Effects of Mixed Air on the Performance and Stiffness of a Viscous Fluid Damper

Abstract

Viscous fluid dampers are widely used for mechanical vibration reduction to ensure the stability and safety of structures and systems. However, when a small amount of air (less than 10%) is mixed into the fluid, the compressibility of the fluid increases, leading to a decrease in the physical series stiffness of the damper. Consequently, under dynamic excitation, the proportion of elastic force in the total output force rises, resulting in an increase in the equivalent parallel additional stiffness—a concept often conflated with the series stiffness in the literature. This paper aims to demonstrate these two aspects of stiffness change by investigating the dynamic characteristics of air-mixed viscous fluid dampers through nonlinear modeling, finite element simulation, and experimental validation. Starting from a nonlinear series model comprising nonlinear damping and a nonlinear fluid spring (series stiffness), the energy dissipation and physical series stiffness under different air mixtures are simulated using a finite element model. To further explore the influence of air, an equivalent linear parallel model is established based on the equal energy principle, yielding an equivalent parallel additional stiffness. The results reveal that the energy dissipation effectiveness and the dynamic stiffness of viscous fluid dampers decrease as the air mixture increases. Nevertheless, the additional stiffness is increased with the air content. When the amount of air mixing is the same, the energy dissipation characteristics of the viscous fluid damper under different excitation frequencies vary. Both the damper efficiency and the additional stiffness are increased with the increase of the excitation frequency. The proposed equivalent linear model effectively captures the coupled effects of air mixture and excitation conditions on damper performance.

1. Introduction

A damper is a device that provides resistance to motion and dissipates kinetic energy. Dampers are widely used in the vibration damping and anti-vibration of building structures to ensure the stability and safety of the structure [1]. It has also been used to reduce earthquake damage to engineering structures in many construction projects in recent years [2]. Fluid viscous damper (FVD), a velocity-dependent damper, can generate a large damping force as the internal high-viscosity fluid flows through the specially designed orifices and interspace [3,4]. This is shown in Figure 1. The outstanding and steady damping performance enables FVDs to consume a large amount of outside input energy fv [5]. Besides, another benefit is that velocity-related characteristic of FVDs makes them adapt to excitations of a wide range of frequencies [6]. In order to analyze the characteristics of fluid dampers, numerous researchers proposed various mechanical models, such as Maxwell model with stiffness and damping in series [7], Kelvin model with stiffness and damping in parallel [8], and fractional derivative model [9]. In recent years, numerous experiments and analytic investigations have focused on linear FVD [10,11,12], but the performance of most dampers is nonlinear. In the study of nonlinear damper performance, the Maxwell model is used widely, which has high accuracy but complex calculation. Constantinou and Symans, and later Seleemah and Constantinou, adopted the Maxwell model—a constitutive equation commonly used to describe viscoelastic materials—to model nonlinear dampers [12]. The damper damping aperture and its shape, viscous fluid type and excitation frequency have a significant influence on the performance of the damper. Usually, the resisting forces of a viscous fluid damper is nonlinearly related to its operation velocity. Hou, et al. [13] studied this nonlinear behavior and found that higher test frequency, smaller annular orifice width, and larger zero-shear-rate viscosity respectively resflect the effects of damper working status, geometry, and filled fluid property on the damper behavior. Chen’s test data showed that dampers with simply an annular orifice also exhibit the nonlinear behavior [14].

The above model analyzes the nonlinear characteristics of the damper and explains the influence of structural parameters on its performance. However, it does not consider the additional stiffness that arises from changes in fluid compressibility after air mixing. Additional stiffness is derived from the nonlinear series model in Figure 1 via the equal energy principle and is distinct from the physical series stiffness which represents the compressibility of the fluid. An increase in additional stiffness indicates that a larger portion of the damper’s output force is elastic rather than purely viscous.

However, in the usual model, the fluid is considered to be an incompressible rigid body [15], additional stiffness due to compressibility is often overlooked or mostly based on experimental data, the performance of fluid dampers mixed with air is not studied deeply enough. In fact, the fluid can be compressed, dynamically generating additional stiffness, especially when the damper is mixed with air, which has a great impact on its performance [16]. The main reason is that after the viscous fluid in the damper is mixed with air, its compressibility changes, the damper coefficient decreases while the stiffness coefficient increases, result in the elastic force increases and the damping force decreases, which are used to measure the elastic force and damping force. Taylor stated that stiffness is caused by the fluid compression, but without any quantitative analysis [17]. Beyond the effects of air mixture, the performance of viscous fluid dampers is also influenced by other coupled factors under complex loading conditions. Recently, Zhang Y. et al. investigated the coupling effects of temperature and pressure on damper performance, proposing an improved mechanical model that accounts for these factors under prolonged excitation [18]. Zoccolini, L., etc., systematically analyzed the influence of these parameters on damper performance, demonstrating that the equivalent damping and stiffness coefficients are highly sensitive to variations in geometric and material properties [19]. More recently, Bouayad Agha et al. introduced a novel numerical model incorporating the Murnaghan equation of state to simulate fluid compressibility in viscous dampers and systematically investigated how rheological and geometrical parameters govern nonlinear damping behavior [20]. Complementing this line of inquiry, Jiao et al. developed a reduced-order model for three-parameter fluid viscous dampers that explicitly accounts for fluid compressibility and bellows volume deformation, achieving high prediction accuracy across a wide frequency range (1–300 Hz) and revealing the critical influence of orifice parameters and nonlinear characteristics on isolation performance [21]. The dynamic properties of viscoelastic materials, such as silicone oil, are known to be strongly frequency-dependent, which directly affects the additional stiffness of fluid dampers under varying excitation frequencies. Recent work by Shi et al. demonstrated that dynamic-bond-mediated chain reptation in PDMS elastomers enables precise control of ultra-high damping performance across a wide frequency range (10⁻²–10² Hz), achieving over 300% higher energy dissipation than commercial silicone rubbers [22]. Their work provides a valuable framework for understanding how design choices affect the trade-off between energy dissipation and additional stiffness generation. While these studies provide valuable insights into environmental, parametric, frequency and compressibility effects, the specific influence of entrained air—particularly its quantitative effect on the evolution from physical series stiffness to equivalent parallel additional stiffness—remains insufficiently explored. The dynamic interplay between air content, fluid compressibility, and the resulting changes in damping and stiffness has not been systematically characterized. Therefore, investigating the influence of the air-induced compressibility changes on the a additional stiffness of viscous fluid dampers is of both practical and theoretical significance.

The study aims to characterize the dynamic behavior and stiffness evolution of air-mixed viscous fluid dampers under different amount of air and frequencies. Starting from the nonlinear theory and equivalent linearization theory, this paper establishes finite element model and linearized parallel model to simulate the energy consumption characteristics and stiffness by changing the air content in the damper. By comparing the simulation and experimental results, the model is good agreement with the experiment as a whole, which further demonstrates the credibility of the theoretical model and can meet the research requirements. Our study could better explore the performance and stiffness changes of viscous fluid damper with less air content inside. Remedy the investigative deficiency of performance in the state of less air content in damper. The models can be used to monitor energy consumption characteristic of viscous fluid damper in the mechanical structure, which provided a good predict method for ensuring systematic security and stability, which has important engineering practical value.

