Let M be a smooth compact connected manifold, on which there exists an effective smooth circle action preserving a positive smooth volume. On M, we construct volume-preserving diffeomorphisms that are metrically isomorphic to ergodic translations on the torus, translations in which one given coordinate of the translation is an arbitrary Liouville number. To obtain this result, we explicitly construct the sequence of successive conjugacies in Anosov-Katok's periodic approximation method, with suitable estimates of their norm. To visualize the construction, we include numerous graphics.