System elements modeling – the resistive element

The behavior of some mechanical systems is identical to that of certain electrical systems. Therefore, we can have an unified modeling approach when dealing with mechanical and electrical system modeling.

Mechanical and electrical lumped parameters systems can be modeled with basic elements. These elements are: the inertial element, the compliant element and the resistive element.

The resistive element is resisting sudden (step) change in position, in mechanical systems, or change in electrical charge for electrical systems. The resistive element acts as a damper for the dynamic system.

Mechanical systems

The purpose of a damper is to damp the oscillations in a mechanical system. Think at the damper in an automobile suspension, when the wheel moves, the suspension coil accumulates the mechanical shocks and the damper damps the oscillations.

Translational damper

Image: Translational damper

The damping phenomena occurs naturally when there is viscous friction between two bodies. Viscous friction is created when the surfaces of two moving bodies, in contact, are separated by a layer of liquid (lubricant). The friction between the bodies creates a damping force F [N] which is the product between the viscous friction coefficient c [Ns/m] and the relative speed v [m/s] between the bodies:

\[F = c \frac{dx}{dt}=cv \tag{1}\]
Rotational damper

Image: Rotational damper

Damping occurs also in rotational systems. A shaft rotating in a hub will produce viscous friction, which will generate a damping torque T [Nm] equal with the product between the viscous friction coefficient c [Nms/rad] and the angular speed of the shaft ω [rad/s]:

\[T = c \frac{d \alpha}{dt}=c \omega \tag{2}\]

Electrical systems

The equivalent of mechanical damping in electrical systems is the electric resistance. From Ohm’s law we can write that the voltage drop across a resistor UR [V] is equal with the product between the resistance R [Ω] and the electric current i [A] flowing through the resistor:

\[U_R = R \frac{dq}{dt} = Ri \tag{3}\]
Electric resistor

Image: Electric resistor

We can now write an equivalency table between the mechanical (translational and rotational) and electrical system for the resistive element.

MechanicalElectrical
TranslationalRotational
Force, FTorque, TVoltage, UR
Linear acceleration, aAngular acceleration, εCurrent variation, di/dt
Linear velocity, vAngular velocity, ωElectrical current, i
Linear position, xAngular position, αElectrical charge, q
Friction coefficient, cFriction coefficient, cResistance, R

If we look at the relationships of force (1), torque (2) and voltage (3), we can see that the friction coefficient and the resistance act as a slope of the linear dependencies.

The characteristic of the resistive element

Image: The characteristic of the resistive element

When modeling mechanical and electrical dynamic systems, the resistive element is always present because all real systems have natural damping.

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