System elements modeling – the compliant 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 compliant element is resisting change, in the sense of separation of its end points (mechanical systems) or change of charge for electrical systems.

Mechanical systems

For mechanical elements this means resisting compression or traction (stretching) forces. The stiffness of a spring, named also spring constant, is the compliant element for mechanical systems.

Translational spring

Image: Translational spring

For a given translational spring with the stiffness k [N/m], the relative displacement x [m] between its end points will produce an elastic force F [N], equal to:

\[F = k \cdot x \tag{1}\]

The displacement x is equal with the integral of the speed v. Therefore, we can write the equation of the force as:

\[F = k \int v dt \]
Rotational spring

Image: Rotational spring

The same behaviour applies to torsional springs. The elastic torsion torque T [Nm] is equal with the product of the metallic bar (shaft) stiffness k [Nm/rad] and the angle of torsion α [rad].

\[T = k \cdot \alpha \tag{2}\]

The angle of torsion α can be written as the integral of angular speed ω [rad/s]. In this case, the equation of torsional torque becomes:

\[T = k \int \omega dt \]

The potential energy E [J] stored in a spring is equal to:

\[E = \frac{1}{2} k x^2= \frac{1}{2} k \alpha^2\]

Electrical systems

The equivalent compliant element for electrical systems is the inverse of the capacitance C [F] of a capacitor.

Electric capacitor

Image: Electric capacitor

The voltage drop UC [V] across a capacitor is equal with the product of the inverse of the capacitance 1/C and the electric charge q [C]. The unit of measurement for the capacitance C is Farad [F].

\[U_C = \frac{1}{C} \cdot q \tag{3}\]

The charge q is equal with the integral of the electrical current i [A]. Therefore, the voltage drop across the capacitor can be written as:

\[U_C = \frac{1}{C} \int i dt \]

The electrical equivalent of the potential energy is the energy stored in a capacitor E [J], given by:

\[E = \frac{1}{2 C}q^2\]

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

Force, FTorque, TVoltage, UC
Linear acceleration, aAngular acceleration, εCurrent variation, di/dt
Linear velocity, vAngular velocity, ωElectrical current, i
Linear position, xAngular position, αElectrical charge, q
Stiffness, kStiffness, kInverse of capacitance, 1/C

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

The characteristic of the compliant element

Image: The characteristic of the compliant element

When modeling mechanical and electrical dynamic systems, the compliant element is one of the most common to be found. Therefore, understanding the basic modeling principles is a core requirement for every engineer.

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