Research on Non-Random Vibration Analysis of Concrete Pump Truck Boom Based on Dynamic Excitation

Abstract

When pouring concrete overhead, a pump truck boom’s vibration has a big effect on how accurately the concrete is poured. This is especially true during fixed-point pouring, where the boom’s vibration is likely to cause the pouring position to deviate, which lowers the quality of the construction. It is difficult to forecast the dynamic reaction of the pump truck boom in a construction setting because of the constantly shifting external factors (wind speed, pipeline stress during pumping, etc.), which makes it difficult to guarantee casting accuracy. This study suggests a non-random vibration analysis technique for pump truck booms based on the interval process theory in order to address this issue. A dynamic excitation analysis method based on rigid–discrete coupling is proposed, taking into account the response influence of the material characteristics in the transportation process. The pumping process of concrete materials in the conveying pipeline is simulated using discrete element simulation technology to determine the system’s stress conditions under pumping conditions. The dynamic response characteristics of the pump truck boom under operating conditions are revealed by using non-random vibration analysis with the mathematical model that has been created based on the particular specifications of the pump truck boom. This study employs the Newmark-β technique for numerical computation to solve the dynamic equations and characterize the displacement response envelope under uncertain system parameter settings. The experimental findings demonstrate that the suggested approach may accurately capture the upper and lower bounds of the boom dynamic response, offering a trustworthy way to assess the dynamic behavior while pumping. The technique can reliably predict the dynamic displacement boundary and control the casting position deviation within a predefined range by accurately predicting the dynamic displacement range of the pump truck’s boom end and efficiently constructing the displacement envelope under uncertain dynamic excitation. For numerical computation, use the Newmark-β algorithm. This outcome confirms the substantial enhancement of the proposed method regarding pouring precision in construction settings, offering a novel solution and technical guidance for vibration control in engineering projects.

1. Introduction

High-precision and high-efficiency concrete pumping technology has become a major need in the engineering industry due to the quick growth of contemporary large-scale building construction technology. However, the pumping truck boom’s vibration issue has a major impact on pouring accuracy during the actual construction process, particularly when complex dynamic excitation is present at high altitudes, and the boom structure displays complex non-random vibration behavior [1,2,3]. In addition to causing pouring position deviation, this vibration may lower the overall construction quality and safety. The majority of conventional vibration analysis techniques are grounded in probability theory, which uses random variables to characterize the dynamic uncertainty of parameters including geometric dimensions, material qualities, loads, and structure boundary conditions [4,5].

Researchers have commenced the investigation of non-random vibration analysis techniques to manage uncertainty analysis under constrained sample settings. Conventional stochastic vibration analysis depends on extensive sample data to develop precise probability distribution models. Researchers frequently regard the parameters of a system as stochastic variables and examine its stress, displacement–deformation, and stability attributes under random excitation to evaluate the system’s reliability [6,7,8]. Source [9] presents an effective evaluation technique that integrates finite element analysis (FEA) with time-dependent dependability. The research focused on the impact of stochastic variables on the long-term efficacy and safety of structures, successfully facilitating rapid and precise predictions of structural performance at various temporal intervals through the implementation of agent models and Monte Carlo simulations. Source [10] established a comprehensive analytical framework that integrates probabilistic methods, finite element analysis (FEA), and Monte Carlo simulation techniques to address uncertainties in vibrating systems, thereby enhancing the accuracy of analysis and the prediction of the response of bilaterally vibrating systems subjected to random vibrations. Nonetheless, stochastic vibration analysis utilizing probabilistic models necessitates substantial data support, and the actual application of stochastic models is constrained in engineering due to challenges in data collecting. The dynamic characteristics of boom structures in mechanical equipment, such as concrete pump trucks, cranes, and excavators, are considerably influenced by the intricate working environment and fluctuating operating conditions. During the pumping of concrete, the friction within the conveying pipeline induces a substantial vibrational response, hence complicating the study of structural vibrations. These intricate and multi-source uncertain excitations frequently appear to be beyond the scope of conventional stochastic analytic techniques. The creation of the stochastic process model must be based on a large amount of sample observation data of the dynamic parameters in order to gather the probability distribution information of the parameters. This is challenging or prohibitively expensive for many real-world engineering challenges. However, research has demonstrated that incorrect subjective probability assumptions for these parameters can result in significant variations in the estimation of structural response and reliability. This can easily lead engineers to design structural parameters irrationally and misjudge the actual outcomes. As a result, the non-stochastic analysis approach employed in this article is better able to characterize and forecast the system’s dynamic response, making it easier for engineers to evaluate the structural performance and design parameters. Source [11] examines the dynamic characteristics of extremely flexible pump truck boom structures and provides a methodology utilizing uncertainty measures and reliability assessments to evaluate their performance and stability across various operating situations. Source [12] introduced a binary subinterval method to address vibration issues with multiple uncertain parameters in mechanical systems, thereby diminishing reliance on extensive sample data and enhancing the precision of vibration response predictions through the incorporation of a deterministic excitation model. Source [13] established a multi-body dynamics model that integrates gap effects to simulate nonlinear behavior in joints, employing combined interval analysis and probability distributions to characterize parameter uncertainties and enhance the accuracy and reliability of dynamic analysis. The analysis of the dynamic response boundaries of time-varying uncertain vibration systems has advanced significantly in previous research, but these developments are primarily restricted to the quantitative analysis of system structural parameter uncertainties (such as the boom’s geometric dimensions, material properties, etc.). On the other hand, the problem with predicting system responses due to external excitation uncertainty can be successfully resolved by the analytical approach mentioned in this paper. This paper’s approach makes it possible to estimate dynamic response boundaries while taking the excitation source’s random properties into account. This approach offers a novel theoretical tool for handling the uncertainty analysis of vibration systems under complicated working conditions, in addition to breaking out of the analytical framework of conventional research restricted to structural parameter uncertainty.

