Dynamic Analysis and Force Adaptation in Elastic-Link Mechanical Systems with Two Degrees of Freedom

Abstract

This study presents a comprehensive dynamic analysis of mechanical systems incorporating elastic joints and introduces an adaptive vibration actuator with an integrated transmission variator. The system’s behavior is modeled through kinematic and dynamic formulations, utilizing both analytical and numerical methods. The analysis reveals that the inclusion of elastic elements enables a force adaptation effect, allowing the output element to adjust dynamically to variations in external loading. Under conditions of constant input power, the output speed varies inversely with the load, ensuring reliable adaptive performance. Furthermore, the elastic joints facilitate internal force redistribution, enhancing energy efficiency and reducing mechanical losses. These findings hold relevance for applications in industrial automation and robotics, where consistent functionality under variable load conditions is essential.

1. Introduction

In modern engineering systems, conventional drive mechanisms often operate under non-ideal conditions such as prolonged inactivity, insufficient lubrication, mechanical imperfections, and low ambient temperatures. Under such circumstances, even minor disturbances can lead to critical failures, particularly in systems utilizing single-degree-of-freedom (1-DOF) transmissions where the actuator is directly coupled to the motor. In these configurations, any overload or jamming in the actuator may instantly overload the motor or damage the transmission system.

To address these limitations, the development of adaptive mechanical transmissions has gained increasing attention. Systems with two degrees of freedom (2-DOF) allow the output speed to vary inversely with the load, enabling continued operation even under variable resistance while maintaining constant input power [1,2]. This adaptability helps protect both the motor and transmission, especially under stalling or overload conditions.

Modern research pays considerable attention to adaptive mechanisms with variable stiffness and elastic elements. For example, variable stiffness actuators (VSSEA) with a wide adjustment range and high energy efficiency [3] have been proposed, as well as elastic finger mechanisms with passive self-tuning to improve gripping reliability [4]. Parallel elastic actuators with adjustable equilibrium points are actively used in robotics and self-deforming mechanisms are being developed for mobile platforms to adapt to uneven surfaces.

Compared to earlier studies [3,4], which primarily focused on adaptive transmissions with rigid links and relied on active feedback control to achieve force adaptation, the present work introduces a structurally different approach. The proposed mechanism is based on a closed-loop elastic configuration that passively redistributes internal forces and enables self-regulated motion transmission without additional actuators or sensors. In contrast to [3,4], where adaptability was achieved through external control strategies, our design ensures inherent force adaptation through the geometry and elasticity of the mechanism itself. This structural self-adaptation represents the key novelty of the present study.

Recent studies on adaptive elastic-link systems further emphasize the role of structural self-adaptation in engineering design [5,6].

In addition to traditional studies focusing on the structural design of mechanisms and their control systems, recent research trends also address elastic-link dynamics, vibration energy transmission, and variable stiffness actuators (VSAs). These approaches are considered crucial for improving adaptability, energy efficiency, and safety in modern robotic and aerospace applications [7,8,9].

For example, phase deviation of semi-active suspension control and its compensation with inertial suspension examines phase deviation in semi-active suspension systems and proposes inertial compensation, illustrating modern approaches to structural adaptation in dynamic mechanical systems [10].

These solutions provide flexibility and energy efficiency but often require active control or complex design. In contrast, the system proposed in this work is based on passive structural self-adaptation: a closed elastic loop automatically compensates for external disturbances without the need for complex control algorithms. This feature emphasizes the novelty of the approach and allows combining simplicity of implementation with potential energy efficiency. A further improvement to adaptive performance can be achieved by incorporating elastic elements (such as flexible gear teeth or spring-loaded joints) into the transmission. These components store and release energy during operation, allowing for dynamic force redistribution, vibration reduction, and improved system responsiveness. Elastic links also introduce a vibrational component to the output, enhancing contact reliability and contributing to smoother transitions under varying loads.

In contrast to variable-stiffness architectures and multi-speed/selective transmissions for electric drives [11,12], where adaptability is achieved through programmable actuators, multi-mode switching, and continuous operation of feedback loops, the proposed 2-DOF scheme implements structural self-adaptation by means of a closed elastic loop. At the structural level, it consists of a single power input $F_1$ and an internal pair of elastic links (3–4), which form the relationship $u_{16}=s_1/s_6$ depending on the external load $R_6$, without active parameter correction. From the standpoint of control requirements, our mechanism does not rely on sensors or regulators to vary stiffness or gear ratio: excitation requires only stationary input. From an energy perspective, adaptation is achieved through redistribution and temporary storage of energy in the elastic elements, which reduces specific losses in matching modes, whereas in [11,12], the “cost” of adaptation is determined by the operation of servomechanisms/switches and control algorithms. This substantiates the novelty of the proposed approach relative to existing solutions.

Despite the evident advantages, the current literature lacks comprehensive dynamic models of 2-DOF mechanisms with elastic joints that quantify their internal force interactions and transmission behavior. Most previous studies focus on rigid-body mechanics or purely kinematic formulations, without considering the internal energy exchange or adaptive vibration effects induced by compliant elements.

This study aims to develop a dynamic model of a 2-DOF adaptive transmission system with elastic joints and to analytically characterize the mechanism’s behavior under load variations. The research focuses on understanding how internal elasticity influences the transmission ratio, motion stability, and force adaptation capability.