2. Nonlinear Viscous Fluid Dampers

2.1. Theory

The compressibility of the fluid and air makes the viscous fluid damper have an additional spring in series with the damping. The mechanical model and motion process of the damper can be represented in Figure 2, where the spring and damping are nonlinear.

The fluid flow through the damping hole is calculated as follows [23,24]:

$$F={\displaystyle \frac{p{A}_3}{2C_q^2}}{\nu }_c^2$$

(1)

where $p$ is the fluid density, ${\nu }_c$ is the fluid flow rate within the damped hole and $A_3$ is the piston effective area $\pi \left(D^2 - d^2\right)/4$, Where $D$ is the piston diameter, $d$ is the piston rod diameter, and $C_q$ is the flow coefficient, and its value is related to the speed and other factors [18]. Judging from Equation (1), the damping force is nonlinear between the flow coefficient and the velocity.

The compressibility of the pure fluid can be expressed by the bulk elastic modulus $E_f$:

$${\displaystyle \frac{\Delta V_f}{V_f}} = -{\displaystyle \frac{\Delta p}{E_f}}$$

(2)

In the actual calculation, the volume elastic modulus of pure hydraulic oil is 700 MPa [25]. The bulk elastic modulus of the mixed fluid at pressure $p$ can be calculated by the following equation:

$$\mathrm{E}=\left({\displaystyle \frac{{\mathrm{V}}_{\mathrm{f}}/{\mathrm{V}}_{\mathrm{g}}+{\mathrm{p}}_{\mathrm{a}}/\mathrm{p}}{{\mathrm{V}}_{\mathrm{f}}/{\mathrm{V}}_{\mathrm{g}}+{\mathrm{E}}_{\mathrm{f}}{\mathrm{p}}_{\mathrm{a}}/{\mathrm{p}}^2}}\right){\mathrm{E}}_{\mathrm{f}}$$

(3)

where $p_a$ is the initial pressure, $V_g$ is the volume of mixed air, and the pure fluid volume $V_f$. The ratio of the elastic modulus of mixed fluid and pure fluid with external pressure is shown in Figure 3. The stiffness in the damper nonlinear series model can be calculated by the following equation [26]:

$$k = {\displaystyle \frac{E{A}_3}{L}}$$

(4)

In formula, $L$ is the length of the liquid column of the pressurized fluid. After mixing air, $E$ changes with the external force $p$. So the stiffness $k$ changes dynamically, is the dynamic stiffness and refers to the physical “fluid spring” stiffness in the nonlinear series model, as show in Figure 2. The relationship between dyncmic stiffness $k$ and piston displacement exhibits a non-linear characteristic. This stiffness is a direct measure of the fluid’s physical compressibility. When air is mixed into the fluid, the effective bulk modulus E decreases significantly, as shown in Figure 3. Consequently, this physical series stiffness k decreases with increasing air content. This corresponds to the reduction in dynamic stiffness.

2.2. Finite Element Model

In order to explore the influence of a small amount of air (volume fraction less than 10%) in the viscous fluid on the performance of the damper, a three-dimensional finite element model of the viscous fluid damper is established using ANSYS Fluent 2023 (as show in Figure 4), and the finite element simulation of the viscous fluid inside the damper is carried out. Both the cylinder and the piston are modeled as rigid bodies. Their material properties (Young’s modulus, Poisson’s ratio) do not influence the fluid dynamics solution. Piston movement is simulated using moving mesh technology. The dynamic mesh technique combining spring-based smoothing and local remeshing is employed to simulate the reciprocating motion of the piston, with the piston wall velocity prescribed via a User-Defined Function (UDF). The fluid domain is discretized using an unstructured tetrahedral mesh. The damping orifice region is locally refined with a minimum element size of 0.15 mm owing to the small orifice diameter (2.2 mm) and large velocity gradients, while the cylinder chamber adopts a maximum element size of 1.0 mm. Following a mesh independence study, the total mesh count was established at 486,344 cells. A spring constant factor of 1.0 is applied for spring-based smoothing. Local remeshing is triggered when the cell skewness exceeds 0.6 or the cell size falls outside the range of 0.15–1.5 mm, with a maximum remeshing interval of 5 time steps. The time step is set to Δt = 0.002 s (400 steps per 0.8 s cycle), with a maximum of 30 iterations per step and convergence judged by residuals of the continuity and momentum equations falling below 1 × 10⁻⁴. The SIMPLE algorithm is used for pressure–velocity coupling with a second-order upwind scheme for the convective terms. For air–oil two-phase flow cases, the mixture model is adopted with the initial air volume fraction uniformly initialized. Cell skewness is monitored throughout the simulation, and the maximum value remains below 0.85 at all time steps. In this CFD model the damping effect originates from fluid viscous dissipation and the throttling effect of damping orifices, solved by the Navier–Stokes equations, following the nonlinear velocity-squared damping model.

The dimensions of the damper model in this article are as follows: the diameter of the inner cylinder of the damper is 63 mm, the diameter of the piston rod is 20 mm, the diameter of the symmetrical 2 damping holes is 2.2 mm, the length of the damping hole is 14 mm, and the piston stroke is 50 mm. The viscous fluid material is 350 cSt simethicone with a density of 970 $\mathrm{kg}∕{\mathrm{m}}^3$, and the kinetic viscosity standard is 350 ${\mathrm{mm}}^2/\mathrm{s}$ at 25 °C.

The homogeneous mixture model is adopted because, for low air volume fractions (<10%) and under dynamic excitation, the air is assumed to be finely dispersed as micro-bubbles that move with the fluid. This simplifies the complex physics of bubble dynamics and allows for a tractable numerical simulation to capture the primary effect of increased bulk compressibility. The air volume fraction is initialized uniformly. This model does not capture phenomena like bubble coalescence, segregation, or dissolution. It is most valid for low air fractions and when the bubble size is much smaller than the characteristic flow dimensions. For higher air fractions or static conditions, a more sophisticated model (e.g., Eulerian-Eulerian multiphase model) would be necessary. The model is suitable for studying the initial effects of air entrainment on global damper performance (damping force, stiffness trends) under dynamic loading, which is the focus of this paper.

In order to verify the accuracy and rationality of the model by setting boundary conditions and dynamic excitation, using the moving mesh to simulate the finite element model for a period of one period (0.8 s), the interception velocity contour diagram at a certain time is shown in Figure 5, when the piston boundary moves to both ends, the fluid in the pressure chamber moves to the other chamber through the damping hole due to pressure, due to the viscosity of the fluid, the flow velocity near the wall of the damping hole is smaller, and the flow velocity at the center line is larger, forming a velocity gradient from the center line to the pipe wall. Due to the abrupt expansion of the left port section of the damping hole, the flow velocity decreases.

