Amplitude VS frequency Vibration measurements made on many engines or reciprocating compressors have a "spike" at 2X rotating speed. That's the "secondary" vibration.
Of all the common engine configurations the inline 4 has the worst secondary vibrations by far. That has limited inline 4 cylinder use in "nice" cars to engines under 2 liters, unless the complexity of twin counter-rotating balance shafts spinning at 2X rpm are added (Mitubishi,Porsche, 1953 Diesel FORD tractor). Honda added some rubber mounts when their inline 4 motocycles went up to 900 cc.
As you mentioned the cause is the difference in piston motion between TDC and BDC. The rod length to stroke ratio is the cause. If the con rods were infinitely long the pistons' motion would purely sinusoidal, and their motion at TDC would be the same as BDC, and there would be no secondary vibration. (Opposed twins and 4s are much smoother because they match pairs up so they cancel each other). I visualize when a piston move a few degrees off TDC the crank and rod both get for-shortened, so the piston must "hurry" away from TDC. Conversely at BDC the rod and throw follow a similar arc, so the piston barely moves at all. For a pair of pistons This repeats twice per revolution.
The amplitude of the force is relative to the "reciprocating" weight, so light pistons help reduce it too. When VW went to 1800 cc in the GTI they lengthened the rods and lightened the pistons.
I used to work at a shop that did engine balancing. When I had just begun One of our customers was a speed shop of good local reputation. They built a Hot Rod Opel 4 cylinder engine for one of their customers. We had balanced the rods/pistons/crank/flywheel/clutch. It had been Bored and stroked. Bigger (heavier?) pistons. And if the original rods were used the rod/stroke ratio got shifted in the "wrong" direction. Years later I heard the engine was unusable because the vibration was so severe the intake manifold "shook right off it." Multiple (primary) balance re-checks. A law suit with damages paid. Only years Later did we get familiar with vibration analysis. Now I strongly suspect that the culprit was strong secondary vibration, maybe combined with a manifold whose resonant frequency was inrange. Volvo B-20E engines have a stout triangulated brace between intake and block.
After we had vibration analysis gear a fork lift truck distributor brought in a new fork lift to analyze. It was a new model with a new powerful 2 liter inline 4 engine from a Datsun car. Rigidly mounted in the frame. Operators said the power was nice, but were complaining the seat and controls buzzed un-bearably. Our vibation spectra showed strong 2X vibration. We rigged up some isolation mounts for the seat and floor board. We heard Eventually the factory re-did the truck design.
V-8s sometimes us "flat" cranks that effectively turn the engine into 2 inline 4s. The complaint is vibration. I suspect its at 2X.