In recent years, the enhancement of standards for pouring precision and construction safety at job sites has led to continuous innovation in the analytical methodologies for pump truck boom vibration. Source [14] suggested an uncertain model update optimization strategy using interval overlap and Chebyshev polynomials. This method approximates the system response with Chebyshev polynomials and quantifies uncertainty using interval overlap, resulting in outstanding computing efficiency and accuracy in the optimization process. Source [15] confirmed the efficacy of non-probabilistic models in adverse environments by examining the structural stability of flexible robotic arms at high-temperature conditions. Source [16] introduced an innovative interval model updating framework utilizing correlation propagation and matrix similarity techniques, designed to enhance the accuracy and reliability of uncertain structural models amid parameter uncertainties (e.g., material properties, geometric dimensions, etc.). These studies [17,18,19] demonstrate that the non-stochastic analysis method can compensate for the limitations of the traditional probabilistic model in scenarios of inadequate data while also more effectively characterizing the system response under complex excitation conditions, thereby offering robust theoretical support for the vibration analysis of pump truck booms. The majority of the work that is now available concentrates on conventional research avenues such as the geometric optimization of boom systems and the investigation of material property parameters, despite the fact that numerous studies have produced notable advancements in the field of mechanical booms. This work, on the other hand, concentrates on a more difficult practical engineering issue: the intricate rheological characteristics of fluid concrete as it passes through the delivery pipeline during concrete pumping. In particular, concrete’s physical property parameters (viscosity, mass density, etc.) exhibit notable non-uniform distribution features in both time and space dimensions due to its non-Newtonian nature. Because of this non-uniformity, the external excitation force acting on the system is largely unknown, which causes the system’s dynamic response to exhibit strongly random features. In order to thoroughly examine the method by which these uncertainty factors affect the system’s vibration characteristics, this work attempts to develop a dynamic model of the boom system that takes into account the uncertainty of concrete rheological properties.

This study examines the vibration response issue of the concrete pump truck boom during construction, cites a non-random vibration analysis method based on the interval process model, and suggests a rigid–dispersed body coupling method to investigate the impact of fluid concrete on the boom. The use of unknown excitation modeling diminishes reliance on probability distribution assumptions, and the dynamic equations are resolved by integrating the Newmark-β technique to formulate the displacement envelope under uncertain system characteristics. The experimental findings indicate that the approach can precisely forecast the dynamic displacement range of the boom’s end, effectively regulate the deviation of the pouring location within the specified limits, and markedly enhance the pouring accuracy and dependability. This method offers a novel technological approach for vibration control in engineering construction and holds significant reference value for further optimizing vibration control strategies under construction settings.

2. Boom Dynamics Analysis

This article focuses on the boom of a concrete pump truck, seen in Figure 1, which comprises six components. The mathematical model of the pump truck boom system was formulated utilizing the Denavit-Hartenberg (D-H) approach [20]. The velocity vectors of the center of gravity for each rigid body in the boom system were computed in the base coordinate system using the rotation transformation matrix derived from the D-H approach, in accordance with the principles of mechanical system dynamics equations. Additionally, utilizing these velocity vectors, the formulations for the total kinetic energy and total potential energy of the pump truck system were established, and the dynamic equations of the pump truck boom system were then obtained from these formulations [21]. The dynamic model of the pump truck boom system is ultimately formulated in matrix form, establishing a foundation for the comprehensive investigation of the system’s dynamic characteristics.

The shape of the boom, as presented in

Table 1. The D-H parameters.

Connecting Rod Number	Mass/kg	Length/m
1	1	1
2	3352.785	11.85
3	2241.339	9.25
4	1457.113	9
5	1144.211	11.635
6	457.417	7.31
7	207.387	4.25

, yields the D-H parameters for the boom, which are detailed in

Table 2. D-H characteristics for the flexible boom of concrete pump trucks.

