The Issue of Hydrodynamic Friction in the Context of the Operational Properties of Ring-Shaped Torsional Vibration Dampers

Abstract

Improving the reliability and durability of internal combustion engines in marine vessels is a complex issue. The vibrations generated in these engines significantly affect their proper operation. One of the current research challenges is identifying effective methods to reduce, among other things, torsional vibrations generated within the crank–piston system. To mitigate these vibrations, viscous dampers are commonly used. The selection of a viscous damper for a high-power multi-cylinder engine, such as those in marine power plants, requires a thorough understanding of the thermo-hydrodynamic properties of oil films formed in the spaces between the damper housing and the inertial mass. The description of the phenomena involved is complicated by the variable positioning of the inertial mass center relative to the housing during operation. Most previous studies assume a concentric alignment between these components. The main novelty of this work lies in highlighting the combined effect of the eccentric motion of the inertial ring on both hydrodynamic resistance and thermal characteristics, which has not been fully addressed in existing studies. This article defines the oil flow resistance coefficients and develops static characteristics of the dampers. Additionally, it evaluates the impact of the size of the frontal and cylindrical surfaces of the damper on its heat dissipation capacity. The presented characteristics can be utilized to assess the performance parameters of this type of damper.

1. Introduction

To assess the reliability of crankshafts and other mechanical components, temperature measurement has emerged as a vital tool for monitoring the system performance. Temperature variations, especially in hydraulic systems, have been shown to influence the reliability and efficiency of various mechanical processes. Novaković et al. [1] explore how changes in hydraulic parameters and tank capacity impact the heating of hydraulic oil, highlighting the relationship between temperature fluctuations and operational performance. Similarly, Hryciów et al. [2] investigate how temperature affects the damping characteristics of hydraulic shock absorbers, underscoring the critical role temperature plays in component functionality. Furthermore, Stawiński et al. [3] examine the effect of hydraulic oil temperature on the performance of variable-speed pumps, demonstrating that temperature management is essential for optimal system behavior. Additionally, Chen et al. [4] focus on the heat dissipation characteristics of torsional vibration dampers used in engine crankshafts, further emphasizing the significance of thermal analysis in ensuring mechanical integrity and performance. These studies reflect growing awareness of the importance of thermal conditions in ensuring the reliability and longevity of mechanical systems, including crankshafts, where temperature-based monitoring can play a significant role in early failure detection and prevention.

A properly functioning crankshaft is the cornerstone of reliable operation in many propulsion systems. Despite stringent design principles that account for stress distributions [5], rotational speeds [6], and material standards, crankshafts continue to experience failures. Even advanced methods for monitoring crankshaft conditions [7,8,9,10,11] do not fully prevent damage, which occurs across various applications such as compressors [12], heavy trucks [13], buses [14], and even aircraft [15]. Failures have been closely monitored and documented in numerous studies. For example, Wang et al. [16] describe a failure that occurred after only 20 min of operation, while Wang et al. [17] focus on a stochastic analysis of fatigue processes and develop models to assess the probability of failure. Fonte et al. [18] report a case of failure after just under three years of service, followed by a subsequent failure of the same crankshaft after repair and only 30,000 km of vehicle operation. Similarly, Pandey [19] details a failure of a forged carbon steel crankshaft (with 0.45% carbon content) in a diesel engine, caused by material fatigue wear.

One of the primary causes of crankshaft failure is torsional vibration, which systematically subjects the component to periodic high-frequency stresses. These vibrations, often imperceptible and silent, pose a significant challenge to ensuring reliable propulsion system operation. A key mechanism for mitigating torsional vibration effects and extending the reliable operational lifespan of the crankshaft is the use of a viscous torsional vibration damper.

Wilson’s book [20] offers a foundational introduction to this topic, providing a classic treatment of torsional vibration issues in rotating systems. It covers both theoretical principles and practical guidelines for design and for troubleshooting common challenges associated with shaft vibrations in machinery. Deuszkiewicz et al. [21], on the other hand, present a method for modeling dynamic phenomena in an internal combustion engine’s propulsion system equipped with a torsional vibration damper. Their study discusses the damper’s influence on reducing crankshaft vibration amplitudes and provides simulation results confirming the effectiveness of the solution.

