A number of musical instruments (electric basses, tanpuras, si-tars...) have a particular timbre due to the contact between a vibrating string and an obstacle. In order to simulate the motion of such a string with the purpose of sound synthesis, various technical issues have to be resolved. First, the contact phenomenon, inherently nonlinear and producing high frequency components, must be described in a numerical manner that ensures stability. Second, as a key ingredient for sound perception, a fine-grained frequency-dependent description of losses is necessary. In this study, a new conservative scheme based on a modal representation of the displacement is presented, allowing the simulation of a stiff, damped string vibrating against an obstacle with an arbitrary geometry. In this context, damping parameters together with eigenfrequencies of the system can be adjusted individually, allowing for complete control over loss characteristics. Two cases are then numerically investigated: a point obstacle located in the vicinity of the boundary , mimicking the sound of the tanpura, and then a parabolic obstacle for the sound synthesis of the sitar.