Tactile perception using vibration sensation helps robots recognize their environment's physical properties and perform complex tasks. A sliding motion is applied to target objects to generate tactile vibration data. However, situations exist where such a sliding motion is infeasible due to geometrical constraints in the environment or an object's fragility which cannot resist friction forces. This paper explores a novel approach to achieve vibration-based tactile perception without a sliding motion. To this end, our key idea is injecting a mechanical vibration into a soft tactile sensor system and measuring the propagated vibration inside it by a sensor. Soft tactile sensors are deformed by the contact state, and the touched objects' shape or texture should change the characteristics of the vibration propagation. Therefore, the propagated-vibration data are expected to contain useful information for recognizing touched environments. We developed a prototype system for a proof-of-concept: a mechanical vibration is applied to a biomimetic (soft and vibration-based) tactile sensor from a small, mounted piezoelectric actuator. As a verification experiment, we performed two classification tasks for sandpaper's grit size and a slit's gap widths using our approach and compared their accuracies with that of using sliding motions. Our approach resulted in 70% accuracy for the grit size classification and 99% accuracy for the gap width classification. These results are comparable to or better than the comparison methods with a sliding motion.