Vibration characteristic analysis of a rack-climbing lift platform based on a rigid–flexible coupled model

Abstract

To address the limitations of traditional multi-rigid-body models in accurately analyzing the vibration characteristics of rack-climbing lift platforms, this study proposes a rigid–flexible coupled dynamic equation and develops a flexible-body substructure reduction method to enhance simulation accuracy while reducing computational complexity. A rigid–flexible coupled simulation model is constructed by integrating ANSYS APDL with ADAMS, enabling an organic coupling between flexible components and rigid bodies. Axial vibration characteristics of the platform are simulated using four models—rigid, multi-flexible coupled, flexible-driven coupled, and rigid–flexible coupled. Results indicate that the Z-direction vibration is similar to the X-direction, with maximum amplitudes of 1.03mm and 1.34mm occurring during the startup phase, whereas the Y-direction amplitude reaches 7mm, representing increases of 579% and 422% relative to the Z- and X-directions, respectively. These findings demonstrate the inadequacy of rigid-body models for effective vibration analysis. Compared with the multi-flexible and flexible-driven coupled models, the rigid–flexible coupled model exhibits smoother transition in vibration response and more pronounced amplitudes, confirming its validity. Further parametric analysis under varying operating speed, load magnitude, eccentric loading, and braking time reveals that eccentric loading effects are condition-dependent, while load magnitude and operating speed significantly influence platform vibration. Finally, experimental validation of axial vibration and vibration acceleration confirms the accuracy of the proposed model. This study provides a theoretical foundation and reliable modeling framework for structural optimization, comfort design, and vibration behavior analysis of lift platforms.

Supplementary Information

The online version contains supplementary material available at 10.1038/s41598-026-51186-7.

Introduction

Safety, reliability, and real-time lifting capacity are key foundations for ensuring the efficient and continuous operation of aerial work platforms. The guide frame lifting platform, as a product of the new era combining the functions of construction hoists and aerial work platforms, is widely used in the construction of prefabricated buildings and the façades of high-rise urban buildings due to its simple operation and safe, reliable advantages. It has gradually become an important direction for the future development of aerial work platform technology. In recent years, with the continuous increase in the demand for aerial work, lifting platforms have been developing towards automation, larger sizes, and intelligence, gradually breaking through traditional design frameworks¹. However, most of these devices currently adopt integrated and standardized designs, which leads to significant vibrations and intermittent abnormal noises during practical applications, severely impacting engineering safety and construction quality. Therefore, a systematic analysis of the vibration characteristics of guide frame lifting platforms under different operating conditions is a prerequisite for achieving stable and reliable operation of the equipment.

At present, most scholars mainly focus on the structural characteristics, structural optimization, and static analysis of lifting platforms. Although these studies have improved the stability and safety of the platforms to some extent, research on the multibody dynamics analysis and vibration theory of lifting platforms remains relatively insufficient^2–6. Although these studies have improved the stability and safety of the platform to some extent, the exploration of multi-body dynamic analysis and vibration theory for lifting platforms remains insufficient. However, vibration is a key factor affecting the operational safety, stability, and comfort of operators. Zheng et al.⁷ and others performed static analysis, modal analysis, and random vibration analysis of lifting platforms based on a five-degree-of-freedom dynamic model, determining the maximum deformation, maximum stress, and their locations. However, their research did not involve multi-body dynamics analysis or the impact of vibration parameters on the platform. Researchers such as Yang et al.⁸ and Bi et al.⁹ established dynamic simulation models of lifting platforms based on the principles of theoretical mechanics and engineering thermodynamics, and conducted systematic investigations into their dynamic responses and vibration characteristics. Florentin et al.¹⁰ and others established a flexible-rigid coupling model based on traditional flexible body modeling methods, driving the research on lifting equipment from static analysis to dynamic analysis, and further explored the main factors influencing vibration responses. Han¹¹ and others further studied and derived the vibration differential equations for tower cranes under single operating conditions, providing a theoretical model foundation for solving such problems. Cibicik A et al.¹² and others validated the accuracy of their model based on numerical analysis and multi-condition dynamic models. Han et al.¹³ and others clarified the vibration response laws through modal analysis and transient analysis of the model. Some scholars have explored the influence of rigid force and natural frequency on vibration characteristics based on numerical models and simulation analysis^14–16. Liu et al.¹⁷ and Peng et al.¹⁸, and others analyzed transverse vibration, longitudinal vibration, and transverse-longitudinal coupled vibration one by one, revealing the coupling relationships between different directions of vibration. Yan et al.¹⁹ and others derived the relationship curve between natural frequency and vibration response.

