D’Alembert–Lagrange Principle in Symmetry of Advanced Dynamics of Systems

Abstract

The D’Alembert–Lagrange principle is a fundamental concept in analytical mechanics that simplifies the analysis of multi-degree-of-freedom mechanical systems, facilitates the dynamic response prediction of structures under various loads, and enhances the control algorithms in robotics. It is essential for solving complex problems in engineering and robotics. This theoretical study aims to highlight the advantages of using acceleration energy to obtain the differential equations of motion and the generalized driving forces, compared to the classical approach based on the Lagrange equations of the second kind. It was considered a mechanical structure with two degrees of freedom (DOF), namely, a planar robot consisting of two homogeneous rods connected by rotational joints. Both the classical Lagrange approach and the acceleration energy model were applied. It was noticed that while both approaches yielded the same results, using acceleration energy requires only a single differentiation operation, whereas the classical approach involves three such operations to achieve the same results. Thus, applying the acceleration energy method involves fewer mathematical steps and simplifies the calculations. This demonstrates the efficiency and effectiveness of using acceleration energy in dynamic system analysis. By incorporating acceleration energy into the model, enhanced robustness and accuracy in predicting system behavior are achieved.

1. Introduction

The D’Alembert–Lagrange principle, often referred to simply as the Lagrange principle or sometimes the Lagrangean principle, is a fundamental concept in classical mechanics. It is named after two influential figures in mathematics and physics: Jean le Rond d’Alembert and Joseph-Louis Lagrange. In 1743, D’Alembert introduced the principle of virtual work, laying the groundwork for what would later become known as the D’Alembert–Lagrange principle. This principle states that the sum of the work done by the inertia forces and the external forces acting on a system in equilibrium is zero.

Lagrange, an Italian mathematician and astronomer, further developed the ideas introduced by D’Alembert. In his work Analytical Mechanics, published in 1788, Lagrange formulated the principle in a more general and systematic manner. He introduced the concept of generalized coordinates demonstrating how the principle could be used to derive the equations of motion for mechanical systems. Lagrange’s formulation provided a powerful and elegant method for solving problems in classical mechanics, bringing together the analysis of various physical systems into a unified mathematical model.

Over the centuries, the D’Alembert–Lagrange principle has become a cornerstone of classical mechanics, playing a central role in the analysis and understanding of dynamic systems. It has found applications in various branches of physics and engineering, including celestial mechanics, fluid dynamics, and robotics. The principle continues to be studied and applied by researchers and engineers around the world, contributing to advancements in science and technology. As research in dynamic systems advances, an in-depth investigation of the D’Alembert–Lagrange principle becomes a necessity in understanding its true significance. The essence of the D’Alembert–Lagrange principle lies in the assertion that for any infinitesimal movement of a system in equilibrium, the combined work done by external forces and inertia forces is null. This fundamental concept serves as the foundation for deriving equations of motion for a broad spectrum of mechanical systems, ranging from simple pendulums to sophisticated robotic mechanisms. The historical evolution of research on the D’Alembert–Lagrange principle is marked by contributions from notable figures such as Lagrange, Hamilton, and Euler, whose works represent the foundation of many modern interpretations of this principle.

The Least Action Principle and Lagrange’s Equations, authored by Jean Vanier and Corina Tomescu (Mandache), provides a comprehensive and in-depth examination of key principles in theoretical physics and classical mechanics.

The book’s significant contributions include a rigorous mathematical exposition of the least action principle and Lagrange’s equations, ensuring readers gain a thorough and precise understanding of these fundamental concepts. This detailed treatment is particularly valuable for advanced students and researchers aiming to deepen their knowledge in these areas.

Furthermore, the authors extend the application of these principles beyond classical mechanics, exploring their relevance in electromagnetism, fluid dynamics, and relativity. This broad scope underscores the versatility and applicability of the least action principle and Lagrange’s equations across various domains of physics [1].

Notable references such as Goldstein’s Classical Mechanics have served as indispensable resources for researchers in managing the complex issues that may arise in the analysis of dynamic systems [2].

Goldstein’s Classical Mechanics is a reference book that offers an in-depth study of classical mechanics, including the D’Alembert–Lagrange principle. It covers the theoretical foundations of the principle, its mathematical formulation, and its applications in analyzing various mechanical systems. The book offers detailed explanations, examples, and exercises to aid readers in understanding and applying the principle to find solutions to different problems from classical mechanics.

However, despite its established significance, the D’Alembert–Lagrange principle remains subject to ongoing scrutiny and debate within the scientific community. Recent studies by Wang et al. and Kormushev et al. have challenged conventional interpretations, proposing novel methodologies for applying the principle in analyzing complex systems [3,4]. These pioneering studies highlight the dynamic nature of this field and the necessity for continuous exploration.

Wang et al.’s study focuses on the application of the D’Alembert–Lagrange principle to the dynamic analysis of smart structures. The book offers a novel approach, based on D’Alembert–Lagrange principle, that can be used for analyzing the dynamic behavior of structures equipped with adaptive materials. The study includes theoretical formulations, numerical simulations, and experimental validations to demonstrate the effectiveness of the approach in analyzing and optimizing smart structures.

Kormushev et al.’s paper explores the application of reinforcement learning techniques in robotics, with a focus on the D’Alembert–Lagrange principle for modeling robotic dynamics. They discuss how reinforcement learning algorithms can be used to learn control policies, enabling robots to adapt to uncertain and dynamic environments. The paper highlights the importance of accurate modeling using the D’Alembert–Lagrange principle for effective robotic control.

Classical Dynamics of Particles and Systems by Jerry B. Marion and Stephen T. Thornton is a comprehensive textbook that provides a thorough introduction to the principles and methods of classical mechanics. The fifth edition covers a wide range of topics including Newtonian mechanics, Lagrangean and Hamiltonian formulations, and the dynamics of rigid bodies. It also explores advanced topics such as chaos theory and nonlinear dynamics. It emphasizes problem-solving techniques and includes numerous examples and exercises to illustrate the theoretical concepts. This text is essential for students and researchers seeking a deep understanding of the mechanics governing particle and system dynamics [5].

The references [1,2,3,4,5,6,7,8,9] provide valuable insights into the theoretical foundations, practical applications, and actual research trends related to the D’Alembert–Lagrange principle in various fields of science and engineering.

This article attempts to contribute to the understanding of the D’Alembert–Lagrange principle by conducting a thorough examination of theoretical foundations and its applications in engineering and robotics. By reviewing existing research [8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23], this paper aims to elucidate both the strengths and limitations of this principle.

Through rigorous theoretical analysis, numerical simulations, and empirical validation, the present paper aims to deepen the understanding of complex dynamic systems and facilitate future advancements.

2. Materials and Methods

A material point, subject to a scleronomic and holonomic constraint with negligible friction, according to [14], is associated to a virtual displacement compatible with a constraint. According to the literature [14,15,16,17,18,19,20,21,22,23], virtual work represents the scalar product between force and virtual displacement.

Starting from D’Alembert’s principle and the definition of virtual displacement (instantaneous and infinitesimal displacement), a differential principle will be demonstrated below, in which virtual work, i.e., the scalar product between force and virtual displacement, will play an essential role. The study of this principle will be conducted for the case of scleronomic, and holonomic constraints applied to a material system.

2.1. D’Alembert–Lagrange Principle Applied in Case of a Material Point

First, the case of a material point is analyzed. Thus, in Figure 1, the material point symbolized by $M$ is considered, having mass $\left(m\right)$ and subjected to a system of external and concurrent forces ${\overline{F}}_i\hspace{0.17em}(i=1\to n)$, mechanically equivalent to a single resultant $\overline{R}$.

According to the principle of force action [1], the material point executes a uniformly varied mechanical motion, characterized at the instant $t$ by the following distribution:

$$\left\{\begin{array}{c}M;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}m;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\overline{r}=\overline{r}\left(t\right)={\left[\begin{array}{ccc}x\left(t\right)& y\left(t\right)& z\left(t\right)\end{array}\right]}^T;\\ \overline{R}=\hspace{0.17em}\hspace{0.17em}{\displaystyle \sum _{i=1}^n{\overline{F}}_i};\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\overline{R}}_L=\overline{N}+\overline{T}=\overline{N}-\mu ·N·\overline{\tau };\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\\ \overline{v}\left(t\right)=\dot{\overline{r}}\left(t\right);\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\overline{a}=\ddot{\overline{r}}=\dot{\overline{v}};\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\overline{h}\left(t\right)=m·\overline{v}\left(t\right);\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\dot{\overline{h}}=m·\overline{a};\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\\ \hspace{0.17em}{\overline{k}}_0\left(t\right)=\overline{r}\left(t\right)\times m·\overline{v}\left(t\right);\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\dot{\overline{k}}}_0=\overline{r}\times m·\overline{a}\hspace{0.17em}.\hspace{0.17em}\hspace{0.17em}\end{array}\right\}.$$

(1)

According to [14], D’Alembert’s principle is defined by the vector and differential equation:

$$\overline{R}+{\overline{F}}_j+{\overline{R}}_L=0,$$

(2)

$$\hspace{0.17em}{\overline{F}}_j=-m·\overline{a},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\Rightarrow \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\overline{R}-m·\overline{a}+{\overline{R}}_L=0,$$

(3)

$$\overline{R}-m·\overline{a}={\overline{R}}_p,$$

(4)

where ${\overline{R}}_p$ represents the lost force, ${\overline{F}}_j$ the inertia force, $\overline{R}$ is the resultant of external forces, and ${\overline{R}}_L$ the resultant of link forces.

Thus, the D’Alembert Equation (2) is scaled by the virtual displacement, resulting in the following expressions:

$$\left(\overline{R}-m·\overline{a}+{\overline{R}}_L=0\right)·\left|\delta \overline{r}\right.,$$

(5)

$$\delta L=\overline{R}·\delta \overline{r}-m·\overline{a}·\delta \overline{r}+{\overline{R}}_L·\delta \overline{r}=0.$$

(6)

In Equations (5) and (6), $\overline{r}$ represents the position vector, defined as follows:

$$\overline{r}\equiv \overline{OM}={\left(\begin{array}{ccc}x& y& z\end{array}\right)}^T\equiv {\left(\begin{array}{ccc}{q}_1& q_2& q_3\end{array}\right)}^T,$$

(7)

$$\mathrm{If} \mathrm{the} \mathrm{material} \mathrm{point} \mathrm{is} \mathrm{in} \mathrm{motion}: \overline{r}\left(t\right)=\overline{r}\left[q_j\left(t\right);\hspace{0.17em}\hspace{0.17em}j=1\to 3;\hspace{0.17em}\hspace{0.17em}t\right].$$

(8)

where $\left(q_j=q_j\left(t\right);\hspace{0.17em}\hspace{0.17em}j=1\to 3\right)$ is a curvilinear coordinate (in analytical mechanics it becomes a generalized coordinate), and the position vector becomes an explicit function of time parameter. By applying the real and virtual differentiation operators, $\left(d\right)$ and $\left(\delta \right)$, respectively, to vector Equation (8), the differential vector of real displacements and the differential vector of virtual displacements are obtained:

$$d\overline{r}={\displaystyle \frac{𝜕\overline{r}}{𝜕q_1\hspace{0.17em}}}·d{q}_1+{\displaystyle \frac{𝜕\overline{r}}{𝜕q_2\hspace{0.17em}}}·d{q}_2+{\displaystyle \frac{𝜕\overline{r}}{𝜕q_3\hspace{0.17em}}}·d{q}_3+{\displaystyle \frac{𝜕\overline{r}}{𝜕t\hspace{0.17em}}}·dt={\displaystyle \sum _{j=1}^3\frac{𝜕\overline{r}}{𝜕q_j\hspace{0.17em}}·d{q}_j}+{\displaystyle \frac{𝜕\overline{r}}{𝜕t\hspace{0.17em}}}·dt,$$

(9)

$$\delta \overline{r}={\displaystyle \frac{𝜕\overline{r}}{𝜕q_1\hspace{0.17em}}}·\delta q_1+{\displaystyle \frac{𝜕\overline{r}}{𝜕q_2\hspace{0.17em}}}·\delta q_2+{\displaystyle \frac{𝜕\overline{r}}{𝜕q_3\hspace{0.17em}}}·\delta q_3={\displaystyle \sum _{j=1}^3\frac{𝜕\overline{r}}{𝜕q_j\hspace{0.17em}}·\delta q_j},\hspace{0.17em}\hspace{0.17em}whence\hspace{0.17em}\hspace{0.17em}\Rightarrow \delta \overline{r}\ne d\overline{r}.$$

(10)

According to (9) and (10), it is noticed that in the case of scleronomic constraints (fixed and nondeformable), the virtual differentiation operator does not act on the time parameter, thus the following relationship exists: $\delta \overline{r}\ne d\overline{r}$.