Due to the movement of the boundary, one side of the cavity is compressed, and the viscous fluid flows into the other cavity through the damping hole, therefore, the pressure of the compression cavity is much higher than that of the other side of the cavity, and the pressure of the damping hole pipe pointing from the compression cavity to the direction of the lifting cavity decreases gradually, as shown in Figure 6a, the pressure contour diagram is more in line with the actual situation. In the model simulation, the hybrid model uses air-oil two-phase flow, the air content is set by initialization, and the air volume fraction in mixture model is 0.3%. The contour in Figure 6b reflects the air volume fraction at a point in time during the simulation. Due to the small air content, the air is evenly integrated into the viscous fluid, and with the reconstruction of the moving grid, the air is gradually unevenly distributed, and the injection port of the damping hole is more concentrated, When the piston continues to move to the maximum displacement, the air from the compression chamber flows into the tensioned chamber during compression, resulting in an uneven distribution of air content in the chambers on both sides. After the piston reaches its maximum displacement, it will move in the opposite direction according to the Moving Mesh Custom Function (UDF) and repeat the process until a cycle (0.8 s) is completed.

By applying a given external excitation to the finite element model, after a period (0.8 s) of simulation, the output velocity curve and displacement curve are retrieved and compared with the initially given excitation velocity and displacement to verify the accuracy and rationality of the model. As shown in Figure 7, the velocity and displacement curves of the output lag by approximately 0.05 s relative to a given excitation curve. This is the hysteresis characteristic of viscous fluid dampers, when subjected to external excitation, due to the large viscosity of the internal viscous fluid, the damping aperture is small, which will have a buffering and hindering effect on the abrupt excitation, so as to slow down the impact of the mechanical mechanism from external excitation. This hysteresis is related to the viscosity of the viscous fluid inside the damper, but also to the number and pore size of the damping holes.

Due to the velocity excitation function is given, the moving mesh is reconstructed and divided according to the velocity function, that is the piston wall moves according to the velocity function. As a result, the velocity curve of the output is highly consistent with the excitation function in terms of trend, as shown in Figure 7a, with no attenuation near the peak. In practice, due to the hysteresis characteristics of the damper, it will have a depleting and hindering effect on the abrupt external force, and the research in this paper is based on the given displacement and velocity, so there will be no large difference in the change of the curve.

As shown in Figure 7b, the output displacement curve also lags by about 0.05 s, for the same reasons as above, due to the hysteresis characteristics of the viscous fluid damper. Through the comparative analysis of the contour diagram and the output curve, it can be seen that the finite element model is reasonable and has high reliability.

2.3. The Performance Under Dynamic Incentives

2.3.1. The Performance Under General Incentives

The amount of air mixing in the viscous fluid model is controlled at 0.3%, 0.5%, 0.7%, 3%, 5% and 7%, the above-mentioned models with different air content are simulated in the time domain for one cycle, and the output damping force curves are shown in Figure 8.

When the air mixing volume is 0.3% (as shown in Figure 8a), the output damping force ranges between −4000 N and 4000 N, and its change trend is consistent with the dynamic excitation, and the peak point of the damping force curve corresponds to the peak point of the velocity excitation. With the increase of air content to 0.5% and 0.7%, the damping force output curves under the three different air contents have a high degree of agreement, and the change trend is consistent under dynamic excitation. Therefore, when the gas mixing volume is less than 1% and air amount of increase less than 1% (as shown in Figure 8a), the damping force of its output changes relatively small, and under normal conditions, it can meet the normal working requirements. When the air content is 3% (as shown in Figure 8a), the output damping force decreases slightly at the peak point of the curve, and when the air content reaches 5% and 7%, the damping force curve shows a downward trend at the peak point, $F_{3\%} >{ F}_{5\% }> F_{7\%}$, and the change is relatively flat, and there is no sudden change, and the rest of the part except the peak point does not obvious change. Therefore, when the air content of increase is greater than 1%, the damping force output by the viscous fluid damper shows a decreasing trend at the peak point of the excitation curve, and the air content has an impact on the energy consumption characteristics of the damper.

The dynamic properties of viscous fluid dampers are affected by the entrained air. As shown in the Figure 9 below, When the proportion of mixed air is 0.3%, 0.5%, 0.7%, 3%, 5% and 7%, there are obvious differences in the hysteresis curves. When the entrained air is 0.3%, as shown in Figure 9a, the closed area formed by the hysteresis curve is relatively full, with the increase of the gas mixture to 0.5% and 0.7%, the area of the closed curve continues to shrink, the displacement of the piston also begins to gradually decrease, the damping force in the hysteresis curve also decreases accordingly, and its energy consumption capacity is gradually reduced, but the decreasing trend is relatively moderate, and there is no obvious depressed change, that is the energy consumption characteristics and efficiency of the damper are good, and under general conditions, it can meet the normal working needs.

When the proportion of air in the damper reaches 3%, as shown in Figure 9b, the displacement in the hysteresis curve is less than the air content is less than 0.3%, the closed graphic area formed by the hysteresis curve is greatly reduced relative to the air content of 0.3%, and its energy consumption capacity is greatly reduced. When the proportion of air reaches 5% and 7%, the displacement change in the hysteretic curve is small, the damping force gradually decreases, and the area of the hysteresis curve decreases correspondingly, and its efficiency decreases significantly compared with that when the air content is less than 1%. Therefore, in terms of efficiency, when the air content in the damper is higher than 3%, its efficiency is greatly reduced.

2.3.2. The Performance Under Simple Harmonic Incentives

The input excitation of the finite element model is changed to sinusoidal velocity excitation, the output damping force variation curve is obtained by time-domain simulation, and different damping force curves are obtained by changing the air content in the finite element model, as shown in Figure 10. When the air content in the finite element model is less than 1%, the damping force curve is shown in Figure 10a, and the damping force change trend under the three air values (0.3%, 0.5% and 0.7%) is in good agreement, and there is no obvious change at the peak point and trough, the output damping force of the viscous fluid damper is relatively stable, so it can meet the general work needs. When the air content in the damper is 3%, 5% and 7%, respectively, the output damping force decreases at the peak and trough as the air content increases at equal spacing, but the curve changes in the same trend, as shown in Figure 10b. That is the air growth in the damper exceeds 1% and air amount of increase more than 1%, the damping force output of the damper will have a downward trend.

As the air content increases, so does the energy consumption characteristics of the damper. When the air content is less than 1% and the growth is less than 1% (0.3%, 0.5%, 0.7%).

The curve remains highly consistent, and the displacement and damping force do not change significantly, as shown in Figure 11a, that is the air content is less than 1%, the viscous fluid damper has good performance and can meet the general working needs. When the air content is greater than 1% and the growth is greater than 1%, the damping force of the hysteresis curve shows a decreasing trend, and decreases with the increase of air content, and its energy consumption capacity also decreases, as shown in Figure 11b.

