Research Article  Open Access
Numerical and Experimental Investigations on Dynamic Response of Hydraulic Cylinder with 3D Spatial Joints considering Radial and Axial Clearances
Abstract
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.
1. Introduction
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 highfrequency response and highspeed fields [1]. Ylinen et al. presented a linear hydraulic actuator model for multibody simulations, considering the state variables and the sealing friction, and found that the cylinder would soften the system dynamics in the starting and stopping states [2]. Moreover, Nizhegorodov et al. studied the frequency spectrum of the vibration exposure in the hydraulic cylinder [3]. Hydraulic cylinders experience impulsive shock and various other loading conditions in engineering machinery, such as breakers, drifters, or drills. The impulsive responses in mechanical systems with multiple clearances have been studied when the excitation and system load changed abruptly [4]. Ficarella et al. analyzed the impact energy of the breaker to minimize the required power to the hydraulic system using numerical and experimental data [5, 6]. Li et al. studied a hydraulic rock drill and investigated the overlapped reversing valve to propose a fault diagnosis method [7, 8].
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 [9]. Erkaya and Uzmay substituted clearance joints with a virtual massless link connecting the journal and bearing centers, which have been widely applied to estimate the kinematic error [10]. Furthermore, a series of scholars determined that the clearance can dramatically change the dynamic behavior of a mechanical system, and the interaction force between joint parts exhibits peaks [11, 12]. The point contact is used extensively and commonly referred to as the penalty approach [13], which allows the relative penetration of bodies to be applied to evaluate the reaction contact force. The equations for the dynamic analysis of constrained multibody systems are based on the Lagrange multiplier and Baumgarte stabilization method [14]. Several contact force models depending on different assumptions have been used to describe revolute joint models with clearance. In the nonlinear Hertz model [15], the contact forces between contacting bodies form an ellipse in the contact area, which is evaluated depending on the penetration depth. Moreover, a damping factor was introduced to remedy the disadvantage of energy dissipation during the impact process. The models of Lankarani and Nikravesh [16] and Hunt and Crossley [17] are the most commonly applied approaches for dealing with point contact situations. Dietl et al. analyzed the Hertz pressure distribution and replaced the nonlinear exponent of 1.5 in the Lankarani–Nikravesh model with 1.08, and the results were experimentally validated [18]. Koshy et al. used an experiment to illustrate that an appropriate contact force model with proper dissipative damping has a significant effect on the dynamic response of a mechanical system [19]. Wang et al. excluded the Lankarani–Nikravesh model’s restitution coefficient and replaced it with a damping coefficient [20]. The contact scenarios in spatial revolute joints with clearance appear as line contacts between cylindrical contact surfaces, and the shape of the line contact is rectangular. Different models have been developed for the relationships between the force and penetration. The model presented by Johnson [21] was based on the Hertz theory and subsequently accepted in the ESDU Tribology Series [22], which provided a comprehensive review covering the contact force and penetration expressions for engineering applications. Meanwhile, other contact force models are available to describe the contact between the two cylinders, including the Goldsmith [23], Radzimovsky [24], and Dubowsky–Freudenstein [25] models. Pereira et al. compared the precisions of the above line contact models and concluded that the Johnson and Radzimovsky models are suitable for describing the contact involving the cylinder length in most practical applications [26]. However, the method for solving the line contact force model using the Lambert function is excessively complicated [27]. Thus, it is necessary to determine a suitable method for solving the line contact model rapidly and precisely. The literature mentioned above mainly focused on clearance joint models. However, actual mechanisms have multiple joints, and thus, multiple joints with clearance make the dynamics of practical mechanisms more complex. It is therefore necessary that much attention should be paid to mechanisms with more than one clearance joint. Lu et al. [28, 29] and Wei et al. [30] analyzed the dynamics of a vehicle shimmy system model with multiple clearance joints. Moreover, several factors have been considered for decreasing the vibration dynamics owing to clearances.
