Dynamic Modeling and Response Analysis of a Landing Gear Retraction and Extension System Considering Irregular Wear Clearance

Abstract

Over the course of long-term operation, wear to moving parts can significantly affect the dynamic behavior, reliability and service life of landing gear retraction and extension systems. The primary innovation of this paper is the proposal of a multi-body rigid-body dynamics modeling method for LGRES that accounts for irregular wear clearances, along with an analysis of its dynamic response under different system parameters. First, an exact dynamic model of the LGRES with joint clearance is developed. Secondly, the Archard wear model is introduced to characterize the wear evolution of the joint surfaces. Finally, the dynamic behavior of the mechanism under different wear cycles, initial clearance values, and drive speeds is compared to analyze the impact of these system parameters on wear characteristics. The results indicate that as these system parameters increase, wear significantly amplifies the impact forces on the joint and further exacerbates wear between the hinge pin and the bearing, as well as motion errors.

1. Introduction

With the continuous advancement in aerospace engineering towards higher reliability, safety and performance, aircraft landing gear systems have become increasingly critical to the operational integrity of modern aircraft [1,2,3]. Moreover, landing gear systems are subjected to highly transient and complex loading conditions during landing and taxiing, including severe impact forces, cyclic stresses, and broadband vibration. These conditions require not only sufficient load-bearing capacity and efficient energy dissipation, but also the ability to maintain structural durability and dynamic stability over prolonged cycles [4,5,6]. LGRES must achieve accurate and reliable motion transmission within a confined environment, usually under strict constraints on weight, geometry, and actuation performance [7,8]. To meet these requirements, such systems are typically designed as multi-link mechanisms with strong kinematic coupling and multiple degrees of freedom. These systems are particularly sensitive to local nonlinearities, and even minor variations in joint conditions can significantly alter the global dynamic behavior [9,10]. This sensitivity poses significant challenges for accurate modeling and analysis.

In theory, landing gear retraction and extension systems are typically modeled as multibody dynamic systems. Many researchers have conducted studies on landing gear retraction systems. Representative studies are as follows: Khapane et al. [11] used a flexible multibody approach to conduct a dynamic simulation of the landing gear. Their dynamic behaviors are determined by the combined effects of inertial forces, elastic deformation, damping effects, and contact interactions. Liu et al. [12] improved the simulation of LGRES by incorporating structural flexibility into the multi-body dynamics; Gao et al. [13] performed a study to ascertain the reliability of retractable LGRES when operating under clearance conditions. Airoldi and Lanzi [14] designed ski landing gear by optimizing a multi-body structure and employed a multibody explicit code. Krüger et al. [15] outlined how to address landing gear vibration problems using numerical simulation methods. Krason et al. [16] conducted a study in which they performed a mechanical analysis of the landing gear of a transportation aircraft.

In addition, nonlinear dynamic phenomena have been extensively studied. Research on nonlinear dynamic phenomena includes impact-induced vibrations, friction-induced instabilities, and flutter oscillations. These research contributions facilitate a more profound comprehension of the dynamic behavior of LGRES in extreme circumstances [17,18]. Nonlinear dynamic phenomena are primarily caused by the inevitable presence of joint clearances in practical engineering applications, resulting from manufacturing tolerances, assembly deviations, thermal effects, and long-term wear and erosion [19,20]. The clearance creates contact conditions which can result in separation and impact during operation. This behavior creates strong nonlinearity, causing forces to become discontinuous. To date, extensive research has been conducted on force discontinuities caused by joint clearances and the resulting complex nonlinear responses: Koshy et al. [21] investigated the impact forces generated in rotating joint clearances based on Hertzian contact theory. In their study, Machado et al. [22] introduced a damping term and investigated key issues related to contact force models based on Hertz’s law. Ravn [23] proposed a method of describing clearance in rotary joints, combining continuous analysis methods with contact force models based on the equations of motion for a multi-body rigid-body system. Corral et al. [24] reviewed nonlinear contact phenomena in many-body systems. Goldman and Muszynska [25] utilized numerical integration to examine the ordered and chaotically responsive characteristics of mechanically structural systems when subject to clearances and impacts. Chen et al. [26] provided a method for the study of planar multi-chain mechanisms’ nonlinear dynamics. These mechanisms possess multiple joint clearances. Wang et al. [27] developed a refined model of nonlinear elastically damped force application to a planar rotating joint with clearance. Zhao et al. [28] presented a model of contact force that includes spring damping and normal friction. This model was then compared with a more traditional model. Flores [29] suggested a continuous-pressure model based on contact circumstances to describe the impact forces generated by joint clearances. The LuGre modeling framework was utilized for the description of tangential friction forces at rotating joints in a two-dimensional multibody system by Muvengei et al. [30].

A substantial body of research on the kinematics of multi-body systems involving clearances has focused on joint clearances, often treated as constant parameters. However, friction, collisions and wear can aggravate joint clearances, shorten mechanisms’ lifecycles and impact precision. This is especially true for landing gear retraction and extension systems, which experience wear at joints due to varied conditions, extreme loads and impacts. Consequently, the examination of the dynamic behavior of mechanical systems under wear conditions has long been a significant topic within the discipline of multibody dynamics. A cylindrical joint model with clearance along both the diameter and axial directions for simulating impact conditions was developed by Qian et al. [31]. Li et al. [32] set out a method for the anticipation of deterioration in flat mechanisms equipped with multiple joints, and set up multibody equations using efficient contact region simplification and the Lagrange multiplier method. Zhu et al. [33] established a model for the dynamic evaluation of nonlinear contact pressure distribution with wear calculations. Using the Archard wear model, a model for predicting the wear properties of mechanical clear-gap joints was developed by Bai et al. [34]. Furthermore, the investigation by Aibin et al. [35] sought to explore the impact of surface stiffness on clearance component deterioration, utilizing an asymmetric Winker surface model to facilitate the calculation of contact pressure distribution. The method developed by Lai et al. [36] for calculating wear in rotating joints of low-speed planar mechanisms is used to study wear changes in clearance joints within multi-body mechanisms. Using the Kring model as a basis, Sun et al. [37] developed a dynamic wear model to forecast the material loss in mechanical devices subject to randomness and cognitive uncertainty, thus laying a theoretical foundation for improving system reliability design. Jing et al. [38] investigated the effects of different ball-joint clearances on the dynamic performance of mechanisms and proposed a method to reduce vibrations caused by multi-joint clearances. Ordiz et al. [39] investigated whether increasing clearance has an effect on mechanical fatigue life under wear conditions. Zhuang et al. [40] examined the correlation between joint wear and the relationship between joint wear and the mechanism’s mechanical output. Hou et al. [41] advanced the proposal of a methodology based on dynamic segmented modeling for the measurement of wear. This method was used to investigate the effect of wear clearance on mechanical properties.

