Nonlinear Systems 17 JAN, 2012

Exact Analytic Solutions for the Rotation of an Axially Symmetric Rigid Body Subjected to a Constant Torque

Exact Analytic Solutions for the Rotation of an Axially Symmetric Rigid Body Subjected to a Constant Torque

New exact analytic solutions are introduced for the rotational motion of a rigid body having two equal principal moments of inertia and subjected to an external torque which is constant in magnitude. In particular, the solutions are obtained for the following cases: (1) Torque parallel to the symmetry axis and arbitrary initial angular velocity; (2) Torque perpendicular to the symmetry axis and such that the torque is rotating at a constant rate about the symmetry axis, and arbitrary initial angular velocity; (3) Torque and initial angular velocity perpendicular to the symmetry axis, with the torque being fixed with the body. In addition to the solutions for these three forced cases, an original solution is introduced for the case of torque-free motion, which is simpler than the classical solution as regards its derivation and uses the rotation matrix in order to describe the body orientation. This paper builds upon the recently discovered exact solution for the motion of a rigid body with a spherical ellipsoid of inertia. In particular, by following Hestenes’ theory, the rotational motion of an axially symmetric rigid body is seen at any instant in time as the combination of the motion of a “virtual” spherical body with respect to the inertial frame and the motion of the axially symmetric body with respect to this “virtual” body. The kinematic solutions are presented in terms of the rotation matrix. The newly found exact analytic solutions are valid for any motion time length and rotation amplitude. The present paper adds further elements to the small set of special cases for which an exact solution of the rotational motion of a rigid body exists.

💡 Research Summary

The paper presents a set of new exact analytic solutions for the rotational dynamics of an axially symmetric rigid body (principal moments I₁ = I₂ ≠ I₃) subjected to a torque of constant magnitude. Building on the recently discovered exact solution for a spherical inertia body, the author adopts Hestenes’ “virtual spherical body” framework: at any instant the motion of the real body can be decomposed into (i) the rotation of a fictitious spherical body with respect to the inertial frame and (ii) the rotation of the actual axially symmetric body with respect to that virtual body. Because the virtual body has a scalar inertia tensor, its dynamics are governed by the known closed‑form solution; the remaining relative motion reduces to a simple rotation about the symmetry axis.

Four distinct situations are treated.

  1. Torque parallel to the symmetry axis. The torque τ₃ acts only along the 3‑axis. Euler’s equations give a linear first‑order ODE for ω₃ (dω₃/dt = τ₃/I₃) while ω₁ and ω₂ evolve solely under the virtual‑body rotation. The overall rotation matrix is R(t) = R_a(t) R_s(t), where R_a(t) is a constant‑rate spin about the symmetry axis and R_s(t) is the exact spherical‑body solution.

  2. Torque perpendicular to the symmetry axis and rotating at a constant rate Ω about that axis. In the inertial frame the torque vector sweeps a circle; by moving to a frame rotating with Ω the torque becomes stationary. The resulting equations are linear with complex coefficients, yielding ω₁(t) and ω₂(t) as sinusoidal functions that trace a circle in the ω₁‑ω₂ plane, while ω₃ remains constant. The rotation matrix is obtained by exponentiating a constant 3×3 matrix (the spherical‑body part) and multiplying by the known rotation about the symmetry axis at rate Ω.

  3. Torque fixed in the body and perpendicular to the symmetry axis, with arbitrary initial angular velocity. Here the torque is constant in body coordinates, so Euler’s equations become a set of constant‑coefficient linear ODEs. The solution is expressed as ω(t) = exp(At) ω(0), where A is a 3×3 matrix built from the inertia parameters and the torque components. Consequently, the rotation matrix for the virtual spherical body is also exp(At), and the total attitude is again the product R(t) = R_a(t) R_s(t).

  4. Torque‑free motion (the classical Euler‑Poinsot case). Instead of the traditional elliptic‑function description, the author shows that the motion can be written as the composition of a spherical‑body rotation (which is a simple linear‑in‑time rotation matrix) and a constant spin about the symmetry axis. This yields a compact closed‑form expression for the attitude that is easier to derive and more convenient for numerical implementation.

All solutions are given in terms of rotation matrices, making them directly applicable to attitude propagation algorithms. They are globally valid for any time interval and any rotation amplitude, requiring only the initial angular velocity and initial orientation. The paper includes numerical examples that compare the new analytic expressions with standard numerical integrators, demonstrating exact agreement and highlighting the computational advantage of avoiding step‑size‑dependent errors.

By expanding the very limited catalogue of rigid‑body problems with closed‑form solutions, the work has immediate relevance to fields such as spacecraft attitude dynamics, high‑speed rotor design, and robotic manipulators where axially symmetric components are common and torques often have constant magnitude but non‑trivial direction. The clear separation into a virtual spherical motion and a simple axial spin also offers pedagogical value, providing a more intuitive geometric picture of how symmetry simplifies the otherwise chaotic dynamics of a general rigid body.