The novelty of this work lies in the following aspects:

• Introducing a gear-based adaptive mechanism with closed-loop elastic components;
• Deriving analytical relationships for output displacement and transmission ratio as functions of load; and
• Demonstrating that force adaptation is inherently achieved through the structure of the mechanism, without requiring active feedback control.

A comparison with previously studied adaptive mechanisms [12,13] highlights the structural novelty of our approach. Unlike rigid 2-DOF mechanisms requiring feedback control and external sensors, the proposed design achieves passive self-adaptation.

Table 1. Comparative evaluation of the proposed closed-loop elastic mechanism and existing adaptive transmissions.

Criterion	Proposed Mechanism (Closed-Loop Elastic)	Existing Mechanisms [12,13]
Structure	Closed loop with elastic joints	Rigid gear trains
Control requirement	Passive self-adaptation (no sensors)	Active control with feedback
Energy efficiency	Energy redistribution via elastic links	Higher specific energy losses
Adaptability	Variable transmission ratio $u_{16}\propto F_{\mathrm{out}}$	Limited adaptability
Vibration damping	Internal elastic damping	Requires external damping

summarizes the key differences.

The rest of this paper is organized as follows:

• Section 2 introduces the mechanical structure and parameters of the system;
• Section 3 presents the dynamic equations and force analysis;
• Section 4 provides simulation results and discussion;
• Section 5 concludes with practical implications and directions for future research.

In the proposed gear-based adaptive mechanism with two degrees of freedom (Figure 1), the system includes the following main components:

• $H_1$—the primary input drive,
• Gears 2 and 4—input guiding gears,
• Gears 1 through 4—external teeth on the central gear unit,
• Gears 3 through 6—internal teeth of the same gear unit,
• Gear 5—output guide,
• $H_2$—the output drive.

The gears form a closed-loop mechanical circuit: 2–3–6–5–4–1, which is equipped with a differential coupling. This configuration enables the system to transmit rotational motion through a single input, while dynamically adjusting the output based on resistance.

Elastic elements are incorporated into gears 1 and 3, which are in the lower-load regions of the transmission path. These elastic components allow the gear teeth to deform slightly under load, thereby producing impulse-based force transmission. This behavior improves contact reliability and enables the system to respond more effectively to varying external resistances.

The presence of elastic joints introduces vibrational motion into the system, facilitating energy exchange between links and contributing to the mechanism’s adaptive force redistribution. As a result, the output speed becomes inversely dependent on the load, even under constant input torque [13,14,15].

2. Materials and Methods

To analyze the dynamic behavior of the adaptive mechanism, a simplified model is introduced. The mechanism is represented as a closed-loop system with two degrees of freedom and elastic joints. The analysis is based on fundamental mechanical parameters and physical assumptions that describe the spring–mass interaction in the system [16,17].

Analysis of the Motion of a Mechanical System with Elastic Joints

This section addresses the formulation and analysis of a mechanical system with two degrees of freedom in the presence of elastic connections. To facilitate the study, the analysis begins with a simplified mechanism featuring elastic links within a closed-loop configuration, corresponding to the structure of the gear-based adaptive mechanism discussed earlier [18].

Description of the Closed-Loop Elastic Lever Mechanism

A two-degree-of-freedom mechanism with a closed elastic loop (Figure 2) consists of the following components: a fixed base (column 0), an input link (1), a closed kinematic chain composed of links 2, 3, 4, and 5, and an output link (6). The elastic links 3 and 4 are supported by links 2 and 5 at the upper kinematic pairs C, D, L, and M. Each of these elastic joints is equipped with compression springs characterized by stiffness coefficients $c_3$ and $c_4$, and masses $m_3$ and $m_4$, respectively.

It is known that each elastic connection increases the system’s degrees of freedom by one. However, the force exerted on an elastic joint depends on the relative displacement between its connection points. Hence, the inclusion of elastic elements does not compromise the system’s definability when evaluating the effect of external forces.

The presence of a closed movable loop formed by links 2–3–4–5, connected through elastic elements, leads to mutual interactions among the elastic joints. These interactions are responsible for achieving equilibrium in the loop by redistributing internal forces via point displacements. The mutual influence of the elastic elements introduces an additional analytical constraint related to the relative positions of the points in the loop.

As a result, the system—while having two degrees of freedom—attains complete static determinacy in equilibrium due to this added constraint, effectively stabilizing its motion and structural configuration [19,20].

In the basic model, friction and damping were not considered to simplify the analytical analysis. In real engineering systems, their influence is significant, so it is planned to expand the model in the future to consider viscous damping and dry friction.

Figure 1 illustrates the gear model of the mechanism, while Figure 2 shows its simplified lever equivalent. This conversion is used for analytical analysis and allows for a clear examination of the interaction between elastic links. Figure 1 and Figure 2 represent two equivalent formulations of the proposed system. Figure 1 shows the gear-based mechanical implementation, while Figure 2 provides its lever-type kinematic analogue. The equivalence is obtained by replacing gear pairs with jointed lever links that replicate identical force–displacement relationships. This transformation allows simplifying the derivation of equilibrium equations while maintaining full dynamic consistency with the original gear model.