Connecting Rod Number	Link Offset D/m	Joint Angle θ/°	Link Length A/m	Twist Angle α/°
1	0	Θ1	0	0
2	−0.6	Θ2	11.85	90
3	−0.3	Θ3	9.25	90
4	0.335	Θ4	9	90
5	−0.235	Θ5	11.625	90
6	0.065	Θ6	7.31	90
7	0.01	Θ7	4.25	90

.

The total kinetic energy of the boom system of a concrete pump truck comprises two components: the translational kinetic energy resulting from the linear motion of the rigid body mass, and the rotational kinetic energy arising from the body’s rotation about the joint. The total kinetic energy of the flexible boom system in a certain kind of concrete pump truck can be articulated as

$$T=T_1+T_2={\displaystyle \sum _{j=1}^{13}\frac{1}{2}}{m}_j{V}_j^T{V}_j+{\displaystyle \sum _{j=1}^{13}\left(\frac{1}{2}{I}_{xj}{\dot{\theta }}_{xj}^2+\frac{1}{2}{I}_{yj}{\dot{\theta }}_{yj}^2+\frac{1}{2}{I}_{zj}{\dot{\theta }}_{zj}^2\right)}$$

(1)

where V_j is the velocity of each mass relative to the base coordinate, $V_j={\left[v_{xj}\hspace{0.33em}{v}_{yj}\hspace{0.33em}{v}_{zj}\hspace{0.33em}0\right]}^T={\displaystyle \frac{d{R}_j}{dt}}$. If the coordinates of the center of mass of the i rigid body are R_i = [x_i, y_i, z_i, 1]^T, then its coordinates relative to the base coordinate R₀ are given by R₀ = ⁰T_i R_i. When j = 1, its calculation expression is $I_1={\displaystyle \frac{1}{2}}m{d}^2$; when j ≠ 1, its calculation formula is $I_2={\displaystyle \frac{1}{3}}m{x}^2$.

The aggregate potential energy of a concrete pump truck boom system is expressed by the equation:

$$V=V_g+V_e={\displaystyle \sum _{i=1}^{21}{m}_i}\cdot g\cdot h_i+{\displaystyle \sum _{i=1}^3\frac{1}{2}}{k}_i{\theta }_i^2+{\displaystyle \sum _{i=4}^{21}\frac{1}{2}}{k}_i{({\theta }_i-{\theta }_{i-3})}^2$$

(2)

where V_g represents the entire gravitational potential energy of the system, and V_e denotes the elastic potential energy of the system.

The Lagrangian technique ${\displaystyle \frac{d}{dt}}\left({\displaystyle \frac{\partial L}{\partial {\dot{q}}_i}}\right)-{\displaystyle \frac{\partial L}{\partial q_i}}=f_i$ is utilized to model the system’s dynamics [22], with f_i computed as f_i = −c_i·θ_i. The following is a mathematical expression for the equation of motion:

$$\left[I\right]\left\{\ddot{\theta }(t)\right\}+\left[C\right]\left\{\dot{\theta }(t)\right\}+\left[K\right]\left\{\theta (t)\right\}=\left\{\tau (t)\right\}$$

(3)

The Lagrange method is employed to resolve the equations of motion for the mechanical system, thereby establishing the dynamic model of the pump truck boom system. This, in turn, enhances comprehension of the dynamic characteristics of the pump truck boom system and aids in the calculations for subsequent analytical methods.

When creating the mechanical model of the boom system, the classical beam bending theory provides a quantitative description of the boom’s stiffness properties. The precise formula is as follows: When creating the mechanical model of the boom system, the classical beam bending theory provides a quantitative description of the boom’s stiffness properties. The precise formula is

$$K={\displaystyle \frac{EI}{L}}$$

(4)

where I = π·d⁴/64 is the moment of inertia of the boom’s cross-section, L is the boom’s length, and E = 206 GPa is the material’s modulus of elasticity. In this case, d stands for the boom’s diameter and L for its overall length. The flexural stiffness of the boom under external loading is described by the stiffness coefficient K, which can be computed using the material’s mechanical parameters and geometry.

Furthermore, the boom system’s damping properties are typically described by a proportionate relationship with mass, as shown by

$$C=\alpha \cdot m$$

(5)

where m is the boom system’s mass, α is the scaling factor, and C is the system’s damping coefficient. The scale factor typically ranges from 0.01 to 10, and the precise value is determined by a number of factors, including the working environment and material qualities.

Table 3. Stiffness coefficient.

C1	C2	C3	C4	C5	C6	C7
0.8	8 × 105	3 × 104	2 × 104	2.9 × 104	3 × 103	1.5 × 103

and

Table 4. Damping coefficient.

K1	K2	K3	K4	K5	K6	K7
0.55125	15,735.5	29,246.8	46,476.7	51,541.595	8147.526	1248.642

in this study contain the data for each of the system’s stiffness and damping coefficients, respectively. While each stiffness is mentioned in Table 4, each column in Table 3 corresponds to a distinct damping coefficient C₁ through C₇.

