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The swinging counterweight trebuchet. On internal forces

Erik Horsdal

TL;DR

The paper develops a rigorous, energy-based derivation of the internal forces in a swinging-counterweight trebuchet, expressing dynamic terms with explicit second-order angular coordinates and their derivatives. Using a Lagrangian framework and Newton's second law, it computes phase-dependent forces, torques, and rates of work for the projectile, counterweight, and throwing arm, highlighting how energy flows from the counterweight to the projectile through the beam. A detailed numerical example quantifies initial accelerations, peak sling tension, hinge bending, and energy transfer, showing that about 90% of the available energy can reach the projectile at release in an ideal case, while friction and structural loads limit real-world performance. The results inform structural design (bearing sizes, trestle rigidity, arm diameter) and underscore the importance of accurately characterizing internal forces to optimize range and efficiency while minimizing wear.

Abstract

The forces that act internally in a trebuchet as it delivers a shot depend on the motions of throwing arm, counterweight and sling. These motions are considered known experimentally or theoretically and given in the form of time-dependent angular coordinates. Explicit expressions in terms of these coordinates and their derivatives to second order are derived for the internal forces. The forces that act immediately after a shot is initiated can be extracted from the equations of motion without solving them, and they are compared with static forces just prior to initiation. Required strengths of the different parts of a trebuchet depend on the internal forces, which also determine sliding friction losses. Illustrative results are given for a specific trebuchet.

The swinging counterweight trebuchet. On internal forces

TL;DR

The paper develops a rigorous, energy-based derivation of the internal forces in a swinging-counterweight trebuchet, expressing dynamic terms with explicit second-order angular coordinates and their derivatives. Using a Lagrangian framework and Newton's second law, it computes phase-dependent forces, torques, and rates of work for the projectile, counterweight, and throwing arm, highlighting how energy flows from the counterweight to the projectile through the beam. A detailed numerical example quantifies initial accelerations, peak sling tension, hinge bending, and energy transfer, showing that about 90% of the available energy can reach the projectile at release in an ideal case, while friction and structural loads limit real-world performance. The results inform structural design (bearing sizes, trestle rigidity, arm diameter) and underscore the importance of accurately characterizing internal forces to optimize range and efficiency while minimizing wear.

Abstract

The forces that act internally in a trebuchet as it delivers a shot depend on the motions of throwing arm, counterweight and sling. These motions are considered known experimentally or theoretically and given in the form of time-dependent angular coordinates. Explicit expressions in terms of these coordinates and their derivatives to second order are derived for the internal forces. The forces that act immediately after a shot is initiated can be extracted from the equations of motion without solving them, and they are compared with static forces just prior to initiation. Required strengths of the different parts of a trebuchet depend on the internal forces, which also determine sliding friction losses. Illustrative results are given for a specific trebuchet.
Paper Structure (27 sections, 33 equations, 13 figures, 2 tables)

This paper contains 27 sections, 33 equations, 13 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Trebuchet and angular coordinates
  3. Kinematics
  4. Mechanical energies
  5. Static initial forces
  6. Dynamic forces
  7. Phase II
  8. Phase I
  9. Torque
  10. Initial accelerations, forces and torque
  11. Initial angular accelerations
  12. Initial linear accelerations
  13. Initial dynamic forces
  14. Initial torque
  15. Example
  16. ...and 12 more sections

Figures (13)

  • Figure 1: Trebuchet. a) Moving parts and fixed pivot. b) Lengths and masses.
  • Figure 2: Angular coordinates and unit vectors in the directions of beam $\em_\theta$, counterweight $\em_\psi$ and projectile $\em_\phi$. Fixed unit vectors $(\em_x,\em_y)$.
  • Figure 3: Forces on the beam $-\mathbf{F}_H$, $-\mathbf{F}_{CM}$, $-\mathbf{F}_S$ and $\mathbf{F}_R$.
  • Figure 4: External forces on beam.
  • Figure 5: Forces at hinge H, pivot P, center of mass CM and spigot S. Small forces: $\mathbf{F}_{CM}=-m_bg\em_y$ in a) and c). $\mathbf{F}_S=-2.9mg\em_x$ in b) and c).
  • ...and 8 more figures