In the aforementioned studies, scholars have conducted in-depth analyses of the influence of factors such as the structure, statics, multi-body dynamics, and natural frequencies of lifting platforms on their vibration characteristics, and have proposed various vibration models and corresponding optimization solutions. However, most of the research relies on single models, single operating conditions, or finite element simulation methods to solve vibration characteristics. These models often exhibit low accuracy, limited analysis depth, and fail to trace the root causes of vibration. To address this issue, this paper proposes a combined simulation method based on a multi-body dynamics model and finite element simulation, aiming to conduct a comprehensive analysis of the platform’s vibration characteristics. To further validate the model’s rationality and analysis results, an experimental setup for the guide frame lifting platform was designed. The study examined the effects of multiple operating conditions and factors, including startup, steady-state operation, braking, and off-center loading, on the vibration characteristics of the platform. This research provides an effective solution to improve the safety and reliability of lifting platforms and offers experimental and theoretical support for future vibration characteristic analysis of lifting platforms under multiple operating conditions.

Establishment of the flexible-rigid coupling model for the guide frame lifting platform

As a typical multi-body dynamics system, the traditional multi-rigid body dynamics models are insufficient to meet the practical engineering requirements for guide frame lifting platforms. Therefore, the development of new models has become an inevitable choice to address the challenges in such systems. This section constructs a flexible-rigid coupling model and its simulation model for the guide frame climbing lifting platform, based on the flexible-rigid coupling dynamics theory and substructure reduction theory in multi-body system dynamics.

Establishment of the dynamic equations for the flexible-rigid coupling system

For describing large displacements of flexible bodies such as rack-climbing lift platforms, common approaches include the floating reference frame method, the absolute nodal coordinate formulation, and the finite element method. Based on existing research, the floating reference frame method is particularly suitable for systems undergoing large rigid-body motion with small elastic deformations, making it widely adopted for modeling flexible systems. In this method, the flexible deformation coordinate system is not directly attached to the flexible body but is represented in a “floating” state. The kinematic formulation of this coordinate system is illustrated in Fig. 1.

Fig. 1.

Spatial description of flexible bodies²⁰.

The position vector of an arbitrary point of the flexible body in the coordinate system can be expressed as:²¹:

1

where denotes the position vector of the platform’s center of mass in the inertial coordinate system, is the attitude direction cosine matrix describing the platform orientation, represents the local coordinate of a point on the flexible body in the undeformed configuration, and denotes the elastic displacement of the point relative to the undeformed configuration.

The velocity of the arbitrary point can therefore be expressed as:

2

where , and denotes the skew-symmetric angular velocity matrix.

The acceleration of the point in the inertial coordinate system can be expressed as:

3

In the body-fixed coordinate system, only the relative deformation of the flexible body is considered, and the displacement can be expressed as:

4

Let denote the generalized coordinates of the rigid body and represent the flexible modal coordinates. The generalized coordinates of the system can therefore be expressed as:

5

The total kinetic energy of the system can be expressed as:

6

Here, denotes the density.

The total kinetic energy of the system can be further expanded as:

7

Here, represents the kinetic energy of the rigid body, denotes the vibrational kinetic energy of the flexible body, and corresponds to the kinetic energy due to rigid–flexible coupling.

Rigid-body kinetic energy:

8

Where denotes the total mass of the rigid-body part, represents the angular velocity vector, and is the inertia matrix of the rigid body.

Flexible-body vibration kinetic energy:

9

Rigid–flexible coupling kinetic energy²¹:

10

Total Strain Potential Energy of the Flexible Body²²:

11

where ε denotes the strain vector and D represents the material constitutive matrix.

After finite element discretization, the strain energy of the system can be further expressed as:

12

The displacement can be expanded using the modal superposition method:

13

where Φ denotes the modal matrix.

By introducing the modal coordinate expansion, the kinetic energy associated with the vibration of the flexible body can be further expressed as:

14

where represents the modal mass matrix of the flexible body, and denotes the mass matrix obtained from the finite element discretization.