3. Analysis and Simulation of Pumping Pipeline Stresses in Pump Trucks

This work establishes a discrete element model for concrete and a kinetic model for the conveying pipeline to analyze the forces during concrete flow within the pipeline. The discrete element technique simulates the interaction and flow behavior of concrete particles, subsequently analyzing the forces under flow conditions. The pipeline model concurrently elucidates the impact of parameters like the stiffness and friction of the conveying pipeline on the flow characteristics of concrete. The integrated analysis of these two models offers theoretical support for optimizing the design and control of the pumping system.

3.1. Discrete Element Modeling

This study presents a discrete element model for fresh concrete developed using the Engineering Discrete Element Method (EDEM) [23], integrating particle contact theory with the material’s rheological parameters, as detailed in

Table 5. Parameters for material interaction.

Typical Materials	Coefficient of Static Friction	Coefficient of Rolling Friction	Collision Recovery Coefficient	JKR Surface Energy (J/mm2)
Coarse Aggregate–Coarse Aggregate Coarse aggregate–Mortar	0.175	0.15	0.15	10
	0.2	0.15	0.05	3.07
Coarse aggregate–Steel	0.305	0.115	0.2	1.5
Mortar–Mortar	0.36	0.09	0.01	6.69
Mortar–Steel	0.45	0.425	0.03	2

.

Table 6. Characteristics of materials.

Typical Materials (C30)	Shear Modulus (pa)	Poisson’s Ratio	Densities (kg/m)	Radius (mm)
aggregate	109	0.2	2600	7.5
mortar	108	0.15	2200	4
steel	7 × 1010	0.3	7850	62.5

lists the characteristics of the different materials. The model utilizes the Hertz–Mindlin with JKR Cohesion contact framework, proficient in precisely characterizing the visco-plastic behavior of concrete and the adhesive interactions among particles. Concrete was defined as a composite system of mortar and aggregate particles, utilizing the Jenkins–Kirchhoff–Roe (JKR) contact theory to elucidate the usual adhesion characteristics between particles [24]. Furthermore, the contact surface energy and physical property parameters are calibrated using a virtual calibration approach to guarantee that the simulation results align with the actual conditions. The model is especially appropriate for simulating the microstructure and flow characteristics of C30-grade concrete, offering a robust theoretical instrument to further optimize the concrete transport process.

This study first establishes the basic range of model parameters before validating the discrete elements model for fresh concrete. To calibrate these characteristics, “virtual experiments” are frequently used. Figure 2 depicts the collapse experiment’s procedure. The flow and extensibility of fresh concrete are measured using the collapse experiment, which also provides information about the cohesion and rheological characteristics of the material. Collapse is a significant physical characteristic of fresh concrete that can be used to describe how well it works. By measuring the concrete’s collapse pattern under the influence of gravity, the experimental approach is primarily used to assess the fluidity of the material. If the simulation results of the virtual experiment and the real experimental findings are consistent within the predefined error range, then the model parameters were established in an acceptable and efficient manner.

Table 7. Actual experimental material results and collapse degree “virtual experiment” material results.

Simulation Identifier	Degree of Collapse (mm)	Mean of the Experiment (mm)
1	201.2	197.5
2	203.69
3	201
4	201.6
5	199.61
6	200.13
⁝	⁝
20	201.49

displays the outcomes of both the real experimental materials and the simulated “virtual experiment” materials. This approach not only confirms the model’s accuracy but also offers a foundation for additional model optimization and modification, guaranteeing the model’s accuracy and usefulness in real-world engineering applications.

This paper’s validation experiment centers on the collapse degree, determining the model’s validity by comparing the actual measured collapse degree with the simulated findings from the “virtual experiment”. The average error of the simulated collapse degree during the validation procedure is around 3%. The findings indicate that the discrete element model for fresh concrete demonstrates strong flexibility in validating the theoretical collapse degree and accurately represents the rheological features of the concrete mix utilized in this study. Consequently, the model accurately characterizes the flow behavior of concrete and is appropriate for subsequent simulation analyses.

3.2. Dynamic Excitation Analysis of Pump Truck Piping

Owing to the inertia effect, as the concrete alters its flow direction within the conveying pipe, it applies supplementary stress on the pipe wall. The force exerted on the boom increases in relation to the expansion angle when the boom sections extend to a specific angle while concrete pours through the pipe’s elbow. The force analysis depicted in Figure 3 illustrates the concrete within the pipeline exerting pressure on the pipe wall through two equal-sized forces acting in opposing directions at the same point. Due to the pronounced vertical vibration of the pump truck boom, this article exclusively examines the impact of the boom’s load in the vertical plane on its dynamic behavior.

