impedance of each interface needs to be matched in some way that permits the squared term to rise. This is related to the mass of the object, moving part, spring constant, coil inductance, and permeance as well as reactive load and ESR. Impedance is related to the mechanical and electromotive forces.
Now go do the math and define the requirements first. (that's what this section is all about)