Based on the Lagrangian function of the system, , and by applying the Euler–Lagrange equation, the dynamic governing equation of the system can be derived as:

15

Accordingly, the rigid–flexible coupled dynamic equation of the system can be obtained and expressed in the following standard matrix form²¹:

16

Here, is the mass matrix, is the generalized velocity-dependent force matrix, denotes the stiffness matrix, and represents the generalized force vector.

Based on the derivation above, the rigid-flexible coupled dynamic equation of the rack-climbing lift platform can be obtained in the standard form as follows:

17

Here, , denotes the equivalent rigid-body mass matrix, where represents the total mass of the platform. and are the inertial coupling matrices between the rigid and flexible components. denotes the structural damping matrix, while and represent the Coriolis coupling terms. is the equivalent stiffness matrix of the flexible body, and denotes the coupling stiffness matrix induced by boundary constraints.

Modal analysis of the rack-climbing lift platform

Modal analysis is one of the fundamental methods in structural dynamics and is primarily used to determine the natural frequencies and mode shapes of a structural system. These two parameters are essential indicators that characterize the free vibration behavior of the system. For any structural system, a set of inherent vibration frequencies exists under free vibration conditions, which are referred to as natural frequencies or resonant frequencies. Each natural frequency corresponds to a specific vibration pattern. When the excitation frequency approaches or coincides with the natural frequency of the structure, a significant resonance response may occur, and the structure exhibits a vibration pattern corresponding to that frequency, known as the vibration mode.

In modal analysis, the dynamic governing equations of the system are typically established based on Hamilton’s principle^23,24, which can be expressed as follows:

18

In the above equation, M and K denote the mass matrix and stiffness matrix of the system, respectively, which are determined by the material properties and structural parameters of the platform. C represents the damping matrix, which is related to factors such as the connection configuration of the platform, structural stiffness, and energy dissipation characteristics. , , and represent the displacement, velocity, and acceleration vectors, respectively, while denotes the time-dependent external dynamic excitation load vector.

During modal analysis, external excitation forces are typically not applied, since the modal characteristics of each order are determined solely by the inherent properties of the structure and are independent of external excitations. By setting in Eq. (18) and neglecting the damping effect, the governing equation for modal analysis can be expressed as follows:

19

According to vibration dynamics theory, the free vibration of a structure generally exhibits a simple harmonic motion. Therefore, the displacement response of the system can be expressed as a harmonic function of time. The displacement can be written as:

20

Here, denotes the amplitude, and represents the phase.

Substituting Eq. (20) into Eq. (19) yields the characteristic equation of the system:

21

Based on the above theoretical framework, the modal analysis of the rack-climbing lift platform can be performed using the finite element method to obtain the eigenvalues . Taking the square root of the eigenvalues yields the natural angular frequencies , and the corresponding natural frequencies are given by . The eigenvector associated with each represents the corresponding mode shape at the natural frequency , which characterizes the vibration pattern of the system under that specific mode.

Substructure reduction method for flexible bodies

Since the Modal Neutral File (MNF) contains extensive information related to structural deformation, strain, and vibration, MNF files directly generated by finite element software are typically large in size, resulting in significant memory consumption and increased computational complexity. During the process of rigid–flexible body substitution and dynamic simulation, the ADAMS software may encounter insufficient memory, leading to computational failure or even software crashes, which significantly reduces computational efficiency and increases simulation time. To address these issues, this study proposes a flexible body generation method that improves computational efficiency while maintaining computational accuracy.

Specifically, a model reduction approach based on the Component Mode Synthesis (CMS) method implemented in ANSYS Workbench is adopted to simplify the flexible body model. A flexible body dynamic model with six degrees of freedom is established, and the displacement vector is decomposed into master degrees of freedom and slave degrees of freedom using the substructure reduction method. A coupling relationship exists between the master and slave degrees of freedom, which can be expressed as:

22

where denotes the transformation matrix.

In a multi-degree-of-freedom system, the relationship between structural displacement and the applied force can be expressed using the stiffness matrix , which is written as:

23

In the equation, is the stiffness matrix, is the displacement vector, and is the force vector.

Therefore, Eq. (23) can be transformed into:

24

By solving the above equation, the following result can be obtained:

25

Substituting Eq. (25) into Eq. (22) yields:

26

Substituting Eq. (25) into Eq. (24) yields:

27

The above equation can be further simplified as:

28

where denotes the transformed equivalent stiffness matrix, which can be expressed a .

