Consider a hollow circular pipe segment of length L with outer diameter D and inner diameter d. A local coordinate system (x,y,z) is defined for each segment. Within the finite element framework, the pipe is discretized into elements, with each node possessing six degrees of freedom: three translational displacements (u, v, w) along the local axes and three rotational displacements (θx, θy, θz) about these axes. The corresponding mechanical model based on Timoshenko beam theory is illustrated in Fig. 1.

The geometric parameters of the pipe, including outer diameter D, inner diameter d, and length L, are determined based on typical aero-engine fuel and hydraulic pipe specifications. The material properties, such as Young’s modulus E, density ρ, and Poisson’s ratio ν, are selected according to the commonly used stainless steel (e.g., AISI 304) for aerospace piping systems. These values are representative of actual engine components and ensure the model captures realistic dynamic behavior. The shear correction factor κ is set to 0.9 for circular thin-walled pipes, accurately accounting for shear deformation effects in Timoshenko beam theory. The selection of these parameters aligns with industry standards and prior experimental studies on similar pipe systems.

Based on the Timoshenko beam theory, the complete element displacement vector qe for the element connecting nodes 1 and 2 is defined as (1)qe=[u1,v1,w1,θx1,θy1,θz1,u2,v2,w2,θx2,θy2,θz2]T.

The kinetic energy for the pipe segment is theoretically represented as (2)Te=12∫0L[ρA(u˙2+v˙2+w˙2)+ρIyθ˙y2+ρIzθ˙z2+ρJθ˙x2]dx,where ρ is the material density, A is the cross-sectional area, i.e., A=π4(D2−d2), whereas Iy and Iz are the second moments of area about the y- and z-axes, respectively. For the circle-section based pipe element, one has Iy=Iz=π64(D4−d4), and the polar moment of inertia J is given as J=π32(D4−d4), in which d and D define the inner and outer diameters of the pipe segment, respectively.

The strain energy of the Timoshenko beam based pipe element is defined as (3)Ue=12∫0L[EAu′2+EIyθy′2+EIzθz′2+GJθx′2+κGA(v′−θz)2+κGA(w′+θy)2]dx,where E is the Young’s modulus, κ is the shear correction factor. Together with the Poisson ratio parameter ν, one can define the shear modulus as G=E2(1+ν).

Application of Hamilton’s principle, δ∫t1t2(Te−Ue)dt=0, to the pipe segment derives the element-based equation for dynamic motion of the Timoshenko beam based pipe element (4)Meq¨e+Keqe=0.

In this formulation, the element mass and stiffness matrices Me and Ke for the Timoshenko beam element govern the interactions among the twelve local degrees of freedom in qe, and are given by (5)Me=[M11eM12eM21eM22e]andKe=[K11eK12eK21eK22e],where (⋅)11e and (⋅)22e characterize the self-dependency of the degrees of freedom at nodes 1 and 2, respectively, while (⋅)12e=(⋅)21e captures the coupling between the two nodes. Detailed expressions for the mass and stiffness matrices can be found in Appendix A.

The Timoshenko beam element incorporates six degrees of freedom per node: three translational and three rotational displacements. This formulation enables accurate representation of three-dimensional pipe behavior. The consistent derivation of kinetic and strain energy expressions captures rotary inertia effects and transverse shear deformation of the pipe body. This formulation serves as the foundation for subsequent global sensitivity analysis, enabling investigation of how clamp support parameters influence the inherent vibrational characteristics of complex piping systems.

Single clamps are extensively employed in aerospace and industrial piping systems to fix pipe segments to engine casings or structural frames. As critical boundary conditions, they substantially influence the system’s dynamic behavior, particularly its natural frequency distribution and mode shapes. Manufacturing tolerances, installation variability, and geometric asymmetry often lead to directional dependence and non-uniformity in clamp stiffness properties, resulting in anisotropic support behavior. Accurate modeling of these effects is crucial for predicting vibration response, avoiding resonance, and ensuring structural reliability under operational loads. This section presents a finite element based approach to model the elastic support of single clamps, incorporating translational and rotational stiffness across six degrees of freedom to capture their asymmetric mechanical response.

As illustrated in Fig. 2, the elastic support behavior of a single clamp is modeled using two sets of spring elements connected at the node corresponding to the clamp location. These springs represent the translational and rotational stiffness characteristics of the clamp across six degrees of freedom: kxSC, kySC, kzSC for translational stiffness along the x-, y-, and z-axes, and kθxSC, kθySC, kθzSC for rotational stiffness about these axes. The directional variation of these stiffness coefficients reflects the anisotropic nature of the clamp’s geometric asymmetry and bolted mounting configuration, which results from manufacturing tolerances and installation constraints.

