Abstract: Piecewise linear oscillators are a class of strongly nonlinear systems. They are prevalently found in the fields of practical engineering and other interdisciplinary research fields. For example, piecewise linear energy sinks, which are constructed with permanent magnets and cantilever beams, and piecewise linear models, which are generally used to approximate nonlinear oscillators. Such systems always exhibit complex dynamical behavior. However, the present studies on the piecewise linear oscillators mainly consider the cases of single frequency excitation. There are rare studies on the quasi-periodic excitation systems. In practical engineering and applications, external excitations are often of multi-frequencies. In this paper, the global dynamics of a class of piecewise asymmetrical oscillators with quasi-periodic excitations are studied. The conditions for existence of transverse homoclinic points are obtained by the extended Melnikov method, and then the threshold of Smale horseshoe chaos is given. The theoretical results are verified numerically by the time series, the phase diagrams, the Poincaré sections and the largest Lyapunov exponent. It also demonstrates the effects of damping, spring stiffness and external excitation amplitude on the chaotic motion. But more than that, the special dynamical behavior showing that the quasi-periodic attractors of the system evolve into strange nonchaotic attractors through the routes of torus doubling and fractalization is illustrated. The strange nonchaotic attractors are a class of attractors. The most distinctive mark of them is that they have distinguished fractal structure, whilst the largest Lyapunov exponent of them is not positive. Whereby, the strange nonchaotic attractors can be regarded as a kind of special attractors, which occupies the middle ground between quasi-periodic attractors and chaotic attractors. The study on the generation mechanism and topological structure of strange nonchaotic attractors clarifies the inherent complexities, such as bifurcation scenarios and chaos, in multi-frequency excitation systems.