Accordingly, the transformation relationships for the mass matrix and damping matrix can be derived as follows:

After performing the above matrix transformations, the dynamic equation of the flexible body can be expressed as:

29

The above derivation demonstrates that, without affecting the overall structural dynamic performance, the finite element degrees of freedom of the flexible body can be reduced by . Therefore, the selection and definition of master nodes are crucial for accomplishing substructure reduction and constitute one of the key factors influencing simulation accuracy and computational performance.

Establishment and simplification of the structural model

This study focuses on the Shenxi Machinery STC100 guide frame lifting platform, which mainly consists of the working platform, drive system, guide frame, base, and wall attachment frame. The working principle of the platform is to fix the lifting platform onto the guide frame using guide wheels, and the lifting motion is achieved through the drive system. The structural materials of the entire machine are primarily made of 345 steel, with a density of 7.8×106kg/mm³, an elastic modulus (E) of 2.07×105MPa, and a Poisson’s ratio (μ) of 0.3. To simplify the simulation process and reduce simulation time, the system was appropriately simplified. The actual guide frame is composed of standard sections, while the simulation model was simplified to an integral structure. Additionally, considering the minimal impact of the guardrail on vibration characteristics and safety, it was omitted in the simulation model. A comparison between the simplified actual machine and simulation model is shown in Fig. 2.

Fig. 2.

Simplification of the guide frame lifting platform model.

Establishment of the flexible-rigid coupling model

Currently, there are three main methods for generating flexible bodies: ADAMS automatic flexible body generation, Mechanical APDL flexibilization, and ANSYS Workbench command stream flexibilization. In this study, Ansys APDL combined with ADAMS is used to flexibilize the guide frame lifting platform, with the process shown in Fig. 3.

Fig. 3.

Simulation process for establishing the flexible-rigid coupling model.

This method is based on modal analysis in ANSYS Workbench, where the mesh is generated and the material properties are defined. Considering both model accuracy and computational efficiency, a total of 170,555 elements and 340,109 nodes are created, with an overall mesh quality of 0.7. At the same time, rigid external points of the flexible bodies are defined, and APDL is invoked to complete the flexibilization of the structures. After meshing and flexibilization, the modal file is imported into ADAMS to replace the corresponding rigid bodies. To ensure the accuracy of the flexibilization process and the integrity of the model, the natural frequencies of the flexible components obtained in the two software environments are compared and validated, as shown in Table 1. The comparison indicates that the natural frequencies of the flexible platform in both software are essentially consistent, with a maximum deviation of 0.198%, thereby confirming the accuracy of the flexible file transfer.

Table 1.

Comparison of natural frequencies of the flexible body between ANSYS and ADAMS.

Modal order	Work platform
	Natural frequency (Hz)		Error (%)
	ANSYS	ADAMS
1	90.204	90.2153	0.012
2	108.26	108.2426	0.016
3	122.85	122.8877	0.030
4	139.24	139.3085	0.049
5	156.42	156.4867	0.043
6	157.97	157.6564	0.198

By utilizing ANSYS APDL combined with ADAMS to flexibilize the guide frame lifting platform, a series of combined constraint connections were established between the rigid bodies, between rigid and flexible bodies, and between flexible bodies. The flexible-rigid coupling model for the entire lifting platform was then constructed, as shown in Fig. 4. However, flexible-rigid transformation often relies on the settings of the simulation software, which may fail to accurately reflect the actual contact types, leading to significant discrepancies between the simulation structure and the real-world conditions. To address this, this study established three different flexible-rigid coupling models for the lifting platform, which were compared with the rigid model to investigate the impact of the primary flexible components on the vibration characteristics. This aims to build a more accurate flexible-rigid coupling model, providing a solid model foundation for subsequent research. The model is shown in Fig. 5.

Fig. 4.

Flexible-rigid coupling model of the guide frame lifting platform.

Fig. 5.

Rigid body model and three types of flexible-rigid coupling models.

In ADAMS multi-body dynamics simulations, when multiple identical types of constraints are applied to the same rigid body, the system recognizes them as independent constraint units, leading to redundant constraints. These redundant constraints inevitably increase the computational load of the system, reducing simulation efficiency. Therefore, it is necessary to apply the correct constraints, such as fixed joints, kinematic joints, and contact between components, to the flexible-rigid coupling structural model. The types of contact between components of the guide frame lifting platform are shown in Table 2.