Each node-localized spring group is represented by a diagonal stiffness matrix (6)KSC=12×diag(kxSC,kySC,kzSC,kθxSC,kθySC,kθzSC).

In the finite element assembly process, the factor 1/2 distributes the elastic support of a clamp equally between its two adjacent nodes, such as Points A and B in Fig. 2. This represents the spatial distribution of clamp stiffness along the pipe span, with the total stiffness of the clamp assembly divided evenly between its connection points.

In summary, the presented method enables precise capture of the elastic support behavior of single clamps within the finite element model. By integrating experimentally derived stiffness parameters and a physically consistent spring-mass formulation, the method supports accurate prediction of natural frequencies, mode shapes, and sensitivity to parametric variations of the single clamp. This capability is particularly valuable in engineering applications where clamp stiffness and position are treated as uncertain variables, enabling robust dynamic analysis and informed decision-making in the early stages of system design.

Double clamps are widely used in aerospace and industrial piping systems to provide enhanced structural restraint at critical junctions between adjacent pipe segments. To capture their dynamic influence within the finite element environment, double clamps are modeled as discrete connection mechanisms linking two nodes on adjacent pipe segments. The mechanical properties are represented by concentrated masses and stiffnesses distributed at the connection points as depicted in Fig. 3. The superscript CE distinguishes double clamp parameters from those of other components, facilitating clear identification and efficient model management.

The mass contribution of the double clamp is distributed equally among its four attachment points, two per node, reflecting the symmetric nature of its installation. As a result, each nodal mass represents one-quarter of the total double clamp mass mCE. The corresponding mass matrix is constructed as a block-diagonal matrix (7)MCEe=[M11CE06×606×6M22CE],where M11CE=M22CE=14×diag(mCE,mCE,mCE,0,0,0) accounts for the lumped translational inertia at each connection point. The model neglects the rotational inertia of the clamp, a standard simplification in the dynamic analysis of beam-like structures supported by discrete fixtures. This assumption is physically justified because the rotational inertia of a compact clamp mass is typically negligible compared to that of the pipe segments, which possess significantly larger radii of gyration. Consequently, the inertial coupling introduced by clamp rotation has a minimal effect on the global system dynamics, particularly for the lower-order modes which are the primary focus of this sensitivity study. This approach aligns with conventional modeling practices, streamlining the finite element formulation without compromising the reliability of the identified parameter importance rankings.

The stiffness matrix of the double clamp reflects the coupling between the two connected nodes. It is defined in a full 12×12 block form to account for mutual interactions (8)KCEe=[K11CEK12CEK21CETK22CE],with submatrices given by (9){K11CE=12×diag(kxCE,kyCE,kzCE,kθxCE,kθyCE,kθzCE)K22CE=12×diag(kxCE,kyCE,kzCE,kθxCE,kθyCE,kθzCE)K12CE=−12×diag(kxCE,kyCE,kzCE,kθxCE,kθyCE,kθzCE).

Here, kxCE, kyCE, kzCE denote the translational stiffness coefficients, while kθxCE, kθyCE, kθzCE represent the rotational stiffness values, all derived from the mechanical properties of the clamp material and geometric design. The negative off-diagonal blocks indicate that displacement at one node induces an opposing reaction at the other, consistent with the principle of action and reaction.

In summary, this procedure enables accurate representation of double clamp effects on the system dynamic characteristics. By incorporating both mass and stiffness contributions in a physically consistent manner, the model provides reliable prediction of natural frequencies and mode shapes for complex piping systems. The resulting element-level matrices are assembled into the global system equations, enabling comprehensive evaluation of how double clamp installations influence overall pipe dynamic behavior.

The integration of Timoshenko beam finite element modeling with discrete clamp representations enables a comprehensive and physically consistent procedure for analyzing the dynamic behavior of clamp-pipe systems. This section presents the global system formulation that combines contributions from pipe segments, single clamps, and double clamps into unified mass and stiffness matrices. The resulting model captures the influence of both clamp stiffness and position on the natural frequencies and mode shapes of the system, providing a robust foundation for sensitivity analysis and design optimization. In this study, the focus is on the deterministic modal characteristics of the clamp-pipe system, specifically the first five natural frequencies, which are critical for identifying potential resonance risks in engineering applications.