Since the connection is characterized by negligible friction, the following differential expressions are derived from (6):

$${\overline{R}}_L\perp \delta \overline{r},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}whence\hspace{0.17em}\hspace{0.17em}\Rightarrow \delta L_L={\overline{R}}_L·\delta \overline{r}=0,$$

(11)

$$\delta L=\left(\overline{R}-m·\overline{a}\right)·\delta \overline{r}=0.$$

(12)

Therefore, in accordance with (12), the virtual mechanical work of the lost force for a virtual displacement, which is compatible with the physical constraint, is equal to zero. By substituting the expression for the virtual displacement vector in (12), it results in the following:

$$\delta L=\left(\overline{R}-m·\overline{a}\right)·{\displaystyle \sum _{j=1}^3\frac{𝜕\overline{r}}{𝜕q_j\hspace{0.17em}}·\delta q_j}=0,\hspace{0.17em}\hspace{0.17em}from\hspace{0.17em}\hspace{0.17em}where\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\Rightarrow \delta L={\displaystyle \sum _{j=1}^3\left[\left(\overline{R}-m·\overline{a}\right)·\frac{𝜕\overline{r}}{𝜕q_j\hspace{0.17em}}\right]·\delta q_j}=0.$$

(13)

Equation (12) or (13) represents the D’Alembert–Lagrange principle for the material point. If the material point is in absolute static equilibrium, then Equation (13) is particularized as follows:

$$\overline{a}=0\Rightarrow \delta L=\overline{R}·\delta \overline{r}={\displaystyle \sum _{j=1}^3\left(\overline{R}·\frac{𝜕\overline{r}}{𝜕q_j\hspace{0.17em}}\right)·\delta q_j}=0.$$

(14)

Equation (14) is called the principle of virtual work, used to determine the independent parameters that express the equilibrium position of the material point. In (14), a force called the generalized force is introduced:

$$Q_j=\overline{R}·{\displaystyle \frac{𝜕\overline{r}}{𝜕q_j\hspace{0.17em}}}.$$

(15)

The term “generalized force” for the symbol $Q_j$ (see Equation (15)) is obtained by analogy with the generalized coordinate for $q_j$. Below, it will be demonstrated that the generalized force transforms into force or moment of force, depending on the type of generalized coordinate, which can be linear or angular. The principle of virtual work is rewritten using the generalized force as follows:

$$\delta L={\displaystyle \sum _{j=1}^3{Q}_j·\delta q_j}=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\delta L=Q_1·\delta q_1+Q_2·\delta q_2+Q_3·\delta q_3=0.$$

(16)

In the case of holonomic constraints, the number and type of independent parameters are the same in finite and infinitesimal displacements. Therefore, applying the principle of superposition effects in (16), the following results:

$$\left\{\begin{array}{l}{q}_1\ne 0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\delta q_1\ne 0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{q}_2=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\delta q_2=0\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}{q}_3=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\delta q_3=0\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\Rightarrow Q_1=0\\ q_1=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\delta q_1=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{q}_2\ne 0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\delta q_2\ne 0\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}{q}_3=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\delta q_3=0\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\Rightarrow Q_2=0\\ q_1=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\delta q_1=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{q}_2=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\delta q_2=0\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}{q}_3\ne 0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\delta q_3\ne 0\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\Rightarrow Q_3=0\end{array}\right.,$$

(17)

$$\mathrm{from} \mathrm{which} Q_j=\overline{R}·{\displaystyle \frac{𝜕\overline{r}}{𝜕q_j\hspace{0.17em}}}=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}for\hspace{0.17em}\hspace{0.17em}j=1\to 3.$$

(18)

Therefore, the virtual work principle (14)–(16) is equivalent to three scalar equations of the form of Equation (17), the significance of which will be seen later. For this purpose, and by considering (7) and (8), the following attributes of the material point are considered:

$${\displaystyle \frac{𝜕\overline{r}}{𝜕q_1\hspace{0.17em}}}={\left(\begin{array}{ccc}1& 0& 0\end{array}\right)}^T,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\displaystyle \frac{𝜕\overline{r}}{𝜕q_2\hspace{0.17em}}}={\left(\begin{array}{ccc}0& 1& 0\end{array}\right)}^T,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\displaystyle \frac{𝜕\overline{r}}{𝜕q_1\hspace{0.17em}}}={\left(\begin{array}{ccc}0& 0& 1\end{array}\right)}^T,$$

$$Q_1=\overline{R}·{\displaystyle \frac{𝜕\overline{r}}{𝜕q_1\hspace{0.17em}}}=R_x=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{Q}_2=\overline{R}·{\displaystyle \frac{𝜕\overline{r}}{𝜕q_2\hspace{0.17em}}}=R_y=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{Q}_3=\overline{R}·{\displaystyle \frac{𝜕\overline{r}}{𝜕q_3\hspace{0.17em}}}=R_z=0.$$

(19)

Following Equations (19), the virtual work principle (16), equivalent to the three scalar equations of general form (18), converts into the scalar equilibrium equations of the material point. Moreover, because the generalized coordinates are linear, it is noted that the generalized forces become actual forces.

2.2. D’Alembert–Lagrange Principle Applied in Case of Discrete System of Material Points

Subsequently, the dynamic analysis expands to a discrete system of material points.

In Figure 2, a discrete system of material points is considered. According to [2,6], the system performs a uniformly varying motion, whose distribution is known:

$$\left\{\begin{array}{c}{M}_i;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{m}_i;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\overline{F}}_i;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\overline{N}}_i;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\displaystyle \sum _{j=1}^n{\overline{F}}_{ij}};\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\overline{r}}_i={\overline{r}}_i\left(t\right);\\ {\overline{v}}_i={\dot{\overline{r}}}_i;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\overline{h}}_i=m_i·{\overline{v}}_i;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\overline{a}}_i={\dot{\overline{v}}}_i={\ddot{\overline{r}}}_i;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\dot{\overline{h}}}_i=m_i·{\overline{a}}_i;\hspace{0.17em}\\ {\overline{k}}_i={\overline{r}}_i\times m_i·{\overline{v}}_i;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\dot{\overline{k}}}_i={\overline{r}}_i\times m_i·{\overline{a}}_i;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\overline{F}}_{ji}=-m_i·{\overline{a}}_i,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}where\hspace{0.17em}\hspace{0.17em}i\ne \hspace{0.17em}j;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}i=1\to n;\hspace{0.17em}\hspace{0.17em}j=\hspace{0.17em}1\to n\end{array}\right\}.$$

(20)

Thus, for each material point within the system, D’Alembert’s equation is applied:

$${\overline{F}}_i\hspace{0.17em}-\hspace{0.17em}\hspace{0.17em}{m}_i·{\overline{a}}_i\hspace{0.17em}+\hspace{0.17em}{\overline{N}}_i\hspace{0.17em}+{\displaystyle \sum _{j=1}^n{\overline{F}}_{ij}=0\hspace{0.17em},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}where\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}i=1\to n}.$$

(21)

Moreover, for each material point subjected to both scleronomic and holonomic constraints with negligible friction, a virtual displacement compatible with the connection is applied. In this scenario, D’Alembert’s Equation (6) is scaled by the virtual displacement, resulting in the following expressions:

$$\left({\overline{F}}_i\hspace{0.17em}\hspace{0.17em}-\hspace{0.17em}{m}_i·{\overline{a}}_i\hspace{0.17em}+\hspace{0.17em}{\overline{N}}_i\hspace{0.17em}+{\displaystyle \sum _{i=1}^n{\overline{F}}_{ij}\hspace{0.17em}}=0\right)·\left|\delta {\overline{r}}_i\right.\hspace{0.17em},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}where\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}i=1\to n.$$

(22)

For the dynamic study of a material system, Equation (22) is summed up:

$$\delta L={\displaystyle \sum _{i=1}^n\left({\overline{F}}_i-m_i·{\overline{a}}_i\right)}·\delta {\overline{r}}_i+{\displaystyle \sum _{i=1}^n{\overline{N}}_i·\delta {\overline{r}}_i}+{\displaystyle \sum _{i=1}^n{\overline{F}}_{ij}·\delta {\overline{r}}_i}=0.$$

(23)

Since the constraints involve negligible friction, from (22), the following differential expression is obtained:

$${\overline{N}}_i\perp \delta {\overline{r}}_i,\hspace{0.17em}\hspace{0.17em}from\hspace{0.17em}\hspace{0.17em}where\hspace{0.17em}\hspace{0.17em}\Rightarrow \delta L_L={\displaystyle \sum _{i=1}^n{\overline{N}}_i·\delta {\overline{r}}_i}=0,$$

(24)

where $\delta L_L$ represents the virtual work of the constraint forces, and Equation (24) is known as the principle of virtual work, used in statics to determine the constraint forces.

However, based on differential Equation (23), it can be demonstrated that the virtual work of the internal constraint forces is zero:

$$\delta L_{L\mathrm{int}}={\displaystyle \sum _{i=1}^n{\overline{F}}_{ij}·\delta {\overline{r}}_i}=0.$$

(25)

Therefore, by considering (24) and (25), the expression of virtual work becomes as follows:

$$\delta L={\displaystyle \sum _{i=1}^n\left({\overline{F}}_i-m_i·{\overline{a}}_i\right)}·\delta {\overline{r}}_i=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}where\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\overline{F}}_i-m_i·{\overline{a}}_i={\overline{F}}_{ip}.$$

(26)

Therefore, in accordance with (26), the virtual work of the lost forces for virtual displacements, which are compatible with the physical constraints, is equal to zero. Applying some transformations on Equation (26), the following equation results:

$$\delta L={\displaystyle \sum _{i=1}^n\left({\overline{F}}_i-m_i·{\overline{a}}_i\right)}·{\displaystyle \sum _{j=1}^k\frac{𝜕{\overline{r}}_i}{𝜕q_j\hspace{0.17em}}·\delta q_j}=0,$$

(27)

$$\delta L={\displaystyle \sum _{j=1}^k\left[{\displaystyle \sum _{i=1}^n\left({\overline{F}}_i-m_i·{\overline{a}}_i\right)·}\frac{𝜕{\overline{r}}_i}{𝜕q_j\hspace{0.17em}}\right]}·\delta q_j=0.$$

(28)

Equations (26)–(28) represent the D’Alembert–Lagrange principle applied to a discrete (finite) system of material points, subjected only to scleronomic and holonomic constraints with negligible friction. If the material system is in absolute static equilibrium, then Equation (28) is particularized as follows:

$${\overline{a}}_i=0,\Rightarrow \Rightarrow \delta L={\displaystyle \sum _{j=1}^k\left[{\displaystyle \sum _{i=1}^n{\overline{F}}_i·}\frac{𝜕{\overline{r}}_i}{𝜕q_j\hspace{0.17em}}\right]}·\delta q_j=0.$$

(29)

The principle of virtual work is rewritten using the generalized forces, as follows:

$$\delta L={\displaystyle \sum _{j=1}^k{Q}_j}·\delta q_j=0,\Rightarrow \Rightarrow Q_1·\delta q_1+\dots +Q_j·\delta q_j+\dots +Q_k·\delta q_k=0.$$

(30)

In the case of holonomic constraints, the number and type of independent parameters are the same in finite and infinitesimal displacements. Therefore, applying the principle of superposition effects in (30), the following expression results:

$$q_j\ne 0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\delta q_j\ne 0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}{q}_i=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\delta q_i=0,\hspace{0.17em}\hspace{0.17em}where\hspace{0.17em}\hspace{0.17em}j=1\to k,\hspace{0.17em}\hspace{0.17em}for\hspace{0.17em}\hspace{0.17em}j\ne i=1\to k\hspace{0.17em}\hspace{0.17em}\Rightarrow \Rightarrow Q_j=0,$$

(31)

$$from\hspace{0.17em}\hspace{0.17em}where\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\Rightarrow \Rightarrow Q_j={\displaystyle \sum _{i=1}^n{\overline{F}}_i·}{\displaystyle \frac{𝜕{\overline{r}}_i}{𝜕q_j\hspace{0.17em}}}\hspace{0.17em}=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}for\hspace{0.17em}\hspace{0.17em}j=1\to k.$$

(32)

Thus, the principle of virtual work in Equations (27) and (28) is equivalent to a system consisting of a number of $k\le 3·n$ scalar equilibrium equations, in which the generalized coordinates expressing the static equilibrium state are unknown. In accordance with analytical mechanics, the independent parameters,

$$\overline{\overline{\theta }}=\left(q_j;\hspace{0.17em}\hspace{0.17em}j=1\to k\le 3·n\right),$$

(33)

form a virtual space with $\left(k\right)$ dimensions, represented by the generalized coordinates, called the configuration space. Within this space is the static equilibrium Equation (32) determined above.

