A single prismatic constraint is enough to force the whole system to remain horizontal, see the model attached. Due to the nature of the constraint, this will be the case even if the springs/dampers are set with different parameter values.
You can let the system rotate if you introduce a spherical constraint component, Modelica.Mechanics.MultiBody.Joints.Spherical, between the end of the prismatic and the horizontal translation that goes to the body. The rotation will clearly manifest if, for instance, you set the springs/dampers with different parameter values, or you play with the initial values.
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