Wiktor Wise
Mechanical–electrical analogies are used to compare electrical and mechanical systems by finding relationships between variables in one domain that have a similar mathematical form in the other domain. There are two widely used analogies: the impedance analogy and the mobility analogy. In the impedance analogy, force and velocity are analogous to voltage and current, respectively. In the mobility analogy, the analogue of voltage is velocity, and the analogue of current is force. The mobility analogy is often used to model the behaviour of mechanical filters, such as loudspeakers, and to represent mechanical systems in the electrical domain.
| Characteristics | Values |
|---|---|
| Electrical Analog of Velocity | Force |
| Mechanical System Input | Force |
| Electrical System Input | Voltage |
| Mechanical System Output | Velocity |
| Electrical System Output | Current |
| Mechanical System Mass | Inductor |
| Electrical System Mass | Capacitor |
| Mechanical System Displacement | x |
| Electrical System Displacement | v |
| Mechanical System Friction | B |
| Electrical System Friction | Resistance |
| Mechanical System Spring Constant | k |
| Electrical System Spring Constant | Inductor |
| Mechanical System Voltage | Velocity Generator |
| Electrical System Voltage | Voltage Source |
| Mechanical System Current | Force Generator |
| Electrical System Current | Current Source |
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What You'll Learn
Force-voltage analogy
The core idea of the Force-Voltage analogy in control systems engineering is to relate mechanical systems with their electrical counterparts. This is done by assuming that force in a mechanical system is analogous to voltage in an electrical system.
Mechanical systems can be classified into two types based on their motion: translational systems and rotational systems. Translational systems are characterised by movement in straight lines and consist of masses, springs, and dampers. In such a system, an external force causes a displacement in the direction of the applied force.
The mathematical equations of a translational mechanical system can be compared with the mesh equations of an electrical system. This is particularly useful for designing and optimising control systems. For example, in a circuit with a resistor, inductor, and capacitor connected in series, the input voltage and current can be described by mathematical equations.
The Force-Voltage analogy is one of several electrical analogies used in control systems. Another example is the Force-Current analogy, where force in a mechanical system is analogous to current in an electrical system.
It is important to note that the accuracy of the Force-Voltage analogy is limited to linear and/or low-frequency ideal systems. Additionally, when converting between electrical and mechanical systems, sources must also be transformed. For instance, a current source becomes an input velocity, and a voltage source becomes a force generator.
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Translational mechanical systems
The force-voltage analogy involves comparing the mathematical equations of the translational mechanical system with the mesh equations of the electrical system. In this analogy, force in the mechanical system is analogous to voltage in the electrical system.
On the other hand, the force-current analogy involves comparing the mathematical equations of the translational mechanical system with the nodal equations of the electrical system. Here, force in the mechanical system corresponds to current in the electrical system.
In both analogies, velocity in the mechanical system is analogous to current in the electrical system. This relationship allows for the conversion of systems with a single set of translational elements.
By establishing these analogies, mechanical systems can be represented as electrical networks, facilitating the application of electrical theories and techniques to analyze and design mechanical systems. This approach is particularly useful in understanding the dynamics of electromechanical systems and designing mechanical filters.
Additionally, the electrical analogs enable the visualization and prediction of the behavior of mechanical systems, such as resonance, passband, damping coefficient, and time constant. The conversion of a mechanical system into its electrical analog also simplifies model construction and testing due to the ease of modifying electrical components in a circuit.
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Torque-voltage analogy
Mechanical-electrical analogies are developed by finding relationships between variables in one domain that have a mathematical form identical to variables in the other domain. There are two widely used analogies: the impedance analogy and the mobility analogy. The impedance analogy makes force and voltage analogous, while the mobility analogy makes force and current analogous.
In the impedance analogy, force and velocity are analogous to voltage and current, respectively. This is because the power conjugate variables in the analog domain are chosen to resemble force and velocity. In the electrical domain, the effort variable is voltage, and the flow variable is electrical current. The ratio of voltage to current is electrical resistance (Ohm's law).