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 [31]. Marques et al. applied a spatial slidercrank mechanism to study the effects of dry spatial revolute joints with radial and axial clearances [32]. Marques et al. accounted for all possible contact scenarios in the clearance joints and demonstrated that the clearance in the mechanical joints would have a significant impact on the dynamic response of the mechanical system [33]. Yang et al. illustrated that the multiple spatial revolute clearance joint would influence the dynamic behaviors of the openloop mechanism to a greater extent than that of planar mechanisms, and the joints would interact with one another to cause fiercer vibrations, such as frequency, amplitude, and penetration [34]. Cavalieri and Cardona replaced the journal with two connecting spheres to construct a model of 3D revolute joints with clearance [35]. Ambrósio and Pombo proposed a unified formulation using the same kinematic information to combine perfect and clearance/bushing joints. This method enables similar modeling of two joint models and easy permutation of multibody systems [36]. Isaac et al. proposed a 3D finite element analysis approach to deal with nonideal spatial revolute joints [37].
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. [38]. Flores et al. presented an experimental verification of a model of the slidercrank mechanism with revolute clearance joints [39]. Erkaya and Uzmay used three accelerometers and two microcameras to study the vibrations of a mechanism with two clearance joints [40]. Haroun and Megahed compared the results of a wear experiment and simulation, which proved the accuracy and efficiency of the simulation method [41]. Lai et al. validated the numerical wear prediction and experimental results of a planar mechanism with a revolute joint at a low velocity [42]. Brito et al. studied the operating effectiveness of an oil bearing with a twin groove or single groove [43]. The sensors for measuring the relative shaft position inside the bearing were arranged in the orthogonal directions [44–46], while vertical position arrangement was used to measure the displacements in the directions of the X and Yaxes. Tian et al. [47] surveyed the models of planar and spatial multibody mechanical systems with clearance joints and conducted experiments on the mechanical systems with clearance joints. Thereafter, they discussed the phenomena commonly associated with clearance joint models, such as wear, nonsmooth behavior, optimization, chaos, and uncertainty [48]. Erkaya analyzed a flexible connection to minimize the clearanceinduced dynamic deterioration by computational and experimental investigations [49, 50]. This is because the flexible connections have a clear suspension on the dynamics of a mechanism. In another study, Erkaya attempted to address the clearance problem in joints and improve the accuracy of robotic manipulators [51]. Yan et al. proposed a model for 3D revolute joints with radial and axial clearances and verified the model experimentally [27]. In the study, a comprehensive contact model was used to describe the contact scenarios, such as the point, line, and face contacts. The simulation results provided a realistic description of the 3D motion. However, several errors existed owing to the simplification of the potential contact points. The journal axle was misaligned with the bearing and eccentricity emerged. Therefore, the projection of the bearing wall circle on the lateral view was an ellipse, for which the angle error should be studied.
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 2. 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. Moreover, we simplified the line contact force by using the Newton iteration and not the Lambert function as described in a previous study [27]. Numerical and experimental examples of the hydraulic cylinder model with the 3D clearance joint and its parameters are described in Section 3. Subsequently, the dynamic responses of the simulation and experiment are compared. The effects of different inputting force frequencies and clearance sizes were studied, and the results are presented in Section 4. Finally, several conclusions of the study are provided in Section 5.
2. Dynamic Model of Hydraulic Cylinder System with Clearance
The top view of the hydraulic cylinder, including a cylinder barrel, piston, rod, forward camber, and return camber, is presented in Figure 1. On the bilateral side of the hydraulic cylinder, there is one spatial revolute joint, each of which has a journal, bearing, and flange.
2.1. Model of 3D Revolute Joints with Radial and Axial Clearances
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 2. Two flanges exist on the bilateral side of the journal, which limit the range of the bearing axial motion.
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 c_{r} is generated. In the axial direction, the bearing length is smaller than the distance between the two flanges connected to the journal. Thus, an axial clearance c_{a} is created.
