Self-powered vibration sensors have the potential to greatly improve vibration monitoring of remote and hard-to-access structures by providing cost-effective and energy-efficient solutions. One promising type of sensors based on the triboelectric generator mechanism has shown significant advantages over other self-powered sensors, including better energy conversion efficiency and high output voltage. However, the majority of the recently reported triboelectric powered sensors exhibit strong frequency dependence, being mostly effective under low frequency vibrations. To address these issues, herein we present a novel design of self-powered vibration sensors based on the clapping mode of a triboelectric nanogenerator (TENG). The sensor, consisting of two tribo-layers made of regular copy paper and PTFE that are clamped together by springs, is able to achieve a high sensitivity of 10.8   mV/g and a broad frequency range from 50 to 8000   Hz with a flat frequency response of 10   dB, regardless of the inclination angle between the contacting surface and gravity. The sensor exhibits high linearity (0.9999) and stability over six days and 400,000 cycles. Furthermore, computational modelling has been conducted to understand the nonlinear clapping phenomenon and the influence of clamping force, proof mass and spring stiffness on the sensor's performance. The potential of the clapping TENG sensor is demonstrated through two different applications such as music recording and diagnosis of the loosing of bolts in a rotating machine. Its superior performances, along with low cost, compact design, and electromagnetic shielding, make it a promising candidate for battery-free high-frequency vibration sensors in practical vibration measurement applications.