In EDEM (Discrete Element Method Simulation Software), a three-dimensional solid model of the pipe was initially created and a predefined granular material model was imported, as shown in Figure 4. A numerical simulation of the pumping operation in the Z-bend pipe was conducted to assess the force exerted on the pipe during this procedure. The simulation study yielded comprehensive data on the temporal variations in force in the pipe, and force–time graphs were generated, establishing a foundation for subsequent research on the dynamic mechanical response during the pumping operation.

When multi-degree-of-freedom (MDOF) systems are described by an angle θ_i and each degree of freedom is subject to external forces, these external forces must be transformed into generalized forces Q_i in order to be applied to the Lagrangian dynamical equations [25]. To do this, we must first take imaginary displacements into account. For each degree of freedom i, the imaginary displacement δr that corresponds to angle θ_i can be written as follows:

$$\delta \overrightarrow{r}=l\cdot \delta \theta \left(-\sin \left(\theta \right)\cdot \overrightarrow{i}+\cos \left(\theta \right)\cdot \overrightarrow{j}\right)$$

(6)

In the Cartesian coordinate system, i and j are unit vectors, and its l is the length corresponding to the degree of freedom i. According to the theory of virtual displacement, the external force’s contribution to the system’s virtual displacement is

$$\delta W={\overrightarrow{F}}_i\cdot \delta r_i$$

(7)

The generalized force Qi for each degree of freedom can be obtained by using the equation above to determine the external force’s contribution to the virtual displacement.

$$Q_{\theta }=-F_{x}l\sin (\theta )+F_{y}l\cos (\theta )$$

(8)

The force at each joint angle is ultimately determined as

$$\left\{\tau \right\}={[\begin{array}{ccccccc}{Q}_1& Q_2& Q_3& Q_4& Q_5& Q_6& Q_7\end{array}]}^T$$

(9)

The generalized forces operate at different joints within the system, and through computer transformations, the external forces are efficiently turned into generalized forces for further processing in the system analysis. The incorporation of generalized forces enables the correlation of external forces with the system’s degrees of freedom, facilitating precise calculations in the following resolution of equations of motion and vibration analysis.

4. Non-Random Vibration Analysis Method

The flow of fluid-like concrete in the conveying pipeline during concrete pumping is influenced by several factors, including the adjustment of the pumping truck’s gear, variations in pumping pressure, and the inhomogeneity of the concrete’s physical properties. These characteristics result in significant uncertainty in the applied force, hence inducing unpredictability in the system’s output reaction. To guarantee the stability of the pump truck boom during construction and to enhance the precision and efficiency of concrete placement, it is essential to thoroughly evaluate the impact of these uncertainties on the system’s structural vibration. This paper employs the interval process model [18] to quantitatively analyze the dynamic excitation resulting from uncertainty, integrating non-random vibration theory with a comprehensive examination of the vibration characteristics of the multilink structure. The objective is to ascertain the boundary range of the system’s dynamic response, thereby offering a theoretical foundation for the precise regulation of the pumping process.

4.1. Characteristic Parameters of Interval Process Model

In the interval process model, a time-varying uncertain parameter {X(t), t ∈ T} is approximately represented by a bounded closed interval X^I(t_i) = [X^L(t_i), X^U(t_i)] at any time t, where T denotes the parameter set of t, and X^L(t_i) and X^U(t_i) signify the lower and upper bounds of the interval variable, respectively. The relationship between the interval variables at two distinct time points is characterized by the autocorrelation coefficient function [26].

For an interval process X^I(t) characterized by an upper bound function X^L(t) and a lower bound function X^U(t), the midpoint function X^m(t) and the variance function D_X(t) can be delineated as

$$X^m(t)={\displaystyle \frac{X^U(t)+X^L(t)}{2}}$$

(10)

$$D_X(t)={\left({\displaystyle \frac{X^U(t)-X^L(t)}{2}}\right)}^2$$

(11)

For any instances t_i and t_j, the covariance function Cov(X_i, X_j) of the interval process X^I(t) is defined as

$$Cov\left(X_i,X_j\right)=\sin \theta \cos \theta \left(r_1^2-r_2^2\right)$$

(12)

For any instances t_i and t_j, the covariance function ρ_XiXj of the interval process X^I(t) is defined as

$${\rho }_{X_i{X}_j}={\displaystyle \frac{Cov\left(X_i,X_j\right)}{\sqrt{D\left(X_i\right)}\sqrt{D\left(X_j\right)}}}$$

(13)

The correlation matrix P can be formed from the autocorrelation function:

$$P=\left[\begin{array}{cccc}D\left(X_1\right)& Cov\left(X_1,X_2\right)& \cdots & Cov\left(X_1,X_n\right)\\ Cov\left(X_2,X_1\right)& D\left(X_2\right)& \cdots & Cov\left(X_2,X_n\right)\\ ⋮& ⋮& \ddots & ⋮\\ Cov\left(X_n,X_1\right)& Cov\left(X_n,X_2\right)& Cov\left(X_i,X_j\right)& D\left(X_n\right)\end{array}\right]$$

(14)

4.2. Dynamic Response Boundary Solving

This work uses the Monte Carlo simulation method to forecast the dynamic response boundaries of the concrete pump truck boom system under unpredictable excitation. This method effectively estimates the response range of the system under unclear parameters, hence establishing the upper and lower bounds of the system [27].