The coupling interactions of the radial and axial clearances create complex contact scenarios, as illustrated in Figure 3. These scenarios can be classified as four different types of journal motion inside the bearing [33, 52]:(1)Free flight motion, where no contact occurs between the two elements, as illustrated in Figure 3(a)(2)The journal surface contacts the bearing wall at a line, as indicated in Figure 3(e)(3)The contact area is between the journal flange and bearing base, as illustrated in Figure 3(f)(4)The journal contacts the bearing wall at certain points: one point in Figures 3(b) and 3(c), two points in Figures 3(d), 3(j), and 3(k), three points in Figures 3(i) and 3(l), and four points in Figure 3(h)
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Figure 4 illustrates the angle error θ_{err} between points A and B when the eccentricity of the outer ellipse is 0.6, which is obvious and should not be neglected. Thus, it is necessary to study the eccentricity effect of the ellipse.
For the description of the 3D joint model with clearances, several coordinate systems are indicated on parts of the joints in Figure 5. Firstly, the global coordinate system xyz is fixed on the ground and provides a basic reference system for the entire model, as illustrated in Figure 5. Secondly, the local coordinate system O_{i}x_{i}y_{i}z_{i} is positioned at the mass center of the part connected to the bearing, and the matrix describes the transformation from the global coordinate system to the local coordinate system of the bearing. Finally, the local coordinate system O_{j}x_{j}y_{j}z_{j} is located at the mass center of the part connected to the journal. The transformation matrix from the ground to the journal is defined by . Here, and are Euler angle transformation matrices, in which α, β, and γ are Euler angle parameters. In this study, the translation sequence was 313. Note that there are two points O_{a} and O_{b} that can easily be used to describe the position of the journal and bearing. Point O_{a} is the center of one ring of the bearing wall and O_{b} is the journal center. Meanwhile, there exist local coordinate systems on points O_{a} and O_{b}. The axes of these local coordinate systems O_{a} and O_{b} are parallel to the coordinate systems O_{i}x_{i}y_{i}z_{i} and O_{j}x_{j}y_{j}z_{j}, respectively.
Because the journal is mounted on the slider of the vibration bench, only the Ydirection movement exhibits freedom. Thus, the local coordinate system O_{j}x_{j}y_{j}z_{j} provides a simplified description of the 3D joint model. The transformation matrix represents the transformation from the local coordinate system O_{j}x_{j}y_{j}z_{j} to O_{i}x_{i}y_{i}z_{i}, and can be expressed as
The points Q in the global coordinate system xyz can be described as follows:where is the vector from the origin point to the circle center point O_{i} in the global coordinate system, which is subscripted as , while is the vector connecting the circle center O_{i} to any point Q in the journal local coordinate system O_{j}x_{j}y_{j}z_{j}. The point Q can be described aswhere the vector is expressed as
In the lateral view, the circle of the bearing is tilted and the shape is changed into an ellipse. The major radius a and minor radius b of the ellipse can be calculated as follows:where R_{i} is the radius of the inner bearing wall and the angle is the Euler angle in the translation .
In (5), the point Q on the ellipse boundary can be illustrated in the coordinate system O_{j}x_{j}y_{j}z_{j}:
The point O_{a} is the center of one ring of the bearing wall, which is depicted in the same manner as Q. The initial angle is the Euler angle in the translation . Moreover, , , and are the ellipse centers in the journal original system.
The distance from Q to the journal center can be expressed aswhere O_{b} is the journal center and , , and are the coordinate values of the journal center.
