Loaded Vibration Characteristics of Large Statically Deformed Single Parabolic Leaf Spring

In the present paper, a physically insightful semi-analytical framework is developed to analyze the loaded vibration characteristics of single parabolic leaf spring system considering the large deformed configuration due to initial static load. The leaf spring system is modeled as a curved beam with parabolic thickness variation having hinged support at one end and involving a rigid link at the other hinged supported end. The post-static dynamic governing equation is formulated through variational principle based Hamilton’s framework considering non-uniform inertia due to the lumped mass, combined bending–stretching vibrating modes, and pre-vibration large static deformation. The static problem determining pre-vibration large static deformed configuration is derived as a special case of the generalized dynamic problem considering rotational restraints and kinematic constraints due to the presence of the rigid link at the curved beam boundary through a suitable geometry updating technique. Within the developed combined static–dynamic semi-analytical framework, parametric studies on the effects of thickness ratio of parabolically varying thickness combined with height to span ratio of the parabolic leaf on the static deformation measures, and loaded natural frequencies and vibrating mode shapes of first and second modes are carried out. The developed semi-analytical framework considering the mentioned complicating effects provides physical insights into the complicated large deformation mechanics of the leaf spring system which would provide the foundation for efficient utilization of single parabolic leaf spring with reduced weight compared to the conventional multi-leaf spring system. Moreover, the parametric studies would provide avenues for designing optimized system configurations.