## 1 Introduction  
Tall and flexible structures, such as offshore wind turbines, chimneys, and transmission towers, are characterized by high flexibility and low damping. Consequently, these structures suffer from excessive vibrations under wind loads or earthquakes and in some extreme cases may even collapse. Therefore, reducing the vibration of these structures can effectively improve their safety and reliability. There are two main types of dampers used for vibration reduction in tall flexible structures, namely energy dissipation dampers (e.g., metallic energy-dissipation dampers [1, 2, 3], viscous dampers [4, 5, 6], friction dampers [7, 8, 9], and magnetorheological dampers [10, 11, 12]) and linear dynamic vibration absorbers (e.g., suspended mass pendulums (SMP) [13, 14] and tuned mass dampers (TMDs) [15, 16]). Linear dynamic vibration absorbers are widely used in structural vibration control due to their simple structure and ease of installation. To improve their vibration-reduction performance, various forms of nonlinear energy-dissipation components (e.g., bounding components [17, 18, 19], internal resonance components [20, 21], and shape memory alloys [22, 23, 24]) have been introduced to effectively increase the energy-dissipation capacity of dampers. As a typical nonlinear damper, a bounding-tuned mass damper (PTMD) outperforms a TMD in terms of vibration-reduction effect [25; 26] and is robust against the de-tuning effect [27; 28]. [PERSON] et al. [29] installed a PTMD in a pipeline structure and showed that the PTMD effectively reduced the vibration in the pipeline and improved the damping ratio of the structure. [PERSON] et al. [30] experimentally investigated the vibration control ability of PTMDs for underwater structures, demonstrating that PTMDs remained significantly effective under submerged conditions. [PERSON] et al. [31] studied the vibration control of a TV tower with a PTMD. [PERSON] et al. [32] applied a bounding spacer damper (PSD) to the wind-induced vibration control of transmission towers. [PERSON] et al. [33] showed through numerical simulation that PTMDs could effectively reduce bridge deck vibrations induced by moving vehicles. [PERSON] et al. [34] proposed a single-sided PTMD for controlling vortex-induced bridge deck vibrations and experimentally verified its vibration-reduction effect. [PERSON] et al. [35] and [PERSON] et al. [36] investigated the suppression of structural vibration with multiple PTMDs. [PERSON] et al. [37] experimentally verified the vibration-reduction performance of a spring steel-type PTMD and a simple pendulum-type PTMD. [PERSON] et al. [38] carried out wind tunnel tests to verify the good suppression effect of PTMDs on the multimode vortex-induced vibration of stay cables. [PERSON] et al. [39] proposed a bidirectional PTMD for vibration reduction in transmission towers. [PERSON] et al. [40] applied a PTMD with a shape memory alloy to the vibration control of an offshore jacket platform. [PERSON] et al. [41] designed a PTMD for cantilevered traffic signal structures and verified its effectiveness through experiments and numerical simulations. [PERSON] et al. [42] proposed a PTMD installed in a confined space and demonstrated its robustness. [PERSON] et al. [43; 44] verified the vibration control performance of a pendulum PTMD (PPTMD). A spring pendulum (SP) is a nonlinear damper composed of a mass block attached to the free end of a spring [45; 46]. It has two characteristic frequencies, i.e., the frequency of the vibration mode along the radial direction of the spring and the frequency of the swing mode in the tangential direction [47; 48]. Energy is transferred between the two modes when their frequencies satisfy the internal resonance condition [49; 50]. A previous study on internal resonance springs [21] verified that an SP can effectively control the vibration of a structure. However, an SP lacks dedicated damping components, resulting in constrained energy dissipation capacity.