Beatrice Lang
Mechanical-electrical analogies are used to represent mechanical systems as electrical networks, with variables in one domain having identical mathematical forms in the other. This allows for the application of electrical theory to mechanical designs and vice versa, aiding in the analysis of mechanical filters and electromechanical systems. One such analogy is the force-current analogy, where force is made analogous to current, and its variation, the force-voltage analogy, where force is made analogous to voltage. Another variation is the torque-voltage analogy, used for rotational mechanical systems. In the context of these analogies, the electrical analog of mass is diode inductance resistance.
| Characteristics | Values |
|---|---|
| Electrical Analog of Mass | Diode Inductance Resistance |
| Mechanical-Electrical Analogies | Representation of mechanical systems as electrical networks |
| Force-Current Analogy | Force generators are replaced by current sources |
| Friction elements are replaced by resistors | |
| Springs are replaced by inductors | |
| Masses are replaced by capacitors | |
| Force-Voltage Analogy | Mathematical equations of translational mechanical systems are compared with mesh equations of electrical systems |
| Torque-Voltage Analogy | Mathematical equations of rotational mechanical systems are compared with mesh equations of electrical systems |
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What You'll Learn
- The energy of the mass in a mechanical system is analogous to the energy of an inductor in an electrical circuit
- Mechanical motion can be compared to electric circuits, with inductance being analogous to mass
- Force-voltage analogy: mathematical equations of translational mechanical systems are compared with mesh equations of electrical systems
- Torque-voltage analogy: mathematical equations of rotational mechanical systems are compared with mesh equations of electrical systems
- Force-current analogy: mathematical equations of translational mechanical systems are compared with nodal equations of electrical systems
The energy of the mass in a mechanical system is analogous to the energy of an inductor in an electrical circuit
In the impedance analogy, the roles of voltage and current are reversed between the two methods, and the electrical representations produced are dual circuits of each other. The impedance analogy is widely used to model the behaviour of mechanical filters, and it preserves the analogy between electrical and mechanical impedance.
When converting a mechanical system to an electrical circuit in the impedance analogy, each node in the electrical circuit becomes a point in the mechanical system. Ground becomes a fixed location, resistors become friction elements, capacitors become springs, and inductors become masses. Sources must also be transformed—a current source becomes a force generator, and a voltage source becomes an input velocity.
The energy of the mass in a mechanical system is measured relative to a fixed reference, such as a single velocity. Similarly, the energy of the inductor in the analogous electrical circuit is measured relative to a single current. To apply this analogy, every loop in the electrical circuit becomes a point in the mechanical system.
It is important to note that this analogy only works easily for inductors with a single current defined through them. This is because the velocity of the mass can only be defined in absolute terms relative to a fixed reference if there is a single current flowing through the inductor.
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Mechanical motion can be compared to electric circuits, with inductance being analogous to mass
Mechanical motion can be compared to electrical circuits using analogies. These analogies are helpful in studying and analysing non-electrical systems, such as mechanical systems, by drawing comparisons with analogous electrical systems.
One such analogy is the 'Mechanical 1' model, which involves substituting the analogous quantities into the equations for the electrical elements. For example, in 'Mechanical 1', a current source becomes a force generator, a voltage source becomes a fixed location, resistors become friction elements, springs become inductors, and masses become capacitors. The capacitors in this model must be grounded, otherwise the process becomes much more complex.
The 'Mechanical 2' model is another analogy that can be used. In this model, every loop in the electrical circuit becomes a point in the mechanical system. The energy of the inductance is measured relative to a single current, and the voltages around a loop summed to zero are analogous to the sum of forces at a point. This model only works easily for inductors with only one current defined through them.
Through these analogies, mechanical motion can be compared to electric circuits, with inductance being analogous to mass.
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Force-voltage analogy: mathematical equations of translational mechanical systems are compared with mesh equations of electrical systems
The force-voltage analogy is a comparison between the mathematical equations of translational mechanical systems and the mesh equations of electrical systems. This analogy is useful when selecting actuators, analysing sensor data, and designing control algorithms.
In the force-voltage analogy, force in a mechanical system is analogous to voltage in an electrical system. Similarly, mass in a mechanical system is comparable to inductance in an electrical system. Other analogous relationships include viscous damping coefficient (mechanical) and resistance (electrical), spring constant (mechanical) and reciprocal of capacitance (electrical), and displacement (mechanical) and charge (electrical).
