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We set up a multidimensional dynamic fiber Bragg grating (FBG) displacement and strain sensing system. The proposed sensing system allows the FBGs to have the ability to measure transient point-wise displacement (in-plane and out-of-plane) and strain responses simultaneously with high sensitivity. The sensing system employs a long-period fiber grating filter and two FBG filters to dynamically demodulate the responses of the FBG sensors. To demonstrate the dynamic sensing ability and the application of the proposed system, the transient elastic wave propagation in a solid subjected to vertical and lateral impact loadings are investigated. A laser Doppler vibrometer is employed in our experiments simultaneously to verify the transient responses of out-of-plane particle transient motions measured by the out-of-plane FBG displacement sensor. The in-plane displacement measurement ability is demonstrated by the responses obtained by two geometrically orthogonal in-plane FBG displacement sensors. Dynamic strain sensing ability of our all-fiber sensing system is also demonstrated. The experimental results both in time domain and frequency domain show that the proposed FBG sensing system is capable of performing real-time displacement (out-of-plane and in-plane) and strain measurements.