1. Introduction Pulse-tube cryocoolers (PTCs) are extensively utilized in cooling high-precision instruments, such as focal-plane infrared detectors [1], space telescopes [2], gravitational-wave detectors [3], superconducting quantum interference devices [4], and X-ray detectors [5], owing to their light weight, robust reliability, extended lifespan, and minimal vibrational output. As detection sensitivities advance, the demand is increasing for effective vibration reduction in cryocooler systems.
