## 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. [PERSON] 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, [PERSON] et al. [2] investigate how temperature affects the damping characteristics of hydraulic shock absorbers, underscoring the critical role temperature plays in component functionality. Furthermore, [PERSON] 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, [PERSON] 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, [PERSON] et al. [16] describe a failure that occurred after only 20 min of operation, while [PERSON] et al. [17] focus on a stochastic analysis of fatigue processes and develop models to assess the probability of failure. [PERSON] 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, [PERSON] [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. [PERSON]'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. [PERSON] 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.