In the literature (see [22,23,24]), it is noted that as early as the 1920s, torsional vibrations in the crankshafts of marine engines were observed in the shipbuilding industry. According to historical accounts, the first use of a viscous torsional vibration damper occurred on a submarine. This device was installed not only to extend the lifespan of the propulsion system but also to reduce the emission of undesirable acoustic waves and to improve working conditions for the crew. As Lakshminarayanan and Agraval [22] point out, the introduction of the damper was a breakthrough, as it significantly reduced vibration levels that could cause both technical issues (such as shaft fractures and bearing damage) and discomfort for personnel operating the vessel.

Latarche [23] highlights that the experiences gained during work on submarines contributed to the further development of dampers used in various diesel engines, which later led to increased reliability in large marine propulsion systems. Meanwhile, Wilbur and Wight [24] emphasize that the use of viscous dampers was not only crucial for operational safety but also essential for the economical and comfortable operation of marine machinery by mitigating resonance phenomena and the risk of vibrations with amplitudes threatening the durability of the entire system.

Although the issue of torsional vibration damping has a long history, it remains highly relevant today, as evidenced by numerous publications in recent years. Sezgen and Tinkir proposed a hybrid damping approach for optimizing torsional vibration dampers in cranktrain systems, improving system dynamics under varying operating conditions [25]. Similarly, Qiu et al. presented a comprehensive theoretical and application-based framework for designing torsional dampers tailored to diesel engines [26]. Kozytskyi and Kiriian introduced a torsional damper based on dilatant fluid, highlighting its suitability for four-stroke diesel engine crankshafts due to its adaptive viscosity behavior [27]. Kim et al. investigated how marine operation conditions influence viscous damper characteristics in marine diesel engine shafting, revealing a direct impact on torsional vibration levels [28]. Hohl et al. described the design and deployment of torsional vibration dampers in field conditions, providing insights into their real-world application and reliability [29]. Yucesan et al. applied physics-informed neural networks to adjust torsional damper models, achieving more accurate predictions of system behavior [30]. Venczel et al. reviewed both historical and emerging damping strategies in the vehicle industry, offering a broad perspective on current and future technologies for torsional vibration mitigation [31].

Kodama and Honda [32] characterize the properties of a viscous torsional vibration damper filled with silicone oil, emphasizing mathematical modeling. Their work analyzes the crankshaft system dynamics and proposes vibration control methods based on the physical and rheological parameters of the damping medium. Meanwhile, Chmielowiec et al. [33] examine the impact of silicone oil deficiency in a viscous torsional vibration damper on its damping performance. Their experimental results and theoretical analysis reveal how insufficient fluid levels affect damper functionality and the overall operational safety of the engine. Zhai et al. examined how the aging of silicone oil in dampers affects crankshaft vibration, underlining the importance of damping medium longevity in heavy-duty diesel engines [34]. Chen et al. analyzed the heat dissipation performance of torsional vibration dampers, emphasizing its role in crankshaft durability and vibration control [4]. A related study by Qiu et al. focused on the optimization of silicone oil-based dampers for diesel generators, balancing thermal performance with damping efficiency [35].

Most recent studies have also addressed vibration damping, primarily focusing on longitudinal vibrations and the application of computational fluid dynamics (CFD) to analyze and optimize damping systems. For instance, Hu et al. [36] conducted a numerical and experimental study on the damping performance of fluid viscous dampers, considering the gap effect, which is crucial for understanding the damping behavior in various dynamic systems. Emami et al. [37] examined the contribution of fluid viscous dampers to the fatigue life of offshore wind turbines, showing the importance of damping systems in mitigating vibration-induced fatigue. Moreover, Lak and Zahrai [38] explored self-heating phenomena in viscous dampers under different loading conditions, using CFD simulations to enhance the design of damping systems under dynamic loads. Zhang et al. [39] investigated the coupling effects of temperature and pressure on the mechanical behavior of viscous fluid dampers, highlighting the impact of environmental factors on damping performance. Greco and Marano [40] focused on identifying the parameters of generalized models for fluid viscous dampers, improving the precision of CFD simulations in predicting damping behavior. Cucuzza et al. [41] compared different numerical models for fluid viscous dampers, providing a deeper insight into the optimization of damping systems through CFD-based approaches. Finally, Lak et al. [42] applied CFD modeling and full-scale dynamic testing to better understand the performance of viscous dampers, demonstrating the importance of detailed fluid dynamics simulations in the design of efficient damping systems. Although these studies primarily focus on longitudinal vibrations and the performance of dampers in different mechanical systems, they provide valuable insights into the use of CFD to model and optimize vibration damping, offering a broader perspective on the application of fluid-based damping techniques. The concepts and methods presented in these publications also served as guidance for the authors on how to approach the issue of torsional vibration damping in the context of CFD.