2.3. Symmetry in D’Alembert–Lagrange Principle in Case of a Rigid Body

The dynamic study extends to the rigid body and, respectively, to systems of rigid bodies [1,12,13,14,15]. Therefore, in Figure 3, a rigid body with an arbitrary geometric shape in uniformly accelerated absolute motion is considered. The rigid body is characterized by the following parameter distribution:

$$\begin{array}{c}{\overline{r}}_0={\overline{r}}_0\left(t\right)=\left[\begin{array}{ccc}{x}_0\left(t\right)& y_0\left(t\right)& z_0\left(t\right)\end{array}\right]{\hspace{0.17em}}^T,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\overline{v}}_0={\dot{\overline{r}}}_0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\overline{a}}_0={\dot{\overline{v}}}_0={\ddot{\overline{r}}}_0,\\ \overline{\overline{\Omega }}\left(t\right)={\left[\begin{array}{ccc}{\alpha }_x\left(t\right)& {\beta }_y\left(t\right)& {\gamma }_z\left(t\right)\end{array}\right]}^T,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{}_S{}^0\left[R\right]\left(t\right),\hspace{0.17em}\hspace{0.17em}\overline{\Omega }\left(t\right),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\overline{\omega }={\displaystyle \frac{𝜕\overline{\Omega }\left(t\right)}{𝜕t}},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\overline{\epsilon }=\dot{\overline{\omega }}={\displaystyle \frac{d^2\overline{\Omega }\left(t\right)}{d{t}^2}},\\ M,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\overline{r}}_C={\overline{r}}_C\left(t\right),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\overline{R}={\displaystyle \sum _{i=1}^n{\overline{F}}_i}={\displaystyle \int d\overline{F}}={\displaystyle \int {\overline{a}}_M·dm},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\\ {\overline{M}}_O={\displaystyle \sum _{i=1}^n\left({\overline{r}}_0+{\overline{\rho }}_i\right)\times {\overline{F}}_i}={\displaystyle \int {\overline{r}}_M}\times {\overline{a}}_M·dm,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{I}_S^{\prime },\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{I}_{pS}^{\prime }.\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\end{array}$$

(34)

In Equations (34), $\overline{\Omega }=\overline{\Omega }\left(t\right)$ represents the angular vector of resultant rotation and $\overline{\overline{\Omega }}\left(t\right)$ is a column matrix containing the rotation angles. In the same expression, $I_S^{\prime }$ and $I_{pS}^{\prime }$ represent the axial–centrifugal inertia tensor and planar–centrifugal inertia tensor, respectively, with respect to the $\left\{0^{\prime }\right\}$ frame.

D’Alembert’s equations [12,13,14,15] are rewritten for the rigid body in absolute general motion, and by considering it a free holonomic mechanical system:

$$\overline{R}+{\overline{R}}_j+{\overline{R}}_L=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}where\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\overline{R}}_L=0,$$

(35)

$${\overline{M}}_C^{\ast }+{\overline{M}}_j^{\ast }+{\overline{M}}_L^{\ast }=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}where\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\overline{M}}_L^{\ast }=0.$$

(36)

In (36), the symbol $\left(\ast \right)$ shows that the moments are expressed relative to a reference system having the origin in the mass center of the rigid body. In the same expression, ${\overline{M}}_C^{\ast }$, ${\overline{M}}_L^{\ast }$, and ${\overline{M}}_j^{\ast }$ represent the moments of the external forces, constraint forces, and inertia forces with respect to a system attached to the center of mass.

Due to its rigid structure, there are no elemental relative displacements between the particles within the rigid body.

Consequently, we observe an equivalence between infinitesimal real displacements and instantaneous, infinitesimal virtual displacements, both in translation and rotation, as they are both fundamental and cumulative.

At the center of mass, C, an intrinsic point of the rigid body, two mobile and concurrent reference systems are fixed. In practice, the following particularities are applied:

$$\left\{\begin{array}{c}O\equiv C;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\overline{\rho }}_C=0;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\overline{r}}_0\equiv {\overline{r}}_C;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\overline{v}}_0\equiv {\overline{v}}_C;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\overline{a}}_0\equiv {\overline{a}}_C;\hspace{0.17em}\\ {\overline{\rho }}_M={\overline{\rho }}_M^{\ast };\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{I}_S^{\prime }\equiv I_S^{\ast };\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{I}_{pS}^{\prime }\equiv I_{pS}^{\ast }\end{array}\right\}.$$

(37)

Considering Equation (37), the following particularity is introduced in the expression of the theorem of the motion of the mass center:

$${\overline{a}}_C={\dot{\overline{v}}}_C={\ddot{\overline{r}}}_C={\left[\begin{array}{ccc}{\ddot{x}}_C& {\ddot{y}}_C& {\ddot{z}}_C\end{array}\right]}^T,$$

(38)

$$M·{\overline{a}}_C=M·{\left[\begin{array}{ccc}{\ddot{x}}_C& {\ddot{y}}_C& {\ddot{z}}_C\end{array}\right]}^T={\left[\begin{array}{ccc}{R}_x& R_y& R_z\end{array}\right]}^T.$$

(39)

In Equation (39), the theorem of the motion of the center of mass is equivalent to a system of three second-order scalar differential equations containing three unknowns, i.e., the parametric equations of the absolute motion of the center of mass, equivalent to the parametric equations of the resultant translation of the rigid body. As a result, the parametric equations of motion (independent parameters of general motion) are defined as:

$$\overline{\overline{X}}\left(t\right)=\left(\begin{array}{c}\hspace{0.17em}\hspace{0.17em}{\overline{r}}_C\left(t\right)\hspace{0.17em}\hspace{0.17em}\\ \overline{\overline{\Omega }}\left(t\right)\end{array}\right)=\left[\begin{array}{c}{\left[\begin{array}{ccc}{x}_C\left(t\right)& y_C\left(t\right)& z_C\left(t\right)\end{array}\right]}^T\\ {\left[\begin{array}{ccc}{\alpha }_x\left(t\right)& {\beta }_y\left(t\right)& {\gamma }_z\left(t\right)\end{array}\right]}^T\end{array}\right]=\left[\begin{array}{c}{\left[\begin{array}{ccc}{q}_1\left(t\right)& q_2\left(t\right)& q_3\left(t\right)\end{array}\right]}^T\\ {\left[\begin{array}{ccc}{q}_4\left(t\right)& q_5\left(t\right)& q_6\left(t\right)\end{array}\right]}^T\end{array}\right],$$

(40)

$$and\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\overline{\overline{X}}\left(t\right)\equiv \overline{\overline{\theta }}\left(t\right)={\left[q_j\left(t\right),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}j=1\to 6\right]}^T,$$

(41)

where the symbol $\overline{\overline{\theta }}\left(t\right)$ is known as the column vector (column matrix) of generalized coordinates. However, considering the significance of the generalized coordinates (see (40) and (41)), the following operator is implemented:

$${\Delta }_j=\left\{\left(0,\hspace{0.17em}\hspace{0.17em}\mathrm{linear}\hspace{0.17em}\hspace{0.17em}\mathrm{coordinates}\right);\hspace{0.17em}\hspace{0.17em}\left(1,\hspace{0.17em}\hspace{0.17em}\mathrm{angular}\hspace{0.17em}\hspace{0.17em}\mathrm{coordinates}\right)\right\}.$$

(42)

The vectorial equations, specific to the resultant translation and rotation, components of the general motion, become as follows:

$${\overline{r}}_C\left(t\right)={\left[\begin{array}{ccc}{q}_1\left(t\right)& q_2\left(t\right)& q_3\left(t\right)\end{array}\right]}^T={\overline{r}}_C\left[q_j\left(t\right)·\left(1-{\Delta }_j\right),\hspace{0.17em}\hspace{0.17em}j=1\to 6\right],$$

(43)

$$\overline{\Omega }\left(t\right)={\left[\begin{array}{ccc}{q}_4\left(t\right)& q_5\left(t\right)& q_6\left(t\right)\end{array}\right]}^T=\overline{\Omega }\left[q_j\left(t\right)·{\Delta }_j,\hspace{0.17em}\hspace{0.17em}j=1\to 6\right].$$

(44)

The angular vector of resultant rotation can be defined in a matrix form as follows:

$$\overline{\Omega }\left(t\right)=\left[\begin{array}{ccc}1& 0& s{q}_5\\ 0& c{q}_4& -s{q}_4·c{q}_5\\ 0& s{q}_4& c{q}_4·c{q}_5\end{array}\right]·\left[\begin{array}{c}{q}_4\\ q_5\\ q_6\end{array}\right].$$

(45)

Furthermore, when considering the rigid body as a holonomic and free mechanical system in Cartesian space, we associate it with a linear virtual displacement compatible with the resultant translation, $\delta {\overline{r}}_C$, and an angular virtual displacement, $\delta \overline{\Omega }$, compatible with the resultant rotation, and defined below.

$$d{\overline{r}}_C\equiv \delta {\overline{r}}_C={\left[\begin{array}{ccc}d{q}_1& d{q}_2& d{q}_3\end{array}\right]}^T={\displaystyle \sum _{j=1}^6\frac{𝜕{\overline{r}}_C}{𝜕q_j}}·d{q}_j·\left(1-{\Delta }_j\right)\equiv {\displaystyle \sum _{j=1}^6\frac{𝜕{\overline{r}}_C}{𝜕q_j}}·\delta q_j·\left(1-{\Delta }_j\right),$$

(46)

$$d\overline{\Omega }\equiv \delta \overline{\Omega }={\left[\begin{array}{ccc}d{q}_4& d{q}_5& d{q}_6\end{array}\right]}^T={\displaystyle \sum _{j=1}^6\frac{𝜕\overline{\Omega }}{𝜕q_j}}·d{q}_j·{\Delta }_j\equiv {\displaystyle \sum _{j=1}^6\frac{𝜕\overline{\Omega }}{𝜕q_j}}·\delta q_j·{\Delta }_j.$$

(47)

The D’Alembert Equations (35) and (36) are then scaled by $\delta {\overline{r}}_C$ and $\delta \overline{\Omega }$, respectively, resulting in the following expressions:

$$\left(\overline{R}+{\overline{R}}_j=0\right)·\left|\delta {\overline{r}}_C\right.\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\Rightarrow \overline{R}·\delta {\overline{r}}_C+{\overline{R}}_j·\delta {\overline{r}}_C=0,$$

(48)

$$\left({\overline{M}}_C^{\ast }+{\overline{M}}_j^{\ast }=0\right)·\left|\delta \hspace{0.17em}\overline{\Omega }\right.\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\Rightarrow \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\overline{M}}_C^{\ast }·\delta \hspace{0.17em}\overline{\Omega }+{\overline{M}}_j^{\ast }·\delta \hspace{0.17em}\overline{\Omega }=0.$$

(49)

For the dynamic study of the rigid body, the two equations in (48) and (49) are summed up, and then, by applying a series of transformations, the following is obtained:

$$\delta L=\left(\overline{R}-M·{\overline{a}}_C\right)·\delta {\overline{r}}_C+\left[{\overline{M}}_C^{\ast }-\left(I_S^{\ast }·\overline{\epsilon }+\overline{\omega }\times I_S^{\ast }·\overline{\omega }\right)\right]·\delta \overline{\Omega }=0,$$

(50)

$$or\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\delta L=\left(\overline{R}-M·{\overline{a}}_C\right)·\delta {\overline{r}}_C+\left[{\overline{M}}_C^{\ast }-{\displaystyle \frac{d}{dt}}\left(I_S^{\ast }·\overline{\omega }\right)\right]·\delta \overline{\Omega }=0.$$

(51)

By substituting the virtual displacements from (50) and (51), namely, $\delta \overline{r}$ and $\delta \overline{\Omega }$ with (46) and (47), the following expressions for the virtual work are obtained:

$$\delta L=\left(\overline{R}-M·{\overline{a}}_C\right)·{\displaystyle \sum _{j=1}^6\frac{𝜕{\overline{r}}_C}{𝜕q_j}}·\delta q_j·\left(1-{\Delta }_j\right)+\left[{\overline{M}}_C^{\ast }-{\displaystyle \frac{d}{dt}}\left(I_S^{\ast }·\overline{\omega }\right)\right]·{\displaystyle \sum _{j=1}^6\frac{𝜕\overline{\Omega }}{𝜕q_j}}·\delta q_j·{\Delta }_j=0,$$

(52)

$$\delta L={\displaystyle \sum _{j=1}^6\left\{\left(\overline{R}-M·{\overline{a}}_C\right)·\frac{𝜕{\overline{r}}_C}{𝜕q_j}·\left(1-{\Delta }_j\right)+\left[{\overline{M}}_C^{\ast }-\frac{d}{dt}\left(I_S^{\ast }·\overline{\omega }\right)\right]·\frac{𝜕\overline{\Omega }}{𝜕q_j}·{\Delta }_j\right\}}·\delta q_j=0.$$

(53)

Equations (52) and (53) represent the D’Alembert–Lagrange principle applied to a free rigid body in Cartesian space and considered as a holonomic mechanical system in absolute general motion. If the mechanical system is in absolute static equilibrium, then Equation (53) is particularized as follows:

$${\overline{a}}_C=0,\hspace{0.17em}\hspace{0.17em}\overline{\omega }=0,\hspace{0.17em}\hspace{0.17em}\overline{\epsilon }=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\Rightarrow \delta L={\displaystyle \sum _{j=1}^6\left\{\overline{R}·\frac{𝜕{\overline{r}}_C}{𝜕q_j}·\left(1-{\Delta }_j\right)+{\overline{M}}_C^{\ast }·\frac{𝜕\overline{\Omega }}{𝜕q_j}·{\Delta }_j\right\}}·\delta q_j=0.$$

(54)

Differential Equation (54) represents the principle of virtual work, used to determine the independent parameters: the position and orientation of static equilibrium of the rigid body. In Equation (54), the generalized force, $Q_j^{\ast }$, is introduced:

$$\hspace{0.17em}{Q}_j^{\ast }=\overline{R}·{\displaystyle \frac{𝜕{\overline{r}}_C}{𝜕q_j}}·\left(1-{\Delta }_j\right)+{\overline{M}}_C^{\ast }·{\displaystyle \frac{𝜕\overline{\Omega }}{𝜕q_j}}·{\Delta }_j,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}for\hspace{0.17em}\hspace{0.17em}j=1\to 6.$$

