All real-life systems, including mechanical systems, are affected by uncertainties that can occur in the system parameters, in the initial conditions, in the external excitations, in the sensor signal, etc. Such uncertainties must be accounted for when designing a mechanical system; designs that ignore uncertainty often lead to poor robustness and suboptimal performance. Thus, in this study, we develop a novel optimal design framework that focuses on uncertain dynamical systems described by ordinary differential equations. Using prior research conducted in our group, in this work the uncertainties are modeled using a Generalized Polynomial Chaos (gPC) approach and are solved quantitatively using a least-square collocation method. The computational efficiency of this approach allows statistical moments to be explicitly included in the formulations of the objective function(s) and of the constraints to perform optimal designs under uncertainty. The mechanical systems that benefit the most from this research methodology are nonlinear, have active constraints, or conflicting design objective, such as the vehicle systems. Using a constraint-based multi-objective formulation, the direct treatment of uncertainties during the optimization process is shown to shift the resulting Pareto optimal trade-off curve.