Fact-checked by Grok 2 weeks ago

Rigid body dynamics

Rigid body dynamics is a branch of classical mechanics that studies the motion of rigid bodies—idealized solid objects in which the distances between any two points remain fixed, regardless of the forces applied to the body.^[1] This field analyzes both translational motion of the body's center of mass and rotational motion about it, treating the body as an infinite collection of particles constrained to maintain relative positions.^[2] Key to this analysis is the assumption of infinite rigidity, neglecting deformations, which simplifies the description to six degrees of freedom: three for translation and three for rotation.^[3] The kinematics of rigid bodies involves describing orientation using rotation matrices or Euler angles, with angular velocity \vec{\omega} defined as the instantaneous axis and rate of rotation.^[2] For dynamics, Newton's laws extend to rotational forms: the net torque \vec{\tau} = I \vec{\alpha}, where I is the moment of inertia tensor—a symmetric 3x3 matrix characterizing rotational inertia about the center of mass—and \vec{\alpha} is angular acceleration.^[4] The inertia tensor can be diagonalized into principal moments along principal axes, simplifying equations for symmetric bodies like rods or disks.^[2] In the absence of external torques, angular momentum \vec{L} = I \vec{\omega} is conserved, leading to stable rotations for symmetric bodies but complex behaviors like precession or nutation for asymmetric ones, governed by Euler's equations: I_1 \dot{\omega}_1 + (\ I_3 - I_2\ )\omega_2 \omega_3 = \tau_1, and cyclic permutations.^[2] These principles underpin applications in engineering, such as vehicle stability, robotics, and spacecraft attitude control, and in physics, including planetary motion and gyroscope design.^[4] Lagrangian and Hamiltonian formulations further generalize rigid body dynamics for constrained systems or non-inertial frames.^[3]

Fundamentals

Rigid body definition

In classical mechanics, a rigid body is defined as a system composed of multiple particles where the distances between any pair of particles remain constant over time, regardless of external forces applied to the body.^[5]^[3] This implies infinite stiffness, such that the body undergoes no deformation—neither elastic nor plastic—under loading, allowing its motion to be described solely by the translation and rotation of the entire structure as a unit.^[6]^[7] The key assumptions underlying this model include the neglect of any internal deformations, which simplifies analysis by treating the body as an idealization valid primarily in non-relativistic regimes where particle velocities are much less than the speed of light.^[5]^[3] In contrast to deformable bodies, where inter-particle distances can vary due to strain, rigid bodies enforce strict holonomic constraints maintained by idealized internal forces, making them a useful approximation for many engineering and physical systems like solid objects in low-speed collisions or rotations.^[7]^[6] Mathematically, the rigidity condition is expressed through constraint equations that preserve inter-particle distances: for particles with position vectors \vec{r}_i and \vec{r}_j, |\vec{r}_i - \vec{r}_j| = c_{ij} = constant, where c_{ij} is a fixed length for each pair i, j.^[7]^[5]^[3] These constraints reduce the system's degrees of freedom from $3N (for N unconstrained particles) to 6 in three dimensions: three for translational motion and three for rotational orientation. The concept of the rigid body was formalized in the 17th and 18th centuries, with Isaac Newton laying foundational principles in his Philosophiæ Naturalis Principia Mathematica (1687) by applying laws of motion to extended bodies, and Leonhard Euler advancing the analytical treatment in works such as Mechanica (1736) and Theoria motus corporum solidorum (1765), which introduced systematic methods for describing rotational dynamics.^[8]^[9]

Kinematics and dynamics distinction

In rigid body dynamics, kinematics refers to the branch of study that describes the geometric aspects of motion—specifically, the position, velocity, and acceleration of a rigid body—without considering the forces or other physical causes that produce it. For rigid bodies, this involves analyzing rigid transformations, such as translations of the center of mass and rotations about it, which maintain fixed distances between all points on the body. These transformations are typically represented using the position vector of the center of mass \mathbf{x}(t) and a rotation matrix R(t), from which linear velocity \mathbf{v}(t) = \dot{\mathbf{x}}(t) and angular velocity \boldsymbol{\omega}(t) are derived.^[10]^[11] Dynamics, in contrast, builds upon kinematics by incorporating the physical interactions that govern how motion changes over time, primarily through the application of forces and torques. It employs Newton's laws extended to rigid bodies: the net force equals the mass times the acceleration of the center of mass, and the net torque equals the rate of change of angular momentum. This allows prediction of the body's response to external influences, such as how applied forces alter \mathbf{v}(t) and \boldsymbol{\omega}(t). Unlike kinematics, dynamics accounts for mass distribution via properties like the inertia tensor, enabling quantitative analysis of acceleration and rotational behavior.^[10] The interconnection between kinematics and dynamics is fundamental, as kinematic quantities serve as the state variables in dynamic formulations. For instance, the current position and orientation from kinematics define the configuration used to compute torques from forces (e.g., \boldsymbol{\tau} = \sum (\mathbf{r}_i - \mathbf{x}) \times \mathbf{f}_i), which in turn determine the time evolution of velocities. This linkage ensures that dynamic equations, such as \dot{\mathbf{P}} = \mathbf{F} for linear momentum \mathbf{P} = M \mathbf{v}, directly utilize kinematic descriptions to model real-world motion.^[10]^[11] A representative example highlighting this distinction is pure rolling without slipping, such as a wheel moving on a flat surface. Kinematically, this imposes a nonholonomic constraint linking linear velocity \dot{\mathbf{r}}_P and angular velocity \boldsymbol{\omega}, expressed as \dot{\mathbf{r}}_P = R \tilde{n} \boldsymbol{\omega} (where R is the radius and \tilde{n} a skew-symmetric matrix for the contact normal), ensuring the contact point has zero relative velocity. Dynamically, however, friction at the contact provides the necessary torque to enforce this constraint, modeled via Lagrange multipliers \lambda in the equations of motion, with the friction force g_c = D^T \lambda (where D is the constraint Jacobian) limited by the static friction coefficient to prevent slipping. If the required friction exceeds \mu_s f_N (normal force times coefficient), the motion transitions to sliding, illustrating how dynamics reveals the physical limits of kinematic assumptions.^[12]

Planar rigid body motion

Translational and rotational degrees of freedom

In planar rigid body motion, the system is confined to a two-dimensional plane, which imposes constraints that reduce the total degrees of freedom to three: two for translation along the x and y directions, and one for rotation about the axis perpendicular to the plane.^[13]^[14] This limitation arises because the rigid body maintains its shape while undergoing combined translation and rotation within the plane, unlike a free particle in 2D which would have only two translational degrees of freedom.^[13] The configuration of a planar rigid body is fully described by the position of its center of mass—specified by coordinates (x, y)—and the orientation angle θ, which measures the rotation relative to a fixed reference frame.^[14] These three parameters suffice to determine the location and attitude of every point on the body, as the fixed geometry dictates the relative positions of all material points once the center of mass and orientation are known.^[5] The rigidity of the body enforces holonomic constraints, where the distances between any pair of points remain constant, expressible as equality constraints on their coordinates.^[5]^[6] These constraints eliminate relative deformations, reducing the independent coordinates needed to describe the motion from those of an unconstrained system of particles to the three aforementioned degrees of freedom.^[6] To visualize this, consider a uniform disk confined to the xy-plane: its motion can be tracked by the (x, y) position of its center and the angle θ of a radius vector from the positive x-axis, allowing arbitrary paths like rolling without slipping or pure sliding.^[14] Similarly, a thin rod in the plane has its configuration specified by the center of mass coordinates and θ, enabling analysis of scenarios such as pivoting about one end or free translation with rotation.^[13]