The penetration depth between the journal and bearing in Figure 6(a) can be determined as follows:where d is the distance between the center O_{b} and Q_{min}. When the penetration depth is shorter than zero, a potential collision may occur. The contact conditions can be summarized as follows: : free movement between the journal and bearing : contactimpact between the journal and bearing
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Q_{0} is located at the intersection point of the line and bearing wall. However, the point Q_{min} is closest to the center O_{b} of the journal. The following part demonstrates the calculation process for determining point Q_{min} to cause d to be minimal, for which the gradient descent method is applied. The target function of the gradient descent method is defined as
In this function, the coordinate values and of the journal center O_{b} are constant. Then, the derivative equation is calculated as follows:
The flowchart for the gradient descent method is presented in Figure 7. The angle θ is set as the following initial value:
This is because Q_{min} is close to the line connecting O_{a} to O_{b}, which can enable the calculation to converge rapidly. In the gradient descent method, the maximum iteration step is 7 and the learning rate α = 0.0005 for determining θ_{min}. The point Q_{min} coordinate can be calculated as follows:
The contact point P in the journal can be described as
Another contact point Q_{a} exists between the flange and bearing base in the axial direction, as shown in Figure 8. The potential point is at the acme of the ellipse and can be expressed as
The penetration depth δ_{a} is calculated by
Moreover, the axial contact scenarios can be expressed as follows: : axial free movement between the journal and bearing : axial contactimpact between the journal and bearing
The radial relative velocity v_{r} of the contact points is required to calculate the contact and friction forces in the local coordinate system O_{j}x_{j}y_{j}z_{j}, which is described aswhere and are the velocities of contact points P and Q_{min} from previous positions at a simulation step before, respectively, in the local coordinate system O_{j}x_{j}y_{j}z_{j}. The velocity v_{r} can be projected as onto a plane that is tangential to the journal through the collision point Q_{min}, as illustrated in Figure 9. The relative normal velocity is projected onto the normal direction of the above plane.
Alternatively, during the occurrence of axial contact, the axial relative velocity v_{a} can be expressed aswhere and are the velocities of the axial impact points P_{a} and Q_{a} at a simulation step, respectively, in the coordinate system O_{j}x_{j}y_{j}z_{j}. The velocity v_{a} is projected as onto the plane of the flange and in the normal orientation of the flange plane.
Here, is the velocity of the relative penetration and can be calculated as
The direction of the relative normal penetration is calculated as
The magnitude of the relative tangential velocity v_{T} can be obtained as follows:
The direction of the relative tangential velocity is described bywhere and are all in the local coordinate system O_{j}x_{j}y_{j}z_{j}.
The contact force model can be decomposed into the normal and tangential contact forces F_{N} and F_{T}, respectively. Sections 2.1.1 to 2.1.3 demonstrate the calculation of the normal contact forces F_{N}, while Section 2.1.4 presents the solving process for F_{T}.
2.1.1. Point Contact Model
The point contact F_{N} is evaluated using the Lankarani–Nikravesh contact force model:where c_{e} is the restitution coefficient, the exponent n is usually 1.5 for a collision between metal bodies, and δ and are the initial velocities when the impact occurs. The stiffness K can be evaluated as follows:
The value of can be determined bywhere ν_{i}, E_{i} and ν_{j}, E_{j} indicate Poisson’s ratio and Young’s modulus of each collision body.
2.1.2. Line Contact Model
The line contact area between the journal and bearing wall is a rectangular place, which appears between the cylindrical contact surfaces. The distributed load P per unit length is expressed as . In the Johnson model, the relative penetration δ_{l} is defined aswhere the parameter represents the difference between the cylinder radii. To guarantee that, equation (25) satisfies the physical requirements, and the following function must be greater than 1:
Thus, P must follow the following condition:where e represents natural exponential. In the domain (0, ), the Johnson function linking P to δ_{l} increases monotonically. The calculations involved for the line force model using the Lambert function are excessively complicated [27]. Therefore, P has only one value in this domain. We apply the Newton iteration to determine P at a certain penetration δ_{l}. The target function is defined as
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 P is obtained. The line contact force is expressed as [37]where D is the damping coefficient.
2.1.3. Area Contact Model
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 bywhere R_{base} is the radius of the bearing base illustrated in Figure 3.