(55)

Several differential transformations are performed on the expression defining the generalized force (55), resulting in:

$$\overline{R}={\displaystyle \sum _{i=1}^n{\overline{F}}_i},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\overline{v}}_C={\displaystyle \sum _{j=1}^6\frac{𝜕{\overline{r}}_C}{𝜕q_j}}·{\dot{q}}_j·\left(1-{\Delta }_j\right)\Rightarrow \Rightarrow {\displaystyle \frac{𝜕{\overline{r}}_C}{𝜕q_j}}={\displaystyle \frac{𝜕{\overline{v}}_C}{𝜕{\dot{q}}_j}},$$

(56)

$${\overline{M}}_C^{\ast }={\displaystyle \sum _{i=1}^n{\overline{\rho }}_i^{\ast }}\times {\overline{F}}_i,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\overline{\omega }={\displaystyle \sum _{j=1}^6\frac{𝜕\overline{\Omega }}{𝜕q_j}}·{\dot{q}}_j·{\Delta }_j,\hspace{0.17em}\hspace{0.17em}\Rightarrow \Rightarrow {\displaystyle \frac{𝜕\overline{\Omega }}{𝜕q_j}}={\displaystyle \frac{𝜕\overline{\omega }}{𝜕{\dot{q}}_j}}.$$

(57)

By substituting the properties of (56) and (57) into (55), the expression of the generalized force becomes as follows:

$$Q_j={\displaystyle \sum _{i=1}^n{\overline{F}}_i}·{\displaystyle \frac{𝜕{\overline{v}}_C}{𝜕{\dot{q}}_j}}+\left({\displaystyle \sum _{i=1}^n{\overline{\rho }}_i^{\ast }}\times {\overline{F}}_i\right)·{\displaystyle \frac{𝜕\overline{\omega }}{𝜕{\dot{q}}_j}}={\displaystyle \frac{𝜕}{𝜕{\dot{q}}_j}}\left[{\displaystyle \sum _{i=1}^n{\overline{F}}_i}·\left({\overline{v}}_C+\overline{\omega }\times {\overline{\rho }}_i^{\ast }\right)\right]={\displaystyle \sum _{i=1}^n{\overline{F}}_i}·{\displaystyle \frac{𝜕{\overline{v}}_i}{𝜕{\dot{q}}_j}},$$

(58)

$${\overline{v}}_i={\overline{v}}_C+\overline{\omega }\times {\overline{\rho }}_i^{\ast }={\displaystyle \sum _{j=1}^6\frac{𝜕{\overline{r}}_i}{𝜕q_j}}·{\dot{q}}_j·\left(1-{\Delta }_j\right)\Rightarrow {\displaystyle \frac{𝜕{\overline{v}}_i}{𝜕{\dot{q}}_j}}={\displaystyle \frac{𝜕{\overline{r}}_i}{𝜕q_j}},$$

(59)

$$Q_j=\overline{R}·{\displaystyle \frac{𝜕{\overline{r}}_C}{𝜕q_j}}·\left(1-{\Delta }_j\right)+{\overline{M}}_C^{\ast }·{\displaystyle \frac{𝜕\overline{\Omega }}{𝜕q_j}}·{\Delta }_j\equiv {\displaystyle \sum _{i=1}^n{\overline{F}}_i}·{\displaystyle \frac{𝜕{\overline{r}}_i}{𝜕q_j}}.$$

(60)

The term “generalized force” attributed to the symbol $Q_j$ defined with (60) arises from the analogy with the generalized coordinate (see (17) and (19)). Below, it is shown that the generalized force transforms into a force or a moment of force, depending on the generalized coordinate, which can be linear or angular. First, the case where the generalized coordinate is linear is analyzed, according to (40) and (43), namely:

$$\begin{gathered} Q_j=\overline{R}·{\displaystyle \frac{𝜕{\overline{r}}_C}{𝜕q_j}}·\left(1-{\Delta }_j\right)=0, \\ {q}_j\ne 0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\delta q_j\ne 0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{q}_i=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\delta q_i=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}for\hspace{0.17em}\hspace{0.17em}j=1\to 3,\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}j\ne i=1\to 3\hspace{0.17em}, \end{gathered}$$

(61)

$$where\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\displaystyle \frac{𝜕{\overline{r}}_C}{𝜕q_1}}={\displaystyle \frac{𝜕{\overline{r}}_C}{𝜕x_C}}={\left(\begin{array}{ccc}1& 0& 0\end{array}\right)}^T,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\displaystyle \frac{𝜕{\overline{r}}_C}{𝜕q_2}}={\displaystyle \frac{𝜕{\overline{r}}_C}{𝜕y_C}}={\left(\begin{array}{ccc}0& 1& 0\end{array}\right)}^T,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\displaystyle \frac{𝜕{\overline{r}}_C}{𝜕q_3}}={\displaystyle \frac{𝜕{\overline{r}}_C}{𝜕z_C}}={\left(\begin{array}{ccc}0& 0& 1\end{array}\right)}^T.$$

(62)

By applying (61) and (62) for each $j=1\to 3$, the generalized forces are specified as follows:

$$\begin{array}{l}{q}_1\ne 0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\delta q_1\ne 0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}iar\hspace{0.17em}\hspace{0.17em}{q}_2=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\delta q_2=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{q}_3=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\delta q_3=0\Rightarrow Q_1=\overline{R}·{\displaystyle \frac{𝜕{\overline{r}}_C}{𝜕q_1}}·\left(1-{\Delta }_1\right)=R_x=0,\\ q_1=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\delta q_1=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}iar\hspace{0.17em}\hspace{0.17em}{q}_2\ne 0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\delta q_2\ne 0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{q}_3=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\delta q_3=0\Rightarrow Q_2=\overline{R}·{\displaystyle \frac{𝜕{\overline{r}}_C}{𝜕q_2}}·\left(1-{\Delta }_2\right)=R_y=0,\\ q_1=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\delta q_1=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}iar\hspace{0.17em}\hspace{0.17em}{q}_2=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\delta q_2=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{q}_3\ne 0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\delta q_3\ne 0\Rightarrow Q_3=\overline{R}·{\displaystyle \frac{𝜕{\overline{r}}_C}{𝜕q_3}}·\left(1-{\Delta }_3\right)=R_z=0.\end{array}$$

(63)

Therefore, when the generalized coordinate represents a linear displacement, in accordance with (63), it is observed that the generalized forces become the Cartesian components of the resultant vector of active forces. In the following, the case where the generalized coordinate is angular is analyzed:

$$\begin{array}{c}\hspace{0.17em}{Q}_j={\overline{M}}_C^{\ast }·{\displaystyle \frac{𝜕\overline{\Omega }}{𝜕q_j}}·{\Delta }_j=0,\\ q_j\ne 0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\delta q_j\ne 0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{q}_i=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\delta q_i=0,\hspace{0.17em}\hspace{0.17em}for\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}j=4\to 6,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}j\ne i=4\to 6,\end{array}$$

(64)

$$\hspace{0.17em}\hspace{0.17em}\overline{\Omega }\left(t\right)=\overline{\Omega }\left[q_j\left(t\right);\hspace{0.17em}\hspace{0.17em}j=4\to 6\right]=\left[\begin{array}{ccc}1& 0& s{\beta }_y\\ 0& c{\alpha }_x& -s{\alpha }_x·c{\beta }_y\\ 0& s{\alpha }_x& c{\alpha }_x·c{\beta }_y\end{array}\right]·\left(\begin{array}{c}{\alpha }_x\\ {\beta }_y\\ {\gamma }_z\end{array}\right)$$

(65)

$$\begin{array}{c}{Q}_4={\overline{M}}_C^{\ast }·{\displaystyle \frac{𝜕\overline{\Omega }}{𝜕q_4}}·{\Delta }_j={\overline{M}}_C^{\ast }·{\overline{i}}_0·{\Delta }_j=M_x^{\ast }·{\Delta }_j,\\ where,\hspace{0.17em}\hspace{0.17em}{\displaystyle \frac{𝜕\overline{\Omega }}{𝜕q_4}}={\displaystyle \frac{𝜕\overline{\Omega }}{𝜕{\alpha }_x}}=\left[\begin{array}{ccc}1& 0& s{\beta }_y\\ 0& c{\alpha }_x& -s{\alpha }_x·c{\beta }_y\\ 0& s{\alpha }_x& c{\alpha }_x·c{\beta }_y\end{array}\right]·\left(\begin{array}{c}1\\ 0\\ 0\end{array}\right)=\left(\begin{array}{c}1\\ 0\\ 0\end{array}\right),\end{array}$$

(66)

$$\begin{array}{c}{Q}_5={\overline{M}}_C^{\ast }·{\displaystyle \frac{𝜕\overline{\Omega }}{𝜕q_5}}·{\Delta }_j={\overline{M}}_C^{\ast }·{\overline{j}}_1·{\Delta }_j=\left(M_y^{\ast }·c{\alpha }_x+M_z^{\ast }·s{\alpha }_x\right)·{\Delta }_j,\\ where,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\displaystyle \frac{𝜕\overline{\Omega }}{𝜕q_5}}={\displaystyle \frac{𝜕\overline{\Omega }}{𝜕{\beta }_y}}=\left[\begin{array}{ccc}1& 0& s{\beta }_y\\ 0& c{\alpha }_x& -s{\alpha }_x·c{\beta }_y\\ 0& s{\alpha }_x& c{\alpha }_x·c{\beta }_y\end{array}\right]·\left(\begin{array}{c}0\\ 1\\ 0\end{array}\right)=\left(\begin{array}{c}0\\ c{\alpha }_x\\ s{\alpha }_x\end{array}\right),\end{array}$$

(67)

$$\begin{array}{c}\hspace{0.17em}{Q}_6={\overline{M}}_C^{\ast }·{\displaystyle \frac{𝜕\overline{\Omega }}{𝜕q_6}}·{\Delta }_j={\overline{M}}_C^{\ast }·\overline{k}·{\Delta }_j=\left(M_x^{\ast }·s{\beta }_y-M_y^{\ast }·s{\alpha }_x·c{\beta }_y+M_z^{\ast }·c{\alpha }_x·c{\beta }_y\right)·{\Delta }_j,\\ where,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\displaystyle \frac{𝜕\overline{\Omega }}{𝜕q_6}}={\displaystyle \frac{𝜕\overline{\Omega }}{𝜕{\gamma }_z}}=\left[\begin{array}{ccc}1& 0& s{\beta }_y\\ 0& c{\alpha }_x& -s{\alpha }_x·c{\beta }_y\\ 0& s{\alpha }_x& c{\alpha }_x·c{\beta }_y\end{array}\right]·\left(\begin{array}{c}0\\ 0\\ 1\end{array}\right)=\left(\begin{array}{c}s{\beta }_y\\ -s{\alpha }_x·c{\beta }_y\\ c{\alpha }_x·c{\beta }_y\end{array}\right)\hspace{0.17em}.\end{array}$$

(68)

By successively applying (64)–(68) for each $j=4\to 6$, the generalized forces are defined as follows:

$$\begin{array}{c}{Q}_4={\overline{M}}_C^{\ast }·{\displaystyle \frac{𝜕\overline{\Omega }}{𝜕q_4}}·{\Delta }_j={\overline{M}}_C^{\ast }·{\overline{i}}_0=M_x^{\ast }=0,\\ where,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{q}_4={\alpha }_x\ne 0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\delta q_4\ne 0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{q}_5={\beta }_y=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\delta q_5=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{q}_6={\gamma }_z=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\delta q_6=0,\end{array}$$

(69)

$$\begin{array}{c}{Q}_5={\overline{M}}_C^{\ast }·{\displaystyle \frac{𝜕\overline{\Omega }}{𝜕q_5}}·{\Delta }_j={\overline{M}}_C^{\ast }·{\overline{j}}_1·{\Delta }_j=\left(M_y^{\ast }·c{\alpha }_x+M_z^{\ast }·s{\alpha }_x\right)·{\Delta }_j=M_y^{\ast }=0,\\ where,\hspace{0.17em}\hspace{0.17em}{q}_4={\alpha }_x=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\delta q_4=0,\hspace{0.17em}\hspace{0.17em}{q}_5={\beta }_y\ne 0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\delta q_5\ne 0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{q}_6={\gamma }_z=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\delta q_6=0,\end{array}$$

(70)

$$\begin{array}{c}{Q}_6={\overline{M}}_C^{\ast }·{\displaystyle \frac{𝜕\overline{\Omega }}{𝜕q_6}}·{\Delta }_j={\overline{M}}_C^{\ast }·\overline{k}·{\Delta }_j=\left(M_x^{\ast }·s{\beta }_y-M_y^{\ast }·s{\alpha }_x·c{\beta }_y+M_z^{\ast }·c{\alpha }_x·c{\beta }_y\right)·{\Delta }_j=M_z^{\ast }=0,\\ where,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{q}_4={\alpha }_x,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\delta q_4=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}iar\hspace{0.17em}\hspace{0.17em}{q}_5={\beta }_y=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\delta q_5=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{q}_6={\gamma }_z\ne 0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\delta q_6\ne 0.\end{array}$$

(71)

When the generalized coordinate represents an angular displacement, in accordance with (69)–(71), it is noticed that the generalized forces become the Cartesian components of the resultant moment of active forces. The results from (63) and (69)–(71) lead to the static equilibrium equations of the free rigid body, namely:

$$\begin{array}{c}{Q}_1=R_x=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{Q}_2=R_y=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{Q}_3=R_z=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\Rightarrow \overline{R}=0,\\ Q_4=M_x^{\ast }=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{Q}_5=M_y^{\ast }=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{Q}_6=M_z^{\ast }=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\Rightarrow \hspace{0.17em}\hspace{0.17em}{\overline{M}}_C^{\ast }=0.\end{array}$$

(72)

In conclusion, it is noted that the principle of virtual work (53) transforms into Equation (60), which, in accordance with (72), becomes the scalar equations of static equilibrium for a free rigid body in Cartesian space.