Equations of motion in the plane

In planar rigid body motion, the equations of motion describe how external forces and torques govern the translation and rotation of the body, building on its three degrees of freedom: two for the position of the center of mass and one for orientation. The translational equation arises from Newton's second law applied to the center of mass, stating that the net external force equals the mass times the acceleration of the center of mass. In scalar components for the plane, this is expressed as:

F_x = m \ddot{x}_{cm}, \quad F_y = m \ddot{y}_{cm}

where m is the mass, (x_{cm}, y_{cm}) are the coordinates of the center of mass, and F_x, F_y are the components of the net force.^[15]^[13] The rotational equation governs the angular motion about the center of mass, where the net torque equals the moment of inertia times the angular acceleration. For planar motion, this scalar equation is:

\tau_{cm} = I_{cm} \alpha

with \tau_{cm} the net torque about the center of mass, I_{cm} the moment of inertia about that point, and \alpha = \ddot{\theta} the angular acceleration, where \theta is the orientation angle.^[15]^[13] These equations couple through the application of forces: if an external force acts at a point offset from the center of mass by position vector \vec{r}, it contributes to both translation via \vec{F} and rotation via the torque \vec{r} \times \vec{F}, which is perpendicular to the plane and equals r F \sin\phi in magnitude, where \phi is the angle between \vec{r} and \vec{F}. This coupling is essential for general plane motion, where translation and rotation occur simultaneously.^[13]^[14] A representative example is the physical pendulum, a rigid body pivoted at a fixed point and oscillating under gravity in the plane. For a physical pendulum of mass m, with the center of mass at distance l_{cm} from the pivot and moment of inertia I_p about the pivot, the rotational equation about the pivot yields:

I_p \ddot{\theta} = -m g l_{cm} \sin\theta

where \theta is the angular displacement from the vertical; the torque arises from gravity acting at the center of mass, offset from the pivot. For small angles, \sin\theta \approx \theta, this approximates simple harmonic motion with period T \approx 2\pi \sqrt{I_p / (m g l_{cm})}. In this case, translation of the center of mass is coupled to rotation about the pivot, with no net external force in the tangential direction beyond the pivot reaction.^[16]

Three-dimensional kinematics

Orientation representations

In three-dimensional space, the orientation of a rigid body is parameterized using various representations that map the body's coordinate frame to a reference frame, capturing the three rotational degrees of freedom while preserving distances and angles. These methods differ in parameterization, computational requirements, and susceptibility to singularities, influencing their suitability for applications like robotics, aerospace, and computer graphics.^[17] Euler angles describe the orientation through three successive rotations about specific axes, typically denoted as φ (precession), θ (nutation), and ψ (spin). The sequence often follows a 3-2-1 or z-x-z convention, where the body undergoes a rotation by φ about the fixed z-axis, followed by θ about the intermediate x'-axis, and ψ about the body's z-axis. This approach, introduced by Leonhard Euler in the 18th century, provides an intuitive geometric interpretation but suffers from gimbal lock, a singularity where two rotation axes align, reducing the effective degrees of freedom to two (e.g., at θ = π/2).^[18]^[19]^[17] Tait-Bryan angles, a variant of Euler angles, employ asymmetric sequences such as yaw-pitch-roll (ψ, θ, φ) in a 3-2-1 order, rotating about the z-axis (yaw), then y'-axis (pitch), and x''-axis (roll). Named after Peter Guthrie Tait and George H. Bryan, who applied them to aircraft stability in the early 20th century, these angles are prevalent in aerospace and vehicle dynamics due to their alignment with intuitive control inputs like heading and attitude. Like Euler angles, they exhibit singularities, such as gimbal lock at pitch angles of ±π/2, limiting their range and complicating interpolation.^[19]^[17] Rotation matrices represent orientation as a 3×3 orthogonal matrix R with determinant 1, belonging to the special orthogonal group SO(3), where columns (or rows) are the unit vectors of the body's axes expressed in the reference frame. This parameterization uses nine elements subject to six constraints (orthogonality and unit norm), effectively yielding three independent parameters. It avoids singularities and directly facilitates vector transformations via \mathbf{v}' = R \mathbf{v}, but requires normalization to maintain the unit determinant and can be computationally expensive for composition or inversion compared to parametric methods.^[17] Quaternions parameterize orientation using a four-dimensional unit vector \mathbf{q} = (q_0, q_1, q_2, q_3) with \|\mathbf{q}\| = 1, where the scalar part q_0 = \cos(\alpha/2) and vector part (q_1, q_2, q_3) = \mathbf{n} \sin(\alpha/2) encode a rotation by angle α about unit axis \mathbf{n}. Invented by William Rowan Hamilton in 1843 and applied to rigid body rotations by 1848, quaternions eliminate singularities like gimbal lock, enabling smooth interpolation (e.g., slerp) and efficient composition via quaternion multiplication. The unit norm constraint requires periodic renormalization, and their non-commutative nature demands careful handling, though they offer superior numerical stability for simulations.^[20]^[17] Comparisons among these representations highlight trade-offs: Euler and Tait-Bryan angles use three intuitive parameters but introduce singularities that can cause discontinuities in simulations, whereas rotation matrices provide a singularity-free but redundant nine-parameter form suitable for linear algebra operations. Quaternions balance efficiency with robustness, using four constrained parameters to outperform angles in avoiding gimbal lock and reducing computational overhead for angular updates, making them preferred in high-fidelity dynamics like spacecraft attitude control.^[17]

Angular velocity and acceleration

In three-dimensional rigid body kinematics, the angular velocity vector \vec{\omega} characterizes the instantaneous rotational motion of the body, specifying both the axis of rotation (the direction of \vec{\omega}) and the magnitude of the rotation rate (the scalar |\vec{\omega}|). This vector arises from the constraint that all points in a rigid body maintain fixed distances from each other, leading to a velocity field for any point at position \vec{r} relative to a reference point on the body given by \vec{v} = \vec{\omega} \times \vec{r}.^[6] The angular acceleration \vec{\alpha} is defined as the time derivative of the angular velocity, \vec{\alpha} = \frac{d\vec{\omega}}{dt}, representing the rate of change of the rotation axis or speed.^[21] To relate angular velocity to orientation parameters, such as those introduced in prior discussions of rotation representations, explicit expressions connect \vec{\omega} to the rates of change of these parameters. For Euler angles (\phi, \theta, \psi) defining the orientation via successive rotations about body-fixed axes \hat{n}_1, \hat{n}_2, \hat{n}_3, the angular velocity is \vec{\omega} = \dot{\phi} \hat{n}_1 + \dot{\theta} \hat{n}_2 + \dot{\psi} \hat{n}_3.^[18]^[22] Similarly, for unit quaternions \mathbf{q} parameterizing orientation, the quaternion derivative satisfies \dot{\mathbf{q}} = \frac{1}{2} \mathbf{q} \circ \boldsymbol{\omega}, where \boldsymbol{\omega} = (0, \omega_x, \omega_y, \omega_z) is the quaternion form of the angular velocity vector, allowing \vec{\omega} to be recovered from \dot{\mathbf{q}}.^[23] In the body-fixed frame, where vectors are expressed relative to the rotating body axes, the time derivative of any vector \vec{A} follows Poisson's kinematic equation: \left( \frac{d\vec{A}}{dt} \right)_b = \frac{{}^s d\vec{A}}{dt} - \vec{\omega} \times \vec{A}, with the superscript s denoting the derivative in the space-fixed frame; this relation accounts for the frame's rotation and is particularly useful for tracking body-frame quantities like \vec{\omega} itself.^[24]