The contact force is denoted by [44]
The normal impact force F_{N} is obtained as follows:where F_{N} represents the point, line, and area contact forces mentioned previously and e^{N} is the unit vector in equation (19).
2.1.4. Friction Model
When relative sliding occurs, the tangential friction force F_{T} affects the journal and bearing to prevent sliding, which can be described as [42]where c_{f} is the friction coefficient, is the relative tangential velocity, and and are the given tolerances for the tangential velocity, which are selected as and in this case.
The tangential friction force F_{T} is obtained as follows:where e^{T} is the unit vector in equation (24).
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 aswhere is the vector from the potential collision impact point Q to the mass center M, M_{1} and M_{2} are the mass centers of the cylinder barrel and rod, as illustrated in Figures 3 and 10, respectively, and M_{3} is the combination of the slider and moving support:where is the vector connecting the potential collision impact point P on the bearing to the mass center M.
2.2. Hydraulic Cylinder 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 β_{e} of the hydraulic oil causes pressure fluctuation with the change in volume [53]. The pressure p_{1} of the forward camber can be calculated aswhere A_{1} is the area of the piston in the forward camber, V_{10} is the initial volume of the forward camber, p_{10} is the initial pressure of the forward camber, and z_{rod} is the relative displacement of the piston from its initial position.
The pressure p_{2} of the return camber can be determined in the same manner:where A_{2} is the return tank flow, V_{20} is the initial volume of the return camber, and p_{20} is the initial pressure of the return camber.
The force on the piston and rod is described by Newton’s equation [54]:where is the piston acceleration; m_{2} is the rod mass; θ is the angle between the axis and the horizon, as shown in Figure 10; and F_{R} is the decomposition of the impact force on the rod journal along the axial direction of the hydraulic cylinder. The impact force is evaluated as follows:where represents the transformation from the local coordinate system of the journal to the global coordinate system. e_{z} is the Zaxle unit vector in the local coordinate system and is equal to [0, 0, 1]^{T}. Moreover, F_{k} is the interaction force in the left or right joints when contactimpact occurs. The value f in equation (41) denotes the linear friction with stiction between the rod and cylinder barrel, as described in the following equation [55]:where F_{S} is the static friction and F_{C} is the dynamic friction. In this study, F_{S} = 385 N and F_{C} = 340 N. Furthermore, v_{T} is the rod velocity. The tolerances for the velocity in equation (43) were selected as = 0.01 mm/s and = 0.05 mm/s.
2.3. Dynamic Constraint Equations
The kinematical constraint equation in a multibody system can be expressed aswhere q_{r} indicates the position and orientation vectors for describing the bodies in global coordinates and t represents the time. The motivation equations for a constrained multibody system in Cartesian coordinates can be represented as follows:where M is the system mass matrix and the components of M denote the masses and moments of inertia of the bodies in the system. Moreover, represents the transformation matrix, λ is the vector of Lagrange multipliers, and g represents the generalized forces, which include the external driving torques or forces, centrifugal force, Coriolis force, and impact forces of the clearance joints:
Equation (45) is combined with equation (44) and rewritten as the matrix of differential algebraic equations that describes the motion of a multibody system:
3. Numerical and Experimental Examples
3.1. Numerical Example Schema
A schematic overview of the numerical example is presented in Figure 11. The hydraulic cylinder is inclined at 5° in the anticlockwise direction, conforming to the installation of the rockbreaker in our previous study [53]. Table 1 presents the parameters of the numerical example and the material of each part is steel, which has a density of 7800 kg/m^{3}. The other properties of steel are shown in Table 2. The study cases include different amplitudes and frequencies of input force, as well as varying clearance sizes.


There are two similar 3D joints with axial and radial clearances on the bilateral side of the hydraulic cylinder. Table 2 presents the geometrical and physical parameters of the 3D joints. A schematic of the joints is provided in Figure 2.