The widespread popularity of viscous torsional vibration dampers can be attributed to their simple construction (Figure 1) and considerable operational durability. Another key factor is their ability to maintain efficiency across a wide range of engine rotational speeds, facilitating smoother transitions through critical speed ranges. As noted by Homik et al. [43], viscous dampers filled with silicone oil (silicone TVDs) or those utilizing a silicone–rubber combination (silicone–rubber TVDs) effectively reduce vibration amplitudes across the entire range of tested speeds. Conversely, in the absence of a damper or when a rubber damper (rubber TVD) is used, a significant increase in vibration amplitudes can be observed.

In summary, previous studies [43,44] indicate that thermodynamic and hydrodynamic modeling of viscous torsional vibration dampers plays a key role in determining their efficiency and predicting operational characteristics under steady-state conditions. Emphasis is placed on the critical role of temperature-dependent viscosity, which can influence the pressure distribution and velocity fields within the lubrication zones [45]. Equally important are detailed analyses of the temperature distribution within the damping fluid, as overheating or insufficient heat dissipation from the oil films can lead to viscosity changes and consequently a reduction in damping capacity [46]. In light of these issues, it is essential to account for both spatial and temporal variations in the oil film parameters within the viscous damper. This aspect is analyzed in detail later in the article, along with an evaluation of potential improvements in design solutions.

The symbols used in the article are collected and described in

Table 1. Table of symbols used in this study.

Coordinate systems
$x,y,z$	Cartesian coordinate system with the z-axis coinciding with
	the axis of rotation of the damper
$r,\phi$	Polar coordinate system for the $(x,y)$ plane, where
	$x=-rsin\left(\phi \right)$ and $y=rcos\left(\phi \right)$.
Geometrical quantities and the quantities dependent on them
$D_{H,1},R_{H,1}$	Inner diameter, radius of the housing body
$D_{H,2},R_{H,2}$	Outer diameter, radius of the housing body
$D_{I,1},R_{I,1}$	Inner diameter, radius of the inertia ring
$D_{I,2},R_{I,2}$	Outer diameter, radius of the inertia ring
$D_1=(D_{H,1}+D_{I,1})/2$	Average inner diameter
$R_1=(R_{H,1}+R_{I,1})/2$	Average inner radius
$D_2=(D_{H,2}+D_{I,2})/2$	Average outer diameter
$R_2=(R_{H,2}+R_{I,2})/2$	Average outer radius
$b_1,b_2$	Inner and outer width of inertia ring ($b_1=b_2$)
${\delta }_1=R_{I,1}-R_{H,1}$	Inner radial clearance
${\delta }_2=R_{H,2}-R_{I,2}$	Outer radial clearance
$A_B$	External area of the damper housing
e	The eccentricity of the housing and the inertia ring
${\epsilon }_1=e/{\delta }_1$	Inner relative eccentricity
${\epsilon }_2=e/{\delta }_2$	Outer relative eccentricity
$\epsilon =2e/({\delta }_1+{\delta }_2)$	Resultant relative eccentricity
$\beta$	The angle between the y-axis and the line connecting
	the center of the housing and the center of the inertia ring
$h_1\left(\phi \right),h_2\left(\phi \right)$	Functions of the inner and outer oil film thicknesses
$h_{1,min},h_{2,min}$	Minimum of the inner and outer oil film thicknesses
Physical quantities
$\omega$	The angular velocity of the inertia ring relative to the housing
T	Temperature
$T_0$	Ambient temperature
$T_B$	Temperature of the damper housing
$T_{B,lim}$	The maximum allowable operating temperature of the damper
$\rho \left(T\right)$	Silicone oil density
$\eta \left(T\right)$	Silicone oil dynamic viscosity
F	Weight of the inertia ring
$F_{L1},F_{L2}$	Inner and outer hydrodynamic buoyant forces
$f_1,f_2$	The fluid friction coefficients of the inner and outer film layers
$\alpha$	The heat transfer coefficient between the damper housing and
	the surrounding environment
$M_0,M_1,M_2$	The frictional moment generated by the front, inner, and outer
	oil film layers
$c_0,c_1,c_2$	The damping coefficient of the front, inner, and outer
	oil film layers

