A structure with PTMD is modelled in Figure 1B. The mass, stiffness, damping coefficient of structure are respectively denoted by m 1 ,     k 1 , c s . The mass and stiffness of the PTMD are represented by a spring of stiffness m 2 , k 2 , respectively. The displacement of the main structure and tuned mass are x 1 , x 2 , respectively. Considering the coupling between the single-degree-of-freedom structure and the PTMD, the equations of motion for the whole, integrated system can be written as in Eq (1): M x ¨ + C x ˙ + K x = P + Γ F (1) where x ¨ ,   x ˙ ,   x , represent acceleration, velocity, displacement, respectively. M, C, K are mass matrix, damping matrix, stiffness matrix, respectively, P is the load matrix applied to the structure, F is the pounding force, and Γ is the pounding force position. The parameters are expanded in Eq (2) to Eq (4): x ¨ = [ x ¨ 1 x ¨ 2 ] , x ˙ = [ x ˙ 1 x ˙ 2 ] , x = [ x 1 x 2 ] , P = [ p 0 ] (2) M = [ m 1 0 0 m 2 ] , C = [ c s 0 0 0 ] , x = [ k 1 + k 2 − k 2 − k 2 k 2 ] (3) Γ = [ − γ γ ] , γ = { 1 ,     x 2 − x 1 > 0 0 ,     x 2 − x 1 ≤ 0 (4)

A linear viscoelastic model has been proposed to describe the contacting force F of viscoelastic impacts (Goldsmith, 1960; Anagnostopoulos, 1988; Jankowski et al., 1998; Anagnostopoulos, 2004), and is given by Eqs (5,6): F = β δ + c δ ˙ (5) δ = x 1 − x 2 , δ ˙ = x ˙ 1 − x ˙ 2 (6) where δ ,   δ ˙ ,   β represent the relative displacement of the colliding body, the relative velocity, the pounding stiffness, and the pounding damping, respectively. The pounding stiffness can be obtained using the displacement and the pounding force recorded in the impact test. The pounding damping c can be computed by Eq (7): c = 2 ξ β m 1 m 2 m 1 + m 2 (7) where m 1 and   m 2 are the masses of two colliding bodies,   ξ is impact damping ratio, which is related to the coefficient of restitution e in Eq (8): ξ = − ln ⁡ e π 2 + ( ln ⁡ e ) 2 (8) However, the linear viscoelastic model may produce a negative force at the end of impact, which is not consistent with experimental observation (Jankowski, 2005). Subsequently, a modified linear viscoelastic model (Mahmoud and Jankowski, 2011), divides the pounding into two individuals periods and assumes that energy loss only happens when the material experiences compression, is given by Eq (9): F = { β δ + c δ ˙ ,     δ ˙ > 0 β δ ,               δ ˙ ≤ 0 (9) where c has same form as in Eq (7), and the pounding damping ratio   ξ can be calculated in Eq (10): ξ = ( 1 − e 2 e [ e ( π − 2 ) + 2 ] ) (10) Given its simplicity, the modified linear viscoelastic model can be easily implemented for numerical simulations. However, at the beginning of impact, a sudden rise of impact force may occur as a result of nonzero relative velocity. In order to overcome disadvantages of the modified linear viscoelastic model, a nonlinear viscoelastic model was introduced by Jankowski (Jankowski, 2005;2006). And it is expressed in Eq (11): F = { β δ 3 / 2 + c δ ˙ ,     δ ˙ > 0 β δ 3 / 2 ,               δ ˙ ≤ 0 (11) where c has the form in Eq (12): c = 2 ξ β δ m 1 m 2 m 1 + m 2 (12) where the pounding damping ratio   ξ can be calculated by Eq (13): ξ = 9 5 2 1 − e 2 e [ e ( 9 π − 16 ) + 16 ] (13)

The influence of temperature on the behavior of viscoelastic layers, viscoelastic dampers and structures with viscoelastic dampers were analyzed by many researchers (Chang et al., 1992; Park and Min, 2010; Guo et al., 2016). In one of their works, Chang et al. (Chang et al., 1992) tested a simple solid viscoelastic damper at different ambient temperatures and established empirical formulas that described the temperature dependence of the damper and loss factor. Previous studies(Tsai, 1994; Park and Min, 2010; Guo et al., 2016) have shown that different kinds of viscoelastic materials have similar dependencies on temperature, but differ in the specifics. Figure 2 shows the typical changes of shear modulus and loss factor of viscoelastic materials to changes in temperature. Note that T_g is the transition temperature of viscoelastic materials.

