A fiber Bragg grating accelerometer based on L-shaped rigid beam and elastic diaphragm is designed for high- precision detection of low- frequency vibration signals. The structure of accelerometer is analyzed theoretically, and the influence of each parameter on the sensitivity and resonant frequency is discussed by simulation with Matlab. Optimization design is made on the basis of simulation. According to the analysis result, the accelerometer is made. The sensing performances that include amplitude- frequency characteristic, linear response and ability of resisting transverse interference in the cross-axis are tested experimentally. Experimental results show that the structure of accelerometer is stable by using L-shaped rigid beam. Also the chirp effect of grating and the multi-peak of reflective spectrum are avoided. The sensor has good performances in detection of low-frequency vibration signals because it has flat response from 20 Hz to 70 Hz and the sensitivity is about 220 pm/g with a linear coefficient of 99.98% . Elastic diaphragm used as elastic element efficiently eliminate transverse interference while the interference degree can be less than -32.73 dB.