# The Load Spectrum of Axial Bearing of Hydraulics Excavator with Shovel Attachment

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## Abstract:

A detailed investigation into the axial bearing load of the revolving platform in a hydraulic excavator equipped with a shovel attachment was presented in this study. A mathematical model was formulated to assess the forces acting on the bearing under various operational conditions. The analysis focuses on a 100,000 kg excavator with a 6.5 m³ bucket, examining the contributions of kinematic chains and drive mechanisms to axial loads. Simulations of multiple positions within the working range were carried out, calculating the load spectrum, including boundary resistance, to ensure machine stability. An optimization program was developed to refine the bearing selection process by identifying equivalent loads and moments. These calculations were benchmarked against manufacturer capacity diagrams, allowing for precise selection of appropriate bearing sizes. The findings underscore the critical role of accurate load calculations in enhancing the performance, reliability, and design optimization of hydraulic excavators. This approach provides engineers with a framework for selecting bearings that can withstand complex operational stresses, thereby improving the efficiency and longevity of hydraulic machinery.## 1. Introduction

Hydraulic excavators are specialized equipment primarily designed for excavation. They work through a particular kinematic chain configuration, which comprises necessary components such as the revolving segment - platform $\left(L_2\right)$, the moving segment $\left(L_1\right)$, and the flexible multi-member manipulators $\left(L_m\right)$, as shown in Figure 1.

As seen in Figure 1, excavators of all sizes use changeable configurations of kinematic chains in several versions for various functions. The revolving segment (platform) and platform drive mechanism remain unchanged despite these modifications in manipulator members. The tracked and wheeled moving segments $\left(L_{11}\right)$ that excavators are equipped with are contingent upon the specific movement and support conditions. Manipulators can be equipped with an array of tools, such as buckets, tillers, grapples, hooks, hammers, and claws. Excavation work above ground uses a shovel attachment and the “away from oneself" technique. The revolving segment $\left(L_2\right)$, which is linked to the moving segment $L_1$ via a rotary joint $O_2$, is how hydraulic excavators perform spatial manipulation. This joint belongs to the fifth class since it is an axial bearing. This is a large bearing that enables the rotation of the excavator's superstructure. It connects the upper structure of the excavator to the undercarriage. The revolving drive mechanism consists of a hydraulic motor (1) and a reduction gear (2). It transmits the driving force needed to rotate the slewing ring. The hydraulic motor provides the power, while the reduction gear adjusts the speed and torque. The drive shaft of the reduction gear is connected to an inner toothed ring (3.1) of the axial bearing that is fixed to the moving segment $\left(L_1\right)$ and an outer ring (3.2), which is toothless and coupled to the revolving segment $\left(L_2\right)$. Rolling objects, such as balls or rollers, are placed between the rings in one or more races. The process of selecting an axial bearing is the first step in synthesizing the revolving platform drive mechanism. Selecting this bearing is quite difficult since the excavator operates in its working area under a wide range of various situations and in a variety of postures.

Various elements have been investigated when designing the basic excavator systems. Both analytical modeling and experimental investigation of the loading acting during the digging operation were made [1], [2]. Additionally, kinematic and dynamic excavator analysis was performed using mathematical models that were established [3], [4], and drive mechanisms and control systems were developed [5], [6], [7].

A recurring objective in numerous research efforts is to strike an optimal balance between simplicity and accuracy in the calculation of the load distribution on the most heavily loaded rolling elements within roller and ball bearings [8], [9]. The research also pertains to determining the axial bearings' static carrying curve. Applying the conditions to deformation compatibility and force equilibrium, Li et al. [10] established a link between the internal maximum rolling element load of each row and the total external axial load and tilting moment load of the axial bearing. The multirow geometry and a specified irregular geometry of the bearing were examined in order to determine their effects on the raceway contact load distribution. Some research has studied the effect of axial bearing machining errors on load-carrying capacities. A methodology for calculating load distribution in four-point contact axial bearings was presented by many researchers [11], [12], [13]. Load distribution was calculated, accounting for ball preload, ring flexibility and manufacturing errors, introducing a more computationally efficient approach for friction torque calculation. This demonstrates that ring deformations affect load distribution in idle but not under external loads.

Manufacturing errors in rolling elements and races affect load distribution, causing excessive local deformation and potential fatigue failure. Therefore, using the finite element method, Aithal et al. [14] revealed that ball size error and raceway waviness height are significant factors influencing load distribution.

Researchers have also examined the synthesis of the revolving platform's drive mechanism and the creation of hybrid drives for the drive mechanism, which enable energy recovery during platform slewing [15], [16].

This study provides a thorough examination of the load operating on the axial bearing in relation to the excavator's resting position on the base. In this study, a novel method was proposed to analyze the loading of the axial bearing of the excavator’s revolving platform by using the load spectra that cover the entire working range of the excavator, based on the forces covered by the excavator's drive mechanisms and the limit forces under which the excavator is still stable.