.

2. Materials and Methods

The vibrations generated in the crank–piston system of internal combustion engines are the result of periodically varying forces, primarily arising from the combustion of the air–fuel mixture. Figure 2 illustrates the distribution of forces stemming from the piston pressure P acting on the connecting rod and crankshaft during the combustion stroke. The following trigonometric relationships hold among the indicated forces:

$$\begin{array}{cc}\hfill N& =Ptan\left(\gamma \right)\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill S& =Pcos{\left(\gamma \right)}^{-1}\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill F_t& =Scos(\alpha +\gamma )\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill F_r& =Ssin(\alpha +\gamma )\hfill \end{array}$$

(4)

Analyzing the forces at point $O_2$, which describes the instantaneous position of the crankshaft and connecting rod, it can be observed that the radial force $F_r(\alpha ,\gamma )$ will induce transverse vibrations, while the tangential force $F_s(\alpha ,\gamma )$ will induce torsional vibrations. The angles defining the position of the crankshaft and connecting rod depend on time $\alpha \left(t\right),\gamma \left(t\right)$. The transverse vibrations are damped in hydrodynamically lubricated sliding bearings. Meanwhile, torsional vibrations are damped by appropriately selected damper constructions, examples of which are shown in Figure 1.

In the case of viscous dampers, the free spaces between the housing and the inertia ring are filled with silicone oil, as indicated in Figure 3. Torsional vibration damping in this scenario is the result of internal friction forces acting in the oil. This phenomenon occurs both in the external and internal oil films, as well as in the oil located between the end faces of the inertia ring and the housing. As the frictional energy is converted into thermal energy, an increase in the oil temperature inside the damper can be observed. Heat is dissipated into the surroundings through the housing components. Under the influence of the periodically varying tangential force $F_s(\alpha ,\gamma )$, the center of inertia $O_I$ moves in a pendulum-like manner along a specific trajectory, as shown in Figure 4. The limitation for proper damper operation is the eccentricity at which the housing temperature reaches the permissible value determined by the thermophysical properties of the oil. The pendulum motion occurs at a variable angular velocity $\omega$, resulting in a change in the position of the inertia ring center $O_I$ and a corresponding change in the generated heat energy. Consequently, for each point on the trajectory of the inertia ring center, we can assign its position, velocity, the value of the hydrodynamic lift force, and the amount of thermal energy generated in the oil films. After reviewing the available literature, we observed that when assessing the operational parameters of such dampers, the changes in the position of the inertia ring center relative to the housing are often disregarded, treating their mutual position as concentric.

The aim of this article is to investigate the impact of changes in the position of the inertia ring center $O_I$ relative to the housing center $O_H$ and the oil flow velocity on the operational properties of the oil films, taking into account the thermal energy generated by the oil motion in the annular torsional vibration damper. Additionally, it was decided to determine the contributions of the cylindrical and end-face surfaces of the damper housing in dissipating thermal energy. To achieve the aforementioned objectives, suitable mathematical models of the physical phenomena occurring during damper operation were developed.

For the numerical modeling of the operation of a viscous torsional vibration damper, assumptions were made concerning the geometric parameters and physical properties of the housing, the inertia ring, and the silicone oil. The necessary quantities and their descriptions are presented in Table 2.

Table 2. Simulation parameters.

