  Response solutions are quasi-periodic ones with the same frequency as the
forcing term. The present work is devoted to the construction of response
solutions for $d$-dimensional beam equations with nonlocal nonlinear damping,
which model frictional mechanisms affecting the bodies based on the average. By
considering $\epsilon$ in a domain that does not include the origin and
imposing a small quasi-periodic forcing with Liouvillean frequency vector,
which is weaker than the Diophantine or Brjuno one, we can show the existence
of the response solution for such a model. We present an alternative approach
to the contraction mapping principle (cf. [5,33]) through a combination of
reduction and the Nash--Moser iteration technique. The reason behind this
approach lies in the derivative losses caused by the nonlocal nonlinearity.