Table 2.

Constraint types between components of the lifting platform.

Component 1	Component 2	Constraint type	Number
Base	Guide frame	Fixed joint	1
Wall attachment	Guide frame	Fixed joint	2
Rack	Guide frame	Fixed joint	6
Work platform	Drive platform	Fixed joint	6
Gear	Drive platform	Revolute joint	2
Roller	Drive platform	Revolute joint	8
Gear	Rack	Contact Force	2
Roller	Guide frame	Contact FORCE	8

To further improve the simulation results, ensure the matching of experimental conditions, and enhance the simulation accuracy of the coupling model, the stress release of the model was reconfigured. Based on Hertzian contact theory, the contact force parameters, after defining the gear-rack, roller-guide frame, and material properties, are set as shown in Table 3.

Table 3.

Contact force setting parameters.

Paired components	Stiffness coefficient K(N/mm)	Damping coefficient C(N·s/mm)	Force exponent e(N/mm)	Penetration depth d(mm)	Static friction coefficient mus	Dynamic friction coefficient mud
Gear-rack		45	1.5	0.1	0.3	0.25
Roller-guide frame	3807.762	1.52	2	0.1	0.13	0.09

During the operation of the lifting platform, the motor speed is 1500r/min, and the braking deceleration ratio is 84.5, resulting in an angular velocity of approximately 1.85rad/s for the drive wheel. To make the simulation motion trend more closely align with the actual operating conditions, the motion process of the platform is divided into three phases: the acceleration startup phase (0~1s), the steady-state running phase (1~4s), and the deceleration braking phase (4~5s), with a constant running speed of 1.85rad/s. Through the simulation, the angular velocity, angular acceleration, and longitudinal (Y-axis) displacement of the driving gear’s center of mass were obtained, as shown in Fig. 6. Additionally, a driving function was set to accommodate this operating mode.

Fig. 6.

Angular velocity and acceleration curves of the gear’s center of mass.

Vibration characteristics analysis of the lifting platform

Axial vibration characteristics analysis

Under standard operating conditions, vibration characteristic simulation analyses were conducted for four models: the rigid model, multi-flexible coupling model, flexible drive coupling model, and flexible platform coupling model. By studying the relationship between vibration amplitude and operating conditions for each model, vibration curves in the X, Y, and Z directions were obtained, as shown in Figs. 7, 8 and 9.

Fig. 7.

X-Axis vibration displacement diagram.

Fig. 8.

Y-Axis vibration displacement diagram.

Fig. 9.

Z-Axis vibration displacement diagram.

From Figs. 7, 8 and 9, it can be seen that during the startup phase, the guide frame lifting platform must overcome inertia and friction, resulting in a larger driving force. This leads to significant shock loads on the platform, causing larger vibration amplitudes. In the steady-state phase, the inertial forces disappear, and the platform is only affected by constant loads. The vibration energy is dissipated by the damping system, and the vibrations gradually decrease and stabilize. During the braking phase, the acceleration in the direction of motion, combined with inertial and braking forces, causes an increase in vibration. The vibration characteristics of the platform in the Z direction are similar to those in the X direction, with the maximum vibration amplitude in the Z direction being 1.03mm and in the X direction 1.34mm, both occurring during the startup phase. In contrast, the vibration amplitude in the Y direction reaches 7mm, which is 579% and 422% higher than the Z and X directions, respectively. This analysis indicates that the Y-direction vibration is more significant and better represents the vibration characteristics of the platform. Simulation results show that the rigid model is ineffective for vibration analysis. Compared to the multi-flexible coupling model and flexible drive coupling model, the flexible platform-rigid drive coupling model (flexible-rigid coupling model) exhibits a smoother transition in its vibration characteristics curve, with more noticeable vibration amplitudes, thereby validating the correctness of the model presented in this study.

The above simulation analyzed the axial vibration characteristics at the end of the guide frame lifting platform under standard operating conditions. However, in real-world conditions, the vibration characteristics at different locations of the platform vary. To further analyze the vibration characteristics at different positions, this study conducted simulation analyses of the vibration characteristics at the end and at the center of mass, with the results shown in Figs. 10, 11.

Fig. 10.

Vibration displacement diagram of the lifting platform’s center of mass.