The majority of extant studies focused on analyzing the nonlinear dynamics caused by regular clearances, and most examined relatively simple mechanisms. However, relatively little research has been dedicated to the dynamic response of LGRES that account for clearance wear. Therefore, this study investigates the wear characteristics and dynamic response of LGRES under conditions of irregular wear-induced clearance.

The primary research undertaken in this study is outlined as follows: In Section 2, a theoretical model was developed to predict clearance and contact forces, with a view to accounting for wear. In Section 3, a multibody rigid-body dynamics model of LGRES incorporating multiple wear-induced clearances was developed. In Section 4, a simulation was conducted to examine the surface conditions of the hinge pin and bearing after wear in order to reflect the cumulative manifestation of the system’s dynamic response characteristics, as well as the impact of wear-induced clearance on the dynamic behavior of the LGRES, for various numbers of wear cycles, initial clearance values, and drive velocities. The conclusion of the paper is provided in Section 5.

2. Developing a Wear Clearance Model in LGRES

2.1. Development of a Contact Force Model

The clearance model for a planar rotating joint (as shown in Figure 1), the radii of the central shaft and the bearing in the articulated joint are denoted by $R_j$ and $R_i$, respectively, while C1 and C2 represent the centers of hinge pin and bearing, respectively. The geographical coordinates of the centroid of the hinge pin and bearing can be denoted as vectors ${\mathit{r}}_{\mathit{j}}^{\mathit{c}}$ and ${\mathit{r}}_{\mathit{i}}^{\mathit{c}}$. Therefore, the positional deviation between the two components within the joint can be represented by an offset vector.

$${\mathit{e}}_{\mathit{n}}={\mathit{r}}_{\mathit{i}}^{\mathit{c}}-{\mathit{r}}_{\mathit{j}}^{\mathit{c}}$$

(1)

Thus, the unit vector of the offset vector can be determined:

$$\mathit{n}={\displaystyle \frac{{\mathit{e}}_{\mathrm{n}}}{\sqrt{{{\mathit{e}}_{\mathit{n}}^{\mathit{T}}\mathit{e}}_{\mathrm{n}}}}}$$

(2)

Assuming the hinge pin and bearing are coaxial at the start, they are free and the eccentricity is less than the hinge joint clearance. Contact between the pin and bearing equalizes the eccentricity. A collision would make the eccentricity greater than the clearance. The contact conditions are

$$\left\{\begin{array}{c}{\mathsf{\delta }}_{\mathrm{r}}<0, free state\\ \hspace{1em}{\mathsf{\delta }}_{\mathrm{r}}=0, contact state\\ \hspace{1em}{\mathsf{\delta }}_{\mathrm{r}}>0, collision state\end{array}\right.$$

(3)

the embedding depth resulting from the collision is

$${\mathsf{\delta }}_{\mathrm{r}}=\sqrt{{{\mathrm{e}}_{\mathrm{n}}^{\mathrm{T}}\mathrm{e}}_{\mathrm{n}}}-c=\left|r_i^c-r_j^c\right|$$

(4)

where the clearance c can be expressed as

$$c=R_i-R_j$$

(5)

Since the eccentric vector unit vector is a local direction vector that varies with the position of the contact point, it is necessary to perform a transformation between the global and local coordinate systems, thus obtaining the collision point’s position vector:

$${\mathit{r}}_{\mathit{n}}^{\mathit{Q}}={\mathit{r}}_{\mathit{n}}^{\mathit{c}}+R_n·{\mathit{n}}_{\mathit{n}}\left(n=i,j\right)$$

(6)

By differentiating its position vector with respect to time, the velocity at the collision point is obtained:

$${\dot{\mathit{r}}}_{\mathit{n}}^{\mathit{Q}}={\dot{\mathit{r}}}_{\mathit{n}}^{\mathit{c}}+R_n·{\dot{\mathit{n}}}_{\mathit{n}}\left(n=i,j\right)$$

(7)

This analysis decomposes the impact velocity into the radial velocity and tangential velocity at the point of collision:

$$\left\{\begin{array}{c}{v}_n={\left({\dot{\mathit{r}}}_{\mathit{j}}^{\mathit{Q}}-{\dot{\mathit{r}}}_{\mathit{i}}^{\mathit{Q}}\right)}^T\cdot \mathit{n}\\ v_{\tau }={\left({\dot{\mathit{r}}}_{\mathit{j}}^{\mathit{Q}}-{\dot{\mathit{r}}}_{\mathit{i}}^{\mathit{Q}}\right)}^T\cdot \mathit{t}\end{array}\right.$$

(8)

The tangent unit vector is the result of a 90° anticlockwise rotation of the normal unit vector.

2.2. Establishment of Contact Force Model

When a shaft and a bearing come into contact due to clearance, an interaction force is generated at the point of contact. This study uses Hertzian contact theory to calculate forces. Although it was developed for non-conformal contacts, it can reasonably approximate the local contact behavior because the contact interaction is in a small localized region. It has been widely applied to impact models for clearance joints [42,43,44].

During a collision, the high elastic recovery rate of steel results in a significantly lower energy dissipation than that absorbed by elastic deformation. It is therefore evident that the present study adopts the Lankarani–Nikravesh damping model and sets the coefficient of restitution to be close to 1 [45]. Together, these models form a collision force model. Therefore, the normal contact force, as derived from the L-N model combined with Hertzian contact theory, can be expressed as follows [46]:

$$F_n=\left\{\begin{array}{l}{K}_n{\delta }_r^n+C_n{\delta }_r^n\dot{{\delta }_r}, {\delta }_r\ge 0\\ \hspace{1em}0, {\delta }_r<0\end{array}\right.$$

(9)

for the exponent n in Equation (9), based on Hertz’s contact theory. As this study concerns the contact between a shaft and a bearing (rolling bearing), this constitutes point contact; therefore, an exponent of 1.5 is used. ${\delta }_r$ is the penetration depth, and $\dot{{\delta }_r}$ is the relative radial velocity at the contact point. $C_n$ is the normal damping constant; $K_n$ is the coefficient of contact stiffness.