Three-dimensional dynamics

Newton's laws for rigid bodies

In rigid body dynamics, Newton's laws are extended from point particles to systems of particles that maintain fixed relative positions, allowing the treatment of the body as a whole while accounting for both translational and rotational motion. The first law states that a rigid body remains at rest or in uniform motion unless acted upon by external forces or torques, implying zero net force and zero net torque for equilibrium. The third law applies pairwise to interactions between particles within or external to the body, ensuring action-reaction pairs. The second law, central to dynamics, takes vector forms for translation and rotation in three dimensions.^[25]^[26] For translation, the net external force \vec{F} on a rigid body equals the time derivative of its linear momentum \vec{p}, which for constant mass m simplifies to \vec{F} = \frac{d\vec{p}}{dt} = m \vec{a}_{\text{cm}}, where \vec{a}_{\text{cm}} is the acceleration of the center of mass. This equation holds in any inertial frame and treats the body as a point mass located at the center of mass for overall linear motion, independent of internal forces that maintain rigidity. In planar rigid body motion, this reduces to scalar components along the axes.^[25]^[27] For rotation, the net external torque \vec{\tau} about a point equals the time derivative of the angular momentum \vec{L} about the same point: \vec{\tau} = \frac{d\vec{L}}{dt}. This general form applies to three-dimensional motion and holds whether the reference point is fixed in an inertial frame or moving with the body. Unlike the translational case, the rotational equation depends on the choice of reference point, as angular momentum varies with it.^[27]^[26] The equivalence of points for applying these laws is key: the translational equation always uses the center of mass, but for rotation, torques and angular momenta can be computed equivalently about the center of mass or a fixed point, provided consistency is maintained; using other points introduces additional terms from the motion of the reference. This choice simplifies analysis for constrained systems, such as those pivoted at a fixed point.^[26]^[27] A representative example is a free rigid body under uniform gravity, where the gravitational force \vec{W} = m \vec{g} acts effectively at the center of mass, producing translational acceleration \vec{a}_{\text{cm}} = \vec{g} without net torque about the center of mass, resulting in pure translation if initially at rest rotationally. If the body has initial angular momentum, it rotates steadily about the center of mass while the center translates uniformly under gravity.^[28]^[27]

Mass properties and inertia tensor

In rigid body dynamics, the mass properties characterize the distribution of mass within the body, which governs both its translational and rotational behavior. The center of mass, denoted as \vec{r}_{cm}, is the point where the body's mass can be considered concentrated for translational motion, defined as \vec{r}_{cm} = \frac{1}{m} \int \vec{r} \, dm, where m is the total mass and the integral is over the body's volume.^[29] This position is independent of the coordinate system and serves as the reference for computing other mass properties.^[30] The moments of inertia quantify the body's resistance to rotational acceleration about specific axes passing through the center of mass. For a Cartesian coordinate system aligned with the body, the principal moments are given by

I_{xx} = \int (y^2 + z^2) \, dm, \quad I_{yy} = \int (x^2 + z^2) \, dm, \quad I_{zz} = \int (x^2 + y^2) \, dm,

where the integrals are taken over the mass distribution, and these are always positive scalars.^[30] The principal axes are the orthogonal directions along which the inertia tensor diagonalizes, corresponding to the eigenvectors of the tensor, with the principal moments as eigenvalues; these axes simplify rotational calculations by eliminating cross-coupling terms.^[31] The inertia tensor \mathbf{I} encapsulates all moments and products of inertia in a 3×3 symmetric matrix relative to the center of mass:

\mathbf{I} = \begin{pmatrix} I_{xx} & -I_{xy} & -I_{xz} \\ -I_{xy} & I_{yy} & -I_{yz} \\ -I_{xz} & -I_{yz} & I_{zz} \end{pmatrix},

where the products of inertia are I_{xy} = \int xy \, dm (and similarly for others), measuring mass asymmetry.^[30] Due to its symmetry (I_{ij} = I_{ji}), the tensor has only six independent components and is positive definite, ensuring positive principal moments and stability in rotational dynamics.^[31] This tensor relates the angular momentum \vec{H} to the angular velocity \vec{\omega} through \vec{H} = \mathbf{I} \vec{\omega}.^[29] To compute the inertia tensor about an arbitrary point O displaced from the center of mass by \vec{a}, the parallel axis theorem applies:

I_{ij}' = I_{ij} + m (a_k a_k \delta_{ij} - a_i a_j),

where \delta_{ij} is the Kronecker delta, extending the moments and products accordingly.^[30] For common shapes, these properties yield simple forms; for a thin uniform rod of length L and mass m about its center perpendicular to the length, I = \frac{1}{12} m L^2.^[32] For a solid uniform sphere of radius R and mass m about a diameter, the principal moments are I_{xx} = I_{yy} = I_{zz} = \frac{2}{5} m R^2, with off-diagonal terms zero due to symmetry.^[33] A rectangular plate of sides a and b and mass m, about its center, has I_{xx} = \frac{1}{12} m b^2 (about the x-axis parallel to side a) and I_{yy} = \frac{1}{12} m a^2.^[30]

Force and torque equations

In three-dimensional rigid body dynamics, the equations governing the response to external forces and torques combine Newton's second law for the center of mass with the rotational dynamics derived from angular momentum. The translational motion of the rigid body follows the familiar form of Newton's second law applied to the center of mass: the total external force \vec{F} equals the mass m times the acceleration of the center of mass \vec{a}_{cm}, or \vec{F} = m \vec{a}_{cm}.^[34] This equation decouples the overall translation from the rotational behavior, allowing the body's center of mass to accelerate independently of internal rotations.^[27] For rotational motion, the net external torque \vec{\tau} about the center of mass equals the time derivative of the angular momentum \vec{L}, expressed in an inertial frame as \vec{\tau} = \frac{d\vec{L}}{dt}.^[2] The angular momentum of a rigid body is given by \vec{L} = \mathbf{I} \vec{\omega}, where \mathbf{I} is the inertia tensor and \vec{\omega} is the angular velocity vector.^[35] In a body-fixed frame rotating with the body, the time derivative accounts for the frame's motion, yielding the general torque equation \vec{\tau} = \mathbf{I} \vec{\alpha} + \vec{\omega} \times (\mathbf{I} \vec{\omega}), where \vec{\alpha} = \dot{\vec{\omega}} is the angular acceleration.^[2] This nonlinear equation captures the coupling between rotation rates due to the body's geometry. When the body-fixed frame aligns with the principal axes of the inertia tensor—where \mathbf{I} is diagonal with principal moments I_1, I_2, I_3—the equations simplify to Euler's equations:

\begin{align} I_1 \dot{\omega}_1 + (I_3 - I_2) \omega_2 \omega_3 &= \tau_1, \\ I_2 \dot{\omega}_2 + (I_1 - I_3) \omega_3 \omega_1 &= \tau_2, \\ I_3 \dot{\omega}_3 + (I_2 - I_1) \omega_1 \omega_2 &= \tau_3. \end{align}

These were originally derived by Leonhard Euler in the 18th century for the rotational dynamics of rigid bodies.^[8] The cross terms reflect the Coriolis-like effects arising from the changing orientation of the principal axes in the inertial frame. In numerical simulations of rigid body motion, direct integration of Euler angles can lead to singularities such as gimbal lock, so quaternion representations are preferred for orientation to ensure robust, singularity-free propagation of the rotational state alongside the Euler equations.^[36] A classic example is torque-free motion (\vec{\tau} = 0), where angular momentum is conserved, but the angular velocity vector \vec{\omega} precesses around the principal axis with the largest or smallest moment of inertia, resulting in nutation—a wobbling motion observable in the rotation of asymmetric objects like a spinning tennis racket flipped in flight./13%3A_Rigid-body_Rotation/13.20%3A_Torque-free_rotation_of_an_inertially-symmetric_rigid_rotor) This instability, known as the tennis racket theorem, arises from the intermediate moment of inertia being unstable for steady rotation.