When the system experiences a time-varying unknown external force, this force is regarded as an interval process f^I(t). In this instance, explicitly solving the equations is challenging because of the time-dependent uncertainty stemming from both the external force and the system parameters. To address this issue, we initially discretize the continuous time-varying variables, and the dependent interval process can be converted into mutually independent interval processes, referred to as M_I(t), K_I(t), and C_I(t), using the transformed system parameters. The following is a mathematical expression for the equation of motion [28]:

$$m^I(t)\ddot{x}(t)+C^I(t)\dot{x}(t)+K^I(t)x(t)=f^I(t)$$

(15)

x(t), ẋ(t), ẍ(t), and f^I(t) represent displacement, velocity, acceleration, and external excitation, respectively. m^I(t) = [m^m(t) − m^t(t), m^m(t) + m^t(t)], k^I(t) = [k^m(t) − k^t(t), k^m(t) + k^t(t)], c^I(t) = [c^m(t) − c^t(t), c^m(t) + c^t(t)], f^I(t) = [f^m(t) − f^t(t), f^m(t) + f^t(t)]. The Newmark-β algorithm can be employed to solve Formula (15).

The solution statement is reformulated into the following matrix equation:

$$\left[\begin{array}{c}{x}_{i+1}\\ {\dot{x}}_{i+1}\\ {\ddot{x}}_{i+1}\end{array}\right]=B\left[\begin{array}{c}x\\ f\end{array}\right],B=\left[\begin{array}{cc}\begin{array}{c}{(B_x)}_1\\ {(B_x)}_2\\ {(B_x)}_3\end{array}& \begin{array}{c}{(B_f)}_1\\ {(B_f)}_2\\ {(B_f)}_3\end{array}\end{array}\right]$$

(16)

To facilitate description and analysis, the uncertainty domain of the external force can be represented by an ellipsoidal model, expressed as

$${(f-f^m)}^T{P}_f^{-1}(f-f^m)\le 1$$

(17)

The aforementioned equations indicate that determining the upper and lower bounds of the system response can be reframed as two optimization tasks. These two optimization problems are employed to identify the extreme values of the system response under uncertainty, hence establishing the boundary range of the dynamic response. The particular optimization challenges are delineated as follows:

$$\begin{array}{l}\max \hspace{0.33em}{x}_{i+1}={(B_x)}_{l}x+{(B_f)}_{l}f\\ \mathrm{s}.\mathrm{t}. \hspace{0.33em}{(f-f^m)}^T{P}_f^{-1}(f-f^m)\le 1\\ \min \hspace{0.33em}{x}_{i+1}={(B_x)}_{l}x+{(B_f)}_{l}f\\ \mathrm{s}.\mathrm{t}. \hspace{0.33em}{(f-f^m)}^T{P}_f^{-1}(f-f^m)\le 1\end{array}$$

(18)

According to optimization theory, the optimal solution to the problem can be attained when the following equations are formulated:

$${(f-f^m)}^T{P}_f^{-1}(f-f^m)=1$$

(19)

Consequently, the optimization problem can be addressed via the Lagrange multiplier approach [29]. Formulate the expression for the Lagrange function as follows:

$$L={(B_x)}_{l}x+{(B_f)}_{l}f+\lambda {(f-f^m)}^T{P}_f^{-1}(f-f^m)$$

(20)

Obtain the ideal solution:

$$f=f^m\pm {\displaystyle \frac{P_f{(B_f)}_l^T}{\sqrt{{(B_f)}_l{P}_f{(B_f)}_l^T}}}$$

(21)

Consequently, the upper and lower bounds of the displacement at each time interval are expressed as

$$\begin{array}{l}{x}_{i+1}^U={(B_x)}_{l}x+{(B_f)}_l{f}^m+\sqrt{{(B_f)}_l{P}_f{(B_f)}_l^T}\\ x_{i+1}^L={(B_x)}_{l}x+{(B_f)}_l{f}^m-\sqrt{{(B_f)}_l{P}_f{(B_f)}_l^T}\end{array}$$

(22)

Consequently, this method allows for the determination of the upper and lower bounds of the dynamic response related to the time-varying system characteristics at each specific time point. The response’s top and lower limits will vary within a specific range due to the temporal uncertainties in the system characteristics. In practical applications of vibration analysis, it is both desirable and feasible to determine a deterministic response boundary for a vibrating system characterized by time-varying uncertainty. This boundary not only offers dependable data support for vibration control but also markedly enhances the robustness and dependability of system design.