The Timoshenko beam finite element procedure allows one to obtain the global matrices for dynamic modeling of the clamp-pipe system as follows: (10){K=∑∀eTeTKPipeeTe+∑∀eTeTKSCeTe+∑∀eTeTKCEeTeM=∑∀eTeTMPipeeTe+∑∀eTeTMCEeTe.

Herein, KPipee and MPipee denote the element-level stiffness and mass matrices of the pipe segment formulated based on Timoshenko beam theory with appropriate shape functions. The terms KSCe and MSCe represent the stiffness and mass contributions from single clamps embedded at specific nodal locations according to their installation configuration. Similarly, KCEe and MCEe account for the coupled stiffness and inertial effects introduced by double clamps, which span between two distinct nodes and are applied as full 12×12 matrices during assembly. The summation operator ∑∀e(⋅) denotes the standard finite element procedure, which maps each element’s contribution to its correct position in the global system matrices, preserving connectivity and continuity across the entire structure. The transformation matrix Te is constructed by repeating the directional cosine matrix te four times along the block diagonal: (11)Te=blkdiag(te,te,te,te).

The operator blkdiag(⋅) creates a block-diagonal matrix, and te defines the orientation of the local coordinate system relative to the global frame: (12)te=[cos⁡(x,u)cos⁡(y,u)cos⁡(z,u)cos⁡(x,v)cos⁡(y,v)cos⁡(z,v)cos⁡(x,w)cos⁡(y,w)cos⁡(z,w)],where (u,v,w) denote the local coordinate axes and (x,y,z) the global axes. The columns of te are the local unit vectors vu,vv,vw expressed in the global coordinate system.

The natural frequencies of the clamp-pipe system are computed by solving the generalized eigenvalue problem, which establishes an implicit relationship between the system response and clamp parameters. The natural frequency vector Y=[y1,y2,⋯,yn]T depends on the parameter vector x=[x1,x2,⋯,xm]T, which contains clamp stiffness coefficients and spatial coordinates. Each parameter in x represents either a clamp location or a translational or rotational stiffness coefficient, thus capturing geometric installation variations and material property effects. This parametric formulation enables systematic investigation of how uncertainties in clamp configurations propagate through the system and influence the dynamic response.

The modal properties are obtained by solving the generalized eigenvalue problem (13)K(x)ϕ(i)(x)=Ωi2(x)M(x)ϕ(i)(x)(i=1,2,⋯,n),where Ωi(x) and ϕ(i)(x) denote the ith natural frequency and mode shape, respectively, both explicitly dependent on the clamp parameter vector x. The global mass matrix M(x) and stiffness matrix K(x) are assembled according to Section 2.4, with entries that are functions of the clamp configuration parameters. This parametric model provides a comprehensive framework for evaluating the influence of clamp stiffness and positioning uncertainties on the system’s vibrational characteristics, particularly the first several dominant modes that govern dynamic behavior under operational conditions.

In summary, this section presents a dynamic model for natural frequency analysis of clamp-pipe systems, integrating three-dimensional Timoshenko beam theory with discrete clamp representations. Pipe segments are modeled using Timoshenko beam finite elements, capturing bending and shear deformation effects essential for accurate dynamic characterization of aero-engine piping components. Clamps are represented as distributed spring elements that account for directional stiffness variations and anisotropic support behavior. Double clamps are modeled as coupled connection mechanisms introducing stiffness coupling between adjacent nodes. This integrated model establishes an explicit parametric relationship between clamp configuration parameters and system natural frequencies, forming a computational foundation for subsequent sensitivity analysis. The model enables evaluation of how clamp stiffness coefficients and spatial positioning influence the vibrational characteristics of complex piping systems, with emphasis on the dominant modes critical for resonance avoidance.

The natural frequency response of the clamp-pipe system is characterized by the functional relationship yk=ℳk(x) (k=1,⋯,n), where yk is the k-th natural frequency and x=[x1,x2,…,xm]T is the input vector containing clamp stiffness coefficients and positional coordinates. This parametric model enables systematic evaluation of how uncertainties in clamp configurations propagate to the system’s dynamic characteristics.

Sobol’ sensitivity analysis decomposes the output variance into contributions from individual parameters and their interactions. The main-effect sensitivity index for parameter xi quantifies its individual contribution to the total output variance, defined as (14)Si,k=V[Ex−i[ℳk(x)∣xi]]V[ℳk(x)](i=1,⋯,m),where x−i denotes all input variables except xi, and Ex−i[⋅] is the conditional expectation over x−i.