2.4. Dynamic Stiffness in the Nonlinear Series Model

The amount of air mixture in the viscous fluid damper has a great influence on its dynamic stiffness (k) in the nonlinear series model, as shown in Figure 2, because the increase of the amount of air mixture can lead to the decrease of the volumetric elastic modulus of the viscous fluid. The initial position of the piston is located at the midpoint of the stroke, the length of the initial pressure liquid column is the same, and the effective area of the piston remains unchanged, it can be derived from Equation (4), under the action of external force, the piston compresses the viscous fluid, the modulus of elasticity decreases, its dynamic stiffness decreases with the amount of air increase in damper.

As can be seen from Figure 12, when the mixing volume is less than 1% (Figure 12a), the dynamic stiffness shows a upward trend at the peak with the increase of the mixing air volume, due to the given velocity excitation is dynamically changing, the peak point here corresponds to the point where the external excitation velocity value is the largest, in other words, the dynamic stiffness is consistent with the absolute value of the excitation velocity in the change trend (the positive and negative velocity in this paper refers to the opposite direction).

When the mixing air volume of the viscous fluid damper is greater than 1% (Figure 12b), the dynamic stiffness decreases significantly at the peak point, and the downward trend is violent with the increase of the mixing air volume. Therefore, the dynamic stiffness of the viscous fluid damper decreases with the increase of the air mixture, and it is very obvious when the air mixture is greater than 1%, which shows that the air mixture of the viscous fluid damper has a great influence on the dynamic stiffness of the damper.

As shown in Figure 13, the dynamic stiffness of the viscous fluid damper decreases with increasing air content when the external excitation is simple harmonic excitation. When the air content is less than 1% (Figure 13a), the dynamic stiffness has a greater change trend than the dynamic stiffness when the air content is more than 1%, This is because the elastic modulus changes significantly when air is mixed into the pure fluid, and the compressibility of the fluid increases, resulting in a drastic decrease in stiffness. And its variation trend is the same as that of the absolute value of the simple harmonic excitation, and the dynamic stiffness presents the maximum value at the peak point of the velocity. Comparing Figure 12a and Figure 13a, it can be seen that when the air content in the damper is the same, the dynamic stiffness is affected by the excitation amplitude, and increases with the increase of the excitation amplitude. When the air content is greater than 1%, as shown in Figure 13b, when the air content in the viscous fluid increases to a certain amount, the elastic modulus decreases, and the decreasing trend of its stiffness gradually slows down with the increase of air content. Thus, the damper mixed with air will have great influence on the elastic modulus of viscous fluid, and affect the compressibility of the fluid and stiffness, and will affect the damper elastic force. In order to further explore the influence of air content on the damper, it is necessary to optimize the calculation theory of stiffness and damping and verify it through experiments and simulations.

3. Linear Stiffness and Linear Damping

In the previous chapter, the performance and dynamic stiffness of viscous fluid dampers were investigated by varying the air content in the finite element model. Because the finite element model is a nonlinear model of spring and damping in series, the calculation of stiffness and damping coefficient is extremely complex, requires a large amount of data to fitting, and has poor generality. Therefore, in this chapter, in order to further explore the influence of air content on the stiffness and damping coefficient of the damper, the equivalent linear parallel stiffness and damping are established through the principle of equal energy consumption, and the calculation formula of linear parallel stiffness and damping is obtained, which greatly simplifies the calculation process.

3.1. Numerical Calculation Method

The mechanical model of the damper piston movement is established by using the central position of the piston movement as the displacement zero, and the following assumptions were made: (1) Excluding the friction and inertia forces of the piston; (2) The temperature of the moving fluid is constant; (3) The dynamical viscosity of the fluid in motion is constant, and its value takes the dynamical viscosity at the maximum shear rate.

Periodic displacement excitation is applied to the piston rod, the damping force applied to the piston rod is calculated by using numerical method. The piston motion should meet the following force balance conditions and displacement continuous conditions:

$$F=-F_c, F_k=F_c, x=x_C+x_k$$

(5)

where $x$ represent the known displacement excitation of the piston, $x_c$ represents the displacement of two-chamber volume changed due to fluid flowed through the damping hole, and $x_k$ represent the elastic displacement of the piston due to the fluid compression, and $x_c$ simply call damping displacement and $x_k$ elastic displacement, both are unknown quantities, $F$, $F_c$, $F_k$ are the external force acting on the piston, damping force and elastic force respectively, are also unknown. Because it is a series model, the elastic and damping forces are equal. According to the above conditions, the damping displacement and the elastic displacement of the piston can be determined, as well as the damping force and the elastic force.

In the numerical calculation, the step size is a $\mathsf{∆}t$, and the changes of excitation displacement, damping displacement and elastic displacement are defined as:

$$\Delta x_i=x_{i+1}-x_i, \Delta x_{\mathit{ci}}=x_{\mathit{ci}+1}-x_{\mathit{ci}}, \Delta x_{\mathit{ki}}=x_{\mathit{ki}+1}-x_{\mathit{ki}}$$

(6)

The specific calculation steps are as follows:

• Set the damping displacement at the initial time $t_0$ is 0, the elastic displacement and damping force are also 0.
• The change of excitation displacement $\mathsf{∆}{x}_i$ from $t_i$ to $t_{i+1}$ is known, the damping force $F_{\mathit{ci}}$ at $t_i$ time, the damping hole flow rate $v_{\mathit{ci}}$ and stiffness $k_i$ are also known. Let the fluid velocity in the damping hole at time ${\mathrm{t}}_{\mathrm{i}+1}$ be $v_{\mathit{ci}+1}$, and the damping force at time $t_{i+1}$ be determined by Equation (7):

  $$F_{\mathit{ci}+1}={\displaystyle \frac{p{A}_3}{2C_q^2}}{\nu }_{\mathit{ci}+1}^2$$

  (7)
• Calculate the change of damper displacement $\Delta x_{\mathit{ci}}$:

  $$\Delta x_{\mathit{ci}}={\displaystyle \frac{(v_{\mathit{ci}}+{ v}_{\mathit{ci}+1})\mathsf{∆}t}{2}}·{\displaystyle \frac{n{A}_0}{A_3}}$$

  (8)

  where $n$ is the number of damping holes of the same structure, $A_0$ is the area of a single damping hole.
• The stiffness $k_{i+1}$ at time $t_{i+1}$ is calculated by Formula (4). The elastic displacement change ${\mathsf{∆}x}_{\mathit{ki}}$ is determined by the damping force at time $t_i$ and $t_{i+1}$ calculated according to the following Equation (9):

  $$F_{\mathit{ci}+1}-F_{\mathit{ci}}={\displaystyle \frac{\left(k_i+k_{i+1}\right){\mathsf{∆}x}_{\mathit{ki}}}{2}}$$

  (9)
• The convergence condition for judging $i+1$ whether the round calculation is completed is:

  $$\left|{\mathsf{∆}x}_{\mathit{ci}}+{\mathsf{∆}x}_{\mathit{ki}}-{\mathsf{∆}x}_i\right|\le \epsilon$$

  (10)

  where $\mathsf{\epsilon }$ is a very small positive number, which is used to control the accuracy of the computer. During the calculation process, we search for the optimal value of the fluid velocity $v_{\mathit{ci}+1}$ at time $t_{i+1}$ between $\pm 2v_{\mathit{max}}$ (the maximum flow rate is $v_{max}$ in the damping hole during not to consider the fluid compression), so that the convergence condition is satisfied, the final solution of ${\mathsf{∆}x}_{\mathit{ki}}$ and ${\mathsf{∆}x}_i$ is obtained, and the next round of calculation is completed. Flowchart of the numerical iteration algorithm for solving the nonlinear series model can be seen from Figure 14.