As this study focuses primarily on wear between the shaft and the bearing, the contact is of the convex–concave type. The equivalent radius of curvature can therefore be expressed as

$$R_{mn}={\displaystyle \frac{1}{R_j}}-{\displaystyle \frac{1}{R_i}}={\displaystyle \frac{R_i{R}_j}{R_i-R_j}}$$

(10)

The non-linear Hertzian contact stiffness coefficient can be expressed as

$$K_n={\displaystyle \frac{4}{3\left({\epsilon }_i+{\epsilon }_j\right)}}{\left(R_{mn}\right)}^{{\displaystyle \frac{1}{2}}}$$

(11)

${\epsilon }_k$ represents the material compliance of the contacting bodies; it can be formulated as

$${\epsilon }_k={\displaystyle \frac{\left(1-{{\nu }_k}^2\right)}{E_k}}(k=i,j)$$

(12)

where the term ${\nu }_k$ denotes the Poisson coefficient of the material, and $E_k$ denotes the modulus of elasticity.

The damping term in Equation (9) only appears in the normal direction of contact, describing energy dissipation during impact. The L-N model links this coefficient to the Hertzian stiffness and recovery coefficients, and the impact velocity. Damping effects occur only with positive normal penetration and normal relative velocity. In tangential contact, the normal relative velocity is zero and there is no normal damping. The phenomenon of tangential interaction is independently explained by the Coulomb friction model, as outlined in Equation (14). The normal damping constant $C_n$ can be expressed as follows:

$$C_n={\displaystyle \frac{3K_n\left(1-c_e^2\right)}{4{\dot{\delta }}_r^{(-)}}}$$

(13)

where $c_e$ is the restitution constant, and ${\dot{\delta }}_r^{(-)}$ is the relative normal velocity at the moment of initial contact.

The tangential contact force employed in this study is founded upon the modified Coulomb friction model proposed by Ambrósio [47,48,49]. A dynamic correction factor is introduced to ensure the smooth transition of friction when the relative tangential velocity approaches zero. This is achieved by avoiding numerical integration instability caused by sudden changes in the direction of friction. The tangential contact force is expressed as

$$F_t=-c_f{c}_d{F}_n·sgn\left(v_t\right)$$

(14)

in the equation, $c_f$ is the static friction constant, $F_n$ is the radial collision force, $v_t$is the tangential relative velocity at the contact point, and $sgn\left(v_t\right)$ is a sign function used to determine the direction of $v_t$, equivalent to:

$$sgn\left(v_t\right)={\displaystyle \frac{v_t}{\left|v_t\right|}}(v_t\ne 0)$$

(15)

$c_d$ is the dynamic correction factor to smooth the friction transition near zero tangential velocity, which is defined as

$$c_d=\left\{\begin{array}{l}0, \left|v_t\right|<v_0\\ {\displaystyle \frac{\left|v_t\right|-v_0}{v_1-v_0}}, v_0\le \left|v_t\right|\le \\ 1, v_1<\left|v_t\right|\end{array}\right.v_1$$

(16)

where $v_0$ is the static friction velocity threshold. When the point-of-contact relative velocity is less than $c_d=0$, this indicates that friction is ‘attenuated’ or considered to be static friction in the low-speed region. In the range $v_0\le \left|v_t\right|\le v_1$, $c_d$ varies linearly between 0 and 1, representing the gradual transition from adhesion to sliding. When the relative velocity at the contact point exceeds the kinetic friction velocity threshold $v_1$, $c_d$ = 1, and friction enters the stage of full sliding friction generation.

The total contact force vectors acting between hinge pin and bearing can therefore be expressed as

$${\mathit{F}}_{\mathit{i}\mathit{j}}=F_n\mathit{n}+F_t\mathit{t}={\left[F_{ij}^x F_{ij}^y\right]}^T$$

(17)

the total contact moment vectors acting between hinge pin and bearing can therefore be expressed as

$${\mathit{M}}_{\mathit{i}\mathit{j}}=x_{i,j}^r{F}_{ij}^y-y_{i,j}^r{F}_{ij}^x$$

(18)

where the arm of force in the moment is

$$x_{i,j}^r=x_{i,j}^Q-x_{i,j}, y_{i,j}^r=y_{i,j}^Q-y_{i,j}$$

(19)

according to Newton’s law of action and reaction, the interaction force $F_{ij}=-F_{ji}$ and the interaction moments satisfy $M_{ij}=-M_{ji}$.

2.3. Modeling LGRES with Irregular Wear Clearance

In practical engineering, the clearance between rotating components is not constant because of wear. This results in an uneven clearance distribution. To analyze the LGRES under these conditions, an irregular wear-induced clearance model is incorporated into its dynamic modeling process.

As demonstrated in Figure 2, the phenomenon of wear is observed within the contact region between the pin and the bearing (between A and B). The Archard wear model is a popular choice for predicting how things wear out in engineering systems, as it relates wear volume to contact load and sliding distance. Thus, the Archard wear model is employed in this paper to describe the volume of wear in a hinge joint. The proposed model delineates a correlation between wear volume, contact load, and sliding distance, which can be expressed as follows [50]:

$${\displaystyle \frac{V}{s}}=K_w{\displaystyle \frac{F_n}{H}}=K_d{F}_n$$

(20)

In the equation, $V$ represents the wear’s volume, s is the magnitude of the relative displacement traversed during sliding, $F_n$ is the radial impact force, the wear parameter is${ K}_w$, and $H$ is the Brinell hardness of the softer material among the two colliding objects.