The figure indicates that the shear modulus and loss factor of viscoelastic materials change with temperature, and viscoelastic materials have high damping in the characteristic temperature range. Additionally, the changes can be divided into three phases:

(1) At low temperature, the viscoelastic materials are in the glassy state. The material molecules undergo ordinary elastic deformation under the action of external force, and the material possesses high rigidity, high modulus and small loss factor;

Despite the known features described above, the relationship between the pounding stiffness β , pounding, damping ratio ξ and temperature, impact velocity, impact materials are rarely studied and need to be investigated.

Figure 3A shows the experimental device and equipment. A noncontact laser displacement senor (HG-C1400, Panasonic Industrial Devices, China) measured the displacements of the frame. A data acquisition board (NI USB-6363, National Instruments, USA) recorded sensor signals at a sampling frequency of 1 kHz. An eccentric motor (JGB37-3650, Xytmotor, China) fixed to the frame provided the exciting force.

As shown in Figure 3B, the temperature experiment is carried out in the temperature-controlled room (MW-BD1824, Ziweiheng testing equipment, China). The temperature of the room can be adjusted from –20°C to 50°C.

As shown in Figure 4, the portal frame is composed of an aluminum block, two spring steel columns, and an aluminum plate base. The detailed parameters of the portal frame are listed in Table 2. The aluminum block is served as the main structural mass. Spring steels served as structural columns because of its high fatigue performance and high elasticity. The base fixes the portal frame to the ground. The free vibration experiment was conducted by applying an initial displacement to the portal frame and releasing. The experimental result shows that the equivalent viscous damping ratio is 0.29%.

A motor is attached to the top of the frame. A quasi-periodic disturbance is generated when the motor rotates. The generated frequency can be controlled by adjusting the excitation voltage motor. In the forced vibration experiments, the structure was excited at twenty different sweep frequency conditions ranging from 2.6Hz to 5Hz. The experimental results indicate that portal frame reaches the maximum displacement when the excitation frequency is 3.13Hz.

As shown in Figure 5, the single-sided PTMD consists of mass block, spring steel, and a viscoelastic delimiter (EVA sponge, produced by 3M). An aluminum pole connects the PTMD to the portal frame. To avoid affecting the dynamic characteristics of the structure, the pole of the PTMD consisted of aluminum. The optimum frequency of the PTMD, f o p , is designed using following formula(Wang et al., 2018a): f o p = 1 2 f d (14) where f d is the natural frequency of the main structure.

The weight of the tuned masses is 246g, and the corresponding mass ratio is 4.7%. A coupon of spring steel connected the tuned mass to the PTMD body. This design is an update of prior designs in which a nylon rope served as the connector and sometimes suffer from circular, out-of-plane motions (Tan et al., 2019b).

At first, the free vibration and forced vibration experiments with and without PTMD control were carried out at room temperature (18°C).

Figure 6 shows the free vibration of the portal frame with and without PTMD control. With PTMD control, the displacement of the frame rapidly reduced to a very small level compared to the case without control. The time taken to suppress the displacement from 10mm to 2mm reduced from 57.4s to 1.19s. The damping ratio of the system is correspondingly increased from 0.29% to 4.4%.

The displacements during forced vibration experiment are shown in Figure 7. The blue line describes the frequency response of the portal frame with PTMD control. The frequency response curve only has one peak at 3.13Hz, and the maximum displacement is 18.43mm, which occurred on the right side of the uncontrolled resonance frequency. Additionally, the peak displacement of the controlled vibration is 3.26mm, which is 17.7% of the uncontrolled resonant amplitude.

Figure 8 shows the vibration response of the portal frame before and after the PTMD is released. After the release of the PTMDs, the portal frame reached a new steady state in a short time. The displacement amplitude decreased from 16.34mm to 2.61mm.