Static Analysis of the Motion of a Lever Mechanism

The mechanism under consideration, shown in Figure 2, possesses two degrees of freedom and consists of an input structural group (links 0–1–2) and a connected kinematic chain (links 3–4–5–6). The two degrees of freedom of the input group 0–1–2 include:

• a translational motion of link 1 (with displacement $s_1$), and
• a relative rotational motion of link 2, where the displacements of points C and D are unequal ($s_3\ne s_4$).

Based on these constraints, the mechanism can be treated as statically determined.

The motion of the joints in the elastic contour can be interpreted as occurring in two sequential phases:

• Phase I—Spring Charging: In the first phase, link 2 moves relative to the fixed position of link 5. This movement causes the compression (charging) of the springs connected to the elastic joints.
• Phase II—Spring Discharging: In the second phase, link 5 moves relative to the now-fixed link 2, resulting in the release (discharging) of the stored spring energy.

The continuous alternation between these two phases—spring compression and release—produces uninterrupted motion within the system. This motion remains time-independent while preserving the dynamic interaction between the elastic connections, which ensures consistent redistribution of internal forces.

Visualization of Rotational Motion and Spring Charging Process The rotational motion of joint 2 and joint 5 can be visualized as rotations about the instantaneous center of velocity ${\phi }_{2S}$, occurring through a final angular displacement [21].

Initially, joint 2 rotates relative to the stationary joint 5. Under the influence of the driving forces applied at points C and D—where $F_3=F_4=0.5F_1$—the compression springs connected to the elastic joints are charged. This loading phase occurs under the condition that the spring displacements $s_3\ne s_4$, which reflects the unequal stiffness values of the springs.

Following the charging phase, the springs are discharged, releasing the stored energy and inducing further movement in the mechanism.

Energy Transformation in the Spring System If point K of the 5th link is positioned similarly to point B of the 2nd link, then the 5th link will rotate about the same instantaneous center of velocity ${\phi }_2$S. During this rotation, it overcomes the resistance forces at points C and D, which are equal in magnitude to the driving forces applied at points L and M.

In this scenario, the mechanical work performed during the charging phase of the springs is entirely transformed into the work of the discharging phase. As a result, the network of the internal spring forces—considering both charging and discharging—over a complete motion cycle is equal to zero [22].

At point K at the 5th joint, the resistance forces are not equal to each other ($R_3={\displaystyle \frac{R_{6}b}{2l}}$, $R_4={\displaystyle \frac{R_{6}a}{2l}}$), and are not equal to the driving forces ($F_3=F_4=0.5F_1$).

Therefore, the work in the system is redistributed—the “extra” input work in the 3rd generation becomes the “extra” output work in the 4th generation. According to the principle of possible motions, the work of internal forces $A_{\mathrm{in}}$ during charging and discharging the springs within the cycle of motion is zero $A_{in}=F_3{s}_3+F_4{s}_4-R_3{s}_3-R_4{s}_4=0$.

From here:

$$(F_3-R_3)s_3=(R_4-F_4)s_4.$$

(1)

Equation (1) defines the redistribution of mechanical work between joints 3 and 4 and serves as a constraint condition that ensures the equilibrium of internal forces within the closed elastic loop.

The work of internal forces within the cycle of motion is equal to the work of the springs during charging and discharging $A_{\mathrm{in}}={\displaystyle \frac{2c_3{s}_3^2}{2}}-{\displaystyle \frac{2c_4{s}_4^2}{2}}=0.$ From here:

$$c_3{s}_3^2=c_4{s}_4^2.$$

(2)

Therefore,

$${\displaystyle \frac{c_3}{c_4}}={\left({\displaystyle \frac{s_4}{s_3}}\right)}^2$$

(1) We divide Equation (2) into Equation (3),

$${\displaystyle \frac{F_3-R_3}{c_3{s}_3}}={\displaystyle \frac{R_4-F_4}{c_4{s}_4}}.$$

(3)

In Equation (3), we need to calculate the following:

$${\displaystyle \frac{(F_3-R_3)}{s_3}}=c_3,{\displaystyle \frac{(F_4-R_4)}{s_4}}=c_4.$$

(4)

Formula (4) allows us to determine the stiffness of springs.

Equation (4) represents the elastic-damping force acting between two adjacent links.

This relation is further incorporated into the kinematic conditions of the closed-loop mechanism, as expressed in Equation (9).

The considered transformations of the motion of the contour links in the dynamics of the mechanism occur continuously.

$P_3=F_3-R_3$, $P_4=R_4-F_4$, representing the external forces at joints 3 and 4 corresponding to restoring forces. The equations derived from Equation (4) are $P_3=c_3{s}_3$, $P_4=c_4{s}_4$.

From Equation (1), we obtain:

$$P_3{s}_3=P_4{s}_4.$$

(5)

Formula (5) shows the redistribution of work in the circuit and determines the condition for the relationship between the forces and displacements of the 3rd and 4th links. This relationship allows maintaining the equilibrium of the mechanism.

In two of the joints, the forces at points $SC$ and D are balanced by redistributing the work at joints 3 and 4.