In Figure 2, the geometric relationships and, using the law of cosines, the value of the contact angle α are

$$\alpha =arccos{\displaystyle \frac{e_n^2+R_j^2-R_i^2}{2e_i{R}_j}}$$

(21)

Then, the chord AB is

$$\left|AB\right|=2R_j\mathrm{s}\mathrm{i}\mathrm{n}(\pi -\alpha )$$

(22)

In practice, angle α is significantly larger than that illustrated in the schematic diagram; arc $\widehat{AB}$ may be approximated as the length of chord $AB$.

$$A_c=\widehat{AB}·w\approx \left|AB\right|·w$$

(23)

To relate the wear to clearance evolution, the wear volume is converted into wear thickness by introducing the contact area $A_c$ [51]:

$${\displaystyle \frac{h}{s}}=K_{d}p$$

(24)

Joint wear is generally considered to be a process that evolves over time; the depth of the wear varies depending on the force of impact and the sliding distance. To describe the continuous evolution of wear in dynamic systems, the Archard wear model in differential form is adopted:

$${\displaystyle \frac{dh}{ds}}=K_{d}p$$

(25)

The sliding distance is expressed as a function of time:

$${\displaystyle \frac{ds}{dt}}=v_t$$

(26)

Substituting the above expression into the Archard wear model equation yields:

$$dh=K_{d}pds=K_{d}p{v}_{t}dt$$

(27)

The surfaces of hinge pin and bearing are divided into numerous small grids, and a wear analysis is performed on the surface of each small grid. Consequently, the total wear thickness of the hinge pin and bearing surfaces can be obtained by summing the wear thickness of each grid, which can be expressed as

$$h={\displaystyle \sum _{k=1}^n}{h}_k$$

(28)

where $k$ denotes the index of the grid cell on the surface, $n$ is the total number of grid sections, and $h_k$ express the wear thickness of each grid section.

The shaft and bearing are assumed to be uniform materials. In the modeling, the cumulative wear is distributed equally between the two contacting surfaces. Consequently, the expressions for the updated radius and effective contact radius of the hinge pin and bearing are as follows:

$$\left\{\begin{array}{c}{R}_i^{*}=R_i-{\displaystyle \frac{h}{2}}\\ R_j^{*}=R_j+{\displaystyle \frac{h}{2}}\end{array}\right.$$

(29)

Unlike the ideal assumption of a constant clearance, the clearance is described as a time-varying and direction-dependent quantity due to wear between the joint surfaces. Specifically, the clearance in a revolute joint is geometrically defined as the radial difference; it can be equivalently expressed as

$$c=c_0+∆c\left(\theta ,t\right)=R_i^{*}-R_j^{*}$$

(30)

where $c_0$ is the initial clearance, and $\Delta c(\theta ,t)$ represents the additional clearance induced by wear, which depends on the angular position $\theta$, and time $t$.

The clearance value varies due to the uneven distribution of impact forces. By applying finite element analysis, the clearance value for each meshing region can be calculated. Therefore, the clearance value for that meshing region is

$$c\left(k\right)=R_i^{*}\left(k\right)-R_j^{*}\left(k\right)$$

(31)

The embedment depth of this grid area after wear is

$${\mathit{\delta }}_{\mathit{r}}\left(k\right)=\sqrt{{\mathit{e}}_{\mathit{n}}^{\mathit{T}}{\mathit{e}}_{\mathit{n}}}-c(k)$$

(32)

The $K_n$ after wear can be considered a function of the updated contact geometry, reflecting the influence of wear on the effective radii and material interaction:

$$K_n^{*}(k)={\displaystyle \frac{4}{3\left({\epsilon }_1+{\epsilon }_2\right)}}{\left({\displaystyle \frac{R_i\left(k\right)R_j\left(k\right)}{R_i\left(k\right)-R_j\left(k\right)}}\right)}^{{\displaystyle \frac{1}{2}}}$$

(33)

The damping coefficient subsequent to wear is ordinarily expressed as a dependent variable of the updated interaction parameters:

$$C_n^{*}(k)={\displaystyle \frac{3\left(1-c_e^2\right){\left[{\delta }_r^{*}\right(k\left)\right]}^n{K}_n^{*}\left(k\right)}{4{\dot{\delta }}_r^{\left(-\right)}\left(k\right)}}$$

(34)

Therefore, the radial collision force in mesh region k of the rotating pair after wear is

$$F_n^{*}\left(k\right)=K_n^{*}\left(k\right){\left[{\delta }_r^{*}\right(k\left)\right]}^n+C_n^{*}\left(k\right)\dot{{\delta }^{*}}\left(k\right)$$

(35)

Accordingly, the $F_t^{*}\left(k\right)$ after wear can be calculated as

$$F_t^{*}\left(k\right)=-c_f{c}_d{F}_n^{*}\left(k\right)·sgn\left(v_t\right)=-c_f{c}_d{F}_n^{*}\left(k\right){\displaystyle \frac{v_t}{\left|v_t\right|}}$$

(36)

The coupling between wear evolution and system dynamics introduces time-varying nonlinearities.

3. Dynamic Modeling of LGRES with Revolute Pair Clearance

3.1. Structural Characteristics of LGRES

Figure 3 presents the rendered model of the LGRES studied in this paper, which serves as the basis for the subsequent kinematic and dynamic analysis. The mechanism is mainly composed of an actuator cylinder, multiple linkages, a rocker arm, a slide rod, a wheel and a fixed bracket. The actuator cylinder is bolted to the frame and provides the driving force. This force is transmitted through a series of linkages to the rocker arm, which ultimately controls the movement of the wheels. The fixed bracket represents the connection to the airframe. This structure allows for better capture of the fundamental dynamic characteristics of a typical LGRES.

Figure 4 shows a rendering of LGRES, illustrating its structure, which illustrates that the mechanism consists of one driving component and four driven components. These include linkages 1, 2 and 3, the landing gear rocker arm, and the slide rod, and this mechanism has one degree of freedom. Therefore, it exhibits deterministic motion. The drive component is the slide rod, performing a regular, uniform vertical reciprocating motion within the guide rails. This motion transmits the driving force through linkage 3, which drives the rocker arm to move along an arc-shaped path around the fixed hinge mount, thus realizing the retraction and extension motion of the landing gear.