In the experiment and simulation, the force F_{input} is a sine wave in each cycle, which can be expressed aswhere F_{max} is the maximum input force and was set to 1, 2, 3, and 4 kN, while ω is the period parameter for determining the frequency of the force and was selected as 5, 10, 15, and 20 Hz.
3.2. Test Rig
A photograph of the experimental test rig is presented in Figure 12. The hydraulic cylinder was selected from the middle arm cylinder of the XE60D excavator, manufactured by the XCMG group. The Hall sensors were made by Hefei Bangli Electronic Co., Ltd. The signals form the Hall sensors were collected with a PCI9111HR data acquisition card from ADLINK in the data acquisition system, with a sampling rate up to 100 kHz. Thereafter, all signals were calibrated and displayed in the LabView software. The vibration bench, from SuShi Co., Ltd, could generate horizontal excitation. The input force F_{input} of the excitation was a simple harmonic wave and limited to 5000 N. A fixed support was mounted on the ground on the opposite side of the vibration bench. Two connecting links were installed between the fixed support and vibration bench, as illustrated in Figure 12, to provide greater stiffness to the fixed support during the experimental process.
Five Hall sensors were installed around the outer surface of the right bearing, as illustrated in Figure 13. Two Hall sensors were positioned in the orthogonal direction to the journal as a group for detecting one side of the ring center movement of the bearing shown in Figure 2. Two other sensors, which are mutually perpendicular to the first two sensors and the journal, were used to measure the displacement on the other side of the ring center. The fifth Hall sensor was mounted between the two sides for detecting the axial clearance distance and was incorporated with a steel block fixed on the bearing. Magnets were mounted on the back of the Hall sensors and provided magnetic fields. The intension of the magnetic field changed with the distance between the sensor and the steel parts. An additional five sensors were positioned in the same manner on the left joint. The 11th sensor was mounted on the rod to detect the hydraulic cylinder movement, as illustrated in Figure 10, which respected the movement of the rod in the cylinder.
4. Calculation and Analysis of Results
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.
4.1. Effect of Eccentricity
This section presents the effect of the eccentricity on the contact point Q, which was calculated by the gradient descent method. The θ_{err} of the potential point Q in the domain (0, π) when the axial and radial clearances were the same at 0.5 mm is illustrated, while θ_{err} in the domain (0, π) is demonstrated for the symmetric quadrants. As illustrated in Figure 14, the error θ_{err} was nearly a complete sinusoidal curve. Moreover, the error reached maximum value at approximately π/4 and 3π/4. Thereafter, the error decreased to zero with the change in the angle. The error disappeared when the angles were 0, π/2, and π. In the domain (π/2, π), the absolute value of θ_{err} was a mirror of the above curve based on the symmetry, but the signal was negative. The maximum amplitudes of θ_{err} exhibited accelerated growth when the eccentricity of the ellipse increased, as indicated in Figure 14(b). Therefore, we can conclude that the angle error θ_{err} is minuscule when the clearance is very small and the circular model and the elliptical model are very similar to each other. Despite the fact that the angle error θ_{err} is not obvious, this study selects the elliptical model in the simulation for future cases with larger clearances.
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4.2. Experimental Verification
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 , and the integral step was set to s to provide sufficient accuracy. Finally, the experimental and simulation times were each set to 10 s to reach a steady state. The initial conditions were as follows: The input force is 3 kN, and the frequency is 10 Hz. The journal is at the center of the bearing at the start of the simulation, the initial velocity is 0, and the entire test rig is in a 9.8 m/s^{2} gravity field. The axial clearance and radial clearance were both 0.5 mm. After the simulation was complete, we used MATLAB to obtain the simulation results and compiled a few scripts to draw the figures. A comparison of the experimental results with the simulations in the steady state is presented in Figure 15, where two periods are selected as an example, from 5.35 to 5.55 s. These results are similar to those of a previous study [34, 53]. Figures 15(a)–15(d) illustrate the twoside ring center experimental trajectories in red lines, which are detected by sensors 1–4 and 6–9. The trajectory of ideal joint is green points at the centers. The experimental data exhibited similar trends to those of the simulation considering clearance. However, the rod displacement of the simulation considering clearance was significantly different compared to that obtained with the model with ideal joints, which obtained different magnifications and shapes of the rod displacement curve. Therefore, as the experimental data agree well with the simulation results, the simulations are verified.