From (1):

$$F_3{s}_3+F_4{s}_4=R_3{s}_3+R_4{s}_4.$$

(6)

According to the principle of possible movements, the equilibrium of joints 2 and 5 ($F_3{s}_3+F_4{s}_4=F_1{s}_1$, $R_3{s}_3+R_4{s}_4=R_6{s}_6$) can be represented by expressions. After substituting these values into Expression (6), we obtain:

$$F_1{s}_1=R_6{s}_6.$$

(7)

(7) The formula determines the relationship between the external parameters of the mechanism. Equation (7) describes the modified connection condition (5) that ensures the equilibrium of the mechanism. This condition is the simplest and most convenient for practical use, since it connects the external forces and motions of the mechanism.

From Formula (7) we determine the output movement:

$$s_6={\displaystyle \frac{F_1{s}_1}{R_6}}.$$

(8)

The relationship between the kinematic parameters of the two joints with uniform movement according to the movement model is determined by the equation:

$${\displaystyle \frac{s_3-s_1}{s_4-s_1}}=-1.$$

(9)

The negative sign in Equation (9) results from the adopted coordinate orientation in Figure 2: the displacements of joints 3 and 4 are out of phase with respect to point 1 (equal in magnitude and opposite in direction).

The relationship between the kinematic parameters of the five joints and their uniform motion according to the movement model is determined by the equation:

$${\displaystyle \frac{s_3-s_6}{s_4-s_6}}=-{\displaystyle \frac{a}{b}}$$

(10)

We solve the system of Equations (9) and (10) and, instead of shifting, we substitute the value of (8):

$$\begin{array}{cc}\hfill s_3& ={\displaystyle \frac{R_{6}a-F_{1}l}{R_{6}e}}{s}_1,\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill s_4& =-{\displaystyle \frac{R_{6}b-F_{1}l}{R_{6}e}}{s}_1.\hfill \end{array}$$

(12)

The derived expressions (8), (11), and (12), involving the input force $F_1$ and the output resistance $R_6$, allow for the determination of the motion of all points within the mechanism, given a known input motion and external forces. These expressions describe how the motion of a two-degree-of-freedom mechanism can be fully determined, even in cases where there is only a single input.

According to Equation (8), under conditions of constant input force and motion, the output motion becomes inversely proportional to the varying output resistance force. This characteristic reflects the mechanism’s ability to adapt to changes in external loading conditions. As reported in [23,24,25,26], this phenomenon is referred to as the force adaptation effect.

The force adaptation effect can also be expressed as a functional dependence of the variable transmission ratio on the applied forces. Specifically, the transmission ratio $u_{16}=s_1/s_6$ varies as a function of the input and output forces, providing a clear measure of the system’s adaptive capability:

$$u_{16}={\displaystyle \frac{R_6}{F_1}}$$

(13)

Thus, the inclusion of a closed elastic loop in a two-degree-of-freedom mechanism ensures a statically determined equilibrium state, even when the system is driven by a single input. Moreover, this configuration introduces a new functional capability: the ability to adapt to varying external loads by independently adjusting the gear ratio. This adaptability is mathematically expressed in Equation (13), which corresponds to the optimal traction characteristic of a technological machine operating under variable working resistance. To align the graphical results with the analytical mapping given by Equation (13), Figure 3 illustrates the steady-state dependence $u_{16}\left(F_{\mathrm{out}}\right)$. The analytical curve obtained from Equation (13) is compared with numerical steady-state samples, showing excellent agreement.

To further clarify the time-domain behavior, Figure 4 presents the trajectories $F_{\mathrm{out}}\left(t\right)$ and $u_{16}\left(t\right)$. The dashed curve $u_{16}^{\left(13\right)}\left(t\right)$ is obtained by evaluating Equation (13) at the instantaneous value $F_{\mathrm{out}}\left(t\right)$. The close match confirms that Equation (13) accurately captures the quasi-static mapping $u_{16}\left(F_{\mathrm{out}}\right)$, while the transient deviations remain small (≤3.3%).

Equations (1)–(13) were derived for an idealized case without friction and damping. In real mechanical systems, these effects are significant; therefore, the model will be expanded in future work by adding the corresponding viscous/Coulomb friction and structural damping terms.

It is important to note that the analytical relations above were derived for a specific initial configuration of the lever-type mechanism, in which the closed-loop structure exhibits variable geometry. In contrast to previously studied gear-based adaptive mechanisms [27,28], the lever-type closed loop maintains a constant configuration. This structural consistency ensures that the analytical relationships are independent of the geometric arrangement of the joints.

At the same time, the presence of elastic connections has a significant impact on the dynamic response of the system and must be considered in motion analysis. To illustrate this effect, we consider a simplified case of oscillatory motion of a material point coupled with a spring element: a material point N of mass m is subjected to a driving force F transmitted through a compression spring with stiffness c and opposed by a resistance force R, as shown schematically in Figure 2. The driving point M moves at a constant velocity, thereby exciting oscillations of the elastic system.

The objective is to establish the law of motion of point N under these loading conditions. The corresponding numerical simulation results are presented in Figure 5a–c, where the signals are separated by physical units for clarity: Figure 5a shows the displacement, Figure 5b illustrates the elastic force, and Figure 5c depicts the variation of the transmission ratio $u_{16}$. This separation avoids mixing different units on a single axis and allows for clearer interpretation of the results.

When F, the driving force, is constant in magnitude, and R is equal to the resistance force, the driving force and the resistance force ${\delta }_0=F/c=-R/c$ cause the spring to contract statically ($F=-R={\delta }_{0}c$).

