System equivalence
Encyclopedia
In the systems sciences the term system equivalence is the notion that a parameter
or component of a system
behaves in a similar way as a parameter or component of a different system. Similarity means that mathematically the parameters/components will be indistinguishable from each other. Equivalence can be very useful in understanding how complex system
s work.
, electrical
, torsional
, fluidic
, and caloric systems.
Equivalent systems are mostly used to change large and expensive mechanical, thermal, and fluid systems into a simple, cheaper electrical system. Then the electrical system can be analyzed to validate that the system dynamics
will work as designed. This is a preliminary inexpensive way for engineers to test that their complex system performs the way they are expecting.
This testing is necessary when designing new complex systems that have many components. Businesses do not want to spend millions of dollars on a system that does not perform the way that they were expecting. Using the equivalent system technique, engineers can verify and prove to the business that the system will work. This lowers the risk factor that the business is taking on the project.
Chart of equivalent variables for the different types of systems
Flow variable: moves through the system
Effort variable: puts the system into action
Compliance: stores energy as potential
Inductance: stores energy as kinetic
Resistance: dissipates or uses energy
For example:
Mechanical systems
Electrical systems
All the fundamental variables
of these systems have the same functional form.
Parameter
Parameter from Ancient Greek παρά also “para” meaning “beside, subsidiary” and μέτρον also “metron” meaning “measure”, can be interpreted in mathematics, logic, linguistics, environmental science and other disciplines....
or component of a system
System
System is a set of interacting or interdependent components forming an integrated whole....
behaves in a similar way as a parameter or component of a different system. Similarity means that mathematically the parameters/components will be indistinguishable from each other. Equivalence can be very useful in understanding how complex system
Complex system
A complex system is a system composed of interconnected parts that as a whole exhibit one or more properties not obvious from the properties of the individual parts....
s work.
Overview
Examples of equivalent systems are first- and second-order mechanicalMechanical engineering
Mechanical engineering is a discipline of engineering that applies the principles of physics and materials science for analysis, design, manufacturing, and maintenance of mechanical systems. It is the branch of engineering that involves the production and usage of heat and mechanical power for the...
, electrical
Electrical engineering
Electrical engineering is a field of engineering that generally deals with the study and application of electricity, electronics and electromagnetism. The field first became an identifiable occupation in the late nineteenth century after commercialization of the electric telegraph and electrical...
, torsional
Torque
Torque, moment or moment of force , is the tendency of a force to rotate an object about an axis, fulcrum, or pivot. Just as a force is a push or a pull, a torque can be thought of as a twist....
, fluidic
Fluid mechanics
Fluid mechanics is the study of fluids and the forces on them. Fluid mechanics can be divided into fluid statics, the study of fluids at rest; fluid kinematics, the study of fluids in motion; and fluid dynamics, the study of the effect of forces on fluid motion...
, and caloric systems.
Equivalent systems are mostly used to change large and expensive mechanical, thermal, and fluid systems into a simple, cheaper electrical system. Then the electrical system can be analyzed to validate that the system dynamics
System dynamics
System dynamics is an approach to understanding the behaviour of complex systems over time. It deals with internal feedback loops and time delays that affect the behaviour of the entire system. What makes using system dynamics different from other approaches to studying complex systems is the use...
will work as designed. This is a preliminary inexpensive way for engineers to test that their complex system performs the way they are expecting.
This testing is necessary when designing new complex systems that have many components. Businesses do not want to spend millions of dollars on a system that does not perform the way that they were expecting. Using the equivalent system technique, engineers can verify and prove to the business that the system will work. This lowers the risk factor that the business is taking on the project.
Chart of equivalent variables for the different types of systems
| System type | Flow variable | Effort variable | Compliance | Inductance | Resistance |
|---|---|---|---|---|---|
| Mechanical | x, dx/dt, d2x/dt2 | F = force | spring (k) | mass (m) | damper (c) |
| Electrical | i = current | V = voltage | capacitance (C) | inductance (L) | resistance (R) |
| Thermal | qh = heat flow rate | ∆T = change in temperature | object (C) | - | conduction and convection (R) |
| Fluid | qm = mass flow rate, qv = volume flow rate |
p' = pressure, h = height | tank (C) | mass (m) | valve or orifice (R) |
Flow variable: moves through the system
Effort variable: puts the system into action
Compliance: stores energy as potential
Inductance: stores energy as kinetic
Resistance: dissipates or uses energy
For example:
Mechanical systems
- Force F = −kx = C dx/dt = M d2x/dt2
Electrical systems
- Voltage V = Q/C = R dQ/dt = L d2Q/dt2
All the fundamental variables
Variable (programming)
In computer programming, a variable is a symbolic name given to some known or unknown quantity or information, for the purpose of allowing the name to be used independently of the information it represents...
of these systems have the same functional form.
Further reading
- Panos J. Antsaklis, Anthony N. Michel (2006), Linear Systems, 670 pp.
- M.A. Kaashoek & J.H. Van SchuppenJan H. van SchuppenJan H. van Schuppen is a Dutch mathematician and an academic researcher. He has worked on systems theory, particularly on control theory and system identification, on probability, and on a number of related practical applications...
(1990), Realization and Modelling in System Theory. - Katsuhiko Ogata (2003), System dynamics, Prentice Hall; 4 edition (July 30, 2003), 784 pp.