= Beam root rotational compliance
{title2=$C_\theta=1/K_\theta$}
\b[Beam root rotational compliance] is the incremental rotation of a beam's attachment relative to its supporting base per increment of applied root <torque>. For a linear rotational restraint with stiffness $K_\theta>0$,
$$
C_\theta=\frac{d\theta_0}{dM_0}=\frac1{K_\theta},\qquad \theta_0=C_\theta M_0.
$$
The angle is measured in <radians>, so the compliance can be reported in $\mathrm{rad/(N\,m)}$. The root's transverse position is held fixed in this model. The <rotational spring boundary condition for a beam> implements this finite restraint; a <clamped boundary condition> is the zero-compliance limit. End moments and section rotations are conjugate quantities in beam <virtual work>.<The Concept of Stress, Generalized Stresses and Equilibrium by Tomasz Wierzbicki>[§§3.4–3.5, equations 3.46 and 3.52–3.57]{ref}
For an original static derivation, apply a transverse tip <force> $P$ to a beam of length $L$ and constant <beam flexural rigidity> $B$. The root moment has magnitude $PL$, giving $\theta_0=C_\theta PL$. In the <small-angle approximation>, root rotation contributes $L\theta_0$ to tip <displacement>. Bending relative to that rotated attachment gives the usual cantilever contribution, so
$$
\delta=P\left(\frac{L^3}{3B}+C_\theta L^2\right),\qquad
\frac{\delta_{\rm root}}{\delta_{\rm bending}}=\frac{3BC_\theta}{L}.
$$
This derivation assumes a linear elastic beam, no root translation and an effectively rigid supporting base. The second term is fixture motion rather than a change in material <Young's modulus>.
For stipulated $L=0.30\,\mathrm m$, $B=6\,\mathrm{N\,m^2}$, $C_\theta=0.020\,\mathrm{rad/(N\,m)}$ and $P=1\,\mathrm N$, the root rotates through $0.006$ radian. Root motion contributes $1.8\,\mathrm{mm}$, while bending contributes $1.5\,\mathrm{mm}$. Attributing the entire $3.3\,\mathrm{mm}$ to a perfectly clamped beam would infer $B=2.73\,\mathrm{N\,m^2}$ instead of the stipulated value six.
A proposed measurement uses a known force and lever arm to apply a root moment, with optical markers on the attachment and on the supporting base to measure their relative angle. A separate displacement channel can detect unwanted root translation. Fit the small-load slope of angle against moment; using only the beam-tip displacement would mix attachment rotation with beam bending. Load reversal tests whether one linear compliance adequately describes the fixture or whether backlash and <hysteresis> require another model.
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