Radial clearance, particularly the axial clearance in the 3D joint of a mechanism owing to the assemblage, manufacturing tolerances, wear, and other conditions, has become a research focus in the field of multibody dynamics in recent years. In this study, a hydraulic cylinder model with 3D clearance joints was constructed by combining various potential contact scenarios. The novelty of this study is that potential contact points between the bearing wall and journal were calculated when the bearing wall circle was projected to an ellipse owing to misalignment of axes. Moreover, the simulation model considered the effective bulk modulus of the hydraulic oil and applied the Lagrange multiplier method. Subsequently, an experiment was conducted to verify the simulation results. The simulation and experimental results indicated that the dynamic responses of the hydraulic cylinder with 3D clearance joints can be classified as free, rebound, slide, and contact. The effects of input force, frequency, and clearance size on the dynamic behavior of the hydraulic cylinder were also investigated. Increasing the input force and clearance size will degrade the hydraulic cylinder dynamic response; however, the input force frequency can reduce the deterioration of the dynamic response. This study aids in providing improved understanding of the hydraulic cylinder with 3D clearances in the theoretical field and for practical engineering applications.
The asymmetric hydraulic cylinder, which is used as an activator in construction equipment, agricultural machinery, special vehicles, and other heavy loading machinery, has attracted substantial research interest in recent years. Chen et al. analyzed the static and dynamic performance of a hydraulic cylinder with adaptive variable clearance around the piston, which can be applied in the high-frequency response and high-speed fields [
Owing to manufacturing and assembling errors, the actual joints obtain radial clearance. The phenomenon of clearance has received the attention of researchers since the 1970s [
However, the abovementioned existing studies have only focused on the radial clearance, omitting the axial clearances existing in the joints to simplify 3D revolute joint models as planar models. The 3D revolute clearance joint models are more complex than planar ones, and additional degrees of freedom are incorporated into systems with spatial revolute joints [
Several experiments have been conducted to verify the effects of clearance joints on the multibody dynamics. The surface wear on the axle and bushing of the revolute joint were investigated by Tasora et al. [
Nevertheless, the effects of the revolute 3D joint with axial and radial clearances on the dynamic properties of hydraulic cylinders in practical engineering applications have rarely been reported. Under actual working conditions, the clearances existing at the joints interact with the hydraulic cylinders, which will deteriorate the dynamic characteristics of the cylinder. Therefore, understanding the interaction effect between hydraulic cylinders with 3D clearance joints has scientific and industrial significance. In this study, we focused on a hydraulic cylinder with a 3D joint with axial and radial clearances considering the contact force model and hydraulic cylinder model in Section
The top view of the hydraulic cylinder, including a cylinder barrel, piston, rod, forward camber, and return camber, is presented in Figure
Model of the hydraulic cylinder with two clearances on bilateral sides.
In this section, the geometric model for the 3D revolute joints with radial and axial clearances is presented. The 3D revolute joints contain a journal and bearing, as illustrated in Figure
Structure of 3D revolute joint showing clearances.
The bearing wall is on the inner side of the bearing, and the cylindrical surface of the journal faces the corresponding bearing wall. The radius of the bearing wall is greater than that of the journal surface in the radial direction. Therefore, a radial clearance
The coupling interactions of the radial and axial clearances create complex contact scenarios, as illustrated in Figure Free flight motion, where no contact occurs between the two elements, as illustrated in Figure The journal surface contacts the bearing wall at a line, as indicated in Figure The contact area is between the journal flange and bearing base, as illustrated in Figure The journal contacts the bearing wall at certain points: one point in Figures
Potential contact scenarios in 3D revolute joint with clearances.
Figure
Schematic overview of angle error when the bearing ellipse contacts the journal circle.
For the description of the 3D joint model with clearances, several coordinate systems are indicated on parts of the joints in Figure
Coordinate systems in the 3D revolute joint with clearances.
Because the journal is mounted on the slider of the vibration bench, only the
The points
In the lateral view, the circle of the bearing is tilted and the shape is changed into an ellipse. The major radius
In (5), the point
The point
The distance from
The penetration depth
Schematic of 3D joint geometry in
In this function, the coordinate values
The flowchart for the gradient descent method is presented in Figure
Flowchart of the gradient descent method.
This is because
The contact point
Another contact point
Schematic view of 3D joint in axial contact.
The penetration depth
Moreover, the axial contact scenarios can be expressed as follows:
The radial relative velocity
Relative velocity in 3D joint with clearances.
Alternatively, during the occurrence of axial contact, the axial relative velocity
Here,
The direction of the relative normal penetration is calculated as
The magnitude of the relative tangential velocity
The direction of the relative tangential velocity is described by
The contact force model can be decomposed into the normal and tangential contact forces
The point contact
The value of
The line contact area between the journal and bearing wall is a rectangular place, which appears between the cylindrical contact surfaces. The distributed load
Thus,
The derivative is obtained as follows:
The iteration algorithm can be applied iteratively to calculate
The iteration error is set to less than 0.1 N, and the maximum iteration step is 7. Then, the final value of the load
When the journal and bearing axles are parallel to one another and the bearing penetrates the journal flange, area contact will occur between the journal flange and bearing base. The contact area is a ring and can be represented by
The contact force is denoted by [
The normal impact force
When relative sliding occurs, the tangential friction force
The tangential friction force
As the contact forces between the journal and bearing have been calculated in different scenarios, it is necessary to combine the interaction forces into the dynamic system. Thus, the forces acting on the journal can be expressed as
Hydraulic cylinder and forces model.