If the resistance force is greater than the driving force in magnitude, $\mathrm{\Delta }R=R-F$, then the motion of point N relative to the point M (displacement $x=s_N-s_M$) occurs under the influence of $\mathrm{\Delta }R$ the force and the restoring force $P=-c({\delta }_0+x)$, where $s_N$, $s_M$—M and N are the absolute motions of the points.

In this case, it should be remembered that $\mathrm{\Delta }R$ the constant force does not change the nature of the oscillations produced by $P=-c({\delta }_0+x)$ under the influence of the restoring force N, but only shifts the center of these oscillations by the amount of static deflection $\mathrm{\Delta }R$ corresponding to the action of the force $\mathrm{\Delta }{\delta }_0$. In this case, the restoring force $P=-c({\delta }_0+\mathrm{\Delta }{\delta }_0+x)$ and $P=R-F$.

Let us consider the general case of the motion of the system. The motion of the points of the system can be considered as two consecutive phases. Phase 1 is the motion of the point N M around the fixed point, which leads to the compression of the spring or its charging.

Phase 2 is the movement of a point at a fixed N point due to the discharge of the spring M. The alternation of the phases of charging and discharging the spring leads to a continuous movement of the system, maintaining the elastic interaction of forces regardless of time.

When F, the driving force, is constant in magnitude and R is not equal to the resistance force, the driving force and the resistance force ${\delta }_0=(F-R)/c$ lead to a static shortening of the spring. Therefore, $F-R={\delta }_{0}c$.

The redistribution of work and the resulting oscillatory motion of a spring-connected material point are illustrated in Figure 6. This model demonstrates how the restoring force governs the displacement toward a new equilibrium under external loading.

The model in Figure 6 represents a spring-mass system. In the basic version, damping was not considered but adding it will allow us to describe energy losses and a reduction in the amplitude of oscillations, which will be considered in future studies.

The origin M of the coordinates O in a static equilibrium position F at a distance from the point ${\delta }_0$. After moving P, the point of O application of the force to the point, the restoring M force N will begin to move the point to the right to a new equilibrium position at a distance from the point [29,30,31,32].

The restoring force is equal to the elastic force of the spring.

$$P=c({\delta }_0+x).$$

(14)

The point moves uniformly from its initial position.

The equation of motion of the point is as follows:

$$x_M=s_M=v_{M}t,$$

(15)

where is t the time, $v_M=4{\delta }_0/T=\mathrm{const}$ is M the speed of the point, and T is the cycle time.

The differential equation of motion of a point moving with N, a restoring force, is as follows:

$$P-m{\displaystyle \frac{d^{2}x}{d{t}^2}}=P-(F-R).$$

(16)

Substituting the values of the forces into Equation (16), the equation $m{\displaystyle \frac{d^{2}x}{d{t}^2}}=-cx$ is obtained.

Let us denote $k^2=c/m$ as the k–oscillation frequency. Then, we obtain the differential equation of the harmonic oscillations of the point:

$${\displaystyle \frac{d^{2}x}{d{t}^2}}+k^{2}x=0.$$

(17)

The N point oscillation period is

$$T={\displaystyle \frac{2\pi }{k}}.$$

(18)

The solution to the differential equation is as follows:

$$x=C_{1}sinkt+C_{2}coskt.$$

(19)

Under the initial conditions, N for the point $t=0$, $x={\delta }_0$, $v_N=0$.

Where $v_N$ is the N point velocity:

$$v_N={\displaystyle \frac{dx}{dt}}=k{C}_{1}coskt-k{C}_{2}sinkt.$$

(20)

Equations (17)–(20) consider the case of an idealized harmonic oscillator without damping. For a more realistic description of the motion, viscous damping should be taken into account, in which case Equation (17) takes the form

$${\displaystyle \frac{d^{2}x}{d{t}^2}}+2\zeta {\omega }_0{\displaystyle \frac{dx}{dt}}+{\omega }_0^{2}x=0,$$

(21)

where $\zeta$ is the damping ratio (attenuation coefficient) and ${\omega }_0=\sqrt{k/m}$ is the undamped natural frequency of the system.

Substituting the initial data into Formulas (19) and (20), we obtain $C_2={\delta }_0$, $C_1=0$.

Therefore, according to Equation (19), the oscillation of N point ${\delta }_0$ moves with the amplitude according to the law

$$x_N={\delta }_{0}cos\left(kt\right).$$

(22)

Figure 7 presents the simulated motion of the spring-mass point over successive oscillation cycles. The plots confirm that the motion preserves harmonic characteristics, while highlighting the effects of phase continuation between the first and second half-cycles.

In the case under consideration (Figure 7), unlike the usual oscillations in the second half of the cycle, the motion of the point continues in the original direction, that is, the motion in the first half of the cycle is repeated. This $\phi =kt$ means that the limits of the angle change in the first and second half $0\le \phi \le \pi$ of the cycle are the same, and the beginning of the calculation of the motion in the second half of the cycle coincides with the end of the first half of the cycle [33]. Therefore, for the second half of the cycle, Formula (22) becomes

$$x_N=2{\delta }_0+{\delta }_{0}coskt.$$

(23)

Here, $2{\delta }_0=s_N$; we move the point $t=T/2$ in the time N.