This work employs Monte Carlo simulation to ascertain the dynamic response boundaries of the system under uncertainty by calculating the upper and lower response limits. Monte Carlo simulation produces several samples via random sampling techniques, thereby emulating the uncertainty distribution of system parameters and deriving the statistical characteristics of the response through statistical analysis. This method may accurately measure the impact of system parameter uncertainty on dynamic response and furnish data support for subsequent system optimization and vibration control.

5. Experimental Design and Result Analysis

5.1. Conveying Pipeline Force Results

This paper employs the D-H model to represent the joints of the concrete pump truck boom system, which comprises a multi-section boom and a rotary table featuring two rotational degrees of freedom, facilitated by elastic joints around the z-axis, y-axis, and x-axis of the pump truck’s base coordinate system. Utilizing the D-H parameters (joint angles, offsets, distances, and torsion angles), we delineate the relative movements between the joints of the boom and formulate a comprehensive mathematical model incorporating the precise rigid body length and mass characteristics of the boom. This not only facilitates the precise analysis of the dynamic response characteristics of the boom system under varying operational conditions but also offers a robust theoretical foundation for the optimization of the subsequent control approach. The dynamic model is ultimately derived as

$$\left[I\right]\left\{\ddot{\theta }(t)\right\}+\left[C\right]\left\{\dot{\theta }(t)\right\}+\left[K\right]\left\{\theta (t)\right\}=\left\{\tau (t)\right\}$$

(23)

The simulation analysis reveals six forces acting on the pipeline during the pumping operation, specifically at the joint points of the two booms. To illustrate the fluctuation of these forces, we exhibit the data graphically. Figure 5 illustrates the time–force curves of the six forces, distinctly demonstrating the dynamic properties of each force over time, encompassing their size, direction, and interrelationship during the pumping operation. Analyzing these curves enables a more profound comprehension of the mechanical behavior during the pumping process, hence offering substantial data support for subsequent vibration analysis and system optimization.

5.2. Analysis of Boom Vibration Response

Boom vibration not only results in material loss and increased costs during concrete pumping but may also cause early wear, fatigue damage, and reduced service life of boom components. The boom experiences complex loads during operation and is frequently exposed to unstable conditions, resulting in its vibration response significantly affecting the performance and structural integrity of the pump truck. Consequently, to guarantee the safety and reliability of the equipment and the precision of the concrete placement process, a thorough analysis and prediction of the boom’s vibration characteristics is essential. This study optimizes the boom design, mitigates vibration-related issues, and offers maintenance and operational personnel timely alerts of failures, thereby preventing potential safety risks.

This study addresses the issue of vibration in pump truck booms. The dynamic equations of this system are shown in Equation (24), determined by integrating the vibration characteristics of the pump truck boom, structural stiffness, mass distribution, and external loads. By modeling and analyzing this vibration system, the vibration response data of the boom under various operational situations may be acquired. These data not only enhance our comprehension of the potential vibration modes of the boom during operation but also establish a theoretical foundation for subsequent design optimization. The dynamic response of the boom during the pumping process significantly influences the concrete flow rate, pouring precision, and operational smoothness; therefore, comprehensive vibration analysis and optimization are essential in the initial design phase to guarantee the equipment’s efficient and safe operation.

$$\left[I\right]\left\{\ddot{\theta }(t)\right\}+\left[C\right]\left\{\dot{\theta }(t)\right\}+\left[K\right]\left\{\theta (t)\right\}=\left\{\tau (t)\right\}$$

(24)

I, C, and K are 7 × 7 matrices among them. Following analysis and simplification, each matrix element’s specific forms are as follows:

$$\left[I\right]=\left[\begin{array}{cccc}\begin{array}{ccc}\begin{array}{cc}\begin{array}{c}\begin{array}{c}{\displaystyle \frac{1}{2}}m{d}^2\\ \\ \\ \end{array}\\ \\ \\ \end{array}& \begin{array}{c}\begin{array}{c}\\ {\displaystyle \frac{1}{3}}m{x}^2\\ \\ \end{array}\\ \\ \\ \end{array}\end{array}& \begin{array}{c}\begin{array}{c}\\ \\ {\displaystyle \frac{1}{3}}m{x}^2\\ \end{array}\\ \\ \\ \end{array}& \begin{array}{c}\begin{array}{c}\\ \\ \\ {\displaystyle \frac{1}{3}}m{x}^2\end{array}\\ \\ \\ \end{array}\end{array}& \begin{array}{c}\begin{array}{c}\\ \\ \\ \end{array}\\ {\displaystyle \frac{1}{3}}m{x}^2\\ \\ \end{array}& \begin{array}{c}\begin{array}{c}\\ \\ \\ \end{array}\\ \\ {\displaystyle \frac{1}{3}}m{x}^2\\ \end{array}& \begin{array}{c}\begin{array}{c}\\ \\ \\ \end{array}\\ \\ \\ {\displaystyle \frac{1}{3}}m{x}^2\end{array}\end{array}\right]$$