3.2. Equivalent Linearization of the Damping and the Stiffness

Based on the numerical solution of the nonlinear tandem model, the nonlinear modes convert into a linear series model with equivalent linear series damping coefficient $c_s$ and equivalent linear series stiffness coefficient $k_s$.

The equivalent linear series damping $c_s$ is calculated according to the principle of equivalent energy consumption. Count the area surrounded by a periodic stagnation curve in the nonlinear simulation, the energy consumed $W$, $W$ is represent for $c_s$: $c_s{v}_{i }$ represents damping force of certain period time. The displacement within this time period is $v_i\mathsf{∆}t$, add together after multiplying of damping force and displacement is equal to $W$ and the equivalent damping is $c_s$. It is expressed as:

$$c_s={\displaystyle \frac{W}{\sum _T{v}_i^2\mathsf{∆}t}}$$

(11)

The equivalent linear series stiffness $k_s$ is calculated according to the principle of equivalent stored energy, and the energy $W_K = {\displaystyle \frac{\sum \mathsf{∆}{x}_{\mathit{ki}}^2{k}_i}{2}}$ stored in a compression stroke of the spring is calculated in the nonlinear numerical solution. The equivalent linear stiffness $k_s$ is used to express the equivalent stored energy $W_{k }= {\displaystyle \frac{k_s\sum {\mathsf{∆}x}_{\mathit{ki}}^2}{2}}{k}_i$, and the equivalent linear stiffness $k_s$ is calculated.

$$k_{s }={\displaystyle \frac{2W_k}{\sum \mathsf{∆}{x}_{\mathit{ki}}^2}}$$

(12)

Because it is not easy to analyze and judge intuitively the influence of elastic force and damping force on the system, people are used to use the parallel mode of stiffness and damping to transform the nonlinear series model into linear parallel model on the basis of equivalent linear damping and equivalent linear stiffness. The displacement continuous equation and the force balance equation for the tandem model are:

$$x=x_k+x_c, F_k=k_s{x}_k=F_c=c_s{\dot{x}}_c$$

(13)

where $x$ is the total displacement, $x_k, x_c$ are elastic displacement and damping displacement, respectively. $F_k$ and $F_c$ are the elastic forces and the damping forces. The expression for the damping displacement from Equation (14):

$$x_c = {\displaystyle \frac{x\left(k_s^2-c_s{k}_{s}i\omega \right)}{k_s^2+c_s^2{\omega }^2}}$$

(14)

And the expression of the external force $F$:

$$F = {\displaystyle \frac{c_s^2{\omega }^2{k}_{s}x}{k_s^2+c_s^2{\omega }^2}} + {\displaystyle \frac{k_s^2{c}_{s}i\omega x}{k_s^2+c_s^2{\omega }^2}}$$

(15)

The force for the parallel model can be expressed: $F=k_p{x+c}_{p}i\omega x$. Based on Equation (15), the linear series model can further be converted to an equivalent linear parallel model. Thus, obtained the equivalent linear parallel stiffness $k_p$ (additional stiffness) and equivalent linear parallel damping coefficient $c_p$ of a linear parallel model [27]:

$$k_p={\displaystyle \frac{c_s^2{\omega }^2{k}_s}{k_s^2+c_s^2{\omega }^2}}, c_p={\displaystyle \frac{k_s^2{c}_s}{k_s^2+c_s^2{\omega }^2}}$$

(16)

To sum up, the additional stiffness k_p/equivalent parallel stiffness is the stiffness coefficient in the equivalent linear parallel model, derived by linearizing the nonlinear series model using the equal energy principle. This is not a direct physical spring but rather an equivalent parameter representing the in-phase component of the force under dynamic excitation. When air is introduced, the dynamic stiffness k/nonlinear series stiffness and its equivalent linearized series stiffness k_s decrease. As a result, the denominator in Equation (16) decreases, causing the calculated equivalent parallel stiffness k_p to increase. Physically, the entrained air increases the system’s compressibility, enhancing the “balloon effect.” This means a larger portion of the total output force becomes an elastic restoring force, which is captured as an increased k_p in the parallel model.

3.3. Validation of Equivalent Linear Models

In the previous section, on the basis of the nonlinear series model of the viscous fluid damper, the nonlinear model is transformed into an equivalent linear parallel model by using the equal energy principle, and the calculation expressions of linear parallel stiffness (additional stiffness) and linear damping coefficient are obtained, which greatly simplifies the calculation process of stiffness and damping.

In this subsection, in order to further verify the correctness of the linear parallel model, the simulation results of the finite element model in the previous chapter are compared with the simulation results of the linear parallel model, and the correctness and rationality of the linear parallel model are verified by changing the excitation frequency and comparing the agreement of the hysteresis curves of the two models at different excitation frequencies. As shown in Figure 15, the air content is 0.3%, when the simple harmonic excitation frequency is 2 Hz (Figure 15a), the damping force of the linear parallel model and the nonlinear model is between −2000 N and 2000 N, and the displacement is between −2.65 mm and 2.65 mm, and the change trend of the curve has a high degree of agreement, the damping force reaches the maximum at the peak point of the excitation speed, and the nonlinear model is slightly larger than the linear model near the peak point of the output damping force, which is because the linear parallel model is established on the basis of the numerical calculation of the series model by using the principle of equal energy, so there is a slight difference near the peak point.

Other parameters remain unchanged, when the excitation frequency is changed to 4 Hz (Figure 15b), it can be seen that the output damping force of the linear and nonlinear models is significantly larger, the damping force ranges from −3000 N to 3000 N, and the hysteresis trend is consistent, with only a small difference near the peak point, for the same reasons as above. It can be seen that the energy consumption of the linear parallel model is consistent with that of the nonlinear model, and the linear parallel model has high confidence in studying the performance of viscous fluid dampers.

It is instructive to compare the proposed energy-based equivalent linearization with alternative methods reported in the literature. The direct least-squares fitting approach (in time or frequency domain) is purely descriptive, as it calibrates parameters to match specific experimental data and lacks predictive capability for untested conditions. The describing function method is powerful for frequency-domain analysis but typically assumes a fixed input amplitude and does not directly reflect the damper’s energy dissipation and storage mechanisms. In contrast, our method derives the equivalent linear parameters c_s, k_s, and subsequently c_p, k_p from the physics-based nonlinear series model before any experimental comparison. It links c_p and k_p explicitly to the energy dissipated and stored over a cycle, offering predictive ability, physical interpretability, and computational simplicity—making it particularly suitable for parametric studies and preliminary damper design.

