The behavior of manganese-copper alloys are investigated over a wide range of temperatures by means of mechanical testing, measurements of the internal friction, and muon spin rotation (μSR). The curve of the temperature dependence of all the investigated characteristics have anomalies at temperatures in the regions of the structural and magnetic transformations and below. It is shown that subtle changes in the exchange interaction cause noticeable changes in the mechanical behavior of these alloys and their damping ability.