Remark 1. 

Due to the random nature of virtual displacements, the D’Alembert–Lagrange principle transforms into several scalar differential equations identical to the number of generalized coordinates. These independent parameters uniquely characterize the law of motion of the material system to which constraints, considered to be scleronomic and holonomic with negligible friction, are applied.

2.4. Acceleration Energy of First-Order and Gibbs–Appell Equations

Considering the aspects analyzed above, the dynamic study extends to the rigid body and then to systems of bodies [13,14,15]. The Lagrange equations of the first kind are applied to a rigid body in absolute general motion, and according to [8], it becomes a free holonomic mechanical system in Cartesian space, characterized by the following differential expressions, rewritten, and presented below:

$$M·{\overline{a}}_C·{\displaystyle \frac{𝜕{\overline{r}}_C}{𝜕q_j}}·\left(1-{\Delta }_j\right)+\left(I_S^{\ast }·\overline{\epsilon }+\overline{\omega }\times I_S^{\ast }·\overline{\omega }\right)·{\displaystyle \frac{𝜕\overline{\Omega }}{𝜕q_j}}·{\Delta }_j=Q_j,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}for\hspace{0.17em}\hspace{0.17em}j=1\to 6,$$

(73)

$$Q_j=\overline{R}·{\displaystyle \frac{𝜕{\overline{r}}_C}{𝜕q_j}}·\left(1-{\Delta }_j\right)+{\overline{M}}_C^{\ast }·{\displaystyle \frac{𝜕\overline{\Omega }}{𝜕q_j}}·{\Delta }_j\equiv {\displaystyle \sum _{i=1}^n{\overline{F}}_i}·{\displaystyle \frac{𝜕{\overline{r}}_i}{𝜕q_j}}\hspace{0.17em},$$

(74)

$$Q_{jö}=-M·{\overline{a}}_C·{\displaystyle \frac{𝜕{\overline{r}}_C}{𝜕q_j}}·\left(1-{\Delta }_j\right)-{\displaystyle \frac{d}{dt}}\left(I_S^{\ast }·\overline{\omega }\right)·{\displaystyle \frac{𝜕\overline{\Omega }}{𝜕q_j}}·{\Delta }_j.$$

(75)

In Equations (74) and (75), $Q_j$ represents the generalized active force while $Q_{jö}$ defines the generalized inertia force. A series of differential transformations are performed on the second-order differential Equation (73). First, the transformations are applied to the vectors ${\overline{a}}_C$ and $\overline{\epsilon }$, as follows:

$${\overline{a}}_C={\displaystyle \sum _{j=1}^6\frac{𝜕{\overline{r}}_C}{𝜕q_j}}·{\ddot{q}}_j·\left(1-{\Delta }_j\right)+{\displaystyle \sum _{j=1}^6{\displaystyle \sum _{m=1}^6\frac{{𝜕}^2{\overline{r}}_C}{𝜕q_j·𝜕q_m}}·{\dot{q}}_j·{\dot{q}}_m·\left(1-{\Delta }_j\right)}·\left(1-{\Delta }_m\right),$$

(76)

$$\mathrm{where} \overline{a}=\overline{a}\left(q_j,\hspace{0.17em}\hspace{0.17em}j=1\to k;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\dot{q}}_j,\hspace{0.17em}j=1\to k;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\ddot{q}}_j\hspace{0.17em}j=1\to k;\hspace{0.17em}\hspace{0.17em}t\right).$$

(77)

Considering the condition (77), from (76), the following relation is obtained:

$$\hspace{0.17em}{\displaystyle \frac{𝜕{\overline{a}}_C}{𝜕{\ddot{q}}_j}}={\displaystyle \frac{𝜕{\overline{r}}_C}{𝜕q_j}},$$

(78)

$$\overline{\epsilon }={\displaystyle \sum _{j=1}^6\frac{𝜕\overline{\Omega }}{𝜕q_j}}·{\ddot{q}}_j·{\Delta }_j+{\displaystyle \sum _{j=1}^6{\displaystyle \sum _{m=1}^6\frac{{𝜕}^2\overline{\Omega }}{𝜕q_j·𝜕q_m}}·{\dot{q}}_j·{\dot{q}}_m·{\Delta }_j}·{\Delta }_m.$$

(79)

For holonomic systems, the hypothesis regarding angular accelerations applies:

$$\hspace{0.17em}\overline{\epsilon }=\overline{\epsilon }\left(q_j·{\Delta }_j,\hspace{0.17em}\hspace{0.17em}j=1\to k,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\dot{q}}_j·{\Delta }_j,\hspace{0.17em}\hspace{0.17em}j=1\to k,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\ddot{q}}_j·{\Delta }_j,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}j=1\to k,\hspace{0.17em}\hspace{0.17em}t\right).$$

(80)

As a result of the hypothesis (80), the following differential identity results from (79):

$${\displaystyle \frac{𝜕\overline{\Omega }}{𝜕q_j}}\hspace{0.17em}={\displaystyle \frac{𝜕\overline{\omega }}{𝜕{\dot{q}}_j}}={\displaystyle \frac{𝜕\overline{\epsilon }}{𝜕{\ddot{q}}_j}}.$$

(81)

Applying (78) to the first term on the left-hand side of the differential Equation (73) yields the following expression:

$$M·{\overline{a}}_C·{\displaystyle \frac{𝜕{\overline{r}}_C}{𝜕q_j}}·\left(1-{\Delta }_j\right)=M·{\overline{a}}_C·{\displaystyle \frac{𝜕{\overline{a}}_C}{𝜕{\ddot{q}}_j}}·\left(1-{\Delta }_j\right)={\displaystyle \frac{𝜕}{𝜕{\ddot{q}}_j}}\left[{\displaystyle \frac{1}{2}}·M·{\overline{a}}_C^2\right]·\left(1-{\Delta }_j\right)={\displaystyle \frac{𝜕\hspace{0.17em}{E}_A^{\left(1\right)TR}}{𝜕{\ddot{q}}_j}}·\left(1-{\Delta }_j\right).$$

(82)

Equation (81) is applied to the second term on the left-hand side of Equation (73):

$$\left(I_S^{\ast }·\overline{\epsilon }+\overline{\omega }\times I_S^{\ast }·\overline{\omega }\right)·{\displaystyle \frac{𝜕\overline{\Omega }}{𝜕q_j}}·{\Delta }_j=\left(I_S^{\ast }·\overline{\epsilon }+\overline{\omega }\times I_S^{\ast }·\overline{\omega }\right)·{\displaystyle \frac{𝜕\overline{\epsilon }}{𝜕{\ddot{q}}_j}}·{\Delta }_j,$$

(83)

$$\begin{array}{c}where\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\left(I_S^{\ast }·\overline{\epsilon }+\overline{\omega }\times I_S^{\ast }·\overline{\omega }\right)·{\displaystyle \frac{𝜕\overline{\epsilon }}{𝜕{\ddot{q}}_j}}·{\Delta }_j=I_S^{\ast }·\overline{\epsilon }·{\displaystyle \frac{𝜕\overline{\epsilon }}{𝜕{\ddot{q}}_j}}·{\Delta }_j+\left(\overline{\omega }\times I_S^{\ast }·\overline{\omega }\right)·{\displaystyle \frac{𝜕\overline{\epsilon }}{𝜕{\ddot{q}}_j}}·{\Delta }_j=\\ =\left\{{\displaystyle \frac{𝜕}{𝜕{\ddot{q}}_j}}\left({\displaystyle \frac{1}{2}}·{\overline{\epsilon }}^T·I_S^{\ast }·\overline{\epsilon }\right)·{\Delta }_j+{\displaystyle \frac{𝜕}{𝜕{\ddot{q}}_j}}\left[{\overline{\epsilon }}^T·\left(\overline{\omega }\times I_S^{\ast }·\overline{\omega }\right)\right]·{\Delta }_j-{\overline{\epsilon }}^T·{\displaystyle \frac{𝜕}{𝜕{\ddot{q}}_j}}\left(\overline{\omega }\times I_S^{\ast }·\overline{\omega }\right)·{\Delta }_j\right\}.\end{array}$$

(84)

Because $\overline{\omega }=\overline{\omega }\left(q_j;\hspace{0.17em}{\dot{q}}_j\right),\Rightarrow \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\overline{\omega }\left({\ddot{q}}_j\right)=0$ in (84), it results that

$${\overline{\epsilon }}^T·{\displaystyle \frac{𝜕}{𝜕{\ddot{q}}_j}}\left(\overline{\omega }\times I_S^{\ast }·\overline{\omega }\right)·{\Delta }_j=0$$

(85)

Thus, through Equations (83)–(85), the second term on the left-hand side of Equation (73) becomes:

$$\left(I_S^{\ast }·\overline{\epsilon }+\overline{\omega }\times I_S^{\ast }·\overline{\omega }\right)·{\displaystyle \frac{𝜕\overline{\Omega }}{𝜕q_j}}·{\Delta }_j={\displaystyle \frac{𝜕}{𝜕{\ddot{q}}_j}}\left[{\displaystyle \frac{1}{2}}·{\overline{\epsilon }}^T·I_S^{\ast }·\overline{\epsilon }+{\overline{\epsilon }}^T·\left(\overline{\omega }\times I_S^{\ast }·\overline{\omega }\right)\right]·{\Delta }_j={\displaystyle \frac{𝜕\hspace{0.17em}{E}_A^{\left(1\right)ROT}}{𝜕{\ddot{q}}_j}}·{\Delta }_j.$$

(86)

Substituting the results from (82) and (86) into Equation (73), the following differential expressions are obtained:

$${\displaystyle \frac{𝜕\hspace{0.17em}{E}_A^{\left(1\right)TR}}{𝜕{\ddot{q}}_j}}·\left(1-{\Delta }_j\right)+{\displaystyle \frac{𝜕\hspace{0.17em}{E}_A^{\left(1\right)ROT}}{𝜕{\ddot{q}}_j}}·{\Delta }_j=Q_j,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}where\hspace{0.17em}\hspace{0.17em}j=1\to 6,$$

(87)

$${\displaystyle \frac{𝜕\hspace{0.17em}}{𝜕{\ddot{q}}_j}}\left[E_A^{\left(1\right)TR}·\left(1-{\Delta }_j\right)+E_A^{\left(1\right)ROT}·{\Delta }_j\right]=Q_j,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}where\hspace{0.17em}\hspace{0.17em}j=1\to 6.$$

(88)

In accordance with [7], the expression on the left-hand side of Equations (87) and (88) represents the first-order energy of accelerations, rewritten below as follows:

$$E_A^{\left(1\right)}=E_A^{\left(1\right)TR}+E_A^{\left(1\right)ROT}={\displaystyle \frac{1}{2}}·M·{\overline{a}}_C^2+{\displaystyle \frac{1}{2}}·{\overline{\epsilon }}^T·I_S^{\ast }·\overline{\epsilon }+{\overline{\epsilon }}^T·\left(\overline{\omega }\times I_S^{\ast }·\overline{\omega }\right).$$

(89)

In Equation (89) the following term from the expression of the energy of accelerations of first order was not included in the differential equations:

$$E_A^{\left(1{\omega }^4\right)}={\displaystyle \frac{1}{2}}·{\overline{\omega }}^T·\left[{\overline{\omega }}^T·I_S^{\prime }·\overline{\omega }\right]·\overline{\omega }\equiv {\displaystyle \frac{1}{2}}·{\overline{\omega }}^T·\left[{\overline{\omega }}^T·Trace\left(I_{pS}^{\prime }\right)·\overline{\omega }-{\overline{\omega }}^T·I_{pS}^{\prime }·\overline{\omega }\right]·\overline{\omega }.$$

(90)

The defining expression for the first-order acceleration energy (89), Equation (90) is identical to the one obtained through mass integration, according to [13,14,15,16].

The explanation is provided by (85), which offers information regarding the derivative of angular velocity vector with respect to the generalized accelerations.

