All arrows "flex" when shot from a bow. The trick is to find a happy medium between too much and too little flex for your set up. Flex will depend on how much energy the string is applying to the nock and how much resistance the arrow has to being moved. The amount of flex an arrow is manufactured with is referred to as "spine."
With modern compound bows you no longer need to worry about the arrows flexing around the shelf. This is because you center shot provides enough clearance for the arrow fletching to clear the shelf without flexing.
Your too main concerns are 1. Is the arrow spine within a tunable range, and 2. Is my arrow inline with the power stroke of the string of my bow.
Spine is usually fairly easy to determine using online charts and calculators. You might do some playing around with arrow lengths, point weight, nock weight, and stiff side of the arrow/nock tuning.
Aligning the power stroke is a little more difficult and optimally only includes yoke tuning or cam shimming. A last resort is moving your arrow rest left or right of the factory center shot. But once you have the power stroke set properly, then you will have no issue shooting a bare shaft arrow into a target at 20 yards and hitting perfectly as long you are not torqueing the bow.
So the flex we see in an arrow should be uniform, with the arrow slightly flexing back and forth until the spine of the arrow has recovered and the arrow begins to fly without flex. That is why alignment of the power stroke is so important.
If the power stroke is not aligned, then the arrow will start on an angled path. If you are shooting fixed broadheads then this angle will cause the broadhead to plane and result in your field points and broadheads have different points of impact. The broadhead blades acts like fletching on the front of the arrow and will cause it to plane in what ever direction the arrow was misaligned to. This is why screwing a mechanical broadhead onto an untuned bow is such a bad idea. You could end up hitting an animal with an angled arrow and lose a lot of penetration.
So in short, wobble or flex is normal. To determine if the wobble or flex is not effecting penetration, you only need to confirm that your fixed blade broadhead and field points are hitting exactly together at whatever distance you plan to hunt at. If you want to take it one step further you can confirm perfect arrow flight with bareshaft tuning.
IMPORTANT: Never re-sight in for a fixed broadhead to make up the difference between your field points and broadhead point of impact. The reason being is that if the arrow is misaligned, then it is like shooting a canted bow. So if you sighted in your bow at 30 yard pin for horizontal adjustment and then move back to 50 yards, you are going to be off left or right at 50 yards.
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