Fig. 11.

Comparison of vibration displacement between the end and center of mass of the lifting platform

From Fig. 11, it can be seen that the vibration amplitude differences between the end and the center of mass in the X and Z directions are relatively small, with a difference of about 17.8%, indicating that the vibration characteristics in the X and Z directions are less influenced by the measurement points. However, in the Y direction, the vibration amplitude at the end is significantly larger than at the center of mass, with a variation rate of approximately 43%. This suggests that the analysis of longitudinal vibration characteristics is more significantly influenced by the selection of measurement points, with end-point vibrations being more pronounced.

Vibration characteristics analysis of the guide frame lifting platform under multiple operating conditions

Impact of operating speed on vibration characteristics

The axial vibration characteristics of the platform have been analyzed in previous sections, assuming the same operating speed. However, in real-world conditions, the operating speed of the platform can be influenced by working conditions. The operating speed not only directly affects the lifting efficiency and stability but also alters the vibration characteristics to some extent, which in turn impacts the platforms fatigue life and operational comfort. This section considers the platform under full load with uniformly distributed mass as the baseline operating condition. The operating speeds are set to 5m/min, 10m/min, and 20m/min, and the longitudinal vibration characteristic curves at different operating speeds are obtained. The specific results are shown in the Fig. 12 below.

Fig. 12.

Impact of operating speed on the longitudinal vibration amplitude of the work platform.

As shown in Fig. 12, the vibration is most significant during the startup phase, with amplitudes of 4.26mm, 5.60mm, and 6.14mm at operating speeds of 5m/min, 10m/min, and 20m/min, respectively. As the speed increases, the vibration amplitude shows an increasing trend: when the speed is doubled, the vibration amplitude increases by 31.4%; when the speed is quadrupled, the vibration amplitude increases by 44.7%. The increase in vibration amplitude follows a diminishing trend with the increase in speed. Specifically, the amplitude growth rates during the startup phase, steady-state phase, and braking phase are 44.7%, 25.76%, and 74%, respectively. This indicates that the operating speed has a significant impact on the platform’s vibration characteristics.

Impact of load size on vibration characteristics

In actual operation, the load on the guide frame climbing lifting platform is not constant but is controlled by the operational requirements. To study the sensitivity of vibration characteristics to load size, this section sets the load to 400kg, 800kg, and 1200kg, assuming the load is uniformly distributed across the platform, with a uniform operating speed of 10m/min, while keeping other parameters unchanged. The analysis of the platform is performed, and the results are shown in Fig. 13.

Fig. 13.

Effect of load magnitude on the longitudinal vibration amplitude of the platform.

The results indicate that the platform vibration is most pronounced during the start-up phase, with vibration amplitudes of 1.91mm, 3.026mm, and 5.6mm, respectively. When the load is doubled, the vibration amplitude increases by 58.4%; when the load is increased by twofold, the vibration amplitude rises by 193%. The maximum vibration amplitude growth rates during the start-up, constant-speed, and braking phases are 193%, 38.1%, and 44.7%, respectively, while the vibration period increases by 25%. As the load increases, the growth rate of the vibration amplitude significantly exceeds linear scaling, indicating that load variations exert an increasingly pronounced influence on the vibration characteristics of the platform.

Effect of eccentric loading on vibration characteristics

In practical operations, owing to engineering requirements, improper handling, and environmental factors, the platform load may be unevenly distributed, resulting in eccentric loading conditions. Such non-uniform load distributions can alter the systems equilibrium state and consequently affect its normal operation, leading to disturbances in vibration modes and frequencies. This, in turn, may impair the platforms operational performance and even pose potential safety risks. To investigate the influence of different eccentric loading conditions on the vibration characteristics, this subsection analyzes the load distribution scenarios illustrated in Fig. 14, with the corresponding results presented in Fig. 14. Cases 1 and 2 represent uniformly distributed loading conditions, whereas Cases 3 to 6 correspond to eccentric loading conditions.

Fig. 14.

Effect of eccentric loading conditions on the longitudinal vibration amplitude of the platform.

