A dobradiça da porta ficou silenciosa depois de a Ana a limpar.

Because ficar often means to become or to end up being when it is followed by an adjective.

So:

• ficou silenciosa = became quiet / became silent

It is the pretérito perfeito form of ficar, so it refers to a completed change in the past.

Compare:

• A porta ficou aberta. = The door stayed / ended up open.
• Ele ficou triste. = He became sad.

Here, the idea is that the hinge changed from being noisy to being quiet.

Because silenciosa agrees with a dobradiça, which is:

• feminine
• singular

In Portuguese, adjectives usually agree in gender and number with the noun they describe.

So:

• a dobradiça → feminine singular
• silenciosa → feminine singular

If the noun were masculine, you would get silencioso instead.

Da is the contraction of de + a.

So:

• de = of
• a = the
• da = of the

A dobradiça da porta literally means the hinge of the door, which English normally expresses as the door hinge.

This contraction happens normally because here de is directly linked to the noun phrase a porta.

In European Portuguese, it is very common to use the definite article before personal names:

• a Ana
• o João
• a Maria

So a Ana simply means Ana, but in the normal European Portuguese way.

This is much more typical in Portugal than in many varieties of Brazilian Portuguese.

This is a very important point.

In depois de a Ana a limpar, the de belongs to the expression depois de, and a Ana is the subject of the infinitive clause.

So the structure is basically:

• depois de
  • [a Ana a limpar]

In standard Portuguese, when a preposition introduces an infinitive clause and the article belongs to the subject inside that clause, contraction is normally avoided.

So:

• de a Ana limpar ✓
• da Ana limpar ✗

This is similar to structures like:

• apesar de o João estar cansado
• not apesar do João estar cansado in careful standard usage

That second a is a direct object pronoun meaning it.

It refers back to a dobradiça.

So:

• a Ana = Ana
• a limpar = cleaning it

In other words:

• depois de a Ana a limpar = after Ana cleaned it

Because dobradiça is feminine singular, the direct object pronoun is a.

This is normal in European Portuguese with depois de + infinitive.

With prepositional infinitive structures like this, object pronouns are commonly placed before the infinitive:

• depois de o ver = after seeing him
• antes de a limpar = before cleaning it

So depois de a Ana a limpar follows a normal European Portuguese pattern.

Yes. Limpar is an infinitive here.

After depois de, Portuguese often uses an infinitive where English would use a full clause such as after Ana cleaned it.

Because the subject is expressed (a Ana), many grammarians describe this as an infinitive clause, and you may also hear the term personal infinitive.

In this sentence, the form looks the same as the ordinary infinitive because it is 3rd person singular.

You can see the difference more clearly in the plural:

• depois de os técnicos a limparem

There, limparem shows the personal infinitive ending.

Yes, absolutely.

That version is often easier for learners to understand:

• depois de a Ana limpar a dobradiça

It means the same thing, but instead of replacing a dobradiça with the pronoun a, it repeats the full noun.

Both are correct. The version with the pronoun is more compact because the hinge has already been mentioned.

Yes.

That uses the compound infinitive:

• ter limpado = to have cleaned

So:

• depois de a Ana a ter limpado = after Ana had cleaned it / after Ana having cleaned it

This version puts a bit more emphasis on the cleaning being completed before the hinge became quiet.

But the original sentence is also fine, because depois de already makes the time sequence clear.

There are two different parts:

Main clause

• A dobradiça da porta ficou silenciosa
• Subject: A dobradiça da porta
• Verb: ficou

Infinitive clause

• depois de a Ana a limpar
• Subject: a Ana
• Verb: limpar
• Object: a = the hinge

So the hinge is the thing that became quiet, and Ana is the person who cleaned it.