The total-effect sensitivity index accounts for both the main effect of xi and all its interactions with other parameters, defined as (15)STi,k=Ex−i[Vxi[ℳk(x)∣x−i]]V[ℳk(x)](i=1,⋯,m),where Vxi[ℳk(x)∣x−i] is the conditional variance of the response given x−i.

These sensitivity indices enable quantitative assessment of how clamp stiffness and position uncertainties influence the natural frequency characteristics of piping systems. The first-order index Si,k measures the direct effect of xi on the k-th natural frequency, while the total-effect index STi,k captures all interactions involving xi, providing comprehensive identification of dominant clamp parameters governing system dynamic behavior.

Exact computation of Sobol indices requires evaluating high-dimensional conditional expectations, posing significant computational challenges for complex pipe system models. For clamp-pipe systems with numerous uncertain parameters representing clamp stiffness coefficients and positional coordinates, direct Monte Carlo simulation becomes computationally prohibitive due to its inherent double-loop sampling structure.

To estimate the sensitivity indices via Monte Carlo simulation, two independent sample matrices X(1) and X(2) are generated from the input parameter space, each of dimension N×m, where m is the number of uncertain clamp parameters. For each sample, the natural frequency response yk,j(i)=ℳk(xj(i)) is computed using the dynamic model from Section 2, requiring 2N finite element evaluations.

The main computational burden arises from evaluating the conditional expectation Ex−i[ℳk(x)∣xi=xi,j], which requires constructing modified sample matrices. In these matrices, the i-th column is fixed at xi,j while all other columns are taken from X(2). This double-loop structure requires N additional finite element analyses per parameter, leading to 2N+mN model evaluations for the first-order sensitivity indices of all m parameters for a single natural frequency. The conditional expectation estimator is (16)E^x−i[ℳk(x)∣xi=xi,j]=1N∑l=1Nℳk(xi,j,xl,−i(2)),where xl,−i(2) is the l-th row of X(2) with the i-th element removed.

The main-effect sensitivity index is estimated as (17)S^i,k=1N∑j=1N(E^x−i[ℳk(x)∣xi=xi,j]−y¯k)21N∑j=1N(yk,j(1)−y¯k)2,with y¯k=1N∑j=1Nyk,j(1) as the sample mean of the first response set.

For the total effect, the conditional variance estimator requires computing (18)V^Ti,k=1N∑j=1N(ℳk(xj(1))−E^x−i[ℳk(x)∣xi=xi,j])2,and the total sensitivity index is estimated as (19)S^Ti,k=V^Ti,kV^yk,where V^yk=1N∑j=1N(yk,j(1)−y¯k)2 is the sample variance.

This computationally intensive procedure becomes intractable for realistic pipe system models, where each finite element analysis involves assembling and solving the full eigenvalue problem for three-dimensional Timoshenko beam elements with complex clamp boundary conditions. For typical aerospace piping systems with m uncertain clamp parameters and sample size N, the total computational cost for evaluating the first-order index is N(1+m) finite element analyses. For instance, with m=10 and N=104, the cost exceeds 105 finite element analyses. Such requirements exceed practical engineering limitations, as each finite element analysis may require several seconds to minutes, making brute-force Monte Carlo simulation computationally prohibitive for sensitivity analysis of realistic aerospace piping applications.

To overcome the computational challenges associated with direct integration in global sensitivity analysis, the multiplicative dimension reduction method provides an efficient alternative for estimating Sobol indices in complex pipe system models.

The multiplicative dimension reduction method leverages a multiplicative decomposition of the original model response ℳk(x) as [33] (20)ℳk(x)≈ℳk(x0)1−m⋅∏i=1mℳk,i(xi,x0,−i).

Herein, x0 is a reference point in the input space, and x0,−i represents all components of x0 except xi.

The component function ℳk,i(xi,x0,−i) is defined as the one-dimensional function obtained by fixing all other variables at their reference values x0,−i while varying only xi. Mathematically, ℳk,i(xi,x0,−i)=ℳk(xi,x0,−i), where (xi,x0,−i) denotes the vector with the i-th component equal to xi and all other components equal to those in x0. This construction transforms the m-dimensional integration problem into m separate one-dimensional integrals, each evaluated along a single parameter dimension while keeping other parameters fixed at their reference values. The accuracy of this decomposition relies on the assumption that the response function can be approximated as a product of univariate functions, which holds well for natural frequency functions with moderate parameter interactions. This multiplicative structure transforms high-dimensional integrals into products of one-dimensional components, enabling efficient evaluation of the required statistical moments for clamp stiffness and position parameters.