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Figure 16 depicts the twoside simulation clearance trajectories. This result was similar to that presented in the literature [27]. The axles are illustrated as black lines that are nearly parallel to the horizon. The angle β of the Euler translation represents the angle between the journal and bearing axles, as illustrated in Figure 17. The angle β was almost equal to zero most of the time and only changed in the free state, which means that the two clearance trajectories were the same at most times. Hence, the center point trajectory of the journal, namely, the middle of the twoside point trajectories, can demonstrate the detailed dynamics of the hydraulic cylinder system.
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 18. These results are similar to those obtained in previous studies [27, 35, 53]. The slider moved toward the inward direction, and the journal mounted on the slider moved in the same direction. The hydraulic cylinder was also involved in the shrink process. The journal slid on the inner surface of the bearing indicated by the red line, as illustrated in Figures 18(b) and 18(d). The two parts of joint were maintained in contact each other without separation owing to the coupling of the hydraulic cylinder gravity and contact force. This mode is defined as the slide contact mode. The slide contact force, indicated by the red line and points, increased gradually in the slide mode, as shown in Figures 18(a) and 18(e). Meanwhile, the force on the piston in Figure 18(c) was similar to the slide contact force and showed certain friction peaks. The contact mode demonstrated in Figures 18(b), 18(d), 18(g), and 18(i), in which the journal reached the bearing wall boundary and remained in contact with a small area of the bearing. The contact force of the contact mode, indicated by green, was larger than that of other modes, as shown in Figures 18(a), 18(e), 18(f), and 18(g); this was the same as the force on the piston. Subsequently, the journal moved toward the outward direction of the bearing wall. When the journal moved away from the bearing wall, the trajectories were inside the clearance circles and remained in the free mode. The contact force, indicated by blue, was equal to zero as illustrated in Figures 18(a), 18(e), 18(f), and 18(g), and the force on the piston showed some fluctuation as demonstrated in Figure 18(h). The journal did not contact the bearing wall, and no contact force was imparted to the two joint parts. The rod simultaneously moved in a simple harmonic state owing to the springmass system of the hydraulic cylinder, which was derived from the bulk modulus β_{e} of the oil [53]. When the free mode was completed, the journal reached the bearing wall, and an impact force peak emerged, indicating the rebound state. Thereafter, the contact mode emerged as the contact area was limited to a small arc section.
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The contact mode is analyzed in Figure 19. The rod displacement z_{rod} and the centers of bearing distance were adjusted near to zero. The clearances in the joints were 0.5 mm each. Thus, the sum boundary of the two clearances was 1 mm, as indicated by the red dashed line. When the rod reached the bearing wall and was maintained in a contact mode, the rod displacement added or subtracted the sum boundary of 1 mm (red dashed line) and then corresponded to the bearing center distance.
Figure 20 illustrates the journal center as a point cloud distribution, which is similar to the results obtained by Yan et al. [27]. One in every twenty points was selected for representation to enable easier viewing of the points. The points exhibited greater contact opportunities on the bottom and bilateral sides of the cylinder, whose radius was the radial clearance and the length was the axial clearance. Therefore, it can be speculated that the bilateral sides of the bearing wall will experience more serious wear.
Figure 21 illustrates the axial displacement of the journal center for the entire experimental and simulation period. When the bearing contacted the journal flanges, the contact force emerged on the corresponding side. It can thus be concluded that the displacement was irregular and contact occurred occasionally. Moreover, the axial contact force was less than the radial contact force. Thus, it is speculated that the wear on the journal flanges is not serious.
