For the integration of flexible bodies into the multibody simulation the Component Mode Synthesis (CMS) has been established as a standard reduction tool. In the case of flexible bodies containing joints with a large spatial distribution the CMS is not suitable. In order to capture the local nonlinearities inside the joints the CMS reduction base is extended with problem-oriented trial vectors which are computed based on trial vector derivatives. The focus of this contribution is on the extension of the mentioned reduction method to general multibody dynamic systems. It will be reported how the trial vectors need to be modified in order to remove the rigid body content. Also the consideration of the contact forces in the vector of generalized forces will be discussed. Finally the method will be demonstrated on a car pendulum system with a pendulum designed as a multilayer sheet structure. The time integration is performed with a modified HHT solver. It can be reported, that the suggested approach gives an excellent balance between accuracy and computational effort.