The mechanism has two sets of clearances. Joint selection is based on two things. Firstly, the slide rod is driven by the hydraulic system, while the rocker arm is connected to the wheel. This means these joints have a bigger impact on the LGRES’s response. Second, these joints are on the main force transmission path, so they are more affected by impact and wear. Therefore, choosing these joints as clearance joints shows how clearance affects the mechanism.

3.2. Dynamic Model of LGRES Considering Irregular Wearing Clearance

Figure 4 shows the LGRES as a planar mechanism. Its components have three generalized coordinates, which determine their position. The five moving components of the LGRES have the following generalized coordinates:

$$\mathit{q}={({\mathit{q}}_1,{\mathit{q}}_2,{\mathit{q}}_3,{\mathit{q}}_4,{\mathit{q}}_5)}^T$$

(37)

$${\mathit{q}}_{\mathit{i}}={(x_i, y_i, {\theta }_i)}^T(i=1, 2, 3, 4, 5)$$

(38)

In a general coordinate system, $x_i$ and $y_i$ represent the coordinates of the component’s center of mass, while ${\theta }_i$ represents the component’s rotation angle.

The LGRES includes six revolute joints and one prismatic joint. Considering the clearances in two revolute joints, each revolute joint introduces two kinematic constraints, yielding 11 independent constraint equations in total, consisting of 10 ideal kinematic constraints and 1 driving constraint. This is expressed by the following equation:

$$\mathsf{\Phi }\left(\mathit{q},t\right)=\left(\begin{array}{c}{x}_1-L_{s1}\cos {\theta }_1\\ y_1-L_{s1}\sin {\theta }_1-H_1\\ x_2-x_1-L_{s2}\cos {\theta }_2-L_{s1}\cos {\theta }_1\\ y_2-y_1-L_{s2}\sin {\theta }_2-L_{s1}\sin {\theta }_1\\ x_3-L_{s3}\cos {\theta }_3\\ y_3-L_{s3}\sin {\theta }_3\\ x_4-x_3-L_{s4}\cos {\theta }_4+(L_{s3}-a)\cos {\theta }_4\\ y_4-y_3-L_{s4}\sin {\theta }_4+(L_{s3}-a)\sin {\theta }_4\\ x_5-a\\ {\theta }_5-{\displaystyle \frac{\pi }{2}}\\ y_5-0.895-0.09(1-\cos \mathsf{\omega }t)\end{array}\right)=0$$

(39)

The velocity constraint equation for this system can be determined by deriving the constraint equation $\mathsf{\Phi }\left(\mathit{q},t\right)$, and can be expressed as

$${\mathsf{\Phi }}_{\mathit{q}}\dot{\mathit{q}}=-{\mathsf{\Phi }}_{\mathit{t}}\equiv \mathit{v}$$

(40)

where ${\mathsf{\Phi }}_{\mathit{q}}$ is the displacement constraint equation’s Jacobian matrix, which can be expressed as ${\displaystyle \frac{\partial \mathsf{\Phi }}{\partial q}}$, the derivative of the displacement constraint is designated as ${\mathsf{\Phi }}_{\mathit{t}}$, which can be expressed as ${\displaystyle \frac{\partial \mathsf{\Phi }}{\partial t}}$, and $\dot{\mathit{q}}$ is the first-order differential equations in general coordinates.

The derivation of the acceleration constraint equation can be achieved through the application of the time derivative operator to the constraint equation of velocity.

$${\displaystyle \frac{d}{dt}}\left({\mathsf{\Phi }}_{\mathit{q}}\dot{\mathit{q}}+{\mathsf{\Phi }}_{\mathit{t}}\right)={\mathsf{\Phi }}_{\mathit{q}}\ddot{\mathit{q}}=-{{(\mathsf{\Phi }}_{\mathit{q}}\dot{\mathit{q}})}_{\mathit{q}}\dot{\mathit{q}}-2{\mathsf{\Phi }}_{\mathit{q}\mathit{t}}\dot{\mathit{q}}-{\mathsf{\Phi }}_{\mathit{t}\mathit{t}}\equiv \mathit{\gamma }$$

(41)

As illustrated above, ${\mathsf{\Phi }}_{\mathit{t}\mathit{t}}$ indicates the time-derivative of the displacement constraint equation, calculated to the second order, while ${\mathsf{\Phi }}_{\mathit{q}\mathit{t}}$ indicates the differentiation of the displacement with respect to time, and $\ddot{\mathit{q}}$ denotes the generalized acceleration vector.

LGRES is a multi-body system with one degree of freedom and multiple kinematic constraints. The equations of equilibrium for each rigid body can be derived and unified in matrix form by applying the dynamics of multi-body systems equation and incorporating a Lagrange multiplier. The dynamic equations for LGRES are:

$$M\ddot{\mathit{q}}+{\mathsf{\Phi }}_{\mathit{q}}^{\mathit{T}}\lambda =\mathit{G}$$

(42)

where the $\lambda$ is Lagrange multiplier, $M$ represents the mass matrix of system, and the matrix $G$ denotes the generalized forces acting on the system, which can be expressed as:

$$G=\left[\begin{array}{c}\begin{array}{cc}0;& -m_{1}g; 0;\\ F_{C{x}_1};& -m_{2}g+F_{C{y}_1};{ T}_{23};\end{array}\\ {-F}_{C{x}_1}; -m_{3}g-F_{C{y}_1};{ T}_{32}; \\ { F}_{C{x}_2}; -m_{4}g+F_{C{y}_2};{ T}_{45}; \\ {-F}_{C{x}_2}; -m_{5}g-F_{C{y}_2};{ T}_{54}; \end{array}\right]$$

(43)

where ${\mathit{F}}_{\mathit{C}{\mathit{x}}_{\mathit{i}}}$, ${\mathit{F}}_{\mathit{C}{\mathit{y}}_{\mathit{i}}}(i=1, 2)$ in the matrix denote the collision forces in hinge clearances.

