An effective formulation is performed to approximate calculate the vibration of the typical composite fuselage battery compartment structures of the hydrogen-electric aircraft, which are highly sensitive to hygrothermal circumstances. The strain energy, kinetic energy and hygrothermal potential energy of CLOCSRC are deduced in frame of Donnell's shell assumption and a series of hypothetical elastic springs operating on the edges of divided components are introduced to simulate coupling connection and the elastic supports of the open shell, which contribute to total system energy in the form of elastic spring potential energy. The equations of motion for CLOCSRC subjected to distinct hygrothermal circumstances and arbitrary boundaries are eventually obtained in frame of the Rayleigh-Ritz method by means of the modified orthogonal polynomial as the displacement admissible function. After confirming the convergence and accuracy of the presented formulation by experiment and finite element method, several numerical examples are presented to investigative the effects of geometric parameters, elastic boundaries and hygrothermal circumstances on the vibration of CLOCSRC. Differ from traditional investigation, the present paper offers an effective approach to insight the hygrothermal mechanism of open shell with rectangular cutouts creatively. The relevant investigation demonstrated that the current strategy is beneficial for further research on the vibration characteristics of CLOCSRC.