The boom is analyzed using Euler–Bernoulli beam theory, modeled as a cantilever beam with an end point load. This first-order analysis evaluates boom-only bending stiffness and does not yet account for pivot compliance, joint flexibility, or torsional deformation.
The tip deflection of a cantilever beam with an end load is given by:
$$ \delta = \frac{P L^3}{3 E I} $$
Where $P=\text{point load at free end},\quad L=\text{length of beam}, \quad E=\text{Young’s modulus},$ and $I=\text{moment of inertia of the beam’s cross-sectional area}$.
Solving for the required bending stiffness $EI$:
$$ EI \ge \frac{P L^3}{3 \delta} $$
Substituting design values (from the previous page):
$$ P=2.5\text{ lb},\quad L=60\text{ in},\quadδ=0.06\text{ in} \\[16pt] EI≥\frac{(2.5 \text{ lb})(60\text{ in})^3}{3(0.06\text{ in})}\newline \\[16pt] \boxed{=3.0×10^6\text{ lb}\cdot \text{in}^2} $$
This value defines the minimum required bending stiffness of the boom section and serves as the baseline constraint for material and cross-section selection.
At the current stage, this beam equation assumes a massless boom. While this is incorrect in the real-world, it will only be used in the beginning to achieve a point of reference. This starting nature of these calculations will allow for smooth integration of complexity: fully understand the problem, forces, loads, and dynamics in a perfect realm, then gradually add components. The initial stiffness sizing of the boom considers the camera system as a concentrated end load, which dominates deflection behavior at long cantilever lengths.
In the model, the boom will be analyzed as a cantilever beam subjected to combined loading, consisting of:
A point load at the free end (representing the camera system):
$$ \delta_P = \frac{P L^3}{3 E I} $$
A uniformly distributed load (representing the self-weight of the boom arm):
$$ δ_{w}=\frac{wL^4}{8EI} \newline _{\text{Where } w\ =\text{ distributed load (self-weight per unit length)}} $$
The total tip deflection will be evaluated using linear superposition:
$$ δ_{total} = δ_{P}+ δ_{w} = \frac{P L^3}{3 E I} + \frac{wL^4}{8EI} $$