The system’s motion is determined by the dynamic equations, while the constraint equations describe the kinematic relationships. Combining the acceleration-level constraint Equation (40) with the dynamic Equation (41) forms a closed system of equations. The resulting coupled equations are as follows [52]:

$$\left(\begin{array}{cc}\mathit{M}& {\mathsf{\Phi }}_{\mathit{q}}^{\mathit{T}}\\ {\mathsf{\Phi }}_{\mathit{q}}& \mathbf{0}\end{array}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{\ddot{\mathit{q}}}{\mathit{\lambda }}}\right)=\left({\displaystyle \genfrac{}{}{0pt}{}{\mathit{G}}{\mathit{\gamma }}}\right)$$

(44)

Employing the aforementioned equation enables a solution to be found for the two variables $\ddot{\mathit{q}}$ and $\mathit{\lambda }$. However, due to numerical errors, the constraint equations may gradually deviate from their ideal conditions, resulting in constraint drift. To address this issue, the constraint equations were modified by introducing position and velocity feedback terms:

$${\mathit{\gamma }}^{\mathit{*}}=\mathit{\gamma }-2\alpha \dot{\mathsf{\Phi }}-{\beta }^2\mathsf{\Phi }$$

(45)

where $\dot{\mathsf{\Phi }}={\mathsf{\Phi }}_{\mathit{q}}\dot{\mathit{q}}+{\mathsf{\Phi }}_{\mathit{t}}$ is the velocity-level constraint; $\alpha$, $\beta$ are stabilization parameters to suppress constraint drift during the numerical integration process, and the stabilization coefficients are α = 50, β = 50. Therefore, the final equations of mechanical dynamics are

$$\left(\begin{array}{cc}\mathit{M}& {\mathsf{\Phi }}_{\mathit{q}}^{\mathit{T}}\\ {\mathsf{\Phi }}_{\mathit{q}}& \mathbf{0}\end{array}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{\ddot{\mathit{q}}}{\mathit{\lambda }}}\right)=\left({\displaystyle \genfrac{}{}{0pt}{}{\mathit{G}}{{\mathit{\gamma }}^{\mathit{*}}}}\right)=\left({\displaystyle \genfrac{}{}{0pt}{}{\mathit{G}}{\mathit{\gamma }-2\alpha \dot{\mathsf{\Phi }}-{\beta }^2\mathsf{\Phi }}}\right)$$

(46)

4. The Analysis of Dynamics of LGRES with Irregular Wearing Clearance

4.1. The Dynamic Model Verification and the Establishment of Parameters

4.1.1. Structure Parameters and Simulation Process

The LGRES mathematical simulation was performed in MATLABR2023a, using the ODE15S solver (with a relative tolerance of ${10}^{-6}$). The time step was set to ${10}^{-4}$. The wear-grid number was set to 1000 based on a related study [53]. This study focuses on the influence of different wear conditions and operating parameters on the nonlinear dynamic response of the LGRES, so the selected grid resolution is sufficient for capturing the overall wear distribution characteristics and dynamic response trends. The model workflow diagram is illustrated in Figure 5.

The mass, moments of inertia, and geometric parameters of the various components in LGRES are provided in

Table 1. Dimension parameters of the various components in LGRES.

Component	Length of Component (m)	Centroid Position Length (m)	Mass (kg)	Moment of Inertia (${10}^{-3}\mathbf{k}\mathbf{g}\times {\mathbf{m}}^2$)
Rod1	1.00	0.5	6.32	2.35
Rod2	1.12	0.56	12.07	1.54
Rod3	1.29	0.645	12.54	9.69
Rocker arm	0.57	0.285	5.59	0.36
Slide rod	0.64	0.32	6.93	0.45

, while the specific parameters for the clearance between moving pairs that account for wear are provided in

Table 2. Dynamics simulation parameters of the various components in LGRES.

Parameter	Values
Joint A’s bearing radius ($R_1$)	0.05 m
Joint B’s bearing radius ($R_2$)	0.08 m
Hinge width (w)	0.1 m
Restitution coefficient ($c_e$)	0.9
Friction coefficient ($c_f$)	0.15
Static friction velocity threshold ($v_0$)	0.0001 m/s
Kinetic friction velocity threshold ($v_1$)	0.001 m/s
Poisson ratio (${\nu }_i,{ \nu }_j$)	0.3
Elastic modulus ($E_i, E_j$)	200 GPa
Liner wear coefficient ($K_d$)	$1.8\times {10}^{-13} {\mathrm{m}}^2/\mathrm{N}$
Brinell hardness of the softer material (H)	$1.716\times {10}^9$ Pa

.

4.1.2. Dynamic Model Verification of Landing Gear Retraction and Extension System

In this study, the effectiveness and physical rationality of the proposed dynamic model was verified by performing comparative multibody dynamics simulations in ADAMS2020 software. This is because ADAMS is founded upon the principles of multibody system dynamics theory and utilizes established numerical formulations to solve the governing equations of constrained rigid-body motion. It can accurately reproduce the kinematic and dynamic behavior of complex mechanical systems. The ADAMS environment utilized the same structural parameters, initial clearance values, material properties, and drive speeds as the MATLAB model. The comparison primarily concentrated on the dynamic response of the rocker arm.

Figure 6 shows that the model response and ADAMS simulation show similar fluctuations in the displacement, velocity, and acceleration curves. Although there are slight differences in the amplitude of local peaks, the overall dynamic trends remain highly consistent. These discrepancies are primarily attributed to differences in the numerical integration algorithms and contact parameter implementation methods between MATLAB and ADAMS. Therefore, this comparison verifies the effectiveness and physical rationality of the LGRES dynamic model.

4.2. Dynamic Responses Analysis of LGRES Considering Wearing Clearance

4.2.1. Wear Characteristics Under Different Wear Cycles

To investigate and analyze the dynamic characteristics of the LGRES, this section examines the effect of different numbers of wear cycles on the wear characteristics. It assumes an initial clearance of 0.5 mm, a drive speed of 60 rpm and a coefficient of friction of 0.15. In order to investigate wear on hinge surfaces during long-term operation, it is assumed that wear occurs in two stages, with each stage lasting 2 million cycles. Since, in the early stages of the wear process, wear is still limited to a minor wear zone, the wear accumulation predicted by the Archard model is roughly proportional to the number of operating cycles. Therefore, this study magnifies the wear thickness observed over 100 simulation cycles by a factor of 20,000 to approximate the wear thickness after 2 million cycles, which shortens the simulation time and treats these 2 million cycles as a single wear cycle [54].