## 2. Mathematical Modeling for the Excavator

To facilitate the load analysis of axial bearings, a comprehensive mathematical model of an excavator was developed. This model integrates both the kinematic chain and the machine drive system. The model is composed of five distinct segments that accurately represent the kinematic configuration of the excavator. These segments include segment $L_1$ - support and movement mechanisms (subgraph (a) of Figure 2), segment $L_2$: revolving segment - platform, and the three-segment plane manipulator, which is further subdivided into segment $L_3$ - boom, $L_4$ - arm and the last segment $L_5$ - bucket. The spatial orientation and coordination of the excavator model were defined within the $O X Y Z$ coordinate system, utilizing unit vectors to precisely delineate the spatial configuration of the model.

The axial bearing kinematic pair's vertical axis $O Y$ coincides with the support and movement segmentrevolving segment (platform) axis of the substrate-reliant excavator, which is situated in the horizontal plane $O X Z$ with respect to the global coordinate system. The kinematic chain excavator's segments construct slewing joints with one degree of freedom, or kinematic pairs of the fifth class.

The normal irruption joint vertical axis passes through the horizontal plane at the center of the joint $O_2$ kinematic pair, support and movement segment- revolving segment, and the centers of the axial bearing rolling elements that connect the support and movement segment- revolving segment of the chain. The places where the kinematic chain manipulator excavator's horizontal axis irruption passes across the symmetry plane are known as the manipulator joints' centers ($\left.O_i, i=3,4,5\right)$. The bucket cutting edge's center, $O_w$, was defined as the penetration point of the cutting edge through the manipulator's plane.

In their local coordinate systems $O_i X_i Y_i Z_i$, models of the kinematic chain segments $L_i$ were defined as follows:

where, $\vec{e}_i=\left\{e_{i x}, e_{i y}, e_{i z}\right\}$ is the vector of the joint axis unit section $L_i$, which is linked to the preceding section $L_{i-1}$ through $O_i ; \vec{s}_i=\left\{s_{i x}, s_{i y}, s_{i z}\right\}$ is the position vector of two adjacent joints of the member; $\vec{t}_i=\left\{t_{i x}, t_{i y}, t_{i z}\right\}$ is the position vector of the segment's center of mass; and $m_i$ is the segment mass.

The driving mechanisms of the boom, arm, and bucket, which are activated by hydraulic cylinders $C_3$, $C_4$, and $C_5$ (subgraph (a) of Figure 2), were included in the model of the excavator's driving system. The following value sets characterize the driving mechanics of the excavator manipulator:

where, $d_{i 1}, d_{i 2}$ represents the hydro-cylinder's piston and piston rod diameters; $c_{i p}, c_{i k}$ denotes the hydrocylinder's beginning and ending lengths; $\vec{a}_i, \vec{b}_i$ represents the joint center position vectors where the hydro-cylinders are connected to the kinematic chain segments; $c_i$ is the transmission lever length if a drive mechanism is present; $m_{c i}$ is the hydro-cylinder mass; and $n_{c i}$ is the number of hydro-cylinder driving mechanisms.

The kinematic chain models of the excavator are based on the following assumptions:

The substrate dependence and individual segments of the excavator kinematic chain are modeled using rigid bodies;

Digging resistance $W$ acts in the center $O_w$ of the side of the bucket that penetrates the soil; the excavator's work is stable during manipulative tasks, meaning there is no possibility of displacement in zero-joint;

Gravitational force and digging resistance act on each member of the excavator chain during a manipulative task.

The following equation defines the vector of digging resistance:

where, $\varphi_w$ is the angle directions of potential digging resistance; $W_z$ is the lateral resistance to digging; and $W_{x y}$ is the potential resistance to digging, which acts in the manipulator's plane. The potential digging resistance is as follows [1]:

where, $W_s$ is the maximum digging resistance established for the excavator's stable operation; and $W_3$, $W_4$, and $W_5$ represent the maximum digging resistance it can tolerate for the boom, arm, and bucket driving mechanisms.

The following is the definition of the lateral digging resistance:

where, $m$ is the the excavator's mass; $L$ is the length of the crawler moving mechanism; $\mu_o$ is the resistance coefficient swivel of caterpillars in relation to the base on which the excavator rests; and $x_w$ is the horizontal coordinate the center of cutting edge of bucket.