Principle of virtual work

Virtual displacements and work

In rigid body dynamics, a virtual displacement refers to an infinitesimal change \delta \vec{r} in the position of a point that is compatible with the geometric constraints of the system at an instant in time, independent of the actual time evolution of the motion.^[37] This concept allows analysis of equilibrium or motion without specifying the path taken, focusing instead on possible variations consistent with rigidity and other constraints.^[38] For a rigid body, the virtual displacement of any point P can be decomposed into the virtual translation of the center of mass \delta \vec{r}_{\rm cm} and the contribution from an infinitesimal virtual rotation \delta \vec{\theta}. Specifically,

\delta \vec{r}_P = \delta \vec{r}_{\rm cm} + \delta \vec{\theta} \times \vec{r},

where \vec{r} is the position vector from the center of mass to P. This expression ensures that all points in the body maintain their relative positions during the virtual variation, preserving the body's rigidity.^[37]^[38] The virtual work \delta W performed by the applied forces acting on the rigid body is defined as the dot product sum over all points,

\delta W = \sum_i \vec{F}_i \cdot \delta \vec{r}_i,

where \vec{F}_i are the forces at each point. For a rigid body, this integral over the body simplifies to the equivalent work done by the net force \vec{F} at the center of mass and the net torque \vec{\tau} about the center of mass:

\delta W = \vec{F} \cdot \delta \vec{r}_{\rm cm} + \vec{\tau} \cdot \delta \vec{\theta}.

This reduction highlights how translational and rotational effects contribute separately to the total virtual work.^[37] Ideal constraints, such as frictionless contacts or rigid links, exert forces that perform no virtual work because the virtual displacements are orthogonal to these constraint forces. This orthogonality condition enables the exclusion of constraint forces from the virtual work calculation, reducing the problem to a smaller set of independent variables or generalized coordinates that inherently satisfy the constraints.^[38]

Generalized forces and coordinates

In rigid body dynamics, generalized coordinates provide a minimal set of independent parameters that fully describe the configuration of the system, reducing the complexity of analysis by accounting for constraints inherent to rigid bodies. For a three-dimensional rigid body, six generalized coordinates are typically required: three to specify the position of a reference point, such as the center of mass, and three to describe the orientation, often using parameters like Euler angles.^[39] This choice ensures the coordinates are complete and independent, capturing all possible configurations without redundancy.^[40] The principle of virtual work, when expressed in terms of these generalized coordinates q_j, takes the form \delta W = \sum_j Q_j \delta q_j, where Q_j are the generalized forces corresponding to each coordinate. These generalized forces are derived from the applied forces acting on the body as Q_j = \sum_i \vec{F}_i \cdot \frac{\partial \vec{r}_i}{\partial q_j}, with \vec{r}_i denoting the position of the point of application of force \vec{F}_i. This formulation projects the physical forces onto the coordinate directions, facilitating the analysis of equilibrium or motion.^[40] In a planar rigid body example, with generalized coordinates consisting of the translational position x and rotation angle \theta, the generalized force for translation simplifies to the x-component of the net force Q_x = F_x, while for rotation it equals the net torque about the reference point Q_\theta = \tau. Extending to three dimensions, the generalized forces incorporate vector components for all six coordinates, such as linear forces for positional coordinates and torque components for orientational ones, allowing consistent treatment across translations and rotations.^[40] Generalized forces Q_j naturally include contributions from non-conservative forces, such as dissipative effects like friction or external control inputs, which cannot be derived from a potential function. These are incorporated by computing the virtual work done by such forces during infinitesimal displacements in the generalized coordinates, ensuring the framework handles real-world applications involving energy loss or actuation.^[41]^[42]

D'Alembert's principle

Inertia forces in virtual work

D'Alembert's principle, introduced in 1743, extends the principle of virtual work from static equilibrium to dynamic systems by incorporating inertial effects as fictitious forces. In his Traité de dynamique, Jean le Rond d'Alembert proposed treating the inertia of particles as additional "forces" opposing acceleration, allowing dynamic problems to be reformulated as static ones where the total virtual work vanishes.^[43] This approach unifies the analysis of motion under constraints by adding these inertia terms to the applied forces. For a system of particles, the principle states that the virtual work done by the applied forces \delta W plus the virtual work of the inertia forces \delta W^{in} equals zero: \delta W + \delta W^{in} = 0. The inertia force on each particle i is defined as -\vec{F}_i^{in} = -m_i \vec{a}_i, where m_i is the mass and \vec{a}_i the acceleration. Thus, \delta W^{in} = \sum_i (-m_i \vec{a}_i) \cdot \delta \vec{r}_i, ensuring that for admissible virtual displacements \delta \vec{r}_i consistent with constraints, the system behaves as if in equilibrium.^[44] In rigid body dynamics, the inertia forces simplify due to the body's rigidity. The translational inertia contributes a term -m \vec{a}_{cm} \cdot \delta \vec{r}_{cm}, where m is the total mass and \vec{a}_{cm} the acceleration of the center of mass \vec{r}_{cm}. For rotation, the virtual work of inertia forces includes the angular acceleration term -\vec{\alpha} \cdot \mathbf{I} \delta \vec{\theta} and the centrifugal term -\vec{\omega} \times \mathbf{I} \vec{\omega} \cdot \delta \vec{\theta}, where \vec{\alpha} is the angular acceleration, \vec{\omega} the angular velocity, \mathbf{I} the inertia tensor, and \delta \vec{\theta} the virtual angular displacement. These terms can be equivalently represented by a resultant inertia force at the center of mass and an inertia torque about it.^[45] When using generalized coordinates q_j to describe the system's configuration, the inertia effects are captured by generalized inertia forces Q_j^{in} = -\sum_i m_i \vec{a}_i \cdot \frac{\partial \vec{r}_i}{\partial q_j}, where the sum is over all particles and \vec{r}_i is the position of particle i. This formulation projects the inertia forces onto the directions of the generalized displacements, facilitating the transition to equation of motion derivations without explicitly resolving constraint forces.^[46]

