What does pre-damped mean in the pre-damped switched-inductor piezoelectric energy-harvesting charger
Simple answer:

It means adding electrical alternating LC tuned DC switched voltage capacitance to a mechanical induced capacitor (piezo) oscillating square wave current pump that generates only 33uW and boost it > 700% in power.

A really long answer

This is an electro-mechanical dynamic impedance matching scheme. It boosts the energy harvesting from 33uW by 780% from gains in efficent energy conversion with 1 Ω ESR at V&lt;20V. THe voltage gain &lt;10 is stored energy but the impedance matching improves coupling efficiency. This matching is performed by the switched partly resonant LC capacitive DC voltage applied returned as current from the end-stop cantelever "jerk" on the crystal. (Jerk is a unit of measure for rate of change of acceleration)

REF

This PZT piezo-electric energy harvesting chip that is a switched charge current source in a vibration mode with end stops that applies small square waves.  By synchronously applying a pre-charge voltage, the PZT can dump  more power thru the action of stepped current.

Essentially it means a precharged capacitance voltage is applied to the piezo cantelever transducer so that greater mechanical piezo energy transfer efficiency can be performed.  This is based on matching the impedance of a switched discrete capacitance voltage ( and some inductive current) to a mechanically excited alternating Norton current source and thus mechanical energy is absorbed by the dwitched electrical capacitance.

When this occurs,  pre-charging the crystal C(pz) to some applied Vpd(pre-damp v) between half cycles allows both end stops of the cantelever mounting to have higher DC voltage charges switched.  In other words Vpd raises the damping force against which motion supplies power, which in turn allows more energy to be harvested.

Basically it is improving energy transfer by matching the impedance of the mechanical alternating step force with an electrical alternating step load impedance to maximize the rms P(pzt) = current * square wave voltage by synchronous impedance matching to a square wave mechanical-to electrical current source.

So a switched Vpd cap load collects more energy from a "pico-bang-bang" ( my terminology) alternating cantelever PZT current source.  THis must be matched and optimized as component ESR*I^2 losses rise exponentially with gains from pre-dampening voltages applied.

What I'm not sure about yet is the optimal LC resonant frequency vs the mechanical vibration range of frequencies and the ratio of Cpc/Cpz.  THe other thing not stated is how this affects the longevity or reliablity of the crystal. 

Anecdotal

As I know from previous car experience, if replacement shock absorbers are too stiff, the McPherson strut popped thru the hood (bonnet) of my Audi CD5000 Turbo on a washboard road.  Exceptionally "smooth" but self-destructive.