The experiment on the effect of temperature on PTMD control was carried out in the temperature-controlled room. The forced vibration test in Preliminary test of PTMD at room temperature was repeated at different temperatures (listed in Table 1). The maximum displacements of the structure with PTMD at different temperatures are shown in Figure 9. As can be seen in Figure 9, the control performance of PTMD can be divided to three phases. In Phase 1 (–20°C to –8.5°C), the displacement of portal frame decreased slightly from 4.4mm to 4.34mm, and the reduction ratio increased from 76.1% to 76.4%. In Phase 2 (–8.5°C to 20°C), the maximum displacement of portal frame decreased first before decreasing. The corresponding reduction ratio increased first before decreasing. The maximum displacement decreased to 2.5mm at 10°C. In the meantime, the maximum reduction ratio reached 86.4%. In Phase 3 (20°C to 45°C), the displacement and reduction ratio remained almost constant (3.26mm and 82.3%, respectively) despite further changes in temperature.

At glassy stage, the damping effect of PTMD almost does not change obviously with temperature variation. For the viscoelastic region (–8.5°C to 20°C), the damping effect of the PTMD changes with temperature, and there is an optimal control temperature, which is around 10°C for this viscoelastic material. Between 20–45°C the material behaved elastically, and the damping effect of PTMD remained in a steady state.

Although the damping effect is affected at low and high temperatures, the maximum displacement of the structure can still be reduced beyond 75%. As the thermal influence on PTMD control performance can be divided to three phases, results of each phase for the forced vibration experiments are shown in Figure 10. The results shows that PTMD has a good temperature robustness.

An impact test at 18°C was carried out to study the performance of the viscoelastic material during the impact. The experimental setup is shown in Figure 11. The spring steel-mass system was the same as in Experimental device and equipment, and the thickness of the viscoelastic materials was also set to 7mm (2 layers). An initial displacement of 35mm was applied to the mass and then released, allowing the mass pounds the viscoelastic material freely. The displacement of the mass was recorded by a laser displacement sensor, and a force sensor was used to record the pounding force during the experiment.

The results of the impact test are shown in Figure 12. Figure 12A, B are the time history records of mass displacement and impact force, respectively. Calculated from the initial displacement (35.00mm) and the rebound displacement (15.33mm) after the first impact, the coefficient of restitution e is 0.434. Furthermore, obtained by eq (13). the pounding damping ratio ξ is 0.882. Through the force peak value (6.24N) in Figure 12B and corresponding displacement (3.12mm), the pounding stiffness β is 3.6 × 10⁴N/m^1.5.

Ignoring the damping of the spring steel, motion equation of the mass in Figure 11 can be expressed as m x ¨ + k x = F (15) where F is the impact force, calculated by eq (11) to eq (13), m and k are mass and stiffness of the spring steel-mass system. The initial x ( 0 ) was set to 35.00mm, and the pounding parameters were set as values obtained in the impact test. The fourth-order Runge-Kutta method was utilized to solve eq (15) with a time step of 1 × 10^-6s. The comparison of experimental result and simulation result is shown in Figure 12. It can be seen that the pounding model is highly accurate to predict displacement and impact force of the mass.

To further validate the accuracy of the PTMD model, a simulation analysis was performed on the experimental structure with PTMD in Experimental setup. The motion equations of the structure with PTMD were expressed in eq (1) to eq (4), and the pounding force model was expressed in eq (11) to eq (13). In the free vibration, the excitation force P ( t ) = 0, in the forced vibration experiment, the excitation force can be calculated as follows: P ( t ) = m e r 2 ⁡ sin ( ω t ) (16) where m e , r ,   ω are the eccentric mass, eccentricity, excitation frequency. In the article, m e = 100 P ( t ) = m e r 2 ⁡ sin ( ω t ) g, r = 55mm.

The parameters of the simulation at 18°C were derived from the results of the previous experiments and were summarized in Table 3. Similarly, Runge-Kutta method was utilized to solve the coupled equations, and both free vibration case and forced resonant case were simulated. The simulation results are plotted in Figure 13. As shown in Figure 13A and Figure 13B, the proposed coupled equations can high accurately predict dynamic response of structure with PTMD in both cases.

In this study, a single-sided pounding tuned mass damper (PTMD) was designed to mitigate the vibration of a portal frame. From experimental results, the conclusions are as follows:

(1) A single-sided PTMD can reduce the structural dynamic response and improve the damping ratio of structure effectively;

The changes of parameters in pounding force model at different temperatures are vital factors for simulation and theoretical analysis of PTMD. The exploration of a wide range of temperatures impact tests to establish the damping factor vs. temperature curve will be delegated as a future work.