(25)

$$\left[C\right]=\left[\begin{array}{ccccccc}{c}_1& & & & & & \\ & c_2& & & & & \\ & -c_3& c_3& & & & \\ & & -c_4& c_4& & & \\ & & & -c_5& c_5& & \\ & & & & -c_6& c_6& \\ & & & & & -c_7& c_7\end{array}\right]$$

(26)

$$\left[K\right]=\left[\begin{array}{ccccccc}{k}_1& & & & & & \\ & k_2+k_3& -k_3& & & & \\ & -k_3& k_3+k_4& -k_4& & & \\ & & -k_4& k_4+k_5& -k_5& & \\ & & & -k_5& k_5+k_6& -k_6& \\ & & & & -k_6& k_6+k_7& -k_7\\ & & & & & -k_7& k_7\end{array}\right]$$

(27)

In the methods presented in this study for solving the arm dynamics equations, the influence of the first joint on the system dynamics equations is disregarded, as it merely serves as a directional change and is not susceptible to external pressures. The condition of the first joint is disregarded, and it is presumed to have no impact on the system’s vibration characteristics. Consequently, as illustrated in Figure 5, the analysis commences at the second joint, namely the joint connecting the first boom, from which the dynamic response curves for the vibration angles of each joint are derived.

In Figure 6, the gray lines depict the sample response curves produced by the Monte Carlo simulation approach, while the red and blue curves illustrate the maximum and lower limits of the dynamic response derived from the computational method presented in this paper, respectively. The analysis results indicate that when the uncertain excitation is represented by the interval process model, the interval curves derived from the numerical simulation fully encompass the Monte Carlo sample response, thereby confirming the validity and reliability of the computational method proposed in this paper for analyzing uncertain dynamic responses. This outcome demonstrates that the proposed method can yield more precise upper and lower limits when analyzing the dynamic response under uncertainty, hence facilitating the determination of dynamic response intervals for the end displacements.

$$\Delta x_i=l_i\cdot \sin ({\theta }_i\cdot \pi /180)$$

(28)

The examination of the boundary curves for each response of the boom vibration model, as derived from the methodology presented in this study, indicates that each response variable progressively attains a steady state following an initial non-stationary phase. Owing to the inadequate comprehension of the alteration of the angle, the magnitude of the angle change is transformed into the quantity of displacement change by utilizing Equation (28). This conversion allows for a clearer observation of the angular displacement range of the six-section boom, as illustrated in

Table 8. The amount of offset distance at the joint.

Boom Rods	Angular Offset	Boom End Offset
1	[0, 0]	[0, 0]
2	[−0.07, 0.07]	[−0.002, 0.002]
3	[−2.07, 2.07]	[−0.33, 0.33]
4	[−1.72, 1.72]	[−0.27, 0.27]
5	[−1.62, 1.62]	[−0.32, 0.32]
6	[−1.01, 1.01]	[−0.13, 0.13]
7	[−0.12, 0.12]	[−0.001, 0.001]

. This signifies that the peak amplitude of this response variable during the pump truck boom’s pouring operation is [−1.05, 1.05].

6. Conclusions

This study proposes a research method utilizing non-random vibration analysis to address the vibration issues of concrete pump truck booms in construction environments, examining the influence of fluid concrete forces within the conveying pipeline on the boom’s vibration characteristics during the pumping process. The force within the conveying pipeline significantly influences the dynamic reaction of the boom, hence directly affecting the precision of pouring. This paper employs an interval process model to quantitatively analyze the uncertain dynamic excitation and integrates the Monte Carlo simulation method to ascertain the upper and lower bounds of the system response, thereby providing essential data support for the vibration response analysis of the boom system. The findings indicate that the response interval of the boom’s end displacement is around [−1 m, 1 m] when pipeline force is accounted for, suggesting that vibration amplitude and response precision significantly affect construction accuracy. The suggested non-random vibration analysis method demonstrates strong adaptability under uncertain conditions, establishing a robust theoretical foundation for the analysis of vibration characteristics and the optimization of construction accuracy for the pumping truck boom, while also offering a significant reference for the design and control strategies of future engineering machinery. Future research will further investigate the nonlinear dynamic excitation scenario, integrated with intelligent algorithms to provide real-time monitoring and the predictive reaction of the construction process, thereby enhancing construction precision and operational efficiency.

This study possesses certain limitations. This research uses interval process modeling and Monte Carlo simulation techniques to analyze the system’s dynamic response; nevertheless, these methods may exhibit limitations in computational accuracy when addressing uncertainties in harsh working conditions. Consequently, the introduction of more efficient numerical optimization algorithms is deemed necessary to enhance the computational accuracy and efficiency of the model when addressing extreme uncertainty excitations, thereby facilitating a more comprehensive evaluation of the dynamic response of the pump truck boom under complex operational conditions.