4. Viscous Fluid Damper Experiment

In the previous chapter, the tandem stiffness and damping of the damper were converted into equivalent linear parallel stiffness (additional stiffness) and linear damping by the principle of equal energy consumption. By comparing with the nonlinear model, it can be seen that the two have a high degree of agreement. In this chapter, the additional stiffness and linear damping will be verified by experimental and nonlinear models, the effect of air content on stiffness and damping will be further investigated by experimental and equivalent linear parallel models.

4.1. Experimental Setup and Input Excitation

The experimental setup is shown in Figure 16. The test principle of the damper is shown in Figure 17. The trench cam mechanism is connected with the cam driven rod and the damper piston rod for a simple harmonic curve with an amplitude of 3 mm. The excitation frequency is determined by the cam speed, which is controlled by the variable frequency speed control motor. 350 cSt dimethyl silicone oil (mean viscosity 350 cSt at 25 °C), the damping hole diameter of 2.2 mm, and two equally sized damping holes were opened on the piston.

The following information on measurement uncertainty, calibration, and repeatability is provided to support the experimental results. The force sensor (CL-YD-312, Jiangsu Lianneng, Yangzhou, China) has a measurement range of ±5000 N with a manufacturer-specified accuracy of ±0.5% full scale. The displacement sensor (LVDT, Helie Jiangsu, Beijing, China) has a range of 0–10 mm with an accuracy of ±0.01% full scale. Both sensors were calibrated before the test series: the force sensor was calibrated using known dead weights (0–4000 N in 500 N increments), and the displacement sensor was calibrated using a micrometer stage (0–10 mm in 1 mm increments). The maximum deviation observed during calibration was within the specified accuracy limits. To assess repeatability, each test condition was repeated three times under identical environmental conditions (temperature maintained at 2 °C). The cycle-to-cycle variability in peak damping force was within ±3.5%, and the variability in energy dissipation per cycle was within ±5%. All reported results are representative of the median cycle from the three repeated tests.

Force and displacement data were collected during the test, and FFT was transformed to obtain displacement $X\left(o\right)$ and force $F\left(o\right)$ in the frequency domain. Figure 18a shows the force and displacement time courses. Figure 18b shows the frequency spectrum of the force, concentrated on the excitation frequency of 3.5 Hz, and the other frequency components are basically 0. In theory, the phase advance displacement of the pure damping force is 90°, but due to the influence of fluid elasticity, there is an elastic force with the displacement, so the phase advance of the force is less than 90°.

The linear parallel model has the relationship between force and displacement in the frequency domain:

$$F\left(\omega \right)=k_{p}X\left(\omega \right)+c_{p}i\omega X\left(\omega \right)$$

(17)

The damping coefficient $c_p$ and the stiffness coefficient $k_p$ can be obtained from Equation (18) are:

$$c_p=\mathit{Im}\left({\displaystyle \frac{F\left(\omega \right)}{\omega X\left(\omega \right)}}\right), k_P=\mathit{Re}\left({\displaystyle \frac{F\left(\omega \right)}{X\left(\omega \right)}}\right)$$

(18)

4.2. Additional Stiffness and Damping at Different Air Content

In order to further investigate the effect of air content on the performance and stiffness of the damper, controlled experiments with varying air mixtures were conducted. This section will extract a part of the liquid from the well-vented fluid damper to obtain a controlled mixture proportional working condition experiment. The air was introduced as discrete micro-bubbles. The procedure involved first thoroughly degassing the damper. Then, a precise volume (2 mL and 6 mL) of fluid was extracted using a syringe through the filling port. Under atmospheric pressure, air was drawn in and mechanically mixed into the viscous fluid through the shearing action generated by manually reciprocating the piston. To promote a more uniform distribution of bubbles within both chambers, the damper was cycled several times at low speed before each formal data acquisition session.

2 mL and 6 mL (Volume of Fluid Extracted, V_E) of fluid were extracted in 200 mL (Total Cylinder Volume, V_T), the mixed air was 1% and 3%, respectively, and the fully exhausted fluid damper experiment was fitted, the mixture ratio was 0.7% (Initial Residual Ratio, R_I), and the mixture ratio after pumping was about 1.7% and 4.7% (Final Ratio, R_F). The calculation was: R_F = R_I + V_E/V_T. The “initial residual ratio R_I (0.7%)” for the well-vented condition was not known a priori. It was obtained by fitting the experimental data from the well-vented case with simulation results from our nonlinear model for various assumed air contents. This section still uses 350 cSt dimethicone with a density of 970 kg/m³ and a damping hole of 2.2 mm. The following Figures show the experimental result curve.

It should be noted that the identified thresholds are specific to the damper geometry and excitation conditions used in this study (e.g., specific damping hole diameter, fluid viscosity, frequency range). The thresholds appear where the effect of air becomes measurable with our setup. These thresholds might shift with different parameters: A smaller damping hole diameter (higher damping force) might make the system more sensitive to air, potentially lowering the threshold where effects become significant. Conversely, a larger gap might raise the threshold. Higher frequencies amplify the effect of air on additional stiffness, so the “significant effect” threshold might occur at a lower air content at higher frequencies.

After the equivalent linearization of the nonlinear model, the simulation results of the linear parallel model are compared with the experimental results, and the hysteresis curves at different frequencies are shown in Figure 19, Figure 20 and Figure 21, the main data obtained are shown in

Table 1. Experimental and simulated data of a damper at 2 Hz.

Mixed Air	Variable	Experiment	Linear Model	Error (%)
0.7%	$k_p/(\mathrm{K}\mathrm{N}·{\mathrm{m}}^{-1})$	1.16	11.09	856%
	$c_p/(\mathrm{K}\mathrm{N}·\mathrm{s}·{\mathrm{m}}^{-1})$	29.05	30.92	6.4%
	$\mathrm{z}$	0.003	0.03	900%
1.7%	$k_p/(\mathrm{K}\mathrm{N}·{\mathrm{m}}^{-1})$	15.76	29.78	89%
	$c_p/(\mathrm{K}\mathrm{N}·\mathrm{s}·{\mathrm{m}}^{-1})$	28.56	29.65	3.8%
	$\mathrm{z}$	0.043	0.080	86%
4.7%	$k_p/(\mathrm{K}\mathrm{N}·{\mathrm{m}}^{-1})$	62.65	82.79	32%
	$c_p/(\mathrm{K}\mathrm{N}·\mathrm{s}·{\mathrm{m}}^{-1})$	24.91	26.25	5.4%
	$\mathrm{z}$	0.2	0.251	26%

