We numerically study a novel scheme of generating linearly chirped signal utilizing the nonlinear dynamics of an optically injected semiconductor laser. With proper adjustment, the optically injected semiconductor laser can be operated in an instable region where the output of the laser exhibits periodic oscillation. The oscillation frequency of the injected laser can be controlled by simply varying the strength of the injection light. By sweeping the injection strength in time, desired chirped signal with very high linearity can readily be obtained. Without modulating the frequency of the laser through either direct current modulation or external frequency modulation, large modulation frequency exceeding 7 GHz is achieved. To suppress the amplitude modulation, a cascaded scheme is further considered that the chirp light generated from the second laser is further injected into a third laser. By tuning the third laser to an injection-locked state, the third laser reproduces the chirp signal injected but with an amplitude modulation greatly suppressed. In this paper, chirp bandwidths, chirp rates, and linearity of the chirp signals generated are studied. A chirp rate of more than 100 GHz/μs is obtained, while the bandwidth of the chirp signal exceeds 7 GHz. The relation between amplitude suppression and chirp rate is also presented. Moreover, the dependence of peak-to-peak intensity modulation suppression on the injection strength is investigated as well.