The analysis indicates that the vibration responses of the platform under Cases 1 and 2 are highly similar, suggesting that, under uniformly distributed loading conditions, the load placement has a negligible effect on platform vibration. In contrast, under Cases 3 to 7, the maximum vibration amplitude reaches up to 11.74mm, while the average vibration amplitude during the constant-speed phase ranges from 4.5 to 7mm. Compared with the uniformly distributed loading conditions, the vibration amplitude under eccentric loading increases by 25.9%, and the vibration amplitude during the constant-speed phase increases by approximately 350%. The influence of eccentric loading on the platforms vibration characteristics is governed by the specific eccentric loading condition, and the effects of load magnitude and operating speed on system stability and safety become increasingly pronounced.

Effect of braking time on vibration characteristics

Braking duration is a primary human-controlled factor influencing the vibration characteristics of the platform. During the braking process, the system undergoes a dynamic response, and the vibration amplitude varies due to inertial effects and the release of kinetic energy, thereby affecting system stability as well as operator comfort and safety. Consequently, analyzing the influence of braking duration on platform vibration characteristics is of significant importance for optimizing braking system design and control strategies, reducing vibration amplitudes, and improving the braking safety of lifting platforms. To investigate the effects of different braking durations on platform vibration characteristics, this section considers a baseline operating condition with the platform under full-load uniformly distributed loading and a constant operating speed of 10m/min. The braking duration is set to 0.1s, 0.25s, 0.5s, 1s, 2s, and 4s, respectively, with a total platform operating time of 8s. The corresponding simulation results are shown in Fig. 15.

Fig. 15.

Effect of braking time on the longitudinal vibration amplitude of the platform.

As shown in Fig. 15, when braking is activated for 4s, the platform vibration is significantly amplified, with the maximum amplitude occurring at approximately half of the braking duration, reaching 3.18mm. Across the six braking schemes, the vibration patterns are similar, and the amplitude variations are relatively small. The optimal braking duration is determined to be 0.5s.

Experimental design and validation

Experimental platform setup

The hardware configuration of the experimental platform is shown in Fig. 16. It primarily consists of a rack-climbing lift platform, a PC workstation, and a DH5920 32-channel dynamic signal acquisition system. Supporting equipment includes a 1394 data acquisition card, a DH5857-1 charge amplifier, and DH131 accelerometers, among others.

Fig. 16.

Hardware composition of the experimental platform.

Comparative analysis of simulated and experimental longitudinal vibrations

In this experiment, a piezoelectric accelerometer DH131 was used to measure the vibration amplitude of the platform. The sensor has a diameter of 10mm, a height of 13mm, a measurement range of 5000m·s², a frequency response range of 1~10000Hz, and an axial sensitivity of 1.06pC/(m·s²). The installation method is illustrated in Fig. 16. Based on the previous analysis, the primary vibration of the platform occurs along the Y-axis at the end, and therefore only Y-axis vibration data were collected and analyzed in this experiment. The results are presented in Fig. 17.

Fig. 17.

Comparison of vibration amplitudes between the coupled model of the lifting platform and experimental results.

As shown in Fig. 17, the maximum vibration amplitudes for the multi-flexible coupled model, flexible-drive coupled model, flexible-platform coupled model, and the experimental measurements are 5.6mm, 1.075mm, 3.295mm, and 5.28mm, respectively. Comparison of the time-domain waveforms indicates that the vibration response curve of the multi-flexible coupled model closely matches the experimental results. Further quantitative analysis shows that the absolute error in amplitude is ≤0.5mm during the start-up phase, ≤0.2mm during the constant-speed phase, and ≤0.3mm during the braking phase, with the relative error of all effective peak points across all phases controlled within 25%. Overall, these results validate the rationality of the proposed rigid–flexible coupled model. The model effectively captures the vibration characteristics of the rack-climbing lift platform under various operating conditions in simulation, providing a solid modeling foundation for subsequent simulation studies.

Comparative analysis of vibration acceleration between simulation and experiment

Vibration acceleration serves as a primary indicator for assessing the vibration frequency of the platform. By analyzing the vibration acceleration, the vibration characteristics of the platform under different operating states and conditions can be obtained, providing data support for improving platform comfort and stability. The comparative analysis results of the lifting platforms vibration acceleration and its frequency-domain characteristics between simulation and experimental measurements are presented in Fig. 18.

Fig. 18.

Comparison of vibration acceleration between the coupled model of the lifting platform and experimental results.