Parameters of the torsional vibration viscous damper
1. The inner radius of the inertia ring	$R_{I,1}=78.5$ mm
2. The outer radius of the housing interior	$R_{H,2}=130$ mm
3. Inner radial clearance	${\delta }_1=R_{I,1}-R_{H,1}=0.14$ mm
4. Outer radial clearance	${\delta }_2=R_{H,2}-R_{I,2}=0.52$ mm
5. Width of the inertia ring	$b_1=b_2=33$ mm
6. External surface of the damper	$A_B=0.105154$ m2
7. Weight of the inertia ring	$F=89.6$ N
8. The heat transfer coefficient	$\alpha =20$ W·m−2K−1
The remaining input values for calculation of the damper operating parameters
9. Type of silicone oil	M30000
10. Silicone oil density	$\rho \left(T\right)=990-0.8·T$ kg·m−3 (Figure 5b)
11. Silicone oil dynamic viscosity	$\eta \left(T\right)=\rho \left(T\right){10}^{\left({\displaystyle \frac{793.1}{273+T}}-4.082\right)}$ kg ·m−1s−1
	(Figure 5a)
12. Ambient temperature	$T_0=70$ °C
13. Relative angular velocity	$\omega \in [0.2;2.0]$ rad·s−1

Table 2. Simulation parameters.

Parameters of the torsional vibration viscous damper
1. The inner radius of the inertia ring	$R_{I,1}=78.5$ mm
2. The outer radius of the housing interior	$R_{H,2}=130$ mm
3. Inner radial clearance	${\delta }_1=R_{I,1}-R_{H,1}=0.14$ mm
4. Outer radial clearance	${\delta }_2=R_{H,2}-R_{I,2}=0.52$ mm
5. Width of the inertia ring	$b_1=b_2=33$ mm
6. External surface of the damper	$A_B=0.105154$ m2
7. Weight of the inertia ring	$F=89.6$ N
8. The heat transfer coefficient	$\alpha =20$ W·m−2K−1
The remaining input values for calculation of the damper operating parameters
9. Type of silicone oil	M30000
10. Silicone oil density	$\rho \left(T\right)=990-0.8·T$ kg·m−3 (Figure 5b)
11. Silicone oil dynamic viscosity	$\eta \left(T\right)=\rho \left(T\right){10}^{\left({\displaystyle \frac{793.1}{273+T}}-4.082\right)}$ kg ·m−1s−1
	(Figure 5a)
12. Ambient temperature	$T_0=70$ °C
13. Relative angular velocity	$\omega \in [0.2;2.0]$ rad·s−1

Based on [43,47], the following assumptions were also made:

• Housing temperature, $T_{\mathrm{B}}$.
• Minimum height of inner and outer oil film, $h_{\min ,1}$ and $h_{\min ,2}$.
• The damper’s construction elements are non-deformable, perfectly smooth, axial parallel, cylindrical surfaces, and their material is homogeneous.
• Oil is an incompressible Newtonian fluid; its viscosity is a function of temperature $\eta =\eta \left(T\right)$; the pressure generated in the oil film may take values greater than or equal to zero; the pressure in the direction of the radial variable y is constant and the ambient pressure is constant.
• Oil flow is laminar for ${\mathrm{Re}}_i={\displaystyle \frac{\rho ·\omega ·D_i·{\delta }_i}{4\eta }}\le {\mathrm{Re}}_{i,\mathrm{CR}}\approx {\displaystyle \frac{41.3}{\sqrt{{\psi }_i}}}$ [48]; the velocity of the oil film near the surface is equal to the velocity of the surface rotating inertia ring or housing; the oil flow rate in the axial direction is much greater than the flow rate of oil flowing in the circumferential direction.
• Fluid heat exchange takes place by convection [49,50,51].
• Ambient temperature $T_0=\mathrm{const}.$
• The heat is absorbed by conduction through the damper housing and discharged to the environment.
• The quantity describing the heat transfer intensity is the heat transfer coefficient $\alpha =20$ W·m⁻²K⁻¹ [48].

