The atomic nucleus, governed by short-range nuclear force, is a quantum many-body system that plays a vital role in the visible energy-mass dynamics of the universe and significantly influences the sustenance, development of society, and the security of nations. There have been numerous discoveries in the past decades concerning exotic structures and properties of short-lived nuclei. These findings have sparked breakthroughs in our understanding of nuclear structures and have given rise to a new field called radioactive ion beam physics, which focuses on the study of unstable nuclei. For more than 30 years, the Beijing Tandem-Accelerator Nuclear-Physics National Laboratory has provided a basic research platform for low-energy nuclear physics experiments. The experimental nuclear physics team at Peking University has continuously developed a dedicated experimental apparatus, conducted a series of physics experiments at the Beijing HI-13 tandem accelerator, and achieved important results related to exotic nuclear structures. In this article, we present several notable experimental achievements of our team at the HI-13 accelerator. These include the investigation of the shape evolution of germanium isotopes (around A=70) using in-beam γ-spectroscopy, the exploration of cluster structures in light neutron-rich nuclei through direct nuclear reactions, and the development and commissioning of collinear laser spectroscopy experiments at the Beijing Radioactive Ion-beam Facility.