Research on Vibration Load Identification Methods for Piping Systems Based on Limited Measurement Points

Affiliations: School of Aerospace Engineering, Xi'an Jiaotong University; China Nuclear Power Engineering Co., Ltd.; Xi'an Aerospace Propulsion Institute.

Abstract

This study proposes and validates a method for identifying vibration load parameters in piping systems using information from a limited number of measurement points. First, a forward finite element (FE) model of a typical engine pipeline was developed, and its validity was verified through experimental testing. Subsequently, load reconstruction methods were proposed and evaluated using Singular Value Decomposition (SVD), Genetic Algorithms (GA), and Tabu Search (TS) algorithms to identify unknown vibration load information. These methods enabled load reconstruction based on both simulated signals and experimental vibration measurements. The results indicate that while load reconstruction based on the Least Squares method and SVD offers high computational speed, it is highly susceptible to signal noise. In contrast, the Genetic Algorithm and Tabu Search algorithm exhibit superior robustness, achieving effective vibration load reconstruction for both simulated and measured vibration signals.

Keywords: Singular Value Decomposition (SVD); Genetic Algorithm (GA); Tabu Search (TS); Load Identification; Pipeline Vibration

1. Introduction

In complex mechanical systems such as aerospace engines and nuclear power facilities, piping systems are frequently subjected to intense vibration loads. Accurate identification of these loads is critical for structural health monitoring, fatigue life prediction, and vibration control. However, in practical engineering environments, it is often impossible to place sensors directly at the source of the vibration due to space constraints or extreme operating conditions. Consequently, identifying vibration loads based on response data from a limited number of accessible measurement points—an inverse problem in structural dynamics—has become a significant area of research.

The piping system is often referred to as the "blood vessels" of a liquid rocket engine and is a critical component for ensuring reliable engine operation. Excessive vibration can lead to malfunctions such as the loosening of clamps or pipe joints, and may even cause pipe wall cracking or fluid leakage, resulting in catastrophic accidents. To address the issue of intense vibrations, researchers have analyzed vibration sources and identified parameters affecting vibration response. However, traditional load identification methods often require the construction of a transfer matrix, which can significantly compromise the authenticity of reconstruction results due to ill-posedness.

This study transforms the load reconstruction problem into an optimization problem that minimizes the residual function between measured and calculated vibrations. The unknown load parameters are reconstructed using both deterministic and stochastic algorithms, and the performance of different inversion methods is analyzed and compared.

2. Finite Element Modeling and Experimental Validation

To establish a foundation for load identification, a forward vibration analysis of a typical engine pipeline was conducted. A high-fidelity finite element model was constructed to simulate the dynamic characteristics of the piping system.

2.1 Finite Element Modeling

The pipeline geometry, material properties, and boundary conditions were modeled using standard FEA procedures. The governing equation for the structural dynamics of the system is expressed as:
$$M\ddot{x}(t) + C\dot{x}(t) + Kx(t) = f(t)$$
where $M$, $C$, and $K$ represent the mass, damping, and stiffness matrices, respectively, and $f(t)$ denotes the external vibration load.

The pipeline assembly consists of straight sections and elbows, incorporating a pipe clamp at the midsection and flanges for fixation. Modeling was conducted using PIPE289 straight pipe elements and ELBOW290 elbow elements. The pipe clamp was modeled as equivalent radial linear spring elements.

2.2 Experimental Verification

To ensure the accuracy of the FE model, modal testing and vibration response experiments were performed. The natural frequencies obtained from the simulation were compared with experimental data. The first three natural frequencies of the single-ended fixed pipeline were measured at 7.08 Hz, 31.10 Hz, and 111.27 Hz, while the simulation yielded 7.37 Hz, 31.33 Hz, and 105.92 Hz, respectively. The error is less than 5%, verifying the validity of the established finite element model.

[FIGURE:1]
[TABLE:1]

3. Vibration Load Identification Algorithms

The core of this research involves reconstructing the unknown input load $f(t)$ from the measured response $y(t)$ at specific points. This is formulated as an optimization problem:
$$\epsilon(\mathbf{c}) = \sum_{m=1}^{N} |X_m(\mathbf{c}) - X_{obs,m}|$$
where $X_m(\mathbf{c})$ represents the calculated characteristic parameters and $X_{obs,m}$ represents the measured values.

3.1 SVD-Based Least Squares Method

In the frequency domain, the relationship between the load and the response is:
$$\mathbf{Y}(\omega) = \mathbf{H}(\omega)\mathbf{F}(\omega)$$
where $\mathbf{H}(\omega)$ is the frequency response function (FRF) matrix. Using Singular Value Decomposition (SVD), the matrix is decomposed as $\mathbf{H} = \mathbf{U} \mathbf{\Sigma} \mathbf{V}^H$. The unknown excitation is solved as:
$$\mathbf{P} = \mathbf{V} \mathbf{\Sigma}^{-1} \mathbf{U}^T \mathbf{x}$$
While computationally efficient, this method is highly sensitive to observation noise due to the ill-conditioning of the coefficient matrix.

3.2 Genetic Algorithm (GA)

The Genetic Algorithm is a heuristic stochastic optimization algorithm characterized by strong robustness. The load parameter vector is selected as the search space. The process involves:
1. Population Initialization: Randomly generating load vectors within a defined search space.
2. Individual Evaluation: Calculating fitness based on the reciprocal of the residual between calculated and measured responses.
3. Genetic Operations: Applying selection (tournament method), crossover (hybrid crossover), and mutation to evolve the population toward the optimal solution.

3.3 Tabu Search (TS) Algorithm

Tabu Search improves upon local search methods by using a "tabu list" to avoid cycles and escape local optima. The algorithm explores the neighborhood of a current solution and selects the optimal neighbor that is not "tabu" (unless it meets aspiration criteria). This method provides a balance between local refinement and global exploration.

4. Results and Discussion

4.1 Numerical Simulation Results

Load reconstruction was performed for a typical piping structure fixed at both ends. A harmonic excitation of 60 Hz with an amplitude of 200 N was applied. To simulate environmental interference, 5% white noise was added to the signals.

The results (Table 3) show that while SVD is fast, its accuracy drops significantly with noise. In contrast, GA and TS algorithms successfully reconstructed the load parameters even in the presence of noise, demonstrating superior robustness.

[FIGURE:4] [FIGURE:5]

4.2 Experimental Signal Results

Under laboratory conditions, a 20 Hz harmonic excitation was applied to a single-end fixed pipeline. Displacement signals were measured and used for reconstruction.
- SVD Method: Required less than 0.1s but showed poor noise immunity, with results deviating from the true values.
- GA and TS Methods: Exhibited excellent robustness. Although the computational burden is higher due to repeated FEA iterations, the reconstructed results remained within a reasonable error range compared to measured values.

[TABLE:6] [TABLE:7] [TABLE:8]

5. Conclusion

This study established a forward FE model and implemented several inversion optimization algorithms for vibration load reconstruction in piping systems.
1. The FE model was validated with experimental modal data, showing errors under 5%.
2. The SVD-based method is computationally efficient but sensitive to noise, making it suitable only for high-precision measurements.
3. Stochastic algorithms (GA and TS) provide high robustness and are effective for both simulated and experimental signals, though they require more computational time.
Future research will focus on hybrid algorithms to balance reconstruction accuracy with computational efficiency.