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Shape memory alloy (SMA) is one of the novel advanced functional materials that has an increasing range of current and potential applications, including smart materials and structures, bio-medical and nanotechnologies. This range includes also applications of SMA for control and vibration tuning of various structures, seismic response mitigation, and others. In vibration tuning in many of these applications, it is often necessary to apply supplementary oscillators to absorb the vibration energy input into the primary system. Moreover, when supplementary oscillators are used in these applications, we often have to deal with a situation where the primary vibration frequency is not known a priori. In such cases, we have to design a robust supplementary oscillator such that it is able to operate in a rather wide range of frequencies. A SMA-based oscillator is an ideal candidate for these purposes. In the present paper, we propose a dynamic nonlinear model and its numerical realization for using SMA oscillators as vibration absorbers. The system under consideration consists of a SMA rod and an end-mass. We demonstrate that due to the thermo-mechanical coupling, the vibration characteristics of the supplementary oscillator can be tuned by changing its temperature. The dynamic nonlinear model of the SMA oscillator is simplified for the vibration analysis and an efficient numerical methodology is proposed to evaluate the performance of the oscillator. It is demonstrated that the vibration of the primary system can be tuned within a rather wide frequency range by using the SMA oscillator. It is also shown that at high temperatures the performance of the oscillator is close to that of a linear oscillator, while at low temperatures, the SMA oscillator behaves as a regular damper by using its dissipation due to mechanically-induced phase transformations.