Derivation of Lagrange's equations

The derivation of Lagrange's equations for rigid body dynamics begins with D'Alembert's principle, which states that the total virtual work performed by the applied forces and the inertia forces on the system is zero for any virtual displacement consistent with the constraints.^[47] For a rigid body, this principle is expressed in terms of generalized coordinates q_j that describe the configuration, such as the position of the center of mass and orientation parameters (e.g., Euler angles). The virtual work equation takes the form \sum_j (Q_j + Q_j^{in}) \delta q_j = 0, where Q_j are the generalized applied forces and Q_j^{in} are the generalized inertia forces.^[47] Since the variations \delta q_j are arbitrary and independent, each coefficient must vanish individually, yielding Q_j + Q_j^{in} = 0 for each j.^[47] To connect this to the Lagrangian formulation, the inertia terms are related to the kinetic energy T of the rigid body. The kinetic energy is given by T = \frac{1}{2} m v_{cm}^2 + \frac{1}{2} \vec{\omega} \cdot \mathbf{I} \vec{\omega}, where m is the mass, v_{cm} is the velocity of the center of mass, \vec{\omega} is the angular velocity, and \mathbf{I} is the inertia tensor evaluated at the center of mass.^[48] In generalized coordinates, T depends on the q_j, \dot{q}_j, and possibly time, and the generalized inertia force is Q_j^{in} = -\frac{d}{dt} \left( \frac{\partial T}{\partial \dot{q}_j} \right) + \frac{\partial T}{\partial q_j}.^[47] Substituting this into the virtual work equation gives \sum_j \left[ Q_j - \frac{d}{dt} \left( \frac{\partial T}{\partial \dot{q}_j} \right) + \frac{\partial T}{\partial q_j} \right] \delta q_j = 0, and again, each term must separately be zero: Q_j = \frac{d}{dt} \left( \frac{\partial T}{\partial \dot{q}_j} \right) - \frac{\partial T}{\partial q_j}.^[47] For systems with potential energy V, the Lagrangian is defined as L = T - V. If the applied forces are conservative (derivable from V), they contribute to Q_j = -\frac{\partial V}{\partial q_j}; otherwise, Q_j represents non-conservative generalized forces Q_j^{nc}. The equations then become the Euler-Lagrange equations: \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}_j} \right) - \frac{\partial L}{\partial q_j} = Q_j^{nc}.^[47] In rigid body dynamics, the coordinate-dependent form of T (e.g., expressing \vec{\omega} in terms of rotational coordinates like Euler angles) ensures the equations capture both translational and rotational motion while respecting the rigidity constraint.^[48] This approach offers significant advantages in rigid body dynamics, as it naturally incorporates holonomic constraints by selecting generalized coordinates that satisfy them, thereby eliminating the need to explicitly account for constraint forces.^[44] Additionally, it readily distinguishes scleronomic constraints, which are time-independent and lead to a time-invariant Lagrangian, from rheonomic constraints, which explicitly depend on time and introduce time-varying terms in the equations.^[49]

Conservation laws

Linear and angular momentum

In rigid body dynamics, the linear momentum \vec{p} is defined as the product of the body's total mass m and the velocity of its center of mass \vec{v}_{cm}, yielding \vec{p} = m \vec{v}_{cm}. This quantity represents the total translational momentum of the body, analogous to that of a point particle with the same mass and velocity at the center of mass. The time evolution of linear momentum follows Newton's second law for the system, where the net external force \vec{F} equals the rate of change: \frac{d\vec{p}}{dt} = \vec{F}.^[50] The angular momentum \vec{L}_O about an arbitrary point O provides a measure of the body's rotational momentum relative to that point. For a rigid body, it decomposes into the orbital contribution from the center of mass motion and the spin contribution due to rotation about the center of mass, expressed as \vec{L}_O = \vec{r}_{cm} \times \vec{p} + \mathbf{I}_{cm} \vec{\omega}, where \vec{r}_{cm} is the position vector from O to the center of mass, \mathbf{I}_{cm} is the inertia tensor evaluated at the center of mass, and \vec{\omega} is the angular velocity vector. The time derivative of angular momentum equals the net external torque \vec{\tau}_O about O: \frac{d\vec{L}_O}{dt} = \vec{\tau}_O. This relation generalizes the rotational analog of Newton's second law to extended bodies.^[51] When analyzing rigid body motion in different reference frames, the transport theorem accounts for the rotation of the body-fixed frame relative to the inertial frame. For the angular momentum \vec{L}, the inertial-frame derivative relates to the body-frame derivative by \left( \frac{d\vec{L}}{dt} \right)_I = \left( \frac{d\vec{L}}{dt} \right)_B + \vec{\omega} \times \vec{L}, where the cross-product term arises from the frame's rotation. This equation is essential for deriving equations of motion in body coordinates, such as Euler's equations.^[52] In the absence of external torques (\vec{\tau}_O = 0), the angular momentum \vec{L}_O about a fixed point O remains constant in the inertial frame, illustrating conservation. For example, a symmetric rigid body, such as a spinning satellite in free space, maintains a fixed \vec{L}_O vector, with its \vec{\omega} precessing if the principal moments of inertia differ, as governed by the body's geometry.^[53]

Impulse and momentum theorems

In rigid body dynamics, the impulse-momentum theorems provide integral forms of the equations of motion, relating impulsive forces or torques acting over a short time interval to the resulting changes in linear and angular momentum. These theorems are particularly useful for analyzing sudden events such as collisions or impacts, where forces vary rapidly and may be difficult to integrate directly over time.^[54]^[55] The linear impulse-momentum theorem states that the linear impulse, defined as the time integral of the external force, equals the change in linear momentum of the rigid body. Mathematically, this is expressed as

\int_{t_1}^{t_2} \vec{F} \, dt = \Delta \vec{p} = m (\vec{v}_2 - \vec{v}_1),

where \vec{p} = m \vec{v} is the linear momentum, m is the mass, and \vec{v}_1, \vec{v}_2 are the velocities before and after the impulse, respectively. This relation holds for the center of mass motion and assumes the body remains rigid during the impulsive event.^[54]^[56] Similarly, the angular impulse-momentum theorem relates the angular impulse to the change in angular momentum about a fixed point or the center of mass. It is given by

\int_{t_1}^{t_2} \vec{\tau} \, dt = \Delta \vec{L},

where \vec{\tau} is the torque and \vec{L} = \mathbf{I} \vec{\omega} is the angular momentum, with \mathbf{I} the inertia tensor and \vec{\omega} the angular velocity. For a rigid body, this equation captures rotational changes induced by impulsive torques, often arising from off-center forces.^[54]^[55] In collisions involving rigid bodies, these theorems are extended using the coefficient of restitution e, which quantifies the elasticity of the impact along the line of contact. Defined as e = -\frac{v'_{n2} - v'_{n1}}{v_{n1} - v_{n2}}, where v_{n1}, v_{n2} are the pre-collision normal components of velocity of the points of contact on bodies 1 and 2, respectively, v'_{n1}, v'_{n2} are the post-collision components, and the normal direction is chosen such that the pre-collision relative velocity v_{n1} - v_{n2} > 0 for approach, e ranges from 0 for perfectly inelastic collisions to 1 for perfectly elastic ones. Combined with the impulse equations, e allows determination of post-collision velocities, assuming no friction or treating tangential components separately.^[57]^[58]^[59] For closed systems of particles that coalesce into a rigid body after collision, the impulse-momentum theorems apply by conserving total linear and angular momentum prior to impact, then using the post-collision rigid body properties to compute the unified motion. This reduction assumes internal impulses cancel in pairs, leaving only external effects to alter the overall momentum.^[56]^[57] Conservation principles follow directly: in a closed system with no external forces, linear momentum \vec{p} remains constant, as \Delta \vec{p} = 0. Likewise, with no external torques, angular momentum \vec{L} is conserved about any point. These hold for rigid bodies as integrated forms of Newton's laws, applicable to isolated systems like free-floating spacecraft during orbital transfers, where angular momentum conservation governs attitude stability without thruster inputs.^[60]^[61] An illustrative example is a ballistic impact, such as a projectile striking a rigid target: the linear impulse from the impact force alters the target's momentum, while any eccentricity induces angular impulse, potentially causing rotation. Solving involves equating impulses to momentum changes and applying the coefficient of restitution for the normal velocity reversal.^[57]^[56]

