We study the excitation and damping of transverse oscillations in a complex multi-stranded model of a coronal loop. By numerically solving the time-dependent magnetohydrodynamic (MHD) equations in two dimensions, we show how the global motion of the whole bundle of tubes, produced by an external disturbance, is converted into localised motions due to the process of resonant absorption. At any location in the structure two dominant frequencies are found, the frequency of the global mode (different from the kink frequency of the individual strands) and the local Alfvén frequency. The mechanism of mode conversion is not affected by the complicated geometry of the system and for certain configurations the energy conversion does not only take place at the external edge of the composite loop but also inside the structure.