. The hysteretic curves under the three working conditions are in good agreement and the change trend is also the same, and the experimental results are compared with the linear model again through numerical calculations, and the shape and change trend of the hysteretic curves of the three are also highly consistent, which can prove the correctness and rationality of the model. After the proportion of the mixed air increases, the hysteresis curve of the damper changes significantly, the tilt is serious, and the F-v curve gap increases, indicating that the mixed air will increase the additional stiffness. As shown in the Figure 19, Figure 20 and Figure 21 at 2 Hz, the maximum damping force of the damper with a 0.7% mixture ratio is 1300 N, and the maximum damping force is reduced to 1200 N by 1% for the mixed air, the maximum damping force is only about 1100 N when the mixed air is 4.7%, so it can be obtained that the amplitude of the damping force decreases after the proportion of gas mixture increases. After 4.7% of the air content, the energy consumption curve also decreased from 9.46 J at 0.7% to 7.52 J, a decrease of 21%. As shown in the Figure 22, by comparing the additional stiffness, linear damping and $z$ (the ratio of elastic force to damping force) of the experiment and the linear model, it can be seen that the air content in the damper mainly has an effect on the additional stiffness, and with the increase of air content, the additional stiffness gradually increases (Figure 22a), while the air content has less influence on the linear damping, and the linear damping coefficient decreases slightly with the increase of air content (Figure 22b), and the change amplitude is small relative to the additional stiffness. By observing $z$ (the ratio of elastic force to damping force), it can be seen that the air content in the damper mainly affects the elastic force, and as the air content increases, the elastic force of the viscous fluid damper also increases, as shown in Figure 22c. Therefore, through the comparison of experiments and simulations, it can be seen that the air content in the damper has a great influence on the additional stiffness, and has little influence on the linear damping, and mainly affects the elastic force of the viscous fluid damper.

The relative errors between the linear model predictions and experimental results are summarized in Table 1. For the equivalent damping coefficient c_p, the relative errors range from 3.8% to 6.4% across all tested air contents and frequencies, indicating excellent predictive accuracy. For the equivalent stiffness k_p, the relative error is 856% at 0.7% air content, which is primarily because the experimental value (1.16 kN/m) is extremely close to zero—within the measurement noise floor—while the model predicts a small positive value (11.09 kN/m). In absolute terms, the difference is only 9.93 kN/m, which is negligible for engineering applications. At 1.7% and 4.7% air content, where k_p becomes practically significant, the relative errors are 89% and 32%, respectively. The error decreases as air content increases, indicating that the model’s predictive accuracy for k_p improves in the parameter range where additional stiffness matters most for structural design. This larger discrepancy in k_p is primarily attributed to the sensitivity of the stiffness calculation to minor uncertainties in the phase angle between force and displacement, as well as the influence of unmodeled friction and inertial effects at low damping force levels. Despite these absolute errors, the trends in k_p with increasing air content are consistently captured by the model, as shown in Figure 21. The z-value errors follow a similar pattern to k_p, as z is directly proportional to k_p.

It is important to note that the 856% error at 0.7% air content does not indicate model failure. Rather, it reflects the fact that the experimental k_p (1.16 kN/m) is essentially zero within measurement uncertainty. The model correctly predicts that additional stiffness is negligible below 1% air content, which is the key engineering insight. The absolute error (9.93 kN/m) is practically insignificant. Therefore, we do not consider this data point as a meaningful validation target for k_p.

Physical interpretation: The increase in additional stiffness (k_p) with air content arises from the increased compressibility of the air-fluid mixture. When air bubbles are entrained, the effective bulk modulus (E) of the fluid decreases substantially (see Equation (3) and Figure 3). Under dynamic compression, a portion of the piston displacement is absorbed by compressing the air rather than forcing fluid through the damping orifice. This delay in fluid flow manifests as an in-phase elastic force component in the total output force, which is captured as an increased k_p in the equivalent parallel model. Notably, the effect is more pronounced on stiffness than on damping because the damping force depends on the square of the flow velocity (Equation (1)), while the elastic force is directly proportional to the bulk modulus. A small reduction in E thus causes a proportionally larger relative increase in the elastic term, explaining why z = k_p/(c_pω) grows rapidly with air content.

4.3. Additional Stiffness and Damping at Different Frequencies

In the previous section, the performance and additional stiffness variation of viscous fluid dampers at same frequencies for the different air content was explored. In this section, experiments and simulations will be carried out on models with the same air content by changing the excitation frequency. 350 cSt dimethicone with a density of 970 kg/m³ damped bore diameter of 2.2 mm and fluid mixed with 2.6% air. Figure 22, Figure 23 and Figure 24 shows the test result curve, and the test data are listed in

Table 2. Experimental and simulation data of gas mixture 2.6% damper.

Frequency	Variable	Experimental Results	Linear Model	Error (%)
2 Hz	$k_p/(\mathrm{K}\mathrm{N}·{\mathrm{m}}^{-1})$	34.4	43.9	31.4%
	$c_p/(\mathrm{K}\mathrm{N}·\mathrm{s}·{\mathrm{m}}^{-1})$	26.0	26.8	3.1%
	$\mathrm{z}$	0.105	0.130	23.8%
3 Hz	$k_p/(\mathrm{K}\mathrm{N}·{\mathrm{m}}^{-1})$	77.4	93.0	20.2%
	$c_p/(\mathrm{K}\mathrm{N}·\mathrm{s}·{\mathrm{m}}^{-1})$	26.9	28.0	4.1%
	$\mathrm{z}$	0.153	0.176	15.0%
4 Hz	$k_p/(\mathrm{K}\mathrm{N}·{\mathrm{m}}^{-1})$	137.8	154.0	11.8%
	$c_p/(\mathrm{K}\mathrm{N}·\mathrm{s}·{\mathrm{m}}^{-1})$	28.3	29.0	2.5%
	$\mathrm{z}$	0.194	0.211	8.8%

.

At the same air content, the simulation results of the linear parallel model with different frequencies are compared with the numerical results, and then the two are compared with the experimental results, as shown in Figure 23, Figure 24 and Figure 25. At different frequencies, the shape and trend of the hysteresis curves and F-v curves of the three results are in agreement, which shows that the equivalent linear parallel model has high reliability. As can be seen from Figure 23, Figure 24 and Figure 25, when the air content is the same, with the increase of the excitation frequency, the hysteretic curve gradually tilts, and the spacing generated by the F-v curve gradually increases, which is due to the presence of air in the viscous fluid, and the presence of air enhances the influence of the compressive properties of the fluid, resulting in an increase in additional stiffness. As the frequency increases, the value of the additional stiffness grows larger and larger, from 33.4 KN/m to 77.4 KN/m, and at 4 Hz, the additional stiffness is 137.8 KN/m. As can be seen from Table 2, the $c_p$ calculated by linear model simulation is close to each experimental frequency, $c_p$ with the biggest difference at 3 Hz, but there is only a 4% difference, within a reasonable range. This indicates that the model can predict damper properties at different frequencies. Moreover, at different frequencies, by comparing the additional stiffness, linear damping coefficient and z-value of the linear model with the experiment, this is shown in Figure 25, the $k_p$ increasing significantly with increase of frequency (Figure 26a), and the $c_p$ also increased with frequency (Figure 26b). but its increased trend is less than additional stiffness $k_p$ The value of $z$ used to measure the ratio of elastic force to damping force increases significantly with increased frequency (Figure 26c). The relative errors for k_p decrease with increasing frequency (from 31.4% at 2 Hz to 11.8% at 4 Hz), indicating that the model’s predictive accuracy for additional stiffness improves under higher-frequency excitations—conditions where additional stiffness is most relevant to structural dynamics. Therefore, through the comparison of experiments and simulations, it can be seen that the frequency in the damper has a great influence on the additional stiffness, and has little influence on the linear damping, and mainly affects the elastic force of the viscous fluid damper.