The torque-voltage analogy is a variation of the impedance analogy used for rotational mechanical systems, such as in electric motors. In this analogy, the mathematical equations of the rotational mechanical system are compared with the mesh equations of the electrical system.
In a rotational mechanical system, the equation for torque is:
> $T=J\frac{\text{d}^2\theta}{\text{d}t^2}+B\frac{\text{d}\theta}{\text{d}t}+k\theta$
By comparing this equation to the equation for an electrical system, we can find the analogous quantities between the two systems.
For example, consider an electrical system consisting of a resistor, an inductor, and a capacitor connected in series. The input voltage applied to this circuit is $V$ volts, and the current flowing through the circuit is $i$ Amps. The equation for this electrical system is:
> $V=R\frac{\text{d}q}{\text{d}t}+L\frac{\text{d}^2q}{\text{d}t^2}+\frac{q}{C}$
> $\Rightarrow V=L\frac{\text{d}^2q}{\text{d}t^2}+R\frac{\text{d}q}{\text{d}t}+\left (\frac{1}{C} \right )q$
By comparing this equation to the equation for the rotational mechanical system, we can determine the analogous quantities between the two systems.
In general, when drawing a mechanical analog of an electrical circuit, we can sum the voltages around each loop and equate these to the forces being applied at a point. This allows us to define the velocity of the mass in absolute terms relative to a fixed reference.
The torque-voltage analogy is a powerful tool for understanding the relationship between rotational mechanical systems and electrical systems. By comparing the mathematical equations of these systems, we can identify analogous quantities and gain insights into the behaviour of complex mechanical and electrical systems.
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Power conjugate variables
Mechanical-electrical analogies are developed by finding relationships between variables in one domain that have a similar mathematical form to variables in the other domain. There are two widely used analogies: the impedance analogy and the mobility analogy. The impedance analogy associates force with voltage, while the mobility analogy associates force with current.
To fully define the analogy, a second variable must be chosen. A common choice is to make pairs of power conjugate variables analogous. Power conjugate variables are pairs of variables whose product is power. In the electrical domain, the power conjugate variables are invariably voltage (V) and current (I). In the mechanical domain, the usual choice for a translational mechanical system is force (F) and velocity (u).
In the impedance analogy, force and velocity are analogous to voltage and current, respectively. This results in a quantity analogous to electrical impedance. The mobility analogy does not preserve this analogy between impedances across domains, but it has the advantage of preserving the topology of networks.
The electrical analog of velocity in the mechanical domain is current (I) in the electrical domain. This is because velocity is the flow variable in the mechanical domain, and current is the flow variable in the electrical domain. The effort variable in the electrical domain is voltage, which is analogous to force in the mechanical domain.
Conjugate variables are pairs of variables that are related through Pontryagin duality and are part of a symplectic basis. They are also related by Noether's theorem, which states that if the laws of physics are invariant with respect to one of the conjugate variables, then the other conjugate variable will remain constant over time. In thermodynamics, conjugate pairs are defined with respect to a thermodynamic state function, such as entropy, where the product of the conjugate pairs yields an entropy.
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Mechanical impedance
The impedance analogy, also known as the Maxwell analogy, classifies the two variables making up the power conjugate pair as an effort variable and a flow variable. The effort variable in the energy domain is analogous to force in the mechanical domain, and the flow variable in the energy domain is analogous to velocity in the mechanical domain. In the impedance analogy, force and velocity are analogous to voltage and current, respectively.
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Frequently asked questions
The electrical analog of velocity is voltage. This is known as the mobility analogy, one of the two main mechanical-electrical analogies.
The other main analogy is the impedance analogy, where force is made analogous to voltage.
There are two main methods for forming electrical analogies of mechanical systems: the force-voltage analogy and the force-current analogy. In the force-voltage analogy, the current is regarded as the output, while in the force-current analogy, voltage is regarded as the output.
