  In this paper we research global dynamics and bifurcations of planar
piecewise smooth quadratic quasi--homogeneous but non-homogeneous polynomial
differential systems. We present sufficient and necessary conditions for the
existence of a center in piecewise smooth quadratic quasi--homogeneous systems.
Moreover, the center is global and non-isochronous if it exists, which cannot
appear in smooth quadratic quasi-homogeneous systems. Then the global
structures of piecewise smooth quadratic quasi--homogeneous but non-homogeneous
systems are studied. Finally we investigate limit cycle bifurcations of the
piecewise smooth quadratic quasi-homogeneous center and give the maximal number
of limit cycles bifurcating from the periodic orbits of the center by applying
the Melnikov method for piecewise smooth near-Hamiltonian systems.