The reference point x0 is typically chosen as the mean or median of the input distribution. The accuracy of the multiplicative dimensional reduction method approximation depends on the smoothness of the response function and the distance of x0 from regions of strong nonlinearity. For the natural frequency functions in this study, which are generally smooth with respect to clamp parameters, selecting x0 at the mean of the input distribution provides good accuracy, as validated by comparison with direct Monte Carlo simulation in our numerical examples.

The first-order moment of the i-th component for the k-th natural frequency is defined as (21)ρi,k=EXi[ℳk,i(xi,x0,−i)](i=1,⋯,m;k=1,⋯,n)and the second-order moment is (22)θi,k=EXi[ℳk,i2(xi,x0,−i)](i=1,⋯,m;k=1,⋯,n).

These moments can be efficiently computed using Gauss-quadrature integration over the one-dimensional domain of xi, avoiding the need for expensive multidimensional numerical integration in the sensitivity analysis of pipe natural frequencies.

The numerical integration employs Gaussian quadrature rules with appropriately selected abscissas and weights for evaluating the one-dimensional component functions arising from the multiplicative decomposition. For a p-point Gauss-Legendre quadrature scheme, the integral of the component function ℳk,i(xi,x0,−i) over the standard interval is approximated as (23)∫Xiℳk,i(xi,x0,−i)dxi≈∑l=1pwlℳk,i(zl,x0,−i)(i=1,⋯,m),where zl and wl denote the quadrature points and corresponding weights, respectively. Variable transformations are applied to map the original parameter domains to the standard integration interval, enabling consistent application of the quadrature rules across different parameter types [45].

Therefore, the main effect sensitivity index for the i-th parameter corresponding to the k-th natural frequency is estimated as (24)Si,k=θi,k/ρi,k2−1(∏j=1mθj,k/ρj,k2)−1(i=1,⋯,m;k=1,⋯,n).

Similarly, the total effect sensitivity index, which accounts for both main and interaction effects involving xi, is given by (25)STi,k=1−ρi,k2/θi,k1−(∏j=1mρj,k2/θj,k)(i=1,⋯,m;k=1,⋯,n).

The computational efficiency of the multiplicative dimensional reduction method is achieved through systematic reduction of finite element reanalysis requirements. For a system with m uncertain clamp parameters and p-point quadrature, the total number of finite element evaluations required is mp for computing the first-order and total Sobol’ sensitivity indices of all parameters [41]. This represents a significant improvement compared to the direct Monte Carlo simulation approach, representing a computational speedup of approximately 103 times compared to direct Monte Carlo simulation with 104 samples (1.1×105 evaluations). This efficiency gain enables practical sensitivity analysis of complex three-dimensional pipe systems in practical engineering applications.

In summary, the multiplicative dimensional reduction method approximation provides an efficient computational method for global sensitivity analysis of clamp-pipe systems by transforming high-dimensional integrals into products of one-dimensional components. This approach significantly reduces the computational burden associated with traditional Monte Carlo based methods, enabling practical sensitivity analysis for complex three-dimensional piping systems with multiple uncertain clamp parameters. The resulting computational efficiency, with speedup factors reaching three orders of magnitude compared to direct Monte Carlo simulation approaches, makes the multiplicative dimensional reduction method based sensitivity analysis particularly valuable for engineering applications where computational resources are limited and rapid design iterations are required.

The computational procedure for global sensitivity analysis of clamp-pipe systems quantifies the influence of clamp stiffness coefficients and installation position coordinates on natural frequency characteristics. The algorithm integrates three-dimensional Timoshenko beam finite element modeling with multiplicative dimensional reduction method based sensitivity analysis to efficiently evaluate the impact of clamp parameters. The procedure encompasses parameterized model construction, Timoshenko beam finite element based natural frequency analysis, and multiplicative dimensional reduction method approximations for Sobol’ sensitivity indices.

