Dynamic modeling and optimization of freight vehicle suspension systems under vibration-induced fatigue

2.1. Overview of the approach

The study applies a coupled dynamic–fatigue modeling framework for a freight vehicle suspension system.

The workflow includes:

– mathematical representation of vertical vibrations using quarter-car and half-car models;

– simulation of stochastic road excitation;

– estimation of stress time histories in suspension elements;

– fatigue damage assessment through rainflow counting and the Palmgren-Miner rule;

– optimization of stiffness and damping parameters via Particle Swarm Optimization (PSO).

This integrated approach allows the prediction and improvement of fatigue life under realistic vibration conditions without physical prototyping.

2.2. Dynamic model formulation

The suspension dynamics are described by the Kelvin-Voigt model combining elastic and viscous behavior.

For the quarter-car model, the governing equations are:

where $m_s$ – sprung mass; $m_u$ – unsprung mass; $k_s$ – suspension stiffness; $c_s$ – damping coefficient; $k_t$ – tire stiffness, $z_s$, $z_u$$z_r$ – displacements of sprung mass, unsprung mass, and road profile.

The half-car model extends the analysis by introducing pitch motion $\theta$ and two suspension pairs (front / rear):

where $I_y$ – moment of inertia of the sprung body; $l_i$ – distance from the center of gravity to suspension $i$.

2.3. Road excitation modeling

Road roughness excitation is modeled as a stationary Gaussian random process defined by its power spectral density (PSD) $G_q\left(n\right)$ according to ISO 8608:

where $n_j$ – spatial frequency, $V$ – vehicle speed, and ${\varphi }_j$ – random phase.

This method reproduces the frequency content of real road irregularities, ensuring realistic dynamic loading for fatigue assessment.

2.4. Fatigue damage evaluation

Stress-time histories obtained from the dynamic model were processed using the rainflow counting algorithm in accordance with ASTM E1049-85 [24] to identify load cycles.

The cumulative fatigue damage DDD was evaluated according to the Palmgren-Miner linear damage rule:

where $n_i$ – number of cycles at stress level $i$; $N_i$ – number of cycles to failure derived from S-N (curve) data. Fatigue life is reached when $D=$1.

This approach enables accurate prediction of the fatigue strength and service life of critical suspension components under variable-amplitude loading, providing a reliable basis for optimizing design parameters and maintenance intervals.

2.5. Optimization procedure

A multi-objective optimization problem is formulated to minimize both vibration acceleration and fatigue damage.

Decision variables: suspension stiffness $k_s$, damping $c_s$, and tire stiffness $k_t$.

Objective function:

subject to constraints: $k_{s,min}\le k_s\le k_{s,max}$, …, $c_{s,min}\le c_s\le c_{s,max}$.

Particle Swarm Optimization (PSO) is selected for its efficiency and convergence stability in nonlinear systems.

Each particle represents a possible suspension configuration; iterations continue until the fitness value converges below a set tolerance.

2.6. Model validation

The developed model was validated through comparison with experimental vibration data from freight vehicle suspension tests.

Acceleration and stress responses measured with tri-axial accelerometers and strain gauges were compared to simulation outputs under identical excitation conditions.

The root-mean-square (RMS) deviation between experimental and simulated accelerations did not exceed 7 %, and the correlation coefficient of response curves exceeded 0.93.

This confirms that the model accurately reproduces real dynamic behavior and is suitable for optimization and fatigue-life prediction.

Fig. 1 presents the comparison between the simulated and measured acceleration responses, showing close agreement and confirming the adequacy of the model.

Fig. 1Comparison of simulated and measured acceleration responses of the freight vehicle suspension system

3.1. Dynamic response analysis

The frequency-domain analysis of the suspension system showed dominant vibration modes within the range of 1-8 Hz, corresponding to low-frequency excitation that strongly influences fatigue life. The comparative S-N curves for tested materials are presented in Fig. 2, illustrating the influence of surface roughness on fatigue life.

Fig. 2Comparison of S-N curves for trailer frame materials with different surface finishes (3.2 µm and 25 µm)

The smoother surface (3.2 µm) exhibits higher fatigue strength, particularly in the low-cycle region (10³-10⁵ cycles), while the rougher surface accelerates early crack propagation.

At higher frequencies (above 10 Hz), vibration amplitudes decreased, primarily affecting ride comfort rather than fatigue.

The calculated root-mean-square (RMS) acceleration for the baseline suspension reached 2.4 m/s², exceeding the comfort threshold recommended by ISO 2631 for long-term operation.

3.2. Influence of damping and stiffness parameters

Parametric sensitivity analysis revealed that damping and stiffness significantly influence both vibration isolation and structural durability.

A reduction of the damping coefficient by approximately 20 % resulted in a 12 % increase in vertical displacement but reduced stress amplitude in suspension links by 30-35 %.

Conversely, excessive damping improved ride comfort but intensified local stress peaks, accelerating fatigue damage.

The ratio between front and rear suspension stiffness also played a major role in balancing comfort and fatigue performance.

The optimized configuration achieved by PSO ensured the best compromise between these competing factors.

The algorithm converged within 25 iterations, resulting in optimized parameters that reduced RMS acceleration to 1.6 m/s² and decreased the cumulative fatigue damage index by approximately 52 % compared to the baseline configuration.

This corresponds to an increase in fatigue life by nearly twofold.

3.3. Optimization efficiency and convergence

The Particle Swarm Optimization algorithm demonstrated stable convergence behavior with low sensitivity to initial conditions.

Each iteration updated particle velocity and position toward the global best solution, while maintaining feasible design limits.

The objective function value stabilized after approximately 20-25 iterations, confirming the robustness of the proposed optimization framework for nonlinear vibration systems.

3.4. Validation and discussion

Similar optimization frameworks have been applied in other transport and mechanical systems to improve durability and resource efficiency [15].

The optimized suspension parameters were validated through experimental comparison.

Measured and simulated vibration responses showed close agreement, with the difference in peak acceleration values not exceeding 7 %, confirming the accuracy of the dynamic model.

This consistency validates the model’s ability to capture the real operational dynamics of freight suspensions and supports its use for fatigue-life prediction.

The results demonstrate that the proposed integrated approach effectively mitigates vibration-induced fatigue through systematic optimization of stiffness and damping parameters.

The methodology enables engineers to enhance durability and reduce maintenance costs without redesigning the entire suspension system.

In practical application, implementing optimized damping-stiffness ratios can extend service intervals, lower vibration energy transfer to structural joints, and improve operational safety of freight transport systems.

The results of vibration and fatigue analysis are consistent with recent findings in transport system modeling and diagnostics.