An approach to generating periodic spiking pulses in a broadband opto-electronic oscillator (OEO) without external injection is proposed and demonstrated. Through biasing the electro-optic Mach-Zehnder modulator (MZM) in the OEO cavity at its nonlinear working point, spiking pulses are excited by the self-generated chaos under the combined action of the loop filter response time and the nonlinear excitation effect. An interaction force is introduced between pulses. As a result, the pulse number in a single roundtrip duration and the temporal distribution of the generated spiking pulses can be tuned by adjusting the loop gain and the direct-current (DC) bias voltage of the MZM. Both numerical simulation and experiment are carried out to demonstrate the spiking pulse generation mechanism, where ultra-short spiking pulse trains with even and uneven temporal distribution are generated.