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The phrase "after some experimentation" is correct and usable in written English.
It can be used when discussing the process of trying out different methods or approaches to achieve a desired outcome.
Example: "After some experimentation, we found the optimal solution to the problem."
Alternatives: "following some trials" or "after conducting tests".
Exact(47)
After some experimentation, we determined that 6″ letters looked best.
By 1938, after some experimentation, he built one that worked.
After some experimentation we were able to determine that 10K resistors were the optimum choice.
After some experimentation the alphabet was reduced to 22 signs for consonants.
After some experimentation by my wife, she came up with a reasonable facsimile but it was only that, reasonable.
After some experimentation, he altered the color scheme of pattern No. D-1063 to suit his more psychedelic taste.
Similar(13)
But after some research and experimentation, I was already able to make it by my own and it was so fun, that I decided to create a page to share it all.
It's easier to imagine that, after years of experimentation in some rickety shack in the Loire Valley, a gifted farmer came up with this style of sauvignon blanc purely to match it with the local cheese, or vice-versa.
In particular, some guidelines of making effective use of computing resources are developed after extensive experimentation.
Prisoners of war were infected with several pathogens, including Y. pestis; >10,000 died as a result of experimental infection or execution after experimentation.
They areunable to board a bus except after fearful experimentation.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com