As shown in Figure 7 and Figure 8, the wear thickness distribution of rotating pair A after one-time wear and two-time wear is concentrated in the angles [$0°$, $210°$], $[250°, 255°]$ and [$320°$, $360°$]. This wear is mainly concentrated in two diagonal directions. For rotating pair B, the wear thickness is concentrated in angles [$80°$, $100°$] and [$250°$, $270°$]. The wear is concentrated in the vertical direction. Moreover, the wear thickness of two-time wear in rotating pairs A and B is greater than that of one-time wear. The peak wear thickness for two-time wear in rotating pairs A and B is ${2.83}^{-5} \mathrm{m}$, ${7.20}^{-5} \mathrm{m}$, and the peak wear thicknesses for one-time wear are${1.50}^{-6} \mathrm{m}$, ${3.58}^{-6} \mathrm{m}$. This is because the irregular contact geometry after one-time wear leads to higher local contact stress and impact enhancement, which accelerates subsequent wear. As shown by the centerline trajectory of the mechanism shaft in Figure 9, the motion of the shaft becomes more erratic after secondary wear. This further demonstrates that secondary wear exacerbates localized wear and nonlinear dynamic responses.

The localization of wear is related to the force transmission and motion constraints of the LGRES joint. Different joints are connected to adjacent links in various ways, resulting in different motion constraints and force transmission paths. Link 2 and 3 are at an angle of 30° in Joint A; therefore, the forces and wear travel diagonally. In Joint B, wear is distributed along the vertical direction as it is constrained by a vertical guide mechanism in the sliding rod drive branch. After two cycles, the wear distribution becomes more irregular due to increased wear-induced clearance in the journal center. This changes the local contact conditions and the non-linear coupling between clearance and impact dynamics, causing more wear.

4.2.2. The Impact of Different Initial Clearances on the Dynamic Response of a Mechanism Under Wear Conditions

To investigate the effect of different initial clearance values on the dynamic response and the wear characteristics of the Landing Gear Retraction and Extension and Release System, considering the irregular wear clearance, a drive speed of 60 rpm was assumed and only the state after one-time wear was considered. Initial clearance values of 0.2 mm and 0.5 mm were selected for this investigation.

Figure 10 compares the motion characteristics of swing arm 3 in the x-axis and y-axis. For a 0.5 mm clearance, the deviations from the ideal values are$2.1\times {10}^{-3}$ m, $0.149 \mathrm{m}/\mathrm{s}$ and $320.18 \mathrm{m}/{\mathrm{s}}^2$ along the x-axis; and $1.91\times {10}^{-3} \mathrm{m}$, $0.195 \mathrm{m}/\mathrm{s}$ and $830 \mathrm{m}/{\mathrm{s}}^2$ along the y-axis. However, for a 0.2 mm clearance, the peaks are $8.90\times {10}^{-4} \mathrm{m}$, $6.51\times {10}^{-2} \mathrm{m}/\mathrm{s}$ and $164.79 \mathrm{m}/{\mathrm{s}}^2$ in the x-direction; and$8.74\times {10}^{-4} \mathrm{m}$, $0.155 \mathrm{m}/\mathrm{s}$ and $395.41 \mathrm{m}/{\mathrm{s}}^2$ in the y-direction. It is revealed that as the clearance value increases, the amplitude of fluctuations in displacement, velocity, and acceleration also increases. This is because a larger clearance allows the shaft to attain a higher relative impact velocity. The resulting impacts weaken the joint’s kinematic constraints and exacerbate nonlinear contacts, amplifying fluctuations in the dynamic response.

In Figure 11, the results indicate that the peak collision forces for hinges A and B are$15.86 \mathrm{k}\mathrm{N} \mathrm{a}\mathrm{n}\mathrm{d} 35.70 \mathrm{k}\mathrm{N}$ at an initial clearance of 0.2 mm. However, the peak collision forces for hinges A and B are $25.26 \mathrm{k}\mathrm{N} \mathrm{a}\mathrm{n}\mathrm{d} 47.71 \mathrm{k}\mathrm{N}$ at a clearance of 0.5 mm. Consequently, it can be deduced that the contact impact force within the clearance joint escalates considerably as the clearance increases. This is because a larger clearance allows the pin to attain a higher collision velocity, thereby intensifying the instantaneous impact and resulting in a significant increase in the peak collision force.

In the wear thickness graph shown in Figure 12, the wear in rotating pair A showed a peak wear depth of ${1.50\times 10}^{-5} \mathrm{m}$ at clearance of $0.5 \mathrm{mm}$, and a peak wear depth of ${3.45\times 10}^{-6} \mathrm{m}$ at a clearance of $0.2 \mathrm{mm}$. With an initial clearance of $0.5 \mathrm{mm}$ for rotating pair B, the peak wear depth is ${3.58\times 10}^{-5} \mathrm{m}$ and the maximum wear depth is ${2.69\times 10}^{-5} \mathrm{m}$ at $0.2 \mathrm{mm}$. For 0.2 mm, the wear in rotating pair A is mainly concentrated in [$50°$, $200°], \mathrm{a}\mathrm{n}\mathrm{d} [350°, 360°]$, the wear in rotating pair B is mainly concentrated in [${85}^{°}$, ${95}^{°}], \mathrm{a}\mathrm{n}\mathrm{d} [260°, 265°]$. Compared to the wear area when the wear clearance is 0.5 mm (as shown in Section 4.2.1), the wear area is smaller. The results indicate that larger clearances lead to more severe wear and a wider wear distribution. This phenomenon occurs because a larger initial clearance increases the range of free movement at the center of the journal, resulting in a wider distribution of contact points, more frequent impact events and higher collision force, which ultimately leads to an expansion of the significant wear area and increase in the depth of wear.

As shown in Figure 13 and Figure 14, when the clearance value is larger, the central trajectory becomes more chaotic, and the system’s nonlinear response becomes more pronounced. The wear region closely matches the region of the center trajectory that extends beyond the clearance circle, thereby validating the model.

