A precision table-leveling scheme shown in Fig. 2.52 relies on thermal expansion of actuators under two corners to level the table by raising or lowering their respective corners. The parameters are as follows:

Assume that (1) the actuator acts as a pure electric resistance, (2) the heat flow into the actuator is proportional to the electric power input, and (3) the motion d is proportional to the difference between Tact and Tamb, due to thermal expansion. Find the differential equations relating the height of the actuator d versus the applied voltage vi.
Figure 2.52 (a) Precision table kept level by actuators; (b) side view of one actuator