Coordinates.
Let the pivot (cart) translate along \(x\) with position \(x(t)\). A rigid rod of length ℓ\ellℓ carries a bob of mass \(M\) at angle \(\theta(t)\) from the downward vertical. (Gravity is \(g\) downward.)
Optionally, the cart may have mass \(M\) (leave \(M=0\) to treat \(x(t)\) as prescribed/known).
Bob kinematics.
('Bob' is the object at the end of the pendulum.)
Pivot at \((x,\,0)\); bob at \((x+ℓsin\,θ,\,−ℓcos\,θ)\)
Velocities: \(\dot{x}_b=\dot{x}+ℓ\dot{θ}cos\,θ,{\,\,\,\,\,}\dot{y}_b=ℓ\dot{θ}sin\,θ\)
Speed squared: \(v_b^2=\dot{x}^2+2ℓ\dot{x}\,\dot{θ}cos\,θ+ℓ^2\dot{θ}^2\)
Energies. \(T=\dfrac{1}{2}M\dot{x}^2+\dfrac{1}{2}m\,v_b^2= \dfrac{1}{2}(M+m)\dot{x}^2+mℓcos\,θ\,\dot{x}\,\dot{θ}+\dfrac{1}{2}ℓ^2\dot{θ}^2\)
\(V=mgℓ(1−cosθ)\)
Lagrangian. \(L=T−V\)
Dissipation (Rayleigh) \(D\).
(dissipation refers to total energy losses)
\(D=\dfrac{1}{2}\,c_x\,\dot{x}^2+\dfrac{1}{2}c_θ\,\dot{θ}^2\)
Generalized inputs.
Horizontal force on the cart \(F_x(t)\); torque at the pivot \(\tau(t)\).
For \(q1=x,\,q^2=θ\) :
\[ \frac{d}{dt}\!\left(\frac{\partial \mathcal{L}}{\partial \dot q_i}\right) -\frac{\partial \mathcal{L}}{\partial q_i} +\frac{\partial \mathcal{D}}{\partial \dot q_i} = Q_i\]
with \(Q_x\,=F_x\,,\,Q_θ\,=τ\)
Carrying out the derivatives gives the coupled, nonlinear dynamics:
\[ \boxed{ (M+m)\ddot{x}+c_x\dot{x}+mℓ(\ddot{θ}cos\,θ−\dot{θ}2sin\,θ)=Fx(t) }\]
\[ \boxed{ mℓ^2\ddot{θ}+c_θ\dot{θ}+mgℓsin\,θ+mℓ\,\ddot{x}cos\,θ=τ(t) }\]
These are the classic cart–pendulum equations (a.k.a. “inverted pendulum on a cart” when \(\theta\) is near \(0\) upward, or ordinary pendulum when near \(\pi\) downward). Note the bilinear coupling terms \(\ddot{x}\,cos\,θ\) and \(\dot{θ}\,2sin\,θ\): the pivot’s motion injects/extracts energy; there is no free sinusoid.
Cart Mass refers to the mass of the moving object the pendulum is attached to. An example would be the hip joint during travel.
If the cart motion \(x_p(t)\) is given (not a state), set \(M=0\) and treat \(\ddot{x}=\ddot{x_p}(t)\) as known. Then the angle equation decouples to a forced, damped pendulum:
\[ \boxed{m\ell^2\ddot\theta + c_\theta\dot\theta + m g \ell \sin\theta = -\,m\ell\,\ddot x_p(t)\,\cos\theta + \tau(t)}\]
The term \(−mℓ\ddot{x}\,p\,cos\,θ\) is parametric excitation from pivot acceleration.
With small angles \((sin\,θ\,≈\,θ,\,cos\,θ\,≈\,1)\)
\[ \ddot\theta + \frac{c_\theta}{m\ell^2}\dot\theta + \frac{g}{\ell}\,\theta = -\frac{\ddot x_p(t)}{\ell} + \frac{\tau(t)}{m\ell^2}\]
Even in this linearized form, the solution is not sinusoidal unless the forcing happens to be sinusoidal and losses vanish (which is close to impossible in dancing).