## 3. Bearing Loads

The excavator's revolving platform $L_2$ was placed at joint $O_2$, and a simulated rupture of the kinematic link there leads to the ensuing force (subgraph (b) of Figure 2) to cause all weights, including the discarded components, to be redistributed toward the machine's center and the moment in joint $O_2$.

where, $\vec{r}_w$ is the position vector of the center of the side of the bucket that penetrates the soil; $\vec{r}_2$ is the center joint (axial bearing) position vector $O_2 ; \vec{r}_{c i j}$ is the position vector of the joint where the hydraulic cylinders are attached to the boom, arm and bucket; $r_{t i}$ is the position vector of the center of mass of the chain segments; and $m_z$ is the mass of material in the bucket (assumed that the material's and the bucket's centers of mass coincide).

The force components $F_2$ and moment components $M_2$ of the joint $O_2$ in $X$-, $Y$- and $Z$-direction are given as follows:

Axial $F_{2 a}$ and radial $F_{2 r}$ force and moment $M_{2 r}$ are load components of the axial bearing revolving platforms of the excavator (subgraph (b) of Figure 2):

The moment $M_{2 z}$, whose vector aligns with the bearing's axis, counterbalances the driving moment $M_{p 2}$ of the platform's revolving member. The moment $M_{2 x}$, the vector of which lies in the horizontal plane, must also be considered. The equivalent spectrum of bearing loads was calculated, and the bearing loading capacity diagrams (curves $A L 3, A L 4, A L 5$, and $A L 6$ in Figure 2), available in the catalogs of bearing manufacturers, serve as the basis for selecting the appropriate bearing [3].

The diagram of equivalent loads is characterized by an equivalent force $F_e$ and an equivalent moment $M_e$, which were determined by the following equations for $F_e$ and $M_e$, respectively:

where, $a$ is the axial force influence factor; $b$ is the radial force influence factor; and $f_s$ is a factor that takes into account the working conditions of the excavator. The amount of these factors can be seen in the manufacturer's catalog for each type of bearing. They depend on the type and size of the bearing as well as on the working conditions.

## 4. Selection of Bearing

Only when the equivalent loads of the axial bearing are determined for all possible combinations of the kinematic chain of the excavator as well as all possible tools with which it can be equipped, can a reliably, safe and secure choice of axial bearing be made. These different combinations of manipulator members, excavator tools, and moving mechanisms are not the same as these possible kinematic chain configurations.

Given that there is digging resistance in various directions at each position of the kinematic chain depending on the excavator's operating conditions, it is necessary to ascertain the load spectrum of the axial bearing throughout the entire working space of the excavator for as many positions of the kinematic chain as feasible [17]. A computer program was developed to select the axial bearing of the revolving platform drive in hydraulic excavators and to determine the loading spectrum using the previously mentioned calculation method in order to satisfy all of the aforementioned requirements.

The number of desired positions for the machine's kinematic chains throughout the workspace, the specifications of the excavator manipulators' driving mechanisms, the maximum pressure of the excavator’s hydrostatic system, and the number of resistance digging directions for the excavator enforcement tools were all entered when the program started.

According to the input values, by cyclically changing the set numbers of the kinematic chain position, the program determines: a) increments $\Delta c_i(i=3,4,5)$ of the length of the hydraulic cylinders which drive boom, arm and bucket and the increment of the angle of the digging resistance force $\Delta \theta_w$; b) the length of the hydraulic cylinders $c_i(i=3,4,5)$ and the angle of the digging resistance force with respect to the support base $(\theta w)$; c) geometrical quantities $\left(\theta_i, r_i, r_{t i}, r_w\right)$ that define the position of the center of the joints and the center of mass of the each member of the excavator and the transmission functions of the drive mechanisms of the excavator manipulator; d) load moments and driving moments of drive mechanisms; e) limit digging resistance forces and possible digging resistance force; and f) axial bearing load components. The program allows for a detailed analysis of the equivalent loads of the axial bearing of the revolving platform drive mechanism for the desired number of kinematic chain positions in the working space of the excavator. It also allows for the desired number of changes in the possible action direction of the digging resistance force at each position through the use of spatial simulation.

Throughout the excavator's spatial simulation, the desired shift in the kinematic chain's position was achieved by cyclically altering the hydraulic cylinder lengths of the boom, arm, and bucket of the manipulator. A bearing load spectrum for the whole excavator workspace was provided by the program, corresponding to a load bearing for each kinematic chain machine position and potential digging resistance direction [18], [19]. The analysis uses parameters and diagrams of permissible load capacities of available single-row ball and roller axial bearings in possible sizes marked $A L 3, \ldots, A L 6$, manufactured by RotheErde [20]. To illustrate the spectrum of axial bearing loads, an excavator with a total weight of $100,000 \mathrm{~kg}$, a crawler mechanism length of $5,035 \mathrm{~mm}$, a track width of $3,600 \mathrm{~mm}$, and a bucket volume of $6.5 \mathrm{~m}^3$ was selected. The land being excavated has a density of $1,650 \mathrm{~kg} / \mathrm{m}^3$. Longitudinal and transverse lines of overturning were observed during the load analysis. The same diagrams show the permissible load capacities of the same type but different size bearing curves $A L 3, A L 4, A L 5$, and $AL6$ (Figure 3). The corresponding bearing is the one under whose load-carrying curve there is a constellation of loads. As the diagrams show, the potential longitudinal line of overturning $(x-x)$ (subgraph (a) of Figure 3) is for the bearing size $AL6$, and the transverse line of overturning $(z-z)$ (subgraph (b) of Figure 3) for the bearing size $A L 5$.

