# Influence of Structural Stiffness and Loss Factor on Railroad Vehicle Comfort  
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Similar models have been analyzed in a number of papers [1; 2; 3; 4; 5; 6; 7; 8; 9] with the same goal that drives our research: to assess the influence of structural flexibility on passenger comfort. However, to the best of our knowledge, no previous work on this particular subject has modeled structural dissipation by means of a complex modulus. They all include a viscous term in the equation of motion of the beam that stems from cross-section stresses proportional to strain rate and in-phase with strain rate. This Kelvin-Voigt damping, also referred to as strain-rate damping, is convenient because it complies with the principle of causality, allowing models to be analyzed in the time domain.  
[PERSON] et al. [1] analyzed a Euler-Bernoulli beam with strain-rate damping supported on two bogies. They quantified comfort via the Sperling index and concluded that comfort improves with stiffer beams. They also showed that to maintain comfort levels for higher speeds, structures should be made stiffer. [PERSON] et al. [2], as well as [PERSON] et al. [3], used the same model with a mass under the beam connected with a flexible element to represent the effect of undercarriage equipment. They showed that the equipment could be used as a dynamic vibration absorber when the connecting stiffness and damping characteristics are properly designed. They compared their model results to experimental measurements, which showed good agreement. [PERSON] et al. [4] also used the same basic model but fitted, in this case, with underframe dampers. They showed that properly placed dampers could mitigate the flexural vibration of the car frame. Their model results were compared to finite element results, showing good agreement.  
[PERSON] [5; 6; 7; 8; 9] has conducted similar studies also using a Euler-Bernoulli beam with strain-rate damping supported on two bogies. She first presents the model as a viable tool for virtual certification [5] and later [6] uses the model to show that the consideration of frame flexibility may shift the least comfortable point on the beam from atop the bogie (for a rigid beam) to the beam center (for a flexible beam under particular conditions).  
The ubiquity of the strain-rate damping model notwithstanding, its ability to represent structural damping is questionable. The energy dissipated per cycle grows with frequency in the case of viscous damping, whereas experience shows otherwise. [PERSON] and [PERSON] [10] conducted experimental tests with composite beams, from which the damping parameters of several models they proposed were extracted. The analytical time response obtained with the estimated coefficients was then compared to the experimentally measured time response. Their hysteresis models produced better results than the strain-rate damping model. Dropping the Kelvin-Voigt term in favor of a complex modulus for the beam material is a better option to represent structural damping. In this case, damping stresses are proportional to strain (rather than strain rate), albeit in-phase with the strain rate. The drawback of complex modulus is that the principle of causality is lost, and models are not amenable to time-domain analysis. Fortunately, frequency domain analysis with a complex modulus is not only feasible but quite simple and straightforward as will be shown next. Moreover, mean square accelerations, from which comfort indexes may be estimated, can be easily obtained from frequency analyses.