Around \(\theta\approx 0\) (downward) and modest speeds, linearize \(\sin\theta\approx\theta,\ \cos\theta\approx 1\):
\[ (M+m)\ddot x + c_x\dot x + m\ell\,\ddot\theta \;=\; F_x(t),\]
\[ m\ell^2\ddot\theta + c_\theta\dot\theta + m g \ell\,\theta + m\ell\,\ddot x \;=\; \tau(t)\]
This is the standard coupled linear cart–pendulum model used in control texts.
Vertical pivot motion \(z_p(t)\) adds an extra parametric term: \(\propto \ddot{z}\,p\,(t)\,sinθ\)
Distributed leg mass replaces \(m\ell^2\) by the leg’s rotational inertia \(I_{leg}\)\, about the hip (\(\ell\) then to COM offset).
Planar 2-link (thigh+shin) introduces a 2×2 inertia matrix \(M(q)\), Coriolis \(C(q,\dot{q}\,)\), and the same external-acceleration coupling.
A fixed pivot pendulum \(x\equiv\text{const}\) can swing (nearly) sinusoidally.
A moving pivot pendulum is forced and damped; its shape is set by \(x_p\,(t)\), \(D\), and inputs \((F_x,\tau)\).
Ballroom pivots (hips/ankles) move with travel and rotate; the “free pendulum” metaphor is physically inapplicable.
Now suppose the pivot itself also moves up/down (as it will during Rise and Fall): \(z_p(t)\)
Reminder: “bob” isn’t a dancer — 'bob' is the object (mass) at the end of the pendulum.
Bob position:
\[ x_b = x_p(t) + \ell \sin\theta, \quad z_b = z_p(t) - \ell \cos\theta\]
Velocities (including pivot terms):
\[ \dot{x}_b = \dot{x}_p + \ell \dot\theta\cos\theta, \qquad \dot{z}_b = \dot{z}_p + \ell \dot\theta\sin\theta\]
Squared speed:
\[ v_b^2 = \dot{x}_p^2 + \dot{z}_p^2 + 2\ell\dot\theta(\dot{x}_p\cos\theta + \dot{z}_p\sin\theta) + \ell^2\dot\theta^2\]
Kinetic energy:
\[ T = \tfrac{1}{2} m v_b^2\]
Since there is no \(PE\) in dancing, \(L = T - V\) becomes \(\mathcal{L}=T\) (no stored \(PE\)), dissipation \(\mathcal{D}=\tfrac{1}{2}c_\theta \dot\theta^2\), and treat gravity + pivot acceleration as external loads.
Equation of motion:
\[ m\ell^2\ddot\theta + c_\theta \dot\theta = \tau(t) - m g \ell \sin\theta - m\ell\big[\ddot{x}_p(t)\cos\theta + \ddot{z}_p(t)\sin\theta\big].\]
where:
\(-m g \ell \sin\theta\): gravitational load
\(-m\ell(\ddot{x}_p\cos\theta + \ddot{z}_p\sin\theta)\): pivot forcing from travel (\(x_p\)) and rise/fall (\(z_p\))
\(c_\theta \dot\theta\): viscous loss
\(\tau(t)\): applied torque (e.g. lead/partner input)
For small \(\theta\):
\[ \ddot\theta + \frac{c_\theta}{m\ell^2}\dot\theta + \frac{g}{\ell}\theta = -\frac{1}{\ell}\big(\ddot{x}_p + \theta\,\ddot{z}_p\big) + \frac{\tau(t)}{m\ell^2}.\]
Interpretation:
A fast pivot rise (\(\ddot z_p>0\)) tips the pendulum.
Forward pivot acceleration (\(\ddot x_p\)) drives angular swing.
No natural sinusoid exists unless both accelerations vanish and losses are zero.
Horizontal forcing \(\ddot{x}_p\) = travel along line of dance.
Vertical forcing \(\ddot{z}_p\) = rise & fall of the pivot.
Both couple into leg/torso “pendulum” motion. The pivot never stays still, so the 'pendulum metaphor' is broken by definition.