Applications and examples

Vehicle dynamics

Vehicle dynamics applies principles of rigid body motion to analyze the behavior of ground and air vehicles under forces such as gravity, propulsion, and aerodynamic or contact interactions, emphasizing stability, control, and response in both planar and three-dimensional environments. For ground vehicles, the bicycle model simplifies the dynamics by representing the vehicle as a single front-steered wheel and a rear wheel, capturing planar motion with non-holonomic constraints that restrict lateral slipping at the contact points, thereby enforcing rolling without sliding. This model, rooted in the Whipple framework, incorporates slip angles—the angular difference between the wheel's heading and its velocity vector—to account for tire deformation under cornering, enabling predictions of yaw rate and sideslip for handling analysis.^[62]^[63] Tire forces play a central role in vehicle stability, generating lateral forces proportional to slip angles via semi-empirical models like the Pacejka Magic Formula, which describes nonlinear force-slip relationships for both cornering and braking/acceleration. In three-dimensional extensions, roll dynamics emerge from suspension compliance and load transfer, where lateral accelerations induce body roll moments balanced by tire vertical loads, influencing camber and overall grip in maneuvers like obstacle avoidance. These interactions are modeled in multi-degree-of-freedom frameworks, such as 14-DOF systems, to simulate coupled longitudinal, lateral, and roll responses under varying road conditions.^[63]^[64] For air vehicles, rigid body principles extend to six-degree-of-freedom (6-DOF) equations that couple translational and rotational motions using Euler angles to describe orientation, incorporating aerodynamic forces, thrust, and moments for full 3D trajectory prediction. Stability modes, such as the phugoid—a low-frequency longitudinal oscillation involving speed and altitude variations—and the Dutch roll—a coupled yaw-roll oscillation damped by dihedral effects—arise from linearizations of these equations, guiding control surface designs for damping and response. These modes highlight the interplay of inertia, aerodynamics, and control in maintaining trim during flight.^[65] The Newton-Euler formulation proves efficient for real-time simulation of vehicle dynamics, recursively propagating forces and accelerations through the body hierarchy to compute states like position, velocity, and attitude at high frequencies suitable for hardware-in-the-loop testing and driver-in-the-loop environments. This approach enables integration of complex interactions, such as tire-road friction and aerodynamic loads, in simulations running faster than real-time on standard hardware.^[66] Post-2020 advancements in advanced driver-assistance systems (ADAS) for autonomous vehicles increasingly incorporate rigid body dynamics models to enhance path planning and stability control, with surveys noting the adoption of bicycle and multi-body formulations in perception-action loops for predictive handling in urban scenarios. These integrations leverage sensor fusion to estimate states like slip angles in real-time, improving safety in Level 3+ autonomy by mitigating risks from dynamic obstacles.^[67]^[68]

Spacecraft attitude control

Spacecraft attitude control relies on principles of rigid body dynamics to maintain or adjust orientation in the torque-free or low-torque environment of space, where gravitational gradients, magnetic fields, and atmospheric drag are minimal influences compared to controlled actuators.^[69] The dynamics are governed by the conservation of angular momentum in the absence of external torques, leading to complex rotational behaviors that must be predicted and managed for mission success, such as precise pointing for scientific instruments or communication antennas.^[35] In torque-free motion, the rotational dynamics of a spacecraft are described by Euler's rigid body equations in the principal body frame, where the inertia tensor is diagonal with principal moments I_1, I_2, I_3 (assuming I_1 > I_2 > I_3) and angular velocity components \omega_1, \omega_2, \omega_3:

\begin{align} I_1 \dot{\omega}_1 + (I_3 - I_2) \omega_2 \omega_3 &= 0, \\ I_2 \dot{\omega}_2 + (I_1 - I_3) \omega_3 \omega_1 &= 0, \\ I_3 \dot{\omega}_3 + (I_2 - I_1) \omega_1 \omega_2 &= 0. \end{align}

These equations reveal that pure spin about the intermediate principal axis (I_2) is unstable, as demonstrated by the tennis racket theorem, where small perturbations cause the angular velocity vector to flip between the maximum and minimum inertia axes, resulting in a characteristic "twist" or 180-degree rotation flip.^[35]^[70] This instability arises geometrically from the intersection of the constant angular momentum sphere and the energy ellipsoid in phase space, with the polhode motion tracing closed curves on the inertia ellipsoid that represent the body's angular velocity evolution in the body frame.^[69] Spins about the maximum (I_1) or minimum (I_3) inertia axes are stable, as perturbations lead to nutation that damps toward the original axis in the absence of dissipation, though energy dissipation can destabilize the minimum inertia spin.^[69] Attitude determination estimates the spacecraft's orientation relative to an inertial frame, often using quaternions to represent rotations without singularities.^[71] Star trackers provide high-precision quaternion measurements by identifying star positions in the camera's field of view and matching them to cataloged inertial vectors, achieving accuracies on the order of 0.5 arcminutes (3-sigma).^[71] These measurements are fused with gyroscope data via Kalman filtering, such as the multiplicative extended Kalman filter (MEKF), to propagate attitude estimates and reduce errors from sensor noise and biases; the unscented Kalman filter variant further improves performance by modeling time-varying systematic errors in star tracker outputs using periodic functions.^[71] Control strategies employ actuators to generate torques that counteract disturbances and achieve desired attitudes, with reaction wheels providing momentum exchange for fine adjustments by spinning internal flywheels to conserve total angular momentum, and thrusters delivering direct impulse torques via chemical or electric propulsion for larger maneuvers or desaturation of wheel momentum.^[72] Proportional-derivative (PD) controllers are widely used for their simplicity, generating torques proportional to attitude error and its rate, with global asymptotic stability proven via Lyapunov analysis on the attitude error dynamics, ensuring convergence to the target orientation even under bounded disturbances.^[73] For example, a Lyapunov function V = \frac{1}{2} \mathbf{q}_v^T \mathbf{q}_v + \frac{1}{2} \boldsymbol{\omega}^T \mathbf{J} \boldsymbol{\omega} (where \mathbf{q}_v is the vector part of the quaternion error and \mathbf{J} is the inertia tensor) yields \dot{V} \leq 0, confirming stability when control gains are positive.^[73] Since 2010, the proliferation of CubeSats has driven miniaturized attitude control applications, with over 1,000 launches enabling diverse missions like Earth observation and technology demonstrations.^[72] Reaction wheels in 3U CubeSats like SwampSat II facilitate precise pointing for low-Earth orbit science, while cold-gas thrusters in 8U designs support active debris removal maneuvers with 1-mN thrust.^[72] PD controllers optimized for energy tradeoffs have been applied in solar panel pointing tasks, demonstrating effective stabilization in simulations and on-orbit tests for missions such as LightSail-1, which uses momentum wheels for 90-degree slews.^[72] Advanced electric thrusters, including electrospray and pulsed plasma types, further enhance CubeSat control for formation flying and lunar trajectories, underscoring the scalability of rigid body dynamics principles to small platforms.^[72]