The expression of the first-order time derivative of the acceleration energy, $E_A^{\left(1\right)}$, can also be determined by applying the mass integrals. The general form is presented below:

$$E_A^{\left(1\right)}={\displaystyle \frac{1}{2}}·{\displaystyle \int a_M^2·dm},$$

(91)

where ${\overline{a}}_M$ represents the acceleration of the elemental particle $(dm)$ (see Figure 3):

$${\overline{a}}_M={\overline{a}}_0+\overline{\epsilon }\times {\overline{\rho }}_M+\overline{\omega }\times \overline{\omega }\times {\overline{\rho }}_M.$$

(92)

By developing Equation (91), according to [13,14,15], the following expression representing the first-order acceleration energy, in explicit form, is obtained:

$$\begin{array}{c}{E}_A^{\left(1\right)}={\displaystyle \frac{1}{2}}·M·{\overline{a}}_0^T·{\overline{a}}_0+M·{\overline{a}}_0^T·\left(\overline{\epsilon }\times {\overline{\rho }}_C\right)+M·{\overline{a}}_0^T·\overline{\omega }\times \left(\overline{\omega }\times {\overline{\rho }}_C\right)+\\ +{\displaystyle \frac{1}{2}}·{\overline{\epsilon }}^T·I_S^{\prime }·\overline{\epsilon }+{\overline{\epsilon }}^T·\left(\overline{\omega }\times I_S^{\prime }·\overline{\omega }\right)+{\displaystyle \frac{1}{2}}·{\overline{\omega }}^T·\left[{\overline{\omega }}^T·I_S^{\prime }·\overline{\omega }\right]·\overline{\omega }.\end{array}$$

(93)

Equation (93) is used to define the acceleration energy of the first order that can be applied in case of a rigid body in general motion.

By considering a reference system having the origin in the mass center of the rigid body, and by applying the following particularities:

$$\left\{\begin{array}{c}O\equiv C;\hspace{0.17em}\hspace{0.17em}\left\{0^{\prime }\right\}\equiv \left\{0^{\ast }\right\};\hspace{0.17em}\left\{S\right\}\equiv \left\{S^{\ast }\right\}\hspace{0.17em};\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\overline{\rho }}_C=0;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\overline{r}}_0\equiv {\overline{r}}_C;\hspace{0.17em}\hspace{0.17em}\\ {\overline{\rho }}_M={\overline{\rho }}^{\ast };\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\overline{r}}_M={\overline{r}}_C+{\overline{\rho }}^{\ast };\hspace{0.17em}\hspace{0.17em}{\overline{a}}_0\equiv {\overline{a}}_C;\hspace{0.17em}\hspace{0.17em}{I}_S^{\prime }=I_S^{\ast }\hspace{0.17em}\end{array}\right\},$$

(94)

in (93), the expression of the acceleration energy of first order becomes the following:

$$E_A^{\left(1\right)}={\displaystyle \frac{1}{2}}·M·{\overline{a}}_C^T·{\overline{a}}_C+{\displaystyle \frac{1}{2}}·{\overline{\epsilon }}^T·I_S^{\ast }·\overline{\epsilon }+{\overline{\epsilon }}^T·\left(\overline{\omega }\times I_S^{\ast }·\overline{\omega }\right)+{\displaystyle \frac{1}{2}}·{\overline{\omega }}^T·\left[{\overline{\omega }}^T·I_S^{\ast }·\overline{\omega }\right]·\overline{\omega }.$$

(95)

In Equation (95), $I_S^{\ast }$ represent the axial–centrifugal inertia tensor with respect to a system having the origin in the mass center.

It can be noticed that Equation (95) includes (89), to which the term (90) is added.

Unlike (95), Equation (89) is obtained by applying some differential transforms on the Lagrange’s equations of the first kind.

Because Equation (87) is obtained by applying the partial derivative with respect to the generalized accelerations ${\ddot{q}}_j$, Equation (89) will not include the term (90).

As a result, the first-order Lagrange Equation (73) transform into the Gibbs–Appell equations [14,15,16] and are expressed in the following form:

$${\displaystyle \frac{𝜕\hspace{0.17em}{E}_A^{\left(1\right)}}{𝜕{\ddot{q}}_j}}=Q_j,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}where\hspace{0.17em}\hspace{0.17em}j=1\to 6.$$

(96)

The components of the energy of accelerations of first order are as below:

$$E_A^{\left(1\right)TR}={\displaystyle \frac{1}{2}}·M·a_C^2={\displaystyle \frac{1}{2}}·M·\left({\ddot{x}}_C^2+{\ddot{y}}_C^2+{\ddot{z}}_C^2\right),$$

(97)

$$\begin{array}{c}{E}_A^{\left(1\right)ROT}={\displaystyle \frac{1}{2}}·\left(I_x·{\epsilon }_x^2+I_y·{\epsilon }_y^2+I_z·{\epsilon }_z^2\right)+\left(I_z-I_y\right)·{\epsilon }_x·{\omega }_y·{\omega }_z+\\ +\left(I_x-I_z\right)·{\epsilon }_y·{\omega }_z·{\omega }_x+\left(I_y-I_x\right)·{\epsilon }_z·{\omega }_x·{\omega }_y.\end{array}$$

(98)

By applying and developing Gibbs–Appell-type Equation (96), the following differential equations are obtained:

$$\left\{\begin{array}{l}{\displaystyle \frac{𝜕}{𝜕{\ddot{q}}_j}}\left[{\displaystyle \frac{1}{2}}·M·{\overline{a}}_C^2\right]·\left(1-{\Delta }_j\right)={\displaystyle \frac{𝜕\hspace{0.17em}{E}_A^{\left(1\right)TR}}{𝜕{\ddot{x}}_C}}·\left(1-{\Delta }_j\right)=Q_1=R_x^{\ast },\\ {\displaystyle \frac{𝜕}{𝜕{\ddot{q}}_j}}\left[{\displaystyle \frac{1}{2}}·M·{\overline{a}}_C^2\right]·\left(1-{\Delta }_j\right)={\displaystyle \frac{𝜕\hspace{0.17em}{E}_A^{\left(1\right)TR}}{𝜕{\ddot{y}}_C}}·\left(1-{\Delta }_j\right)=Q_2=R_y^{\ast },\\ {\displaystyle \frac{𝜕}{𝜕{\ddot{q}}_j}}\left[{\displaystyle \frac{1}{2}}·M·{\overline{a}}_C^2\right]·\left(1-{\Delta }_j\right)={\displaystyle \frac{𝜕\hspace{0.17em}{E}_A^{\left(1\right)TR}}{𝜕{\ddot{z}}_C}}·\left(1-{\Delta }_j\right)=Q_3=R_z^{\ast }.\end{array}\right.$$

(99)

The differential Equations (99) are referred to as Gibbs—Appell equations, applied to a rigid body in general and uniformly varied mechanical motion. According to the direct dynamic model, these differential equations introduce as unknown only the generalized variables, representing the independent parameters of the general and uniformly varied mechanical motion of the rigid body.

$$\left\{\begin{array}{l}{\displaystyle \frac{𝜕\hspace{0.17em}{E}_A^{\left(1\right)ROT}}{𝜕{\ddot{q}}_j}}·{\Delta }_j={\displaystyle \frac{𝜕\hspace{0.17em}{E}_A^{\left(1\right)ROT}}{𝜕{\ddot{\alpha }}_x}}=Q_4=M_x^{\ast },\\ {\displaystyle \frac{𝜕\hspace{0.17em}{E}_A^{\left(1\right)ROT}}{𝜕{\ddot{q}}_j}}·{\Delta }_j={\displaystyle \frac{𝜕\hspace{0.17em}{E}_A^{\left(1\right)ROT}}{𝜕{\ddot{\beta }}_y}}=Q_5=M_y^{\ast },\\ {\displaystyle \frac{𝜕\hspace{0.17em}{E}_A^{\left(1\right)ROT}}{𝜕{\ddot{q}}_j}}·{\Delta }_j={\displaystyle \frac{𝜕\hspace{0.17em}{E}_A^{\left(1\right)ROT}}{𝜕{\ddot{\gamma }}_z}}=Q_6=M_z^{\ast }.\end{array}\right.$$

(100)

$$\mathrm{From} \mathrm{where} \mathrm{it} \mathrm{results} \mathrm{that} I_S^{\ast }·\overline{\epsilon }+\left(\overline{\omega }\times I_S^{\ast }·\overline{\omega }\right)={\overline{M}}_C^{\ast }.$$

Therefore, the differential equations of the rigid body in uniform, general, and absolute motion are obtained. Through differential transformations applied to the Gibbs–Appell equations, the following expressions are derived:

$${\displaystyle \frac{𝜕\hspace{0.17em}\stackrel{\left(m-2\right)}{E_A^{\left(1\right)}}}{𝜕\stackrel{\left(m\right)}{q_j}}}=Q_j,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}where\hspace{0.17em}\hspace{0.17em}j=1\to 6.$$

(101)

In Equation (101), $m\ge 2$ represents the derivative order with respect to time.

Consequently, the system of differential equations is obtained. The system includes the higher-order accelerations typical of rapid motions in the rigid body undergoing general and uniformly varied motion.

These equations embody a general form of the Gibbs–Appell equations, customized for advanced system mechanics.

3. Results

In this study, the mathematical model developed was applied to a mechanical system consisting of two rods $O_1{O}_2$ and $O_{2}P$, connected by rotational joints, as shown in Figure 4. The mechanical system can be assimilated to a planar robot (2 DOF).

The rods are homogeneous, of equal lengths $l$ and masses $M$. Initially, the mechanical system is in a state of equilibrium.

A point force ${\overline{P}}_S$ is applied at the characteristic point (see Figure 4). The applied force ${\overline{P}}_S$ is known, with specified magnitude and orientation.

Given the initial conditions of the mechanical system’s motion, $\left\{\hspace{0.17em}{q}_i=0\hspace{0.17em},\hspace{0.17em}\hspace{0.17em}{\dot{q}}_i=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}i=1\hspace{0.17em},\hspace{0.17em}2\hspace{0.17em}\right\}$, the purpose of this study is to determine the expression for the acceleration energy and then of the differential equations of motion using the classical approach (Lagrange equations of second kind) and the Gibbs–Appell equations, with the acceleration energy as the central function.

3.1. Using the Classical Approach to Obtain the Differential Equations and Driving Force

First, the kinetic energy of the mechanical system is calculated using König’s theorem, rewritten in a symbolic form [13], as follows:

$$\begin{gathered} K{E}^i={\left(-1\right)}^{{\Delta }_{\hspace{0.17em}M}}·{\displaystyle \frac{1-{\Delta }_{\hspace{0.17em}M}}{1+3·{\Delta }_M}}·\left\{{\displaystyle \frac{1}{2}}·M_i·{}^i\overline{v}_{C_i}^T·{}^i\overline{v}_{C_i}\right\}+{\Delta }_{\hspace{0.17em}M}^{\hspace{0.17em}2}\hspace{0.17em}·{\displaystyle \frac{1}{2}}\hspace{0.17em}·{}^i\overline{\omega }_i^T·{}^{i}I_i^{*}·{}^i\overline{\omega }_i, \\ {\Delta }_{\hspace{0.17em}M}=\left\{\left\{-1,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathrm{general}\hspace{0.17em}\hspace{0.17em}\mathrm{motion}\hspace{0.17em}\hspace{0.17em}\mathrm{and} \mathrm{rotation} \mathrm{motion} \hspace{0.17em}\right\};\left\{0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathrm{translation}\hspace{0.17em}\hspace{0.17em}\mathrm{motion}\right\}\hspace{0.17em}\right\}. \end{gathered}$$

(102)

Considering that the rod $O_1{O}_2$ executes an absolute rotational motion while the other rod $O_{2}P$ undergoes a planar parallel motion, the expression for the kinetic energy can be detailed as follows:

$$K{E}^1\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\displaystyle \frac{1}{2}}\hspace{0.17em}·{\displaystyle \frac{M·l^2}{3}}·{\dot{q}}_1^2,$$

(103)

$$K{E}^2\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\displaystyle \frac{1}{2}}\hspace{0.17em}\hspace{0.17em}·M·v_{C_2}^2\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\displaystyle \frac{1}{2}}·\hspace{0.17em}{\displaystyle \frac{M·l^2}{12}}·{\dot{q}}_2^2\hspace{0.17em}.$$

(104)

The position of the mass center $C_1$, corresponding to the first rod, as well as the position of the mass center $C_2$ for the second rod, is determined using the relations below:

$${\overline{r}}_{C_1}=\left(\begin{array}{c}{x}_{C_1}\\ y_{C_1}\\ 0\end{array}\right)=\hspace{0.17em}{\hspace{0.17em}}_1^0\left[R\right]·{}^1\overline{p}_{21}=\left[\begin{array}{ccc}c{q}_1& -s{q}_1& 0\\ s{q}_1& c{q}_1& 0\\ 0& 0& 1\end{array}\right]·\left(\hspace{0.17em}\begin{array}{c}l/2\\ 0\\ 0\end{array}\right)=\left(\begin{array}{c}\hspace{0.17em}{\displaystyle \frac{l}{2}}·c{q}_1\\ {\displaystyle \frac{l}{2}}·s{q}_1\end{array}\right),$$

(105)

$$\begin{array}{c}{\overline{r}}_{C_2}=\left(\begin{array}{c}{x}_{C_2}\\ y_{C_2}\\ 0\end{array}\right)=\hspace{0.17em}{\hspace{0.17em}}_1^0\left[R\right]·{}^1\overline{p}_{21}{+}_2^0\left[R\right]·{}^1\overline{r}_{C_{21}},\\ {\overline{r}}_{C_2}=\left[\begin{array}{ccc}c{q}_1& -s{q}_1& 0\\ s{q}_1& c{q}_1& 0\\ 0& 0& 1\end{array}\right]·\left(\begin{array}{c}l\\ 0\\ 0\end{array}\right)+\left[\begin{array}{ccc}c\left(q_1+q_2\right)& -s\left(q_1+q_2\right)& 0\\ s\left(q_1+q_2\right)& c\left(q_1+q_2\right)& 0\\ 0& 0& 1\end{array}\right]·\left(\begin{array}{c}l/2\\ 0\\ 0\end{array}\right)=\left(\begin{array}{c}\hspace{0.17em}l·c{q}_1+{\displaystyle \frac{l}{2}}·c\left(q_1+q_2\right)\\ \hspace{0.17em}l·s{q}_1+{\displaystyle \frac{l}{2}}·s\left(q_1+q_2\right)\\ 0\end{array}\right).\end{array}$$