4.2.3. The Impact of Different Drive Velocities on the Dynamic Behavior of LGRES Under Wear Conditions

This section analyzes the dynamic behavior of LGRES under different driving speeds while considering wear clearances. Thus, it compares the dynamic response at drive speeds of 45 rpm and 105 rpm. The initial clearances for both clearance A and B are set to $0.5 \mathrm{mm}$, and considering only a single cycle of wear, the dynamic simulation was performed on the LGRES. The nonlinear response curve for rocker arm 3 is as follows:

In Figure 15, as the drive speed is 45 rpm, the peak values of velocity and acceleration in the x-direction are $0.77 \mathrm{m}/\mathrm{s}$ and 68.60${\mathrm{m}/\mathrm{s}}^2$, and the peak deviation values are ${5.65\times 10}^{-2} \mathrm{m}/\mathrm{s}$ and $72.52 \mathrm{m}/{\mathrm{s}}^2$. For a drive speed of 105 rpm, the peak values are $1.83 \mathrm{m}/\mathrm{s}$ and $232.45 \mathrm{m}/{\mathrm{s}}^2$, and the deviation values are ${9.34\times 10}^{-2} \mathrm{m}/\mathrm{s}$ and $244.86 \mathrm{m}/{\mathrm{s}}^2$, respectively. Similarly, in the y-direction, the peak values are $1.22 \mathrm{m}/\mathrm{s}$ and $414.50{ \mathrm{m}/\mathrm{s}}^2$, and the deviation values are ${8.13}^{-2} \mathrm{m}/\mathrm{s}$ and $420.17 \mathrm{m}/{\mathrm{s}}^2$ at a drive speed of 45 rpm. For a drive speed of 105 rpm, the peak values are $2.93 \mathrm{m}/\mathrm{s}$ and $616.43 \mathrm{m}/{\mathrm{s}}^2$; the deviation values are ${2.71\times 10}^{-1} \mathrm{m}/\mathrm{s}$ and $641.54 \mathrm{m}/{\mathrm{s}}^2$. When the drive speed increases, the amplitude of the dynamic response curve becomes larger, the vibration speed increases, and the deviation from the ideal state becomes more pronounced. It has been demonstrated that an increase in drive speed results in a rise in relative collision velocity at the joints of the mechanism. This, in turn, has a deleterious effect on the system’s nonlinear dynamic response.

Figure 16 illustrates that at a drive speed of 45 rpm, the maximum clearance forces for rotating pairs A and B are 9.575 $\mathrm{k}\mathrm{N}$ and $31.886 \mathrm{k}\mathrm{N}$. When the drive speed is 105 rpm, the maximum clearance forces for rotating pairs A and B are 21.85 $\mathrm{k}\mathrm{N}$ and $79.88 \mathrm{k}\mathrm{N}$. It can be observed that as the drive speed increases from 45 rpm to 105 rpm, the collisions within the joint clearances intensify, and the collision forces increase. This is because the increased drive speed enhances the relative collision velocity of the mechanism’s joints, thereby intensifying the nonlinear impact contact behavior and resulting in a larger amplitude of the collision force.

As shown in Figure 17 and Figure 18, after a single wear cycle, as the drive speed increases from 45 rpm to 105 rpm, and the maximum wear thickness and wear region increases significantly. When the drive speed is 45 rpm, the maximum wear thickness is ${2.36\times 10}^{-6} \mathrm{m}$ in hinge A, with the wear area mainly concentrated in the range [$0°$, $205°$] and [$330°$, $360°$]. The maximum wear thickness is ${3.28\times 10}^{-5} \mathrm{m}$ in hinge B, and the wear area is mainly concentrated in the range [$85°$, $100°$] and [$250°$, $265°$]. When the drive speed is 105 rpm, the maximum wear thickness is ${1.43\times 10}^{-5} \mathrm{m}$ in hinge A, and the wear occurs across almost the entire surface. In hinge B, the maximum wear thickness is ${4.52\times 10}^{-5} \mathrm{m}$, and the wear area is mainly concentrated in the range [$80°$, $105°$] and [$245°$, $265°$]. As the driving speed increases, the relative sliding speed between components rises significantly, and the peak impact force during collisions also increases accordingly, leading to a higher accumulation of wear per unit time. Furthermore, the increased instantaneous contact impact force causes the pin to generate a more complex motion trajectory within the clearance (see Figure 18), and also causes the shaft and bearing to make contact at more angular positions, thereby expanding the actual contact area.

As shown in Figure 19, the centerline curve at higher drive speeds is more densely packed than that at lower drive speeds. This is because higher drive speeds are more likely to cause stronger vibrations and non-stationary motion, resulting in high-frequency impacts between the shaft and the bearing. Consequently, as seen in Figure 17, the wear surface is more uneven at higher drive speeds.

5. Conclusions

This study proposes a dynamic model of the LGRES that accounts for wear-induced clearances. This study focuses on how different system parameters affect the dynamic behavior and wear characteristics of the LGRES. The primary conclusions that can be drawn from this analysis are summarized as follows:

(1)

The present study proposes a dynamic wear prediction method for LGRES, taking into account the clearance of kinematic pairs. The integration of Hertzian contact theory with the Lankarani–Nikravesh damping formulation resulted in the establishment of a model that delineates the interaction forces within a clearance joint. Furthermore, the integration of the Archard wear model and the shaft-bearing reconstruction scheme enables the model to account for the time-varying, irregular distribution of clearance, thus providing a more realistic description than traditional constant-clearance models.

(2)

The analysis focused on the surface reconstruction process of the LGRES hinge pins and bearing surfaces under one-time and two-time wear conditions. The results indicate that the wear evolution exhibits increasingly irregular and localized characteristics, which further complicates the nonlinear dynamic behavior.

(3)

This study was conducted to analyze and compare the effects of different initial clearances and drive speeds on nonlinear dynamic behavior. The results of this study indicated that larger initial clearances significantly increase motion errors, wear severity and the area affected by wear. In addition, higher drive speeds were found to exacerbate nonlinear impact behavior, resulting in more severe wear and a larger wear area.

Overall, the results highlight that the coupling between wear evolution and clearance nonlinearity plays an important role in the long-term dynamic performance of the LGRES and should be considered in both the modeling and design of aerospace mechanisms. However, the study did not account for the connecting rod’s flexibility, so the model’s primary application is for low-frequency rigid-body dynamics. High-frequency structural vibration effects and local elastic modes are not included.