The rod moves in the cylinder barrel, causing the volumes of the forward and return cambers to change. When leakage in the hydraulic cylinder is neglected, the effective bulk modulus
The pressure
The force on the piston and rod is described by Newton’s equation [
The kinematical constraint equation in a multibody system can be expressed as
Equation (
A schematic overview of the numerical example is presented in Figure
Schematic overview of numerical example with 3D clearance joints.
Parameters of hydraulic cylinder.
Item | Value |
---|---|
Mass of slider and moving support (kg) | 325.2 |
Mass of rod (kg) | 26.1 |
Mass of cylinder barrel (kg) | 37.2 |
Moment of inertia of rod (kg·m2) | 2.63 |
Moment of inertia of cylinder barrel (kg·m2) | 4.5 |
Piston area of forward camber (mm2) | 5675 |
Piston area of return camber (mm2) | 3299 |
Initial volume of forward camber (L) | 2.554 |
Initial volume of return camber (L) | 2.144 |
Geometric and physical parameters of revolute clearance joints.
Item | Left joint | Right joint |
---|---|---|
Radius of bearing (mm) | 25 | 25 |
Young’s modulus (GPa) | 207 | 207 |
Poisson’s ratio | 0.3 | 0.3 |
Friction coefficient |
0.01 | 0.01 |
Restitution coefficient | 0.9 | 0.9 |
There are two similar 3D joints with axial and radial clearances on the bilateral side of the hydraulic cylinder. Table
In the experiment and simulation, the force
A photograph of the experimental test rig is presented in Figure
Photograph of test rig.
Five Hall sensors were installed around the outer surface of the right bearing, as illustrated in Figure
Installation positions of five sensors in the right 3D joint.
The submodel of the joint with 3D clearance was one part of the studied mechanism, providing the basis of the simulation and experiment. In the following sections, the effect of the ellipse eccentricity on the contact angle error and its innovation are first investigated. Thereafter, the experimental verification of the simulation and dynamics is analyzed, following which the effects of other parameters are studied.
This section presents the effect of the eccentricity on the contact point
(a) Angle error
The test rig model was a cosimulation with ADAMS and MATLAB. ADAMS used the HHT integral solver and calculated the position, velocity, angle, and angle velocity for implementation in MATLAB. MATLAB then translated the ADAMS data to determine potential contact points. If there was a contact point, MATLAB would provide the impact forces and torques to the parts in ADAMS, which changed the rigid body acceleration. Meanwhile, the hydraulic cylinder force was also calculated in MATLAB. The integration tolerance in the simulation was less than
Trajectories and rod displacements in the numerical and experimental studies with
Figure
Trajectories and bearing axles of 3D joints in simulation.
Euler angle
We investigated one period as an example. The journal trajectories can be classified into four modes: freedom, collision, contact, and occasional rebounding, as illustrated in Figure
Contact forces, trajectories, and forces on the piston in the numerical study with
The contact mode is analyzed in Figure
Displacement
Figure
Point distribution of middle joint in simulation with
Figure
Displacements and contact forces between journal flanges and bearing bases in the axial direction during simulation with
Figures
Trajectories in numerical simulation and experiment with different
Displacements
Statistical results of time in numerical and experimental studies with different
Figures
Trajectories in numerical and experimental studies with different frequencies: (a) left trajectories at 5 Hz, (b) right trajectories at 5 Hz, (c) left trajectories at 10 Hz, (d) right trajectories at 10 Hz, (e) left trajectories at 15 Hz, (f) right trajectories at 15 Hz, (g) left trajectories at 20 Hz, and (h) right trajectories at 20 Hz.
Displacements
Statistical results of time in numerical and experimental studies with different frequencies.
It was necessary to analyze the influence of the clearance size because the clearance increases gradually during the servicing period of a hydraulic cylinder. The maximum value of the input force
Trajectories in numerical and experimental studies with different
Displacements
Statistical results of time in numerical and experimental studies with different
The results of this study demonstrate that the dynamics of the 3D hydraulic cylinder are affected by variations in the parameters. A comparison of the effects of the input force size, frequency, and clearance size in Figures
The present study was concerned with investigation of the dynamic responses of the hydraulic cylinder mechanism with a 3D joint model considering the radial and axial clearances. A simulation model for the hydraulic cylinder with 3D joints was presented, including the potential contact scenarios and points, line, and plate contact force models, and the addition of the effective bulk modulus of hydraulic oil, as depicted in our previous work. We also simplified the line contact force calculation. Subsequently, an experiment verified the simulation results. The dynamics of the hydraulic cylinder model considering 3D clearance in the simulation are closer to the experimental results compared to those obtained by the model with ideal joints. In contrast, the disadvantage is the increased amount of calculations. The main conclusions are summarized as follows: The effect of eccentricity on the contact point was thoroughly analyzed. The results demonstrate that the angle error in the domain (0, The dynamic response of the rod can be classified into four states: free, rebound, slide, and contact. The stiffness of the hydraulic cylinder caused the rod to vibrate rapidly in the free model. The journals slid on the bottom of the bearing wall because of the gravity and low velocity in the slide mode. When the journal reached the boundary together with the bearing at the lines, the joint was in the contact state. An increase in the input force and clearance size will degrade the dynamic response of the hydraulic cylinder, for example, by increasing the time of the free state and the peak of the rod displacement. The frequency of the input force causes the trajectories and displacement to be more regular and can reduce the deterioration of the dynamic response of the hydraulic cylinder.
This work provides greater insight into the changing dynamics of the hydraulic cylinder with 3D clearance joints and offers theoretical support for further studies on the hydraulic cylinder.
The complete constraint equations of the test rig are as follows:
The data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that they have no conflicts of interest regarding the publication of this paper.
This research was supported by a grant from the National Natural Science Foundation of China under research project no. 51801049.