(106)

The velocities of the mass center corresponding to the two rods are calculated as:

$${\overline{v}}_{C_1}=\left(\begin{array}{l}{\dot{x}}_{C_1}\\ {\dot{y}}_{C_1}\end{array}\right)=\left(\begin{array}{l}-{\displaystyle \frac{l}{2}}·{\dot{q}}_1·s{q}_1\\ {\displaystyle \frac{l}{2}}·{\dot{q}}_1·c{q}_1\end{array}\right),$$

(107)

$${\overline{v}}_{C_2}=\left(\begin{array}{c}{\dot{x}}_{C_2}\\ {\dot{y}}_{C_2}\\ 0\end{array}\right)=\left(\begin{array}{c}-l·{\dot{q}}_1·s{q}_1-{\displaystyle \frac{1}{2}}·l·\left({\dot{q}}_1+{\dot{q}}_2\right)·s\left(q_1+q_2\right)\\ l·{\dot{q}}_1·c{q}_1+{\displaystyle \frac{1}{2}}·l·\left({\dot{q}}_1+{\dot{q}}_2\right)·c\left(q_1+q_2\right)\\ 0\end{array}\right).$$

(108)

The squares of the velocities of the mass centers necessary for the calculus of the kinetic and acceleration energy are determined as follows:

$$v_{C_1}^2\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\displaystyle \frac{1}{4}}\hspace{0.17em}·l^2·{\dot{q}}_1^2.$$

(109)

$$v_{C_2}^2\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\displaystyle \frac{5}{4}}·l^2·{\dot{q}}_1^2\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\displaystyle \frac{1}{4}}\hspace{0.17em}·l^2·{\dot{q}}_2^2\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}{l}^2·{\dot{q}}_1^2·c{q}_2+\hspace{0.17em}{\displaystyle \frac{1}{2}}\hspace{0.17em}·l^2·{\dot{q}}_1·{\dot{q}}_2+l^2·{\dot{q}}_1·{\dot{q}}_2·c\hspace{0.17em}{q}_2.$$

(110)

The total kinetic energy of the mechanical system is calculated by considering the motion performed by the two rods. By applying König’s theorem for kinetic energy, the following results:

$$\begin{array}{c}KE={\displaystyle \frac{5}{6}}·M·l^2·{\dot{q}}_1^2+{\displaystyle \frac{1}{6}}·M·l^2·{\dot{q}}_2^2+{\displaystyle \frac{1}{2}}·M·l^2·{\dot{q}}_1^2·\mathrm{c}{q}_2+\\ +{\displaystyle \frac{1}{3}}·M·l^2·{\dot{q}}_1·{\dot{q}}_2+{\displaystyle \frac{1}{2}}·M·l^2·{\dot{q}}_1·{\dot{q}}_2·c{q}_2.\end{array}$$

(111)

The vector ${\overline{r}}_S$ describing the position of point $P$ is defined by scalar components:

$${\overline{r}}_S=\left(\begin{array}{l}{x}_S\\ y_S\end{array}\right)=\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\left(\begin{array}{c}l·c{q}_1+l·c{q}_2\\ l·s{q}_1+l·s{q}_2\end{array}\right).$$

(112)

The motion equations result by applying Lagrange’s equations of the second kind:

$$\left\{\begin{array}{l}{\displaystyle \frac{d}{dt}}\left({\displaystyle \frac{𝜕KE}{𝜕{\dot{q}}_1}}\right)-{\displaystyle \frac{𝜕KE}{𝜕q_1}}=Q_{1ö}^{\left(1\right)},\\ {\displaystyle \frac{d}{dt}}\left({\displaystyle \frac{𝜕KE}{𝜕{\dot{q}}_2}}\right)-{\displaystyle \frac{𝜕KE}{𝜕q_2}}=Q_{2ö}^{\left(2\right)}.\end{array}\right.$$

(113)

The Lagrange’s equations applied for the first component of the mechanical system are determined by applying the first equation of the system (113). By calculating the differentials of the kinetic energy, the following expressions are obtained:

$${\displaystyle \frac{𝜕KE}{𝜕q_1}}=0,$$

(114)

$${\displaystyle \frac{𝜕KE}{𝜕{\dot{q}}_1}}={\displaystyle \frac{5}{3}}·M·l^2·{\dot{q}}_1+M·l^2·{\dot{q}}_1·c{q}_2+{\displaystyle \frac{1}{3}}·M·l^2·{\dot{q}}_2+{\displaystyle \frac{1}{2}}·M·l^2·{\dot{q}}_2·c{q}_2,$$

(115)

$$\begin{array}{c}{\displaystyle \frac{d}{dt}}\left({\displaystyle \frac{𝜕KE}{𝜕{\dot{q}}_1}}\right)={\displaystyle \frac{5}{3}}·M·l^2·{\ddot{q}}_1+{\displaystyle \frac{1}{3}}·M·l^2·{\ddot{q}}_2+M·l^2·{\ddot{q}}_1·c{q}_2+\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\displaystyle \frac{1}{2}}·M·l^2·{\ddot{q}}_2·c{q}_2-M·l^2·{\dot{q}}_1·{\dot{q}}_2·s{q}_2-{\displaystyle \frac{1}{2}}·M·l^2·{\dot{q}}_2^2·s{q}_2.\end{array}$$

(116)

The differential equations of motion corresponding to the first element:

$$\begin{array}{c}{Q}_{1\mathrm{ö}}^1={\displaystyle \frac{d}{dt}}\left({\displaystyle \frac{𝜕KE}{𝜕{\dot{q}}_1}}\right)-{\displaystyle \frac{𝜕KE}{𝜕q_1}}={\displaystyle \frac{5}{3}}·M·l^2·{\ddot{q}}_1+{\displaystyle \frac{1}{3}}·M·l^2·{\ddot{q}}_2+M·l^2·{\ddot{q}}_1·c{q}_2+\\ +{\displaystyle \frac{1}{2}}·M·l^2·{\ddot{q}}_2·c{q}_2-M·l^2·{\dot{q}}_1·{\dot{q}}_2·s{q}_2+{\displaystyle \frac{1}{2}}·M·l^2·{\dot{q}}_2^2·s{q}_2.\end{array}$$

(117)

Further, the differential equations of motion corresponding to the second element:

$${\displaystyle \frac{𝜕KE}{𝜕q_2}}=-{\displaystyle \frac{1}{2}}·M·l^2·{\dot{q}}_1^2·s{q}_2-{\displaystyle \frac{1}{2}}·M·l^2·{\dot{q}}_1·{\dot{q}}_2·s{q}_2,$$

(118)

$${\displaystyle \frac{𝜕KE}{𝜕{\dot{q}}_2}}={\displaystyle \frac{1}{3}}·M·l^2·{\dot{q}}_2+{\displaystyle \frac{1}{3}}·M·l^2·{\dot{q}}_1+{\displaystyle \frac{1}{2}}·M·l^2·{\dot{q}}_1·c{q}_2,$$

(119)

$${\displaystyle \frac{d}{dt}}\left({\displaystyle \frac{𝜕KE}{𝜕{\dot{q}}_2}}\right)={\displaystyle \frac{1}{3}}·M·l^2·{\ddot{q}}_2+{\displaystyle \frac{1}{3}}·M·l^2·{\ddot{q}}_1+{\displaystyle \frac{1}{2}}·M·l^2·{\ddot{q}}_1·c{q}_2-{\displaystyle \frac{1}{2}}·M·l^2·{\dot{q}}_1·{\dot{q}}_2·s{q}_2,$$

(120)

$$Q_{2\mathrm{ö}}^{\left(2\right)}={\displaystyle \frac{1}{3}}·M·l^2·{\ddot{q}}_2+{\displaystyle \frac{1}{3}}·M·l^2·{\ddot{q}}_1+{\displaystyle \frac{1}{2}}·M·l^2·{\ddot{q}}_1·c{q}_2+{\displaystyle \frac{1}{2}}·M·l^2·{\dot{q}}_1^2·s{q}_2.$$

(121)

The generalized driving forces for the two links are obtained using the following general expression:

$$Q_{i\mathrm{ö}}^i+Q_g^{\hspace{0.17em}i}+Q_S^{\hspace{0.17em}i}=Q_m^{\hspace{0.17em}i},$$

(122)

where $Q_g^{\hspace{0.17em}i}$ represents the generalized gravitational force and $Q_S^{\hspace{0.17em}i}$ is the external generalized force [14]. The following are the defining expressions for these two forces:

$$Q_g^{\hspace{0.17em}i}={\overline{P}}_1·{\displaystyle \frac{𝜕{\overline{r}}_{C_1}}{𝜕q_i}}+{\overline{P}}_2·{\displaystyle \frac{𝜕{\overline{r}}_{C_2}}{𝜕q_i}},$$

(123)

$$Q_S^i={\overline{P}}_S·{\displaystyle \frac{𝜕{\overline{r}}_S}{𝜕q_i}}$$

(124)

The generalized gravitational forces for the two links are determined with (123):

$$Q_g^1={\overline{P}}_1·{\displaystyle \frac{𝜕{\overline{r}}_{C_1}}{𝜕q_1}}+{\overline{P}}_2·{\displaystyle \frac{𝜕{\overline{r}}_{C_2}}{𝜕q_1}}={\displaystyle \frac{3}{2}}M·g·l·c{q}_1+M·g·l·c\left(q_1+q_2\right),$$

(125)

$$Q_g^2={\overline{P}}_1·{\displaystyle \frac{𝜕{\overline{r}}_{C_1}}{𝜕q_2}}+{\overline{P}}_2·{\displaystyle \frac{𝜕{\overline{r}}_{C_2}}{𝜕q_2}}=-M·g·{\displaystyle \frac{l}{2}}·c\left(q_1+q_2\right).$$

(126)

The generalized external force $Q_S^i$ is calculated based on (124), as follows:

$$Q_S^1={\overline{P}}_S·{\displaystyle \frac{𝜕{\overline{r}}_S}{𝜕q_1}}=M_S·g·l·c{q}_1,$$

(127)

$$Q_S^2={\overline{P}}_S·{\displaystyle \frac{𝜕{\overline{r}}_S}{𝜕q_2}}=M_S·g·l·c{q}_2.$$

(128)

By substituting (117), (121), and (125)–(128) in Equation (122), the generalized driving force $Q_m^{\hspace{0.17em}i}$ for the two links is obtained:

$$\begin{array}{c}{Q}_m^1={\displaystyle \frac{5}{3}}·M·l^2·{\ddot{q}}_1+{\displaystyle \frac{1}{3}}·M·l^2·{\ddot{q}}_2+M·l^2·{\ddot{q}}_1·c{q}_2-{\displaystyle \frac{1}{2}}·M·l^2·{\dot{q}}_2^2·s{q}_2+{\displaystyle \frac{1}{2}}·M·l^2·{\ddot{q}}_2·c{q}_2-\\ -M·l^2·{\dot{q}}_1·{\dot{q}}_2·s{q}_2+{\displaystyle \frac{3}{2}}M·g·l·c{q}_1+M·g·l·c\left(q_1+q_2\right)+M_S·g·l·c{q}_1,\end{array}$$

(129)

$$\begin{array}{c}{Q}_m^2={\displaystyle \frac{1}{3}}·M·l^2·{\ddot{q}}_2+{\displaystyle \frac{1}{3}}·M·l^2·{\ddot{q}}_1+{\displaystyle \frac{1}{2}}·M·l^2·{\ddot{q}}_1·\mathrm{c}{q}_2+\\ +{\displaystyle \frac{1}{2}}M·l^2·{\dot{q}}_1^2·s{q}_2-M·g·{\displaystyle \frac{l}{2}}·c\left(q_1+q_2\right)+M_S·g·l·c{q}_2.\end{array}$$

(130)

3.2. Using Acceleration Energy to Obtain the Differential Equations and Driving Force

The acceleration energy of first order is calculated according to [13] by using the following expression:

$$E_A^{\left(1\right)}\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}{\displaystyle \sum _{i=1}^2\left[\frac{1}{2}·M_i·{\dot{\overline{v}}}_{C_i}^2+\frac{1}{2}·{\overline{\epsilon }}_i^T·{}^{i}I_i^{*}·{\overline{\epsilon }}_i+{\overline{\epsilon }}_i^T·\left({\overline{\omega }}_i\times {}^{i}I_i^{*}·{\overline{\omega }}_i\right)\right]}\hspace{0.17em}+{\displaystyle \sum _{i=1}^2\frac{1}{2}·{\overline{\omega }}_i^T·\left({\overline{\omega }}_i^T·I_S·{\overline{\omega }}_i\right)·{\overline{\omega }}_i}.$$

(131)