References

[1]
[PDF] Rigid body :
A rigid body is an object for which the distance between any two points on the object remains fixed regardless of the motion of the object.
[2]
[PDF] 3. The Motion of Rigid Bodies - DAMTP
The most general motion of a free rigid body is a translation plus a rotation about some point P. In this section we shall develop the techniques required to ...
[3]
[PDF] Chapter 2 Rigid Body Dynamics - MIT OpenCourseWare
If we know 3 non-collinear points in the body, the remaing points are fully determined by triangulation. The first point has 3 coordinates for translation in 3 ...
[4]
[PDF] Introductory Physics I
This textbook supports teaching introductory physics, including elementary mechanics, for life science, engineering, or potential physics majors, with calculus.<|separator|>
[5]
[PDF] Chapter 4 Rigid Body Motion - Rutgers Physics
In this chapter we develop the dynamics of a rigid body, one in which all interparticle distances are fixed by internal forces of constraint. This is,.
[6]
[PDF] Chapter 4. Rigid Body Motion
Such an abstraction is. called the (absolutely) rigid body; it is a reasonable approximation in many practical problems, including the motion of solid samples.
[7]
[PDF] Classical Mechanics
... constraint x2 + y2 + z2 = a2. Or if the particles make up a rigid body, then the distances between any two particles is constant,. |xi − xj| = const,. (1).
[8]
[PDF] Euler and the dynamics of rigid bodies
7.2. Daniel Bernoulli states in a letter to Euler dated 12 De- cember 1745 that the motion of a rigid bodies is “and extremely difficult problem that will not ...
[9]
[PDF] Euler, Reader of Newton: Mechanics and Algebraic Analysis
and a large number of papers on the same subject, Euler also much contributed to the mechanics of rigid and elastic bodies. (Truesdell, 1960), the mechanics ...
[10]
[PDF] Rigid Body Dynamics (I) - UMD CS
• Definition. • Motivation: forces. (one mass particle:) (entire body:) Image ... Mirtich, “Impulse-based Dynamic Simulation of Rigid Body Systems,”. Ph.D ...
[11]
[PDF] Kinematics - Rigid Body Dynamics
A rigid body is a distributed collection of matter that translates and rotates as a single entity, without deformation or other shape change.
[12]
[PDF] ANALYTICAL MODELING, CONTROL, ENERGY ANALYSIS, AND ...
An example of a kinematic constraint that appears in this dissertation is pure rolling constraint when the system rolls on a surface. Before getting to the ...
[13]
[PDF] Lecture L21 - 2D Rigid Body Dynamics - MIT OpenCourseWare
In particular, the only degrees of freedom of a 2D rigid body are translation and rotation. Parallel Axes. Consider a 2D rigid body which is rotating with ...
[14]
[PDF] General planar motion of a single rigid object
The goal here is to generate equations of motion for general planar motion of a. (planar) rigid object that may roll, slide or be in free flight.
[15]
11.1: Equations of Motion for a Rigid Body in General Plane Motion
May 22, 2022 · Therefore, ⁡ , ⁡ , and ⁡ are called degrees of freedom, and general plane rigid-body motion is called a three-degrees-of-freedom (3-DOF) ...
[16]
[PDF] 8.01SC S22 Chapter 24: Physical Pendulum - MIT OpenCourseWare
Mar 24, 2022 · A physical pendulum consists of a rigid body that undergoes fixed axis rotation about a fixed point S (Figure 24.2). Figure 24.2 Physical ...
[17]
[PDF] Euler Angles, Unit Quaternions, and Rotation Vectors
Oct 20, 2006 · Abstract. We present the three main mathematical constructs used to represent the attitude of a rigid body in three- dimensional space.<|control11|><|separator|>
[18]
[PDF] 3D Rigid Body Dynamics: Euler Angles - MIT OpenCourseWare
Figure 1: Euler Angles. In order to describe the angular orientation and angular velocity of a rotating body, we need three angles. As shown on the figure ...
[19]
The Euler angle parameterization - Rotations
Although the paper [7] dates to the 18th century, it was first published posthumously in 1862. In Euler's papers, he shows how three angles can be used to ...<|control11|><|separator|>
[20]
[PDF] On Quaternions and the Rotation of a Solid Body. By Sir William R ...
Sir William Rowan Hamilton gave an account of some applications of Quaternions to questions connected with the Rotation of a Solid Body.
[21]
Rigid bodies #rkg - Dynamics
A rigid body is an extended area of material that includes all the points inside it, and which moves so that the distances and angles between all its points ...
[22]
[PDF] 7.13 Eulerian Angles - • Rotational degrees of freedom for a rigid body
Since [ψ], [θ], [ϕ] are orthogonal, so is the final matrix [ϕ][θ][ψ]. gives the inverse transformation. • Any possible orientation of the body can be attained ...
[23]
[PDF] Quaternions∗ - Iowa State University
Sep 24, 2024 · The development of quaternions is attributed to W. R. Hamilton [5] in 1843. Legend has it that. Hamilton was walking with his wife Helen at ...
[24]
[PDF] Classical Mechanics - UC Homepages
Nov 30, 2023 · 3.3 Poisson's equation∗. It is sometimes useful to carry over some ... We want to express the angular velocity of the rigid body, ω, w.r.t. to the ...
[25]
Equations of motion for a rigid body - Engineering Mechanics
Note: For a single particle the center of mass and the location of the particle are the same and one recovers Newton's second law. Angular momentum of a ...
[26]
[PDF] A. Rigid Body Kinetics: The Newton-Euler Equations
single particle i (Newton's Second Law and the angular momentum equation): ... • If you choose A to be the center of mass, G, then rG/A = 0. For this case ...<|control11|><|separator|>
[27]
[PDF] Chapter 21 Rigid Body Dynamics: Rotation and Translation about a ...
Jun 21, 2013 · 3) are principles that we shall employ to analyze the motion of a rigid bodies undergoing translation and fixed axis rotation about the center ...Missing: history | Show results with:history
[28]
[PDF] 25. Rigid Body Moving Freely - Galileo and Einstein
The total force on all the particles is a sum of the total external force on the body and the sum of internal forces between particles—but these internal forces ...
[29]
[PDF] An Introduction to Physically Based Modeling: Rigid Body ...
This document introduces rigid body simulation, focusing on unconstrained motion, and the basic structure for simulating rigid body motion.
[30]
[PDF] 3D Rigid Body Dynamics: The Inertia Tensor - MIT OpenCourseWare
Here, r is the position vector relative to the center of mass, v is the velocity relative to the center of mass. We note that, in the above expression, an ...<|control11|><|separator|>
[31]
23. Motion of a Rigid Body: the Inertia Tensor - Galileo and Einstein
Definition of Rigid We're thinking here of an idealized solid, in which the distance between any two internal points stays the same as the body moves around. ...
[32]
10.5: Calculating Moments of Inertia - Maricopa Open Digital Press
Calculate the moment of inertia for uniformly shaped, rigid bodies · Apply the parallel axis theorem to find the moment of inertia about any axis parallel to one ...
[33]
3D Centroid and Mass Moment of Intertia Table - Mechanics Map
Center of Mass and Mass Moments of Inertia for Homogeneous Bodies ; Sphere. Centroid of a Sphere Volume=43πr3 V o l u m e = 4 3 π r 3, Ixx=Iyy=Izz=25mr2 I x x = ...
[34]
https://eng.libretexts.org/Bookshelves/Mechanical_Engineering/Mechanics_Map_(Moore_et_al.)/12%3A_Newtons_Second_Law_for_Rigid_Bodies
[35]
[PDF] Euler's Equations - 3D Rigid Body Dynamics - MIT OpenCourseWare
We now turn to the task of deriving the general equations of motion for a three-dimensional rigid body. These equations are referred to as Euler's equations ...
[36]
Quaternion‐based rigid body rotation integration algorithms for use ...
Aug 6, 2025 · An alternative approach to track the evolution of orientation and angular velocity of a rigid body is to replace the Euler angles by quaternions ...Missing: authoritative | Show results with:authoritative
[37]
[PDF] Chapter 11: Virtual Work
