A Multibody Mathematical Model to Simulate the Dynamic Behavior of Aerial Work Platforms Using Python ^†

Abstract

Understanding and knowing the extent of dynamic loads (forces and moments) acting on critical points of structures from the early stages of new product design allows companies to reduce product time-to-market and cut costs associated with physical validation tests. This is made possible through the use of commercial software that enables the simulation of the dynamic behavior of a wide range of mechanical systems, and beyond. However, the use of such software is not straightforward and often requires a specialist within the company to have in-depth knowledge of both simulations and the software itself. Consequently, many companies give up the use of these tools, compromising the whole product development process, falling into the design and testing iterative loop. The multibody mathematical model developed in this study, simulated using python programming language, allows easy retrieval of all the necessary information without requiring the user to be a specialist in multibody dynamics simulations. This model is intended as a tool specifically designed for the type of product (aerial work platform) that serves, during the conceptual and preliminary design phases, to predict dynamic loads through calculations. The balance between the simplicity of such a tool and the accuracy of the results is the key point for the success of this work.

1. Introduction

The main goal of the engineering profession is to design and manufacture a high-quality marketable product in a short lead time. Today’s industries are utilizing the computer in every phase of the design and manufacturing of their products. The design and manufacturing process, beginning with an idea and ending with a final product, is a closed-loop process. The design process requires a thorough understanding and ability to analyze the product. Computer-aided analysis allows an engineer to simulate and predict the behavior of a product. Based on the analysis results, the product design can be optimized prior to actual production. To simulate the behavior of a product, we must know the individual components that make up the product [1]. Modern mechanical and aerospace systems are often very complex and consist of many components interconnected by joints and force elements such as springs, dampers, and actuators [2]. These systems are referred to, in the modern literature, as multibody systems. The dynamics of such systems are often governed by complex relationships resulting from relative motion and joint forces between the components of the system [2]. For these reasons, the use of commercial simulation software is essential. However, current general CAE software packages are very complex, and it is not obvious for the designer or for an industrial corporation how to fully utilize the potential of these new tools. Several important questions are still under discussion in industry, e.g., whether an integrated environment or a distributed simulation with communication between different types of simulation software should be pictured, if simple geometry models should be created directly in the simulation software, if a CAD solid model should always be used as a master model for all analyses, etc. There is obviously a need for a systematic approach in modeling and simulation [3]. The use of simulation software should be different during the various design phases for a product development project. Design theory and product development models from design research have obvious potential and should be used as a basis for an improved and efficient methodology and approach to modeling and simulation in the design of mechanical products. The optimal combination of simulation and physical prototype testing is another important issue in industry. Simulation results should be used to minimize testing effort and to concentrate the expensive testing on the most relevant cases. Test results, on the other hand, should be used to improve mathematical models of basic or product-specific phenomena and to incorporate empirical knowledge in the simulation models [3].

In industry, there is a clear gap between the availability of modeling and simulation tools and the ability to fully utilize their potential. This forms the foundation of the present work. Here, a mathematical multibody model of an Aerial Work Platform is developed and analyzed. The aim of this research is to create a computational tool capable of predicting dynamic loads on the machine during the conceptual and preliminary design stages. To validate the simulation results produced by the Python package, a comparative ADAMS model of the platform was built, demonstrating a strong correlation between the two models.

2. Test Case

The machine involved in this work is shown in Figure 1.

The model used to perform the simulation is shown in Figure 2, which consists of three main components: the front axle, the rear axle, and the suspended mass, which represents the combined structure of the drive chassis, the slewing chassis, and the boom, previously shown in Figure 1. Two revolute joints are used to connect the axles to the suspended mass. This configuration was chosen to focus on determining the vertical forces at the wheels, as well as the forces and moments transmitted between the axles and the drive chassis through the pins. Consequently, all parts above the axles are treated as a single body, with the center of gravity and the inertia moments calculated starting from the individual mass properties of the bodies that composed the suspended mass. The model’s goal is to capture the spatial dynamic behavior of the machine, although each body typically has six degrees of freedom (DOF), and simplifications have been made to focus on vertical dynamics, as outlined in Figure 2.

3. Non-Linearity

Although the multibody mathematical model has been simplified, some non-linearities have been considered to make the simulation as close to reality as possible.

3.1. Wheel Model

Each wheel has been modeled as a spring–damper element that operates only along the y-axis, with one end connected to a specific point on the axle of the platform and the other end connected to the road profile, as shown in Figure 3.