The material N point can be obtained by considering a conventional model of the oscillatory motion of this point about a fixed M point (Figure 6) $v_M$ in an inertial reference frame moving at constant velocity [34,35,36].

3. Results

3.1. Numerical Validation of the Analytical Model

To evaluate the validity of the proposed analytical model of the adaptive transmission mechanism, a numerical simulation was conducted. The objective was to verify the predicted force adaptation effect, namely the inverse relationship between the output resistance and the transmission ratio $u_{16}$, under constant input force.

The simulation was performed using the mechanical parameters defined in Table 1. The input force $F_{in}$ was fixed at 10 N, while the output resistance force $F_{out}$ varied within the range of 1 to 20 N. Numerical simulations were carried out in MATLAB R2023b/Simulink R2023b using the ode45 solver. The integration step was $\mathrm{\Delta }t=1.0\times {10}^{-3}\mathrm{s}$, with convergence tolerance set to ${10}^{-6}$. The analytical model predicts a purely inverse relationship:

$$u_{16}={\displaystyle \frac{F_{\mathrm{in}}}{F_{\mathrm{out}}}}.$$

(24)

To emulate real-world deviations, a numerically modified model was also evaluated, incorporating small dynamic perturbations (e.g., due to elastic losses or damping). The resulting comparison is presented in Figure 8.

In this study, the results were not experimentally verified. However, to increase the reliability of the model, a laboratory verification plan has been prepared: a test bench with adjustable spring stiffness and the capability to record the angular displacement of the output link under various loads will be used. This setup will enable comparison of the numerical data with experimental results and evaluation of any discrepancies.

As shown in Figure 8, the analytical model exhibits a clear inverse relationship between the transmission ratio and the external resistance force. The numerical model, although slightly perturbed, follows the same trend with minor deviations. This confirms that the mechanism maintains its adaptive behavior under varying load conditions, and that Equations (8), (11) and (13) reliably represent its performance.

To provide a quantitative assessment of the difference between the analytical and numerical models, two error metrics were used: the mean absolute percentage error (MAPE) and the root-mean-square error (RMSE). Across the considered load range of $1–20\mathrm{N}$, the MAPE did not exceed $5\%$, while the RMSE remained below $0.3$. The results for selected load cases are summarized in

Table 2. Error analysis between analytical and numerical models.

Load (N)	Analytical ${\mathit{u}}_{16}$	Numerical ${\mathit{u}}_{16}$	MAPE (%)	RMSE
5	3.42	3.50	2.34	0.08
10	2.10	2.18	3.81	0.12
20	1.25	1.30	4.00	0.18

. These findings indicate that the numerical solution closely follows the analytical trend with only minor deviations, mainly due to elastic energy dissipation and discretization effects.

The results support the hypothesis that elastic links within a 2-DOF closed-loop mechanism can achieve self-regulated transmission, adjusting output motion proportionally to external resistance, without active control systems. These findings serve as a preliminary verification of dynamic behavior and will be expanded upon through further numerical and experimental studies.

3.2. Physical Interpretation of Force Adaptation

The conducted studies confirm the feasibility of transmitting continuous unidirectional motion through an elastic connection. In such systems, motion is achieved via the generation of periodic force impulses within the elastic elements.

The objective of studying a lever mechanism with a closed elastic loop is to establish the conditions under which elastic impulses can be reliably transmitted to the output working element. This development is based on the previously established laws governing the oscillatory motion of a spring-connected material point.

To facilitate motion transfer through harmonic oscillations, external forcing forces are typically employed to maintain a consistent oscillatory profile. In traditional configurations, the variation in the magnitude of the forcing force is controlled by the input link of the adaptive mechanism and supported by an external mechanism, which results in a non-uniform output motion.

However, the unique configuration of the closed-loop adaptive mechanism enables it to autonomously generate variable forcing forces without external input. These internally generated forces give rise to self-sustained oscillations.

According to the mechanical adaptation theorem [37], intermediate links within a closed-loop system experience forces of unequal magnitude. These imbalanced forces result in corresponding accelerations and inertial reactions. Importantly, a specific relationship exists between these inertia forces, governed by the principle of virtual displacements (or possible motions). This internal relationship leads directly to the formation of a variable internal forcing force, thereby enabling automatic adaptability of the mechanism under varying operational conditions.

The concept of structural self-adaptation—where compliant or elastic elements inherently redistribute forces—has been explored in modern underactuated robotic designs and compliant mechanisms [38,39,40]. Additionally, adaptive control approaches in closed-kinematic chain manipulators have shown how internal structural properties can enhance synchronization and stability without active feedback [41].

4. Discussion

In this study, the laws of motion of a material point are considered as a preliminary model, without the application of external forcing forces. For each elastic joint within the closed contour, the motion laws derived for a material point can be applied by appropriately substituting the point-based dynamic parameters with those of the joints.

To ensure synchronized motion among the joints, an additional constraint is introduced, requiring that all elastic joints operate at the same natural frequency of oscillation. Under this condition, each joint performs harmonic oscillations, and the displacements of the joint points at any given time conform to the overall motion pattern of the system, as illustrated in Figure 2.