With such defined assumptions, the shape of the oil gaps can be approximated using the equation

$$\begin{array}{cc}\hfill h_i\left(\phi \right)& ={\delta }_i(1+{\epsilon }_{i}cos\left(\phi \right)),\hfill \end{array}$$

(5)

and the position of the line connecting the geometric center of the housing and the geometric center of the inertia ring is defined by the angle $\beta$, described by the following relationship:

$$\begin{array}{cc}\hfill \beta & =arctan\sqrt{{\displaystyle \frac{\pi (1-{\epsilon }^2)}{4\epsilon }}}.\hfill \end{array}$$

(6)

The resultant relative eccentricity of the inertia ring and the housing is described by

$$\begin{array}{cc}\hfill \epsilon & ={\displaystyle \frac{2e}{{\delta }_1+{\delta }_2}}.\hfill \end{array}$$

(7)

The ability of a viscous damper to attenuate torsional vibrations can be described using the so-called oil flow resistance. If we assume that two surfaces between which the oil is located rotate relative to each other at a certain angular velocity, then the oil flow resistance is defined as the ratio of the generated torque to this angular velocity. In Figure 3, the layers for which the oil flow resistances are defined are indicated. The symbol $c_0$ denotes the flow resistance for a single end-face surface, $c_1$ denotes the flow resistance for the inner cylindrical surface, and $c_2$ denotes the flow resistance for the outer cylindrical surface. For the purposes of further considerations, a polar coordinate system is defined on the $(x,y)$ plane, which is given by the following relationships:

$$\begin{array}{cc}\hfill x& =-rsin\left(\phi \right),\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill y& =rcos\left(\phi \right),\hfill \end{array}$$

(9)

where $\phi$ is the angle denoted in Figure 3, and r is the leading radius.

For laminar flow in an annular torsional vibration damper, the damping fluid is a Newtonian fluid, and the proportionality coefficient between the tangential stresses and the velocity gradient is the dynamic viscosity:

$$\begin{array}{cc}\hfill {\tau }_{r,\phi }& =\eta \left(T\right){\displaystyle \frac{\mathrm{d}{v}_{\phi }}{\mathrm{d}r}},\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill {\tau }_{r,z}& =\eta \left(T\right){\displaystyle \frac{\mathrm{d}{v}_r}{\mathrm{d}z}}.\hfill \end{array}$$

(11)

The flow resistance for the internal and external oil films can be expressed as

$$\begin{array}{cc}\hfill c_i& ={\displaystyle \frac{M_i}{\omega }},\mathrm{where}i\in \{1,2\}.\hfill \end{array}$$

(12)

where $c_1$ corresponds to the flow resistance in the internal oil film, and $c_2$ corresponds to the resistance in the external oil film. The friction torque $M_i$, bearing capacity $F_{Li}$, and fluid friction coefficient $f_i$ in the viscous torsional vibration damper were determined from the Stokes–Navier equation and the oil flow continuity equation:

$$\begin{array}{cc}\hfill M_i& ={\displaystyle \frac{f_i·F_{Li}·D_i}{2}},\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill F_{Li}& ={\displaystyle \frac{1}{8}}\omega {\displaystyle \frac{{\epsilon }_i}{{(1-{\epsilon }_i^2)}^2}}\sqrt{{(1-\epsilon )}^2{\pi }^2+16{\epsilon }_i^2}{D}_i{b}_i^3{\displaystyle \frac{1}{{\delta }_i^2}}\eta ,\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill f_i& =4\pi {\displaystyle \frac{D_i{\delta }_i}{b_i^2{\epsilon }_i}}{(1-{\epsilon }_i)}^2{(1-{\epsilon }_i^2)}^{3/2}{\displaystyle \frac{1}{\sqrt{{(1-\epsilon )}^2{\pi }^2+16{\epsilon }_i^2}}},\hfill \end{array}$$

(15)

where $D_1=(D_{H,1}+D_{I,1})/2$ and $D_2=(D_{H,2}+D_{I,2})/2$. The oil flow resistance between the end-face surfaces of the inertia ring and the housing is described by the following relationship:

$$\begin{array}{cc}\hfill c_0& ={\displaystyle \frac{M_0}{\omega }}={\displaystyle \frac{{\tau }_{r,\phi }·A_c·R_m}{\omega }},\hfill \end{array}$$

(16)

where

$$\begin{array}{cc}\hfill A_c& ={\displaystyle \frac{1}{4}}\pi (D_2^2-D_1^2),\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill R_m& ={\displaystyle \frac{1}{2}}{D}_m={\displaystyle \frac{1}{4}}(D_2-D_1).\hfill \end{array}$$