As shown in Fig. 18, the maximum vibration accelerations for the multi-flexible coupled model, flexible-drive coupled model, flexible-platform coupled model, and experimental measurements are 50614.3mm/s², 8546.3mm/s², 33109.7mm/s², and 33273.4mm/s², respectively. Based on the calculation of effective peak points, the relative errors of the multi-flexible coupled model during the start-up, constant-speed, and braking phases are 17.75%, 16.05%, and 17.78%, respectively. This model exhibits the highest agreement with the experimental data. Overall, the analysis demonstrates that the multi-flexible coupled model effectively captures the dynamic behavior of the platform and serves as an ideal model for analyzing the influence of factors such as operating speed and braking duration on the platforms vibration characteristics.

As shown in the Fig. 19, the dominant frequencies of vibration acceleration for the multi-flexible coupled model, flexible-drive coupled model, flexible-platform coupled model, and experimental measurements are 8.98Hz, 8.87Hz, 11.66Hz, and 8.95Hz, respectively. Frequency deviation analysis indicates that the dominant frequency of the multi-flexible coupled model is closest to the experimental result, with a relative deviation of 6.65%. The frequency analysis reveals that when the excitation frequency approaches 8.9Hz, the systems vibration response increases significantly, indicating a high sensitivity of the system to excitations in this frequency range and a substantial influence of the excitation on vibration amplitude. Therefore, in vibration characteristic analysis, it is recommended to focus on the dynamic response within the frequency range of 8.9±0.3Hz.

Fig. 19.

Comparison of vibration acceleration in the frequency domain between the coupled model of the lifting platform and experimental results.

Conclusions

1. The rack-climbing lift platform is modeled as a multi-body dynamic system. By applying rigid–flexible coupled dynamics theory and substructure reduction theory, a rigid–flexible coupled system dynamics model is constructed. This approach overcomes the limitations of traditional multi-rigid-body dynamic models, such as low computational accuracy and weak dynamic responses, while retaining the high computational efficiency of rigid-body models.

2. A command stream softening method embedded in ANSYS Workbench is introduced. By replacing the original rigid components in the system with modal neutral files, rigid-flexible transformation is achieved, and a rigid-flexible coupled simulation model of the platform is established. Using modal theory, modal verification of the rigid-flexible coupled model is conducted in ADAMS, and the natural frequencies of the model are compared to ensure the accuracy of the transformation, with a maximum error of 0.198%. This method resolves potential data loss or corruption issues that may occur when importing ANSYS Workbench modal neutral files into ADAMS.

3. Axial vibration characteristics of the platform are simulated and analyzed. The results show that the Z- and X-direction vibrations are similar, whereas Y-direction vibration is more pronounced, reaching amplitudes up to 7mm an increase of 579% and 422% compared with Z- and X-directions, respectively. Significant differences in vibration characteristics are observed between the platforms centroid and its end, indicating that vibration analysis is influenced by the choice of measurement points. The primary factors affecting platform vibration are analyzed under four operating conditions: travel speed, load magnitude, eccentric loading, and braking duration. The maximum vibration amplitudes are 6.14mm, 5.6mm, 11.74mm, and 3.18mm, respectively. When both travel speed and load magnitude are doubled, vibration amplitudes increase by 31.4% and 58.4%, respectively. With increasing load, the growth of vibration amplitude exceeds linear scaling, with eccentric loading having the most significant impact. Compared with other conditions, braking duration notably affects vibration characteristics, with an optimal braking time determined to be 0.5s. An experimental platform for analyzing the vibration characteristics of the rack-climbing lift platform was designed, along with a comparative scheme between axial vibration and vibration acceleration simulations and experiments. The results show that the simulation-experiment error for axial vibration amplitude is approximately 6%, and the average error for vibration acceleration is about 17.19%. These findings validate the effectiveness of the proposed rigid–flexible coupled model. Furthermore, multi-condition analysis of platform vibration characteristics provides a basis for optimizing operational protocols, improving platform structure, and enhancing stability, safety, and operator comfort in subsequent studies.

Author contributions

Mu Lei conducted the experiments, performed the data analysis, and wrote the manuscript. Jinjin Sun processed the experimental data and revised the manuscript. Jie Wu secured the funding and supervised the experimental project. All authors reviewed and approved the final manuscript.

Data availability

• The datasets used and/or analysed during the current study available from the corresponding author on reasonable request. • All data generated or analysed during this study are included in this published article [and its supplementary information files].

Declarations

Competing interests

The authors declare no competing interests.