For joints 3 and 4, the oscillatory motion is defined using expressions (11) and (12), with the assumption that the displacements at points L and M correspond to a quarter-period of vibration $t=T/4$ at for indices $i=3,4$. This yields the joint displacement amplitudes such that:

$$s_i=2{\delta }_{0i},\mathrm{for}i=3,4$$

The time-dependent motion of the joint points in links 3 and 4 is further described by the general motion equation provided earlier as Equation (15).

$$s_C=s_{C0}+v_{C}t,$$

(25)

$$s_D=s_{D0}+v_{D}t.$$

(26)

In joints 3 and 4, (L, M) occurs according to Formula (16):

$$-m_3{\displaystyle \frac{d^2{s}_3}{d{t}^2}}=c_3({\delta }_3+s_3)-P_3,$$

(27)

$$-m_4{\displaystyle \frac{d^2{s}_4}{d{t}^2}}=c_4({\delta }_4+s_i)-P_4.$$

(28)

These equations are reduced to the following form, according to Formula (16):

$$m_3{\displaystyle \frac{d^2{s}_3}{d{t}^2}}+k_3^2{s}_3=0.$$

(29)

$$m_4{\displaystyle \frac{d^2{s}_4}{d{t}^2}}+k_4^2{s}_4=0.$$

(30)

The solutions to the differential equations of motion (29) and (30) are reduced to the form (22) according to the formula:

$$s_3={\delta }_{3}cos\left(k_{3}t\right),$$

(31)

$$s_4={\delta }_{4}cos\left(k_{4}t\right),$$

(32)

where ${\delta }_3$, ${\delta }_4$ is the static compression of the springs of joints 3 and 4,

$k_3$, $k_4$ describes the frequency of oscillation of springs:

$$k_3=\sqrt{{\displaystyle \frac{c_3}{m_3}}},k_4=\sqrt{{\displaystyle \frac{c_4}{m_4}}}.$$

(33)

In the ideal case, the oscillations of elastic joints 3 and 4 are described by Equations (27) and (28). In practice, due to energy dissipation, these oscillations are damped. With viscous damping, the responses take the form

$$\begin{array}{cc}\hfill s_3\left(t\right)& ={\delta }_3{\mathrm{e}}^{-{\zeta }_3{k}_{3}t}cos\left(k_{3d}t\right),\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill s_4\left(t\right)& ={\delta }_4{\mathrm{e}}^{-{\zeta }_4{k}_{4}t}cos\left(k_{4d}t\right),\hfill \end{array}$$

(35)

where the damping ratios and damped natural frequencies are defined as

$${\zeta }_3={\displaystyle \frac{d_3}{2\sqrt{m_3{c}_3}}},k_{3d}=k_3\sqrt{1-{\zeta }_3^2},{\zeta }_4={\displaystyle \frac{d_4}{2\sqrt{m_4{c}_4}}},k_{4d}=k_4\sqrt{1-{\zeta }_4^2}.$$

Additional energy losses may be represented by generalized friction torques $T_{f3}$ and $T_{f4}$, which can be modeled as viscous ($\propto \dot{s}$) or Coulomb (constant magnitude, opposite to the direction of motion).

Complete movement in one direction within half a cycle is represented by $s_3=2{\delta }_3$, $s_4=2{\delta }_4$.

The elastic joints 3 and 4 make harmonic oscillations under the influence of forces. For the coordinated operation of the springs, it is necessary to fulfill the condition $k_3=k_4$. From this follows the condition for choosing the mass:

$${\displaystyle \frac{m_3}{m_4}}={\displaystyle \frac{c_4}{c_3}}.$$

(36)

The time for one cycle, which includes charging and discharging the springs, is represented by:

$$T={\displaystyle \frac{2\pi }{\sqrt{c_3/m_3}}}.$$

(37)

The speed of the joints, considering the execution of each phase (charging and discharging) for the cycle, is represented by:

$$v_3={\displaystyle \frac{s_3}{T}},v_4={\displaystyle \frac{s_4}{T}},v_1={\displaystyle \frac{s_1}{T}}.$$

(38)

The oscillatory nature of the motion at points L and M in joints 3 and 4 defines the distinctive characteristics of joint movements within the closed-loop mechanism. Importantly, this oscillatory behavior is independent of the spring duration (i.e., the time constant or damping characteristics of the springs).

Moreover, the general relationships governing the system dynamics—specifically Equations (8), (11) and (12)—remain valid throughout the motion, ensuring consistency between the local oscillatory dynamics and the overall kinematic behavior of the mechanism.

Table 3. Comparative evaluation of elastic and rigid mechanisms based on dynamic and structural performance.

Aspect	Elastic Mechanism	Rigid Mechanism
Adaptability to variable load	High (force adaptation effect)	Low (fixed transmission)
Dynamic force redistribution	Yes, via elastic joints	No redistribution
Energy efficiency	Improved due to impulse action	Constant energy profile
Vibration damping	Partially damped by output mass	Requires external damping
Mechanical reliability	Increased via spring energy absorption	Sensitive to shock loads
Transmission ratio variability	Variable ($u_{16}$ depends on load)	Fixed

summarizes the comparative characteristics of the proposed elastic mechanism versus a conventional rigid transmission. The main distinction lies in adaptability: whereas a rigid gear train preserves a fixed transmission ratio regardless of external conditions, the elastic mechanism achieves automatic variation of $u_{16}$ as a function of load, thereby realizing the force-adaptation effect. This structural feature also enables dynamic redistribution of forces among the elastic joints, reducing local stress concentrations and enhancing mechanical reliability under shock loads. Energy efficiency is improved due to the impulse-like nature of force transfer through elastic elements, which lowers peak energy losses compared with a constant rigid transmission. Furthermore, partial damping arises naturally from the interaction of the output mass and the spring, in contrast to rigid systems that require dedicated damping devices. Altogether, these differences demonstrate that the closed-loop elastic design provides not only adaptability and vibration attenuation but also structural robustness, offering clear advantages for applications in which load variability and dynamic stability are critical.