(18)

Therefore, the total oil flow resistance is

$$\begin{array}{c}\hfill c=2c_0+c_1+c_2={\displaystyle \frac{2M_0+M_1+M_2}{\omega }}.\end{array}$$

(19)

However, the operating temperature of the damper, given the assumed heat exchange through the damper housing, can be determined from the following equation:

$$\begin{array}{cc}\hfill T_B& ={\displaystyle \frac{P}{\alpha A_B}}+T_0,\hfill \end{array}$$

(20)

where $P={\omega }^{2}c$ is the heat power generated by friction forces, $\alpha =20$ W·m⁻²K⁻¹ is the thermal conductivity of the housing, and $A_B$ is the surface responsible for heat dissipation.

3. Results

The operating temperature calculations for the viscous torsional vibration damper were conducted for the specified parameters presented in Table 2. For the study, it was assumed that the damper was filled with silicone oil M30000, and the weight of the inertia ring was $F=89.6$ N. The partial differential equations were solved numerically using the finite difference method implemented in the Matlab software (Version R2024a). The convergence criterion adopted was the relative error, which should not exceed 1% between iterations.

As a result of solving the system of Equations (5) and (6), which constitute the mathematical model of the damper, parameters describing its operational properties were obtained, including the relative position of the inertia ring center $O_I$ and the housing center $O_H$, expressed by ${\epsilon }_i\left(\omega \right)$; the minimum heights of the oil film $h_i(\omega ,\epsilon )$; the oil flow resistances $c_0\left(\omega \right),c_1(\omega ,\epsilon )$, and $c_2(\omega ,\epsilon )$; the friction torques in the oil films $M_0\left(\omega \right),M_1(\omega ,\epsilon )$, and $M_2(\omega ,\epsilon )$; the housing temperature $T_B(\omega ,\epsilon )$; and the heat generated by friction forces in the oil films $P_t(\omega ,\epsilon )$. The resulting parameters described by these functions are presented in the graphs.

The calculated values of the relative eccentricities for the internal oil film ${\epsilon }_1$ and the external oil film ${\epsilon }_2$ as a function of the relative velocity $\omega$ are presented in Figure 6. These eccentricities take the following values: ${\epsilon }_1\in [0.081;0.260]$ and ${\epsilon }_2\in [0.021;0.070]$. An increase in the relative velocity $\omega$ causes a decrease in the values of ${\epsilon }_i$. As the velocity $\omega$ increases, the center of the inertia ring also shifts toward the center of the housing. For relative velocity values in the range $\omega \in [0.2;2.0]$ s⁻¹, the values of the minimum oil film heights fall within the following ranges: $h_{1,min}\in [0.104;0.124]$ mm and $h_{2,min}\in [0.483;0.504]$ mm.

Figure 7 shows the effect of the resultant relative eccentricity $\epsilon$ on the angle $\beta$, defining the position of the centerline between the inertia ring and the housing. Their calculated values for the given range of relative velocities are as follows: $\epsilon \in [0.11;0.03]$ and $\beta \in [81.9^{\circ };87.5^{\circ }]$.

Analyzing the behavior of the function $T_B\left(\omega \right)$ presented in Figure 8a, it can be observed that an increase in the relative velocity $\omega$ results in a rise in the housing temperature $T_B$ of the torsional vibration damper. Conversely, an increase in the resultant relative eccentricity $\epsilon$ contributes to a decrease in the housing temperature, as shown in Figure 8b. Assuming the permissible operating temperature of the oil to be $T_{B,lim}$ = 90 °C, the maximum value of the relative velocity and the minimum resultant relative eccentricity were determined to be $\omega =0.7$ s⁻¹ and $\epsilon =0.0452$, respectively. From the developed model, it is therefore possible to infer the maximum allowable relative angular velocity $\omega$ and the minimum allowable value of the resultant relative eccentricity $\epsilon$ for a working damper. The use of the numerical model allows for the creation of graphs analogous to those shown in Figure 8 for specific damper dimensions and oil viscosity. This provides the designer with clear information regarding the permissible operating range of the damper.

