We consider the problem of estimating the fundamental initial frequency and frequency rate of a linear chirp with random amplitudes harmonic components. We develop an iterative nonlinear least squares estimator, which involves a large number of computations as it requires high resolution search in the initial frequency and frequency rate parameter space. As an alternative, we suggest two suboptimal low-complexity estimators. The first is based on the high-order ambiguity function, which reduces the problem to a one-dimensional search. The second method applies our recently published harmonic separate-estimate method, which was used for constant-amplitude harmonic chirps. We present modifications of both methods for harmonic chirps with random amplitudes. We also provide a framework for estimating the number of harmonic components. Numerical simulations show that the iterative nonlinear least-squares estimator achieves its asymptotic accuracy in medium to high signal-to-noise ratio, while the two sub-optimal low-complexity estimators perform well in high signal-to-noise ratio. Real data examples demonstrate the performance of the harmonic separate-estimate method on random amplitude real-life signals.