4.3. Effect of Input Forces
Figures 22 and 23 illustrate that the increased input force acting on the slider changed the rod movement; the maximum value of the input force F_{max} was selected as 1, 2, 3, and 4 kN with an input force frequency of 10 Hz. The initial values of the radial and axial clearances were both equal to 0.5 mm. The trajectories of the free state were longer and closer to the clearance circle center. The contact state increased gradually, as indicated in Figure 24. Meanwhile, the slide state descended. In contrast, the proportion of the free mode increased. The z_{rod} peaks became increasingly larger with increasing input forces. Furthermore, the distance z_{rod} exhibited two peaks when the slider moved inward and the space within the hydraulic cylinder shrank. The kinetic energy of the moving parts increased with the change in the input force, thus generating the ascending peaks. Increasing the input force will therefore deteriorate the hydraulic cylinder dynamic response.
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4.4. Effect of Frequency
Figures 25–27 illustrate the effects of the input force frequencies 5, 10, 15, and 20 Hz on the rod movement. The maximum value of the input force F_{max} was 3 kN. The initial values of the radial and axial clearances were the same as those used in the previous simulations (c_{r} = 0.5 mm and c_{a} = 0.5 mm). The trajectories and displacement z_{rod} were more regular, and the free mode was closer to the clearance circle center. Moreover, the proportions of the free and contact modes increased with the frequency change corresponding to a reduction in the slide time, as indicated in Figure 26. When the frequency increased, the peak magnitudes of z_{rod} decreased gradually. The reason for the descending peaks is that the slider movement time was shorter with the kinetic energy of the parts being reduced. One can thus conclude that the input force frequency reduces the deterioration of the dynamic response.
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4.5. Effect of Clearance Size
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 F_{max} was 3 kN, and the frequency of the input force was 10 Hz. The clearance size c_{r} was adjusted by varying the journal radius of the joints, selected as 0.2, 0.5, and 1 mm. The trajectories at c_{r} = 0.2 mm were the most random for the small clearance owing to the rapid bearing collision, as shown in Figure 28. At c_{r} = 0.2 mm, the displacement z_{rod} in the contact mode was minimal and the least vibrations occurred in every period. When the clearance increased to 1 mm, the displacement z_{rod} was maximal and the most vibrations emerged. The displacement and vibration when the clearance was 0.5 mm were within the two clearance values above, as illustrated in Figure 29. The contact state time decreased in the study, while the free mode time and the slide state increased, as shown in Figure 30. The reason for the above phenomenon is that expanding clearance provides more space for the kinetic energy of the slider and rod to be accumulated. This is also the cause for additional vibrations in the free mode and the shorter slide state time. The hydraulic cylinder dynamic response will therefore degrade with increase in the clearance size.
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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 24, 27, and 30 shows that the clearance size could drastically reduce the contact mode time in simulation and experiment results. However, the contact mode time increases with an increase in the input force size and frequency. The free mode exhibits a peak with a frequency change, which increases with the magnitude of input force and clearance size. The slide mode increases gradually with the input force and frequency. However, the time of sliding decreases when the magnitude of the input force and the frequency increase. These findings are of significance for the design and application of hydraulic cylinders.
5. Conclusions
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:(1)The effect of eccentricity on the contact point was thoroughly analyzed. The results demonstrate that the angle error in the domain (0, π) was nearly a complete sinusoidal curve. The error disappeared only when the angle = 0, π/2, π. The maximum amplitudes of the angle error ascended when the eccentricity increased. It can therefore be concluded that the angle error of the contact point is not ignored when the misalignment is large compared to a traditional 3D joint with clearances; in addition, more accurate collision points are provided.(2)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.(3)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.
Appendix
The complete constraint equations of the test rig are as follows:
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest regarding the publication of this paper.
Acknowledgments
This research was supported by a grant from the National Natural Science Foundation of China under research project no. 51801049.
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Copyright © 2019 Ke Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.