Rigid body displacement of P = translation of A + rotation about A ... The principle of virtual work states that if a body is in equilibrium, then the ...
[38]
[PDF] Principle of Virtual Displacements in Structural Dynamics
The work of the constant force F moving through displacement δDx, (F δDx), is called the virtual work of the external force. And the work of the constant bar ...
[39]
https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Variational_Principles_in_Classical_Mechanics_(Cline)/13%3A_Rigid-body_Rotation/13.02%3A_Rigid-body_Coordinates
[40]
[PDF] Lagrangian Dynamics: Virtual Work and Generalized Forces
Apr 4, 2007 · If we fix x and y, we can still rotate in a range with θ. # degrees of freedom = # of generalized coordinates: True for 2.003J. True for.
[41]
[PDF] Lagrange Handout - MIT OpenCourseWare
The technique used to evaluate these non-conservative generalized forces is to assume the system undergoes a small virtual deflection for which we compute the ...
[42]
Traité de dynamique : D'Alembert - Internet Archive
Feb 17, 2009 · 1743. Publisher: J; B; Coignard. Collection: europeanlibraries. Book ... PDF WITH TEXT download · download 1 file · SINGLE PAGE PROCESSED TIFF ...
[43]
[PDF] Lecture #9 Virtual Work And the Derivation of Lagrange's Equations
• Rigid body is a collection of masses. • Masses held at a fixed distance. → Virtual work for the internal constraints of a rigid body displacement is zero ...
[44]
[PDF] 6. THE PRINCIPLES OF DYNAMICS 6.1. D'Alembert's Principle for a ...
a Rigid Body. It follows from the Statics that a system of inertia forces applied to a rigid body can be replaced by a single force equal to and applied at ...
[45]
[PDF] Chapter 1 D'Alembert's principle and applications
Feb 1, 2014 · 1.2.5 Motion of solid bodies in two dimensions. The inertial virtual work in general for a solid body in two-dimensional rotation and trans-.Missing: rigid | Show results with:rigid<|control11|><|separator|>
[46]
6.3: Lagrange Equations from d'Alembert's Principle
Jun 28, 2021 · The Principle of Virtual Work provides a basis for a rigorous derivation of Lagrangian mechanics. Bernoulli introduced the concept of virtual ...
[47]
13.18: Lagrange equations of motion for rigid-body rotation
Feb 27, 2021 · ... the inertia tensor is diagonal in the principal axis frame, the kinetic energy is given in terms of the principal moments of inertia as.
[48]
6.E: Lagrangian Dynamics (Exercises) - Physics LibreTexts
May 22, 2021 · What kind (holonomic, nonholonomic, scleronomic, rheonomic) of constraint acts on $m$? ... Lagrangian function, the equation of constraint, and ...
[49]
[PDF] Lecture D18 - 2D Rigid Body Dynamics: Equations of Motion
Equations of Motion. The equations describing the general motion of a rigid body follow from the conservation laws for systems of particles established in ...
[50]
Calculating angular momentum - Engineering Mechanics
Angular momentum can be calculated using a fixed point, a fixed point at the body's center of mass, or an arbitrary point on the body.
[51]
[PDF] Spacecraft Dynamics and Control - Matthew M. Peet
Angular momentum of a rigid body can be found as. ⃗H = I⃗ωI where ⃗ωI = [p, q, r]T is the angular rotation vector of the body about the center of mass. • p = ωx is ...
[52]
The Feynman Lectures on Physics Vol. I Ch. 20: Rotation in space
If we sum (20.13) over many particles, the external torque on a system is the rate of change of the total angular momentum: τext=dLtot/dt.
[53]
The Impulse Momentum Theorem for a Rigid Body - Mechanics Map
The Impulse-Momentum Theorem states that the impulse exerted on a body will be equal to the change in momentum of that body.Missing: sources | Show results with:sources
[54]
15.2: Impulse and Momentum for a Rigid Body System
Aug 18, 2024 · Impulse and momentum methods are no different, and we will begin this chapter by defining linear impulse, angular impulse, linear momentum, and angular ...Missing: dynamics sources
[55]
[PDF] 2D Rigid Body Dynamics: Impulse and Momentum
In this lecture, we extend these principles to two dimensional rigid body dynamics. Impulse and Momentum Equations. Linear Momentum. In lecture L22, we ...
[56]
Rigid Body Surface Collisions - Mechanics Map
Specifically, the coefficient of restitution relates the velocities before and after the collision in the normal direction at the point of impact. e=− ...Missing: sources | Show results with:sources
[57]
10.2: Surface Collisions and the Coefficient of Restitution
Jul 28, 2021 · The coefficient of restitution is a number between 0 and 1 that measures the bounciness of the body and the surface in the collision.Missing: rigid | Show results with:rigid
[58]
Rigid Body Collisions | J. Appl. Mech. - ASME Digital Collection
This paper presents a general approach to solving rigid body collisions, using coefficients of restitution and the ratio of tangential to normal impulses.Missing: sources | Show results with:sources
[59]
11.4: Conservation of Angular Momentum - Physics LibreTexts
Mar 16, 2025 · The angular momentum of a system of particles around a point in a fixed inertial reference frame is conserved if there is no net external torque around that ...
[60]
11.3 Conservation of Angular Momentum - OpenStax
Sep 19, 2016 · This law is analogous to linear momentum being conserved when the external force on a system is zero. As an example of conservation of angular ...
[61]
[PDF] Linearized dynamics equations for the balance and steer of a ...
We present canonical linearized equations of motion for the Whipple bicycle model consisting of four rigid laterally-symmetric ideally-hinged parts: front ...
[62]
(PDF) Vehicle dynamics and tire models: An overview - ResearchGate
Oct 26, 2021 · This paper presents fundamentals of vehicle dynamics by introducing vehicle models and tire model, which have been widely adopted for vehicle motion control.
[63]
An overview on vehicle dynamics | International Journal of ...
Oct 29, 2013 · Cebon proposed a three-dimensional vehicle model with 14 DOF considering the longitudinal tire force, lateral tire force and the non-linear ...Missing: source | Show results with:source
[64]
[PDF] Flight Stability and Automatic Control - Iowa State University
The rigid body aircraft equations of motion are developed along with ... Phugoid Dynamics / 8.2.3 Roll Dynamics / 8.2.4 Dutch. Roll Approximation.<|separator|>
[65]
Suitable Two Wheeled Vehicle Dynamics Synthesis for Interactive ...
This paper describes a modelling technique for deriving the motorcycles equation of motion. Based on the recursive Newton-Euler approach adapted to tree ...
[66]
https://www.sciencedirect.com/science/article/pii/S1474667016389315
[67]
A Survey of Vehicle Dynamics Modeling Methods for Autonomous ...
Jul 19, 2025 · This paper presents the first survey of vehicle dynamics modeling methods for autonomous racing. Previous surveys have covered dynamics models for standard ...
[68]
[PDF] Attitude Dynamics Fundamentals - Dr. Hanspeter Schaub
However, as will be shown in the study of the polhodes, the spin about the least moment of inertia is only stable in the absence of energy dissipation. 5.3 ...
[69]
Geometric Origin of the Tennis Racket Effect
### Abstract
[70]
[PDF] Filtering Methods for Error Reduction in Spacecraft Attitude ...
Precision attitude determination for recent and planned space missions typically includes quaternion star trackers. (ST) and three-axis inertial reference ...
[71]
[PDF] CubeSat Technology Past and Present: Current State-of-the-Art ...
An extensive and detailed literature review that includes over. 830 citations has been conducted to provide a comprehensive resource on both NASA and non-NASA ...
[72]
[PDF] Three-axis attitude control design for a spacecraft based on ...
The controller output is the torque required to eliminate error. In this study, actuators (reaction wheels) are modeled and the required torque for the attitude.Missing: PD | Show results with:PD