Hovering insects and birds across a wide range of sizes and morphologies, from tiny flies to hawkmoths and hummingbirds, are thought to achieve energy-efficient flapping flight by storing and releasing elastic energy in their muscles, tendons, and thoraxes. These disparate species all seem to operate in regimes where the ratio of inertial to aerodynamic power, N, is in the range of 1 to 10, suggesting that there is some constraint on the energetics of flapping outside of that range. We present an updated model of the dynamics and energetics of flapping flight that includes internal losses associated with structural damping within the insect thorax. Recent work has suggested that structural damping may be as high as 20% of the total energy loss in flapping insects, so it is necessary to understand the impact of such losses on the overall energetics of flapping systems. We perform dimensional analysis and numerical simulation and conduct physical experiments on a robotic flapping wing with tunable elasticity and structural damping. We find that any damping, even in small amounts, fundamentally changes the biomechanical parameter space in which flight has evolved. We show that the upper bound on dynamic efficiency, an important metric of the ability of a wing to transmit muscle effort, monotonically decreases with increasing N in any system that has non-ideal elastic storage, potentially explaining the somewhat narrow range of N in hovering insects and birds. This detailed non-dimensional formulation of the dynamics and energetics of flapping flight is valuable both for understanding the evolution of insect flight biomechanics and for the future design of flapping-wing micro-robots.