System analysis and optimization of combustion engines and engine components are increasingly supported by digital simulations. In the simulation process of combustion engines multi physics simulations are used. As an example, in the simulation of a crank drive the mechanical subsystem is coupled to a hydrodynamic subsystem. As far as the modeling of the mechanical subsystems is concerned, elastic multibody systems are frequently used. During the simulation many equations must be solved simultaneously, the hydrodynamic equations as well as the equations of motion of each body in the elastic multibody system. Since the discretization of the elastic bodies, e.g with the help of the finite element method, introduces a large number of elastic degrees of freedom, an efficient simulation of the system becomes difficult. The linear model reduction of the elastic degrees of freedom is a key step for using flexible bodies in multibody systems and turning simulations more efficient from a computational point of view. In recent years a variety of new reduction methods alongside the traditional techniques were developed in applied mathematics. Some of these methods are introduced and compared for reducing the equations of motion of an elastic multibody system. The special focus of this work is on balanced truncation model order reduction which is a singular value based reduction technique using the Gramian matrices of the system. We investigate a version of this method that is adapted to the structure of a special class of second order dynamical systems which is important for the particular application discussed here. The main computational task in balanced truncation is the solution or large-scale Lyapunov equations for which we apply a modified variant of the low-rank ADI method. The simulation of a crank drive with a flexible crankshaft is taken as technically relevant example. The results are compared to other methods like Krylov approaches or modal reduction.