## 5. Conclusions

To construct the load diagram of axial bearing, a set of points was used to represent the equivalent forces and moments plots. Each point on the diagram corresponds to a specific combination of these parameters, illustrating the overall load distribution across the bearing. These calculations were performed for all manipulators that can equip the excavator for every position of the kinematic chain and all possible directions of the external load-digging forces. A comparison of the load diagrams for available bearings of identical design but varying sizes with the spectrum of operating loads provides critical insights into the bearing selection process. This comparison ensures that the chosen bearing can handle the expected loads efficiently and reliably, thus enhancing the overall performance and durability of the excavator. The selection of axial bearing is the first step in synthesizing the revolving platform mechanism. The mathematical model and software presented in this study provide a comprehensive approach to analyzing equivalent loads. This approach ensures that selected bearings meet the demanding requirements of various operational scenarios, thereby improving the performance and reliability of hydraulic excavators. Understanding and accurately calculating the loads on axial bearings is crucial for optimizing the design and selection of critical components in these complex machines. This approach to load distribution in complex mechanisms offers practical tools for engineers working with hydraulic excavators and similar machinery.

Not applicable.

The authors declare no conflict of interest.

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$\vec{e}_i$ | Joint axis unit vector $O_i$ |

$\vec{s}_i$ | Position vector of the joint center $O_{i+1}$ |

$\vec{t}_i$ | Position vector of center of mass segment |

$m_i$ | Segment mass |

$d_{i 1}, d_{i 2}$ | Diameter of the piston and piston rod of hydro-cylinder |

$c_{i p}, c_{i k}$ | Start and final length of hydro-cylinder |

$\vec{a}_i, \vec{b}_i$ | Vectors of joint center position in which the hydro-cylinders linked to the segments of the kinematic chain |

$\vec{c}_i$ | Length of transmission levers if a drive mechanism is present |

$m_{c i}$ | Mass of hydro-cylinder |

$n_{c i}$ | Number of hydro-cylinder driving mechanism |

$W$ | Resistance to digging |

$W_{x y}$ | Potential resistance to digging |

$W_z$ | Lateral resistance to digging |

$\varphi_w$ | Angle directions of possible digging resistance |

$W_s$ | Maximum limit value digging resistance |

$W_3$ | Maximum limit values for digging resistance that can overcome the driving mechanisms boom |

$W_4$ | Maximum limit values for digging resistance that can overcome the driving mechanisms arm |

$W_5$ | Maximum limit values for digging resistance that can overcome the driving mechanisms bucket |

$\vec{F}_2$ | Resultant force in the joint $O_2$ revolving platform |

$\vec{M}_2$ | Resultant moment in the joint $O_2$ revolving platform |

$\vec{r}_w$ | Position vector of the center of the cutting edge of bucket |

$\vec{r}_2$ | Position vector of the center joint (axial bearing) |

$r_{t i}$ | Position vector of the center of mass the segments of the kinematic chain |

$\vec{r}_{c i j}$ | Position vector of the joint in which are linked hydro-cylinders of driving mechanisms for kinematic chain segments |

$m_z$ | Mass of material seized by bucket |

$F_{2 a}$ | Axial force in the joint $O_2$ revolving platform |

$F_{2 r}$ | Radial force in the joint $O_2$ revolving platform |

$M_{2 r}$ | Moment in the joint $O_2$ revolving platform |

$F_e$ | Equivalent force in the joint $O_2$ revolving platform |

$M_e$ | Equivalent moment in the joint $O_2$ revolving platform |

$a$ | Factor of the axial force influence |

$b$ | Factor of the radial force influence |

$f_s$ | Factor of the bearing working conditions |

*J. Eng. Manag. Syst. Eng.*, 3(3), 175-182. https://doi.org/10.56578/jemse030305

*J. Eng. Manag. Syst. Eng.*, vol. 3, no. 3, pp. 175-182, 2024. https://doi.org/10.56578/jemse030305

*Journal of Engineering Management and Systems Engineering*, v 3, pp 175-182. doi: https://doi.org/10.56578/jemse030305

*Journal of Engineering Management and Systems Engineering*, 3, (2024): 175-182. doi: https://doi.org/10.56578/jemse030305