The components of the acceleration of the mass centers for the two rods are obtained by differentiating, with respect to time, the matrix Equations (107) and (108):

$${\ddot{x}}_{C_1}=-\hspace{0.17em}{\displaystyle \frac{l}{2}}·\left({\dot{q}}_1^2·c{q}_1+{\ddot{q}}_1·s{q}_1\right),$$

(132)

$${\ddot{y}}_{C_1}={\displaystyle \frac{l}{2}}·\left({\ddot{q}}_1·c{q}_1-{\dot{q}}_1^2·s{q}_1\right),$$

(133)

$$\begin{array}{c}{\ddot{x}}_{C_2}=-l·{\ddot{q}}_1·s{q}_1-l·{\dot{q}}_1·{\dot{q}}_2·c\left(q_1+q_2\right)-{\displaystyle \frac{l}{2}}·\left({\dot{q}}_1^2+{\dot{q}}_2^2\right)·c\left(q_1+q_2\right)-\\ -{\displaystyle \frac{l}{2}}·\left({\ddot{q}}_1+{\ddot{q}}_2\right)·s\left(q_1+q_2\right)-l·{\dot{q}}_1^2·c{q}_1,\end{array}$$

(134)

$$\begin{array}{c}{\ddot{y}}_{C_2}=l·{\ddot{q}}_1·c{q}_1-l·{\dot{q}}_1·{\dot{q}}_2·s\left(q_1+q_2\right)-{\displaystyle \frac{l}{2}}·\left({\dot{q}}_1^2+{\dot{q}}_2^2\right)·s\left(q_1+q_2\right)+\\ +{\displaystyle \frac{l}{2}}·\left({\ddot{q}}_1+{\ddot{q}}_2\right)·c\left(q_1+q_2\right)-l·{\dot{q}}_1^2·s{q}_1.\end{array}$$

(135)

By squaring Equations (132) and (133), and (134) and (135), respectively, and then adding them up two by two, the result is as follows:

$${\dot{v}}_{C_1}^2\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}{\ddot{x}}_{C_1}^2+\hspace{0.17em}\hspace{0.17em}{\ddot{y}}_{C_1}^2\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}{\displaystyle \frac{l^2}{4}}·\left({\dot{q}}_1^4+{\ddot{q}}_1^2\right),$$

(136)

$$\begin{array}{l}{\dot{v}}_{C_2}^2\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}{\ddot{x}}_{C_2}^2+\hspace{0.17em}\hspace{0.17em}{\ddot{y}}_{C_2}^2\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}{\displaystyle \frac{5}{4}}·l^2·{\ddot{q}}_1^2+l^2·{\ddot{q}}_1^2·c{q}_2+{\displaystyle \frac{1}{4}}·l^2·{\ddot{q}}_2^2\hspace{0.17em}+{\displaystyle \frac{5}{4}}·l^2·{\dot{q}}_1^4+l^2·q_1^4·c{q}_2+{\displaystyle \frac{1}{4}}·l^2·{\dot{q}}_2^4+\\ \hspace{1em}\hspace{1em}+{\displaystyle \frac{l^2}{2}}·{\ddot{q}}_1·{\ddot{q}}_2+l^2·{\ddot{q}}_1·{\ddot{q}}_2·c{q}_2+{\displaystyle \frac{3}{2}}·l^2·{\dot{q}}_1^2·{\dot{q}}_2^2+l^2·{\dot{q}}_1^2·{\dot{q}}_2^2·c{q}_2+l^2·{\dot{q}}_1^3·{\dot{q}}_2+l^2·{\dot{q}}_2^3·{\dot{q}}_1+\\ \hspace{1em}\hspace{1em}+l^2·{\dot{q}}_1^2·{\ddot{q}}_2·s{q}_2-l^2·{\dot{q}}_2^2·{\ddot{q}}_1·s{q}_2-2·l^2·{\dot{q}}_1^3·{\dot{q}}_2·c{q}_2-2·l^2·{\dot{q}}_1·{\dot{q}}_2·{\ddot{q}}_1·s{q}_2.\end{array}$$

(137)

The angular velocities (${\overline{\omega }}_i$) and accelerations (${\overline{\epsilon }}_i$) relative to the mobile system $\left\{i\right\}$ are determined using the iterative algorithm from kinematics. Additionally, the axial–centrifugal inertia tensor with respect to the mass center, ${}^{i}I_i^{*}$ and the inertia tensor with respect to system $\left\{0^{\prime }\right\}$, denoted with $I_S$ for the two links are determined by considering them as homogenous rods.

For the mechanical system presented in Figure 4, the acceleration energy is written the following form:

$$\begin{array}{l}{E}_A^{\left(1\right)}={\displaystyle \frac{5}{6}}·M·l^2·{\ddot{q}}_1^2+{\displaystyle \frac{1}{2}}·M·l^2·{\ddot{q}}_1^2·c{q}_2+{\displaystyle \frac{1}{6}}·M·l^2·{\ddot{q}}_2^2+\\ \hspace{1em}\hspace{1em}+{\displaystyle \frac{1}{2}}·M·l^2·{\ddot{q}}_1·{\ddot{q}}_2·c{q}_2-M·l^2·{\dot{q}}_1·{\dot{q}}_2·{\ddot{q}}_1·\mathrm{s}{q}_2+\\ \hspace{1em}\hspace{1em}+{\displaystyle \frac{1}{3}}·M·l^2·{\ddot{q}}_1·{\ddot{q}}_2+{\displaystyle \frac{1}{2}}·M·l^2·{\dot{q}}_1^2·{\ddot{q}}_2·s{q}_2-{\displaystyle \frac{1}{2}}·M·l^2·{\dot{q}}_2^2·{\ddot{q}}_1·s{q}_2.\end{array}$$

(138)

Considering Gibbs–Appell Equation (96) in the expression of the acceleration energy, the terms containing ${\dot{q}}^4$ and $\dot{q}·{\dot{q}}^3$ were considered zero [12,13]. The generalized inertia forces ${Q^i}_{iö}$ are obtained by applying the following differential expression:

$$Q_{i\mathrm{ö}}^i={\displaystyle \frac{𝜕E_A^{\left(1\right)}\hspace{0.17em}\hspace{0.17em}}{𝜕{\ddot{q}}_i}}.$$

(139)

The generalized forces for the two links are calculated as follows:

$$\begin{array}{l}{Q}_{1\mathrm{ö}}^1={\displaystyle \frac{𝜕E_A^{\left(1\right)}\hspace{0.17em}\hspace{0.17em}}{𝜕{\ddot{q}}_1}}=\hspace{0.17em}{\displaystyle \frac{5}{3}}·M·l^2·{\ddot{q}}_1+{\displaystyle \frac{1}{3}}·M·l^2·{\ddot{q}}_2+M·l^2·{\ddot{q}}_1·c{q}_2-\\ \hspace{1em}\hspace{1em}-{\displaystyle \frac{1}{2}}·M·l^2·{\dot{q}}_2^2·s{q}_2+{\displaystyle \frac{1}{2}}·M·l^2·{\ddot{q}}_2·c{q}_2-M·l^2·{\dot{q}}_1·{\dot{q}}_2·s{q}_2.\end{array}$$

(140)

$$Q_{2\mathrm{ö}}^2={\displaystyle \frac{𝜕E_A^{\left(1\right)}\hspace{0.17em}\hspace{0.17em}}{𝜕{\ddot{q}}_2}}={\displaystyle \frac{1}{3}}·M·l^2·{\ddot{q}}_2+{\displaystyle \frac{1}{3}}·M·l^2·{\ddot{q}}_1+{\displaystyle \frac{1}{2}}·M·l^2·{\ddot{q}}_1·\mathrm{c}{q}_2+{\displaystyle \frac{1}{2}}M·l^2·{\dot{q}}_1^2·s{q}_2.$$

(141)

By comparing the terms in Equations (140) and (141) with those from Equations (117) and (121), it is evident that the two expressions are identical. Therefore, applying the derivative with respect to ${\ddot{q}}_1$ and ${\ddot{q}}_2$ to the expression of the first-order acceleration energy, it results in the same differential equations of motion as obtained using Lagrange’s equations of the first kind.

For the analyzed mechanical system, the dynamics equations are determined according to [12,13] as follows:

$${\displaystyle \frac{𝜕E_A^{\hspace{0.17em}\left(1\right)}\hspace{0.17em}\hspace{0.17em}}{𝜕{\ddot{q}}_i}}+Q_g^{\hspace{0.17em}i}+Q_S^{\hspace{0.17em}i}=Q_m^{\hspace{0.17em}i},$$

(142)

where $Q_g^i$, $Q_S^i$, and $Q_m^i$ retain the same meaning as in (122) and are defined in the same manner as in (123)–(128).

The generalized driving force $Q_m^{\hspace{0.17em}i}$ for the two links is obtained by substituting the results from (140), (141), and (125)–(128) into general Equation (142):

$$\begin{array}{l}{Q}_m^1={\displaystyle \frac{5}{3}}·M·l^2·{\ddot{q}}_1+{\displaystyle \frac{1}{3}}·M·l^2·{\ddot{q}}_2+M·l^2·{\ddot{q}}_1·c{q}_2-{\displaystyle \frac{1}{2}}·M·l^2·{\dot{q}}_2^2·s{q}_2+{\displaystyle \frac{1}{2}}·M·l^2·{\ddot{q}}_2·c{q}_2-\\ \hspace{1em}\hspace{1em}-M·l^2·{\dot{q}}_1·{\dot{q}}_2·s{q}_2+{\displaystyle \frac{3}{2}}M·g·l·c{q}_1+M·g·l·c\left(q_1+q_2\right)+M_S·g·l·c{q}_1,\end{array}$$

(143)

$$\begin{array}{l}{Q}_m^2={\displaystyle \frac{1}{3}}·M·l^2·{\ddot{q}}_2+{\displaystyle \frac{1}{3}}·M·l^2·{\ddot{q}}_1+{\displaystyle \frac{1}{2}}·M·l^2·{\ddot{q}}_1·\mathrm{c}{q}_2+\\ \hspace{1em}\hspace{1em}+{\displaystyle \frac{1}{2}}M·l^2·{\dot{q}}_1^2·s{q}_2-M·g·{\displaystyle \frac{l}{2}}·c\left(q_1+q_2\right)+M_S·g·l·c{q}_2.\end{array}$$

(144)

Equations (143) and (144) represent the equations of motion corresponding to the mechanical structure presented in Figure 4 obtained based on the acceleration energy.

It can be noticed that the expressions for the driving forces obtained using the classical approach in (129) and (130) are identical to those derived from Gibbs–Appell equations, where the central function is considered the acceleration energy in (143) and (144).

The advantage of using acceleration energy to determine differential equations is evident. While applying Lagrange’s equations involves three differentiation operations, using acceleration energy requires only a single differentiation to achieve the same result.

4. Discussion

The D’Alembert–Lagrange principle plays an essential role in modern engineering, especially in fields that require precise modeling and control of dynamic systems, such as robotics. Lagrange’s formulation of the principle provides a unified mathematical model, using generalized coordinates to obtain equations of motion, which simplifies the solution to complex mechanical problems. The main advantage of applying the D’Alembert–Lagrange principle consists in its ability to convert complex mechanical problems into solvable equations of motion.

This study emphasizes the principle’s application in advanced mechanics, particularly in the design and analysis of complex dynamic systems. A specific instance of the d’Alembert–Lagrange principle is the principle of virtual work, which applies to static equilibrium. This paper demonstrates that the generalized forces represent the static equilibrium equations of the free rigid body.

Using the d’Alembert–Lagrange principle, this paper highlights, through differential transformations, the importance of the expression of first-order acceleration energy and its implementation as a central function in Gibbs–Appell equations.

This paper is a theoretical study that aims to highlight the advantages of using acceleration energy to determine the differential equations of motion and the generalized driving forces, compared to the classical approach based on the Lagrange equations of the second kind. For this purpose, this study considered a mechanical structure with two degrees of freedom (DOFs), specifically a planar robot consisting of two rods connected by rotational joints. The rods are homogeneous, with equal lengths $l$ and masses $M$, initially in a state of equilibrium. A point force was applied at a characteristic point of the structure. Both the classical Lagrange approach and the acceleration energy method were applied to this system.

It was found that while both approaches yielded the same results, the method employing acceleration energy required only a single differentiation with respect to the generalized accelerations ${\ddot{q}}_1$ and ${\ddot{q}}_2$ to reach the desired outcome.

In contrast, the classical approach required three differentiation operations to achieve the same result. This highlights the efficiency and effectiveness of using acceleration energy in dynamic system analysis. The D’Alembert–Lagrange principle proves to be a fundamental tool in classical mechanics, significantly contributing to the precise modeling and analysis of dynamic systems in engineering and robotics.

This study demonstrated that by incorporating acceleration energy into the model, it is possible to achieve enhanced robustness and accuracy in predicting system behavior.

When external forces induce acceleration, accounting for the resultant energy changes is essential for maintaining energy balance and obtaining accurate motion equations. The included case study exemplifies the practical application of the D’Alembert–Lagrange principle, showcasing how acceleration energy can be used for obtaining accurate motion equations.

Future research could explore the implementation of acceleration energy into the mathematical modeling model of complex and high-speed dynamic systems. Moreover, applying this analysis to autonomous vehicles and advanced robotic structures could enhance their performance and safety features.