Abstract: Two harmonically excited systems with symmetrical rigid constraints are considered. Repeated impacts usually occur in such systems due to the rigid constraints. The impact formats that occur in two systems are different, i.e., the components of one system collide with each other, and one of components of the other system collides with rigid obstacles. Dynamics of these systems are studied with special attention to pitchfork and Neimark-Sacker bifurcations associated with several periodic-impact motions. Two important parameters, the clearance and the excitation frequency, are considered, and the influence of these parameters on the symmetrical double-impact periodic motion and bifurcations is analyzed with emphases. With a decrease in the excitation frequency, symmetrical double-impact periodic motion usually undergoes Neimark-Sacker bifurcation for large values of clearances and pitchfork bifurcation for small values of clearances. Neimark-Sacker bifurcations, associated with symmetrical double-impact periodic motion, asymmetrical double-impact periodic motion, single-period four-impact symmetrical motion, double-period four-impact motion and double-period six-impact motion, etc., are analyzed in detail. The studies show that the vibratory systems with symmetrical rigid amplitude constraints may exhibit complex and rich quasi-periodic impact motions associated with several periodic-impact ones.