The reaction force provided by a single wheel can be expressed as:

$$F_{tire}=-(K_{tire}·\Delta s)-(C_{tire}·\Delta \dot{s})$$

(1)

where $\Delta S$ and $\Delta \dot{s}$ represent, respectively, the variation in the length and in the ratio of the single spring–damper element. The $K_{tire}$ and $C_{tire}$ characteristics have been defined in a way to take into account the wheel detachment from the ground. Considering the chain of distances shown in Figure 4, it is possible to retrieve the spatial coordinates of the wheel–ground contact point, and, consequently, it is possible to calculate at each integration time step and for each wheel, the deflection value (${\delta }_{y.th}$) as:

$${\delta }_{y.th}=-[y_1+{\left(u_{1R}\right)}_y-{\left(u_{3R}\right)}_y+{\left({\tau }_{3RR}\right)}_y]+{\left(u_{RR}\right)}_y$$

(2)

When the deflection value becomes negative (i.e., ${\delta }_{y.th}$ < 0), it means that the considered wheel has no more contact with the ground, as show in Figure 5.

Depending on the deflection value (${\delta }_{y.th}$), it is possible to define a non-linear behavior for the stiffness and the damping characteristics (Figure 6).

3.2. Hydraulic Cylinders Model

The simplified model shown in Figure 7 has been adopted for determining the hydraulic cylinders’ dynamic behavior (the involved cylinders are shown in Figure 8).

Considering the two different oil volumes that work inside the cylinder and following the method highlighted by Feng et al. in [5], it is possible to calculate two different stiffness values for a single cylinder, one for the tensile working condition and one for the compressive working condition. This is a clear non-linear behavior. During the evolution of the simulation, in fact, these two values need to be chosen in relation to the operating condition of the cylinder at that moment (i.e., at that integration time step). To avoid this non-linear behavior, the schematic proposed in Figure 8 has been used.

By some simple geometrical relations and through the use of the principle of virtual works, a relation that links the linear hydraulic cylinder stiffness ($K_t$ and $K_c$) values with the torsional one ($K_{txx}$) can be obtained:

$${\displaystyle \frac{1}{2}}·K_c·\Delta S_{AB}^2+{\displaystyle \frac{1}{2}}·K_t·\Delta S_{CD}^2={\displaystyle \frac{1}{2}}·K_{txx}·\Delta {\phi }^2$$

(3)

$$K_{txx}={\displaystyle \frac{K_c·\Delta S_{AB}^2+K_t·\Delta S_{CD}^2}{\Delta {\phi }^2}}$$

(4)

3.3. Contact Model

Contacts between bodies have to be considered inside the model due to the fact that the rear axle is allowed to freely rotate for a few degrees with respect to the drive chassis (i.e., with respect to the suspended mass), as shown in Figure 9.

To simulate the contact between these two bodies, a reaction moment is imposed along the rear pivot axis. This reaction moment is the result of the multiplication between a torsional stiffness, linked to the bodies, and their relative angular variation, as expressed in Equation (5).

$$M_{xx}=-K_{txx}·({\phi }_{RA}-{\phi }_{SM}),$$

(5)

To allow the relative rotation between the bodies, a non-linear stiffness characteristic has been adopted, as illustrated in Figure 10. Defining the $AS$ parameter as:

$$AS=max(\Delta \phi )$$

(6)

it is possible to calculate the $pe{n}_{th}$ value as:

$$pe{n}_{th}=(\Delta \phi -AS)$$

(7)

In relation to the $\Delta \phi$ value, it is possible to choose the correct value for the stiffness constant, as highlighted in Figure 10.

4. Simulation Input

The input for the simulation is shown in Figure 11. It represents a real part of the test truck used to validate the design proposed for a new product and is defined both in terms of displacement and velocity, considering that wheels have been modeled as spring–damper elements.

5. Simulation Results

To compare the obtained results, an ADAMS model of the aerial work platform has been developed under the same assumptions as the simplified model. The DOF responses obtained from the ADAMS model and from solving the equations of motion of the rigid body model, using the mathematical approach defined in Appendix A, are highlighted in Figure 12, demonstrating a strong correlation between the two models.

In Figure 13, instead, are shown two of the four wheel reaction forces. Again, there is a good correlation between the two models. It is especially evident how the non-linear stiffness and damping models developed in this work are able to capture any wheel detachment from the ground (reaction force value = 0), making the simulation very close to reality.

6. Conclusions

In this study, we developed a simplified mathematical model to simulate the dynamic behavior of an elevating work platform. The simplified model discussed here proves to be a good approximation of a more complex Adams model. By incorporating all relevant non-linearities, the simulation accurately captures the dynamic responses of the system. The strong correlation between the results of the Adams model and our Python model demonstrates the reliability of the developed functions. Furthermore, the model’s effectiveness is validated through the degree of freedom (DOF) behavior observed in both simulations. This indicates that our model can reliably predict the platform’s dynamics under various operating conditions. The ability to accurately simulate such complex behavior with a simplified model not only enhances computational efficiency but also provides a valuable tool for preliminary design and analysis. Additionally, the use of the Python-based approach offers flexibility and accessibility, making it easier to implement and modify compared to more proprietary software solutions.