When converting between electrical and mechanical systems, the relationships between Kirchoff's Current Law and D'Alembert's Law, and between Kirchoff's Voltage Law and D'Alembert's Law, are important. For example, when converting a circuit to the Mechanical 1 analog, the electrical version of Ohm's law, e=iR, becomes v=f/B in the Mechanical 1 analog.
The force-voltage analogy is one of two types of electrical analogies of translational mechanical systems, the other being the force-current analogy. In the force-current analogy, the mathematical equations of translational mechanical systems are compared with the nodal equations of electrical systems.
The force-voltage analogy is a powerful tool for designing and analysing control systems that utilise both mechanical and electrical components. By drawing parallels between these systems, knowledge and techniques from electrical systems can be applied to mechanical systems, and vice versa.
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Torque-voltage analogy: mathematical equations of rotational mechanical systems are compared with mesh equations of electrical systems
In physics, analogies are often drawn between mechanical and electrical systems. One such analogy is the torque-voltage comparison, which relates the mathematical equations of rotational mechanical systems to the mesh equations of electrical systems.
The rotational mechanical system can be represented by the equation:
$T=J\frac{d^2\theta}{dt^2}+B\frac{d\theta}{dt}+k\theta$
Where:
- $T$ is the torque
- $J$ is the moment of inertia
- $\theta$ is the angular displacement
- $t$ is the time
- $B$ and $k$ are constants
By comparing this equation with Equation 3 from the source, we can identify the analogous quantities in the rotational mechanical system and the electrical system. For example, in the electrical system, voltage is analogous to torque in the mechanical system, and electrical current is analogous to angular displacement.
This analogy allows us to understand the behaviour of rotational mechanical systems by drawing comparisons with electrical systems. For example, the voltage in an electrical circuit can be thought of as the torque in a mechanical system, with both quantities driving their respective systems. Similarly, the electrical current flowing through a circuit can be likened to the angular displacement of a rotating object.
It is important to note that this analogy has some limitations. For instance, it works most easily when only one current is defined through inductors, allowing for the absolute definition of the velocity of a mass relative to a fixed reference.
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Force-current analogy: mathematical equations of translational mechanical systems are compared with nodal equations of electrical systems
There are two types of electrical analogies of translational mechanical systems: force-voltage analogy and force-current analogy. In the force-voltage analogy, the mathematical equations of the translational mechanical system are compared with the mesh equations of the electrical system. In the force-current analogy, the mathematical equations of the translational mechanical system are compared with the nodal equations of the electrical system.
In the force-voltage analogy, the force in a mechanical system is analogous to voltage in an electrical system, and the velocity in a mechanical system is analogous to the current in an electrical system. The elements connected between two masses in a mechanical system correspond to the elements common between two meshes in electrical systems. For example, the mass in a mechanical system is analogous to inductance in an electrical system, and the spring constant in a mechanical system is analogous to the reciprocal of capacitance in an electrical system.
In the force-current analogy, the mathematical equations of the translational mechanical systems are compared with the nodal equations of the electrical systems. By comparing the equations, analogous quantities of the translational mechanical and electrical systems can be obtained. For example, consider the following electrical system:
> This circuit consists of a current source, resistor, inductor, and capacitor. All these electrical elements are connected in parallel. ... $i=\frac{1}{R}\frac{\text{d}\Psi}{\text{d}t}+\left ( \frac{1}{L} \right )\Psi+C\frac{\text{d}^2\Psi}{\text{d}t^2} \Rightarrow i=C\frac{\text{d}^2\Psi}{\text{d}t^2}+\left ( \frac{1}{R} \right )\frac{\text{d}\Psi}{\text{d}t}+\left ( \frac{1}{L} \right )\Psi$ (Equation 6)
By comparing Equation 1 and Equation 6, we will get the analogous quantities of the translational mechanical system and electrical system.
In general, when drawing a mechanical analog of an electrical circuit, one can sum the voltages around each loop and equate these to the forces being applied at a point. If possible, one should draw currents such that only one current flows through inductors, so that the velocity of the mass can be defined in absolute terms relative to a fixed reference.
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Frequently asked questions
Electrical analogs are representations of mechanical systems as electrical networks.
The electrical analog of mass is diode inductance resistance.
Initially, electrical analogs were used to explain electrical phenomena in familiar mechanical terms. However, as electrical network analysis evolved, it was discovered that certain mechanical issues could be more efficiently solved using electrical analogs.
