PHY.2:5.1 - Build the missing coupling in a mechanical/electrical analogue

Two bodies move along a line, joined by an ideal spring. Let their masses be m1 and m2, velocities v1 and v2, and the spring extension be d. External forces are f1 and f2; linear drag coefficients b1 and b2 are nonnegative. With spring stiffness k>0, the physical laws are:

m1*dv1/dt = f1 - b1*v1 - k*d
m2*dv2/dt = f2 - b2*v2 + k*d
dd/dt = v1 - v2.

The question is how a deformation transfers motion between the bodies. Two independent electrical elements for the masses would omit that interaction. Choose node voltages to represent velocities, injected currents to represent external forces, and charge-accumulating capacitors for the masses. A conductance from each node to the reference node can represent its drag. The missing coupling needs a state whose rate is proportional to the difference of the two voltages. An inductor connected between the nodes supplies that relation.

Let s=r*t be source time, V_i=a*v_i the voltages and I_i=h*f_i the injected currents, with positive dimensional conversion factors a and h and positive time ratio r. Let J be current through the inductor from node 1 to node 2. Current conservation and the ideal inductor law give:

C1*dV1/ds = I1 - G1*V1 - J
C2*dV2/ds = I2 - G2*V2 + J
L*dJ/ds = V1 - V2.

Substitute the conversions and J=h*k*d. All three relations match the mechanical account when:

C_i = r*h*m_i/a
G_i = h*b_i/a
L = r*a/(h*k).

The constructed inductor carries the coupling state. The initial source state must satisfy V_i(0)=a*v_i(0) and J(0)=h*k*d(0). For m1=2 kg, m2=1 kg, k=3 N/m, d(0)=1 m, zero velocities and zero applied forces, the first accelerations are -1.5 and +3 metres per second squared. The prepared inductor current produces the corresponding voltage changes. With J(0)=0, the source would instead remain at rest and miss the deformation-driven motion.

The stored mechanical energy is (m1*v1^2 + m2*v2^2 + k*d^2)/2. Differentiating gives power f1*v1 + f2*v2 - b1*v1^2 - b2*v2^2. The circuit’s stored energy is r*a*h times the mechanical energy; its supplied and dissipated powers have the compatible factor a*h. The correspondence therefore includes storage and transfer under the stated ideal laws.

The first result is a source design and interpreted initial response. Before building it, choose the conversion factors so capacitances, conductances, inductance, voltages and currents lie in the available ranges. Include losses or loading that would alter the requested response.

Now change the target: a controller supplies a force that increases with velocity, producing effective negative damping over a stated range. A nonnegative conductance cannot realize that contribution. The changed physical construction needs an active element and its energy supply, or another means of obtaining the result. The previously constructed passive analogue still answers the original dissipative question.