The total heat dissipation surface area of the vibration damper is $A_B=0.105154$ m², with the end-face surfaces accounting for 68% and the cylindrical surfaces accounting for 32%. The temperature rise in the oil is a result of the conversion of internal energy from the oil friction forces into heat. Figure 9 shows the dependence of the total friction torque on the relative velocity $\omega$. For the assumed permissible operating temperature of the M30000 oil, $T_{B,lim}=90$ °C, and the corresponding relative velocity $\omega =0.7$ s⁻¹, the calculated total torque generated by the friction forces in the oil is $M=59.02$ N·m.

Analyzing the oil films where friction is generated and denoting the friction torques produced in the cylindrical spaces by $M_1$ and $M_2$ and the friction torque generated in the end-face spaces by $M_0$, their contributions to the total torque can be calculated. These contributions are $M_1+M_2=0.18·M$ and $2·M_0=0.82·M$. Specifically, the friction torque generated in the internal oil film is $M_1=0.08·M$, and in the external oil film it is $M_2=0.10·M$. The obtained result clearly shows that, for the analyzed damper, the primary surfaces responsible for dissipating the energy of torsional vibrations are the frontal surfaces. Performing such calculations for dampers with different geometries may lead to the optimization of the damper’s design, which would involve minimizing material consumption and reducing the damper’s weight while maintaining its damping performance. This is, therefore, very important information from the perspective of the designer.

Figure 10 shows the behavior of the function $P_t\left(\omega \right)$, which defines the power dissipated in the form of heat and transferred through the damper housing. As the relative velocity $\omega$ increases, the amount of heat generated by friction forces in the oil films also increases. For the assumed permissible oil temperature $T_{B,lim}=90$ °C, the maximum value of heat generated and dissipated into the surroundings is $P_{t,max}=41$ W. The relationship shown in this graph is of great importance to the designer. By knowing the power of vibrations generated on the crankshaft, it is possible to determine whether the damper will be able to mitigate their presence in real time.

4. Conclusions

To assess reliability or durability, it is necessary to understand the procedures that allow for the calculation of the operational parameters of the device being studied. This issue can be considered for the components of the device that are in a state of static equilibrium or in a state of dynamic equilibrium.

For the purpose of the quantitative evaluation of a damper’s operating parameters, a function $\epsilon (e,{\delta }_1,{\delta }_2)$ was introduced to describe the position of the inertia ring relative to the housing. As a result of the conducted studies, it was found that during the operation of the viscous torsional vibration damper, there is a small displacement of the center of the inertia ring relative to the center of the housing ($\epsilon >0$). As a result, the cylindrical surfaces of the inertia ring and the housing are separated by an oil film. The position of the center of the inertia ring is influenced by the geometry of the damper, the mass of the ring, and the properties of the oil. The damping of torsional vibrations generated by the crankshaft–piston system is also related to the relative velocity between the housing and the inertia ring, which is defined as the difference in angular velocities of both components. The energy of the torsional vibrations is damped through oil films separating both the cylindrical and the end surfaces. As a result of the conversion of torsional vibration energy into heat, the temperature of the oil filling the working spaces of the damper increases. The calculations clearly showed the dominant role of the end surfaces in heat dissipation and reductions in torsional vibration energy. For the assumed geometric and physical properties of the device, it was shown that the end surfaces were responsible for dissipating up to 82% of the vibration energy. For the chosen type of oil, the maximum relative velocity was also determined, beyond which the oil temperature would exceed the permissible value.

According to this study, the temperature of the oil generated by the oil friction forces should serve as a criterion for designers to determine the operational range of the damper.

Increasing the relative velocity while simultaneously lowering the working temperature of the oil requires an expansion of the heat dissipation surface area. A possible solution to this problem could be the use of ribbing on the damper housing. It is also crucial to ensure proper air circulation in the engine compartment. In particular, special attention must be paid to avoid the cooling of the engine block causing the accumulation of air with excessively high temperatures in the immediate vicinity of the damper.

It should also be noted that a near-concentric position of the damper housing and the inertia ring will favor the occurrence of transverse vibrations. The cause of these vibrations may be external forces or the variable radial force generated in the crankshaft–piston system of the internal combustion engine. This issue is an important topic that warrants further investigation.