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I am done doing my homework

This sounds odd to me and I don't understand what "doing my homework" is in this sentence.

I don't understand how the phrase "I am done" (I assume "done is an adjective here, passive voice doesn't make sense to me) can be followed by this -ing phrase. Sentences like "I finished doing my homework." are easily understood and the gerund could be replaced with a noun ("I finished my homework.") This is a different kind of construction of course. But instead of "to be done doing" I would say "I have done my work" or if I want to keep the "doing" maybe "I am done with doing my homework." Otherwise I find it unintelligible.

Also similar constructions use prepositions too for example "I am tired of being alone" "I am fed up with being single" this one does not.

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It's idiomatic. Either of these two forms are commonly used:

Jimmy, what are you up to?

I'm doing my homework. mom.

...

Mom, I'm done doing my homework!

or

Mom, I'm done with my homework!

In the first case, "doing my homework" is the task. In the second case, "homework" is the task. It depends on whether you regard 'homework' as a concrete noun - a collection of papers to be worked with in some way - or as an abstract noun - an assignment to be accomplished.

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Done and doing close together might be confusing, but each has a grammatical role in this sentence.

  1. Done to express completion

    I am done verb-ing

    As you wrote, done here is an adjective indicating the completion of an activity. You appear to understand this already, so I won't spend too much time explaining its meaning, but in terms of grammar the key is that done can take a following -ing clause as a complement.

  2. Do as a light verb

    doing my homework

    In this clause, we have the light verb do, which adds very little meaning of its own. Instead, the bulk of the meaning comes from the noun homework (which is semantically "heavy").

    In English, we sometimes use light verbs with nouns like homework because they have no verb form: *homeworking is not an established English verb, so it's not an option. There are other times when both light (take a shower) and heavy verbs are possible (shower), but this is not one of them.

    This is an -ing clause, so the adjective done can take it as a complement. (These are called gerund-participial clauses by Huddleston and Pullum; they do not distinguish gerund clauses from participle clauses, and we don't need to do so here either.)

Put them together, and you get your sentence:

I'm done doing my homework.

Doing my homework is the activity, and done expresses completion of that activity, taking doing my homework as a complement. It may be confusing to you at first to hear two different forms of do next to each other, but each has its place, and this is a perfectly natural English sentence.

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"I'm done," or "I'm finished," means that you are completely exhausted, spent, and no longer able to continue in the task at hand.

Finally! I'm done with that.

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