The band gap has been used to control the transmission features of acoustic/elastic waves. Roton-like dispersion relations show that the energy and momentum of acoustic waves are inversely proportional to each other at finite region. To modulate the band gap and obtain the unusual roton-like behavior, the Kapitza’s pendulums and nonlocal connection stiffness are introduced into the linear mass-spring periodic system. The frequency range with the roton-like behavior is modulated via the parametric excitation. Moreover, the dispersion relations show some fascinating phenomena (i.e., the negative/zero-group velocity) under special parameters, which indicate the potential application to control the transmission of acoustic/elastic waves and design a negative/zero-refraction or nonpropagating-vibration structure.