As depicted in Algorithm 1, the NaFreSA procedure implementation begins with constructing a parameterized finite element model that explicitly incorporates clamp stiffness coefficients and installation position coordinates as the m-dimensional input parameter vector x. The assembly process generates global mass and stiffness matrices that are explicit functions of these clamp parameters using the formulation in Eq. (10). Solving the resulting parameterized eigenvalue problem in Eq. (13) yields natural frequencies represented as yk(x) that explicitly depend on clamp configurations. The multiplicative dimensional reduction method based sensitivity analysis then efficiently computes both first-order and total-effect sensitivity indices for each of the m parameters by evaluating one-dimensional integrals along each parameter dimension. This computational procedure enables comprehensive identification of critical clamp parameters that most significantly influence the dynamic characteristics of complex piping systems, while maintaining computational efficiency through systematic dimension reduction.

In summary, the proposed NaFreSA algorithm provides a computationally efficient method for global sensitivity analysis of clamp-pipe systems. By integrating parameterized Timoshenko beam finite element modeling with multiplicative dimensional reduction method based sensitivity computation, the algorithm achieves an optimal balance between accuracy and efficiency essential for practical engineering applications. The evaluation of both first-order and total-effect sensitivity indices enables comprehensive assessment of clamp parameter importance for natural frequency predictions. The resulting sensitivity analysis supports informed decision-making in design optimization and uncertainty quantification of engine piping systems, making it a valuable tool for engineering practitioners and researchers in structural dynamics and vibration analysis.

Consider the Z-shaped pipe model illustrated in Fig. 4. Control point coordinates of Z-shaped pipe are given as P1=(0,0,0) mm, P2=(300,0,0) mm, P3=(300,400,0) mm, and P4=(800,400,0) mm. Bending radius of the Z-shape pipe is 24 mm, whereas the outer and inner diameters of the pipe section are 8.0 and 6.4 mm, respectively. Specifically, positions of the single clamps to fix the Z-shape pipe system are defined as C1=(100,0,0) mm, C2=(300,200,0) mm, C3=(500,400,0) mm. This allows one to develop an experimental model for natural frequency analysis of the pipe as presented in Fig. 5.

The Z-shaped pipe represents a typical aero-engine fuel pipe segment. The material is AISI 304 stainless steel with a Young’s modulus of 200 GPa, density of 7800 kg/m3, and Poisson’s ratio of 0.3. The pipe has an outer diameter of 8.0 mm and an inner diameter of 6.4 mm, corresponding to a standard aerospace wall thickness. A bending radius of 24 mm ensures a smooth flow path and mitigates stress concentration. These parameters align with industrial design standards and are validated by experimental data from similar configurations.

The boundary conditions consist of three single clamps (C1, C2, C3) modeled as six-degree-of-freedom elastic supports. At each clamp, distributed spring elements provide translational stiffness in the y- and z-directions (kiy, kiz) and rotational stiffness about the x-axis. The pipe ends are free, creating a statically indeterminate system where clamp positions critically influence natural frequencies by altering effective span lengths and boundary stiffness distributions. The stiffness values are derived from experimental measurements as detailed in Section 2.2, ensuring a realistic representation of clamp behavior in aerospace applications.

Previous analyses of planar pipe systems have demonstrated that clamp position parameters dominate natural frequency characteristics compared to stiffness parameters. This finding motivates the current investigation into positional uncertainty effects in three-dimensional configurations in Fig. 7. The geometric configuration is based on actual routing data in Table 4. A three-dimensional finite element model is developed comprising 681 Timoshenko beam elements, 683 nodes, and 4098 degrees of freedom, enabling accurate simulation of natural vibration modes. Flexible pipe segments are represented by beam elements, while clamp constraints are modeled via spring elements. This approach captures the influence of clamp installation deviations on system stiffness and boundary conditions, providing a reliable basis for assessing the impact of positional uncertainties on dynamic response.

Table 5 compares the first ten natural frequencies from finite element simulation and experimental testing. The relative errors all fall within ±3%, demonstrating the high fidelity of the numerical model. The maximum deviation occurs at the sixth mode fn6, where the simulation underestimates the frequency by 2.93%. Most other modes agree within 1%. This validation confirms the accuracy of the modeling approach in capturing the dynamic characteristics of the spatial series-parallel piping system and its suitability for subsequent sensitivity analysis of clamp position effects.

Fig. 8 displays the experimentally obtained frequency response functions, which identify the first ten natural frequencies of the spatial series-parallel pipe system. The FRF peaks align with the natural frequencies listed in Table 5, validating both the experimental measurements and numerical predictions. The close agreement across a wide frequency range demonstrates that the finite element model accurately captures the physical behavior of the pipe system, including the influence of clamp positioning on its dynamics. This validated model provides a basis for the subsequent global sensitivity analysis of clamp position uncertainties.