Our objective is to model and simulate Cyber-Physical Systems (CPS) such as robots, vehicles, and power plants. The structure of CPS models may change during simulation due to the desired operation, due to failure situations or due to changes in physical conditions. Corresponding models are called multi-mode. We are interested in multi-domain, component-oriented modeling as performed, for example, with the modeling language Modelica that leads naturally to Differential Algebraic Equations (DAEs). This paper is thus about multi-mode DAE systems. In particular, new methods are discussed to overcome one key problem that was only solved for specific subclasses of systems before: How to switch from one mode to another one when the number of equations may change and variables may exhibit impulsive behavior? An evaluation is performed both with the experimental modeling and simulation system Modia, a domain specific language extension of the programming language Julia, and with SunDAE, a novel structural analysis library for multi-mode DAE systems.