Disclosed is a vibration-damping support device, including a housing, and a core shaft having one end passing through the housing to connect with a vibration source. The vibration-damping support device further includes two elastic members arranged within the housing and spaced apart from each other. Each elastic member includes an inner hole, so that it can be mounted on the core shaft. Each elastic member includes an elastic rubber body, and a plurality of metal plates that are embedded in the elastic rubber body and parallel to each other. The elastic rubber body extends beyond the metal plates in a radial direction. With this vibration-damping support device, a variable stiffness can be achieved.