Physical interpretation: The observed increase in additional stiffness with excitation frequency is attributed to the rate-dependent behavior of the air-fluid mixture. At higher frequencies, the pressure within the compression chamber rises and falls more rapidly. The entrained air bubbles, due to their finite size and the time required for gas dissolution or bubble collapse, cannot fully respond to the rapid pressure changes. Consequently, the mixture behaves as a more stiff, less dissipative medium at higher frequencies, leading to a larger elastic force component relative to the viscous damping force. This frequency sensitivity of k_p is a key signature of viscoelastic behavior in aerated fluids and is consistent with the general trends observed in polymer rheology (e.g., increased storage modulus at higher frequencies). The practical implication is that even a small amount of entrained air (e.g., 1–2%) can cause a non-negligible additional stiffness when the damper is subjected to high-frequency vibrations, such as those induced by machinery or high-speed wind loads.

A key limitation of this study is the assumption of a uniform air-bubble distribution within the fluid. In reality, during static periods or low-velocity motion, bubbles tend to coalesce and rise due to buoyancy, leading to a non-uniform spatial distribution. This non-uniformity introduces uncertainty into the effective compressibility and is considered a primary source of the minor discrepancies between our experimental and model results, particularly in the measured additional stiffness (k_p). Despite this, the overall consistency between experimental trends and model simulations confirms that the homogeneous assumption is reasonable for capturing the dominant dynamic effects. Future work should employ advanced techniques, such as high-speed imaging, to directly observe bubble dynamics and enable more sophisticated multiphase modeling.

5. Conclusions

5.1. Summary of Key Findings

This study investigated the effects of entrained air (volume fraction < 10%) on the dynamic performance and stiffness of viscous fluid dampers through a combined finite element, equivalent linear modeling, and experimental approach. The main findings are summarized as follows:

• When the air volume fraction is less than 1%, the entrained air has little effect on the damping force. When the air increase amount exceeds 1%, the damping force decreases noticeably, especially at the peak of dynamic excitation, and the reduction becomes more pronounced with higher air content.
• The energy dissipation capacity of the damper decreases with increasing air content. The hysteresis curve becomes less full and increasingly tilted, and the enclosed area shrinks. The gap in the F-v curve widens with higher air mixture.
• The physical series stiffness (governed by the bulk modulus E) decreases with air content, while the equivalent parallel additional stiffness (k_p) derived from the linearized model increases. This seemingly contradictory behavior is resolved by recognizing that k_p represents the in-phase elastic force component that emerges from the compressibility-induced phase shift, rather than a direct physical spring.
• The additional stiffness k_p is highly sensitive to excitation frequency and amplitude, increasing with both. The damping coefficient c_p is much less affected by air content and frequency. The ratio z = k_p/(c_pω) therefore grows rapidly with air content and frequency, indicating that the elastic force dominates under these conditions.

5.2. Practical Engineering Implications

The results have several direct implications for the design, quality control, and application of viscous fluid dampers:

• The study shows that even a small amount of entrained air (as little as 1–3%) can significantly reduce energy dissipation (by up to 21% at 4.7% air content) and introduce non-negligible additional stiffness. Therefore, degassing procedures during fluid filling should be strictly enforced, and sealed damper designs should be verified for air-tightness to prevent air ingress over time.
• The observed thresholds of 1% and 3% are not universal constants. They depend on damper geometry (e.g., orifice diameter) and excitation conditions (frequency, amplitude). Engineers should conduct parametric analyses using the proposed model to determine the critical air content for their specific applications.
• For dampers subjected to high-frequency vibrations (e.g., machinery isolation, high-speed rail, wind-excited structures), even a 1% air mixture may produce measurable additional stiffness that could alter the natural frequency of the protected structure. In such cases, the model can be used to pre-evaluate the impact of residual air.
• The equivalent linear parallel model provides a simple yet predictive tool for balancing energy dissipation and additional stiffness. For example, reducing orifice diameter increases damping but also amplifies the frequency sensitivity of k_p; this trade-off can be quantitatively assessed during the design phase using the proposed method.

5.3. Limitations and Future Research Directions

Several limitations of the present study should be acknowledged, and they point to clear directions for future work:

• The uniform bubble distribution assumption in the FEM and two-phase model may not hold under static or low-velocity conditions, where bubbles tend to coalesce and rise due to buoyancy. This non-uniformity is a source of uncertainty, particularly for k_p measurements at low air contents (as reflected in the larger relative errors at 0.7% air content).
• The experimental frequency range was limited to 0.25–5 Hz for sinusoidal excitation and up to ~4 Hz for dynamic tests. The trends at higher frequencies (e.g., >10 Hz) were extrapolated from the model and require experimental validation.
• The model does not account for temperature-dependent viscosity or thermal-pressure coupling, which may become important under long-duration or high-energy loading (e.g., seismic events).
• The equivalent linearization method, while predictive, yields large relative errors for k_p at very low air contents (e.g., 856% at 0.7%), because the experimental value approaches the measurement noise floor. At practically significant air contents (≥1.7%), the relative errors for k_p range from 32% to 89%, which are larger than those for c_p (typically <6%). This reflects the greater sensitivity of stiffness to experimental uncertainties and phase measurement errors.

Future research directions:

• Following recent studies (e.g., the temperature-pressure coupling work in Machines, 2024), we plan to extend our model to incorporate temperature-dependent viscosity and thermal expansion, enabling prediction of damper performance under long-duration or cyclic loading.
• Future work will employ more advanced two-phase models (e.g., Eulerian-Eulerian with bubble coalescence and breakup) and experimental techniques (high-speed imaging, ultrasonic probing) to capture non-uniform air distribution and its effect on local compressibility.
• New experiments at higher frequencies (up to 20 Hz or more) and with controlled, independently measured air fractions (e.g., using in-situ pressure-volume-temperature sensors) are needed to further validate the model’s predictive capability for k_p in the high-frequency regime.
• The equivalent linear parallel model will be integrated into system-level simulations of damped structures to quantitatively assess how air-induced additional stiffness affects overall dynamic response, particularly for systems with tight frequency tolerances.