Shock absorbers are essential in enhancing vehicle ride comfort by mitigating vibrations. However, traditional rubber shock absorbers are constrained by their fixed stiffness and damping properties, limiting their adaptability to varying loads and thus affecting the ride comfort, especially under extreme road conditions. Shape Memory Alloys (SMAs), known for their intelligent material properties, offer a unique solution by adjusting stiffness and damping in response to temperature changes or strain rates, making them ideal for advanced vibration control applications. This study builds upon the Auricchio constitutive model to propose an enhanced SMA hyper-elastic constitutive model that accounts for different loading rates. This new model elucidates the impact of loading rates on the stiffness and damping characteristics of SMAs. Additionally, we introduce an innovative circular rubber-based SMA composite vibration reduction structure. Through a parameterized model and finite element simulation, we comprehensively analyze the stiffness and damping properties of the composite damper under various loading rates and harmonic excitations. Our findings suggest a novel approach to improving the vehicle ride comfort, offering significant potential for engineering applications and practical value.