Shear thickening in concentrated suspensions is known to occur via the formation of load bearing hydroclusters from theory, simulations, flow-SANS experiments, and indirect rheological measurements. However, the dynamic behavior of these concentrated suspensions is not well understood. In this work we study model colloidal dispersions of aerosil (Degussa A200) in PPG at concentrations up to 30 wt%, 450nm silica in PEG up to 67wt% and kaolite clay in glycerol up to 61 wt%. Large amplitude oscillatory shear (LAOS) measurements are performed at a variety of frequencies and particle loadings to provide a systematic study of the dynamic response of concentrated suspensions at various volume fractions, while also varying particle size and shape. LAOS measurements in the shear thickening regime exhibit a triangular response function to an applied sinusoidal deformation or stress field. We apply a new method of trial functions [Klein et. al. Macromolecules, 2007] to analyze the strain response in Fourier space to better understand the complex nonlinear response of shear thickening suspensions. The technique is shown to define the onset of shear thinning and shear thickening as well as slip. The results are used to test rheological constitutive equations for concentrated suspensions and to elucidate the nature of the shear thickened state.