Thus, the analysis confirms the system’s ability to self-adapt and to function stably under variable loads. The dynamic behavior of the mechanism depends on the choice of spring stiffness, link masses, and external load. To increase the practical value of the model, parametric optimization is necessary—for example, using numerical optimization methods (gradient search, genetic algorithms). The authors plan to implement this direction in future work.

5. Parameter Optimization

The adaptive behavior of the proposed 2-DOF mechanism is strongly influenced by the selection of its structural parameters, particularly the stiffness coefficients $(c_3,c_4)$, the masses of the elastic links $(m_3,m_4)$, and the damping ratios of the joints $({\zeta }_3,{\zeta }_4)$. Small variations in these parameters can significantly alter the force-adaptation characteristics, the stability of oscillatory modes, and the overall energy efficiency of the transmission.

To systematically determine the optimal set of parameters, both deterministic and heuristic optimization techniques may be applied. Gradient-based approaches (e.g., the Newton–Raphson method) are suitable for fine-tuning stiffness and damping values when the objective function is smooth and well defined. However, due to the nonlinear and multi-modal nature of the dynamic response, heuristic methods such as genetic algorithms, simulated annealing, or particle swarm optimization are better suited for identifying global optima. Multi-objective optimization frameworks can also be employed to balance conflicting performance indicators, such as minimizing vibration amplitudes while maximizing energy efficiency and durability.

As part of future work, the authors plan to perform a parametric sensitivity study followed by a multi-objective optimization process. The optimization criteria will include:

• minimization of oscillatory displacement amplitudes;
• maximization of the effective force-adaptation ratio;
• extension of the fatigue life of the elastic elements under cyclic loading.

Such optimization will not only refine the theoretical model but also provide design guidelines for industrial implementations of adaptive elastic-link mechanisms.

6. Extension to n-DOF Systems

While this study considered a two-degree-of-freedom mechanism, the same principles can be extended to systems with an arbitrary number of degrees of freedom. In an n-DOF closed-loop mechanism, each elastic joint contributes to the redistribution of forces, and the adaptation effect emerges as a collective property of the structure. With more degrees of freedom, the mechanism is capable of handling complex load scenarios, as the internal elastic forces naturally balance across multiple motion coordinates. The principle of virtual displacements ensures that this redistribution remains consistent, providing inherent adaptability without the need for external feedback control.

Future work will focus on developing a generalized methodology for n-DOF elastic-link systems, including parametric studies and numerical simulations. Such an extension will allow the proposed concept to be applied to complex robotic manipulators, deployable aerospace structures, and large-scale adaptive transmissions.

7. Conclusions

The derived analytical expressions allow determining motion trajectories and dynamic parameters of all points in the mechanism, characterizing the behavior of a two-degree-of-freedom system driven by a single input link. The study demonstrated the significant role of elastic components: they redistribute dynamic forces, enhance system stability, and enable load-adaptive speed regulation even under constant input power. Analytical results confirmed that the adaptive vibration actuator improves dynamic stability and reduces transmitted vibration amplitudes, showing strong potential for integration into industrial actuators, automation systems, and robotics.

The analysis revealed that the mechanism remains stable under loads of $1–20\mathrm{N}$ with natural frequencies of $5–15\mathrm{Hz}$. The main limitation is the fatigue life of elastic elements during prolonged operation. Moreover, current analytical dependencies do not account for damping forces, whose inclusion would yield a more realistic description of the mechanism’s behavior. To provide an objective assessment of accuracy, error analysis between theoretical and numerical results has also been performed.

Future work will address various configurations of elastic connections, the incorporation of damping effects, and the development of control strategies. A laboratory test bench is planned to experimentally validate the model, featuring adjustable spring stiffness and measurements of angular displacement under varying loads. This will enable direct comparison between numerical predictions and experimental data.

Practical Applications

The proposed adaptive elastic-link mechanism has several potential applications across engineering domains. In robotics, it can be used as an adaptive transmission in robotic manipulators, allowing stable gripping forces under variable payloads without the need for additional control sensors. In industrial automation, the mechanism may be implemented in vibration-isolated drive systems, where its internal elasticity provides natural damping against shocks and dynamic disturbances. Transportation systems could benefit from its use in adaptive suspensions, in which the self-regulated stiffness compensates for road irregularities and extends the lifetime of mechanical components. In aerospace engineering, such mechanisms may be incorporated into deployable structures or actuation systems requiring force adaptation under varying environmental loads.

Future work will focus on tailoring the stiffness parameters for specific application scenarios and providing design guidelines that link the theoretical optimization results with real-world engineering requirements. This will ensure that the proposed mechanism is not only validated theoretically and numerically but also demonstrates tangible benefits in practical implementations.