The original copy of the U.S. Constitution is written on parchment and resides with the National Archives and Records Administration or NARA. It is in a [protective case] (https://www.archives.gov/preservation/special-projects) and is not taken out for amendments to be physically added.

No amendments have ever been physically appended to the handwritten copy of the Constitution that currently resides with NARA. The Bill of Rights, the first ten amendments are on a separate parchment on display alongside the Constitution and Declaration of Independence.

When a proposed amendment passes both houses of Congress the [enrolled copy] (https://en.m.wikipedia.org/wiki/Enrolled_bill) of the resolution goes to the Archivist of the United States, who heads NARA, and the administration's Office of the Federal Register or OFR for processing and to be sent to the states for consideration.

States send the results of their ratification process back to the Archivist and if 3/4 of states vote to ratify the Archivist certifies the results and publishes the certification.

The original, certified copy of the amendment resides with the OFR as does all signed legislation. Most likely the certified copy of the amendment itself is in a document box in an office building or records storage facility.

[NARA] (https://www.archives.gov/federal-register/constitution) has a much more detailed explanation.