This paper examines the influence of negative work on energy required for the flapping wing flight of birds, or other flapping wing animals or vehicles. Because negative energy is typically required by muscles or actuators to produce negative work, it must be accounted for when determining the most efficient flapping configuration. The present work provides a simple theoretical analysis for determining the influence of negative work by introducing a simple actuator/muscle model, which specifies the amount of input energy required to produce negative or positive work. The influence of aerodynamic, structural, and inertial forces are treated in the study. The aerodynamic forces are modeled using unsteady thin airfoil theory, which is appropriate for the relatively high aspect ratio wings of most birds. The influence of springs on a flapping system are discussed, and theoretical approach to determine the spring stiffness required to minimize negative work is presented. The developed analysis is applied to a Pigeon and a Pied Flycatcher. It is found that the flight speed requiring minimum muscle energy is dependent upon the energy cost of negative work.
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