We investigate the evolution of magnetic excitations in Sr$_3$Ru$_2$O$_7$ using a three band tight binding model that takes into account the influence of Mn and Ti dopant ions. The effect of dopant ions on the Sr$_3$Ru$_2$O$_7$ band structure has been included by taking into account the dopant induced suppression of the oxygen octahedral rotation in the tight binding band structure and changes in electron occupation. We find that the low energy spin fluctuations are dominated by three wave vectors around Q=$((0,0),(pi/2,pi/2))$, and $(pi,0)$ which compete with each other. As the octahedral rotation is suppressed with increasing doping, the three wave vectors evolve differently. In particular, the undoped compound has dominant wavevectors at Q=$((0,0),(pi/2,pi/2))$, but doping Sr$_3$Ru$_2$O$_7$ leads to a significant enhancement in the spin susceptibility at the Q=$(pi,0)$ wavevector bringing the system closer to a magnetic instability. All the features calculated from our model are in agreement with neutron scattering experiments. We have also studied the effect of a c-axis Zeeman field on the low energy spin fluctuations. We find that an increasing magnetic field suppresses the AFM fluctuations and leads to stronger competition between the AFM and FM spin fluctuations. The magnetic field dependence observed in our calculations therefore supports the scenario that the observed nematic phase in the metamagnetic region in Sr$_3$Ru$_2$O$_7$ is intimately related to the presence of a competing ferromagnetic instability.
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