Simple, Linear, and Complex Systems
Simple, linear, and complex systems are foundational concepts that help in understanding how machines and processes operate. Simple machines, such as levers, pulleys, and screws, serve as tools to amplify force and perform tasks efficiently. They are characterized by a clear input and output, providing a mechanical advantage, although they trade off some efficiency. Linear systems consist of components where one device directly influences a single output, while complex systems feature interconnected elements where one component can affect multiple outputs, as seen in machines like bicycles that integrate various simple machines for function.
Feedback and control systems are vital for regulating the actions of these machines, employing mechanisms that adjust inputs based on output measurements to maintain desired performance. Real-life applications of these systems are evident in modern technology, where simple machines combine to create linear and complex systems that underpin daily functions. As systems grow in complexity with the addition of components, understanding these distinctions becomes crucial for optimizing performance and efficiency in various fields.
Simple, Linear, and Complex Systems
FIELDS OF STUDY: Classical Mechanics
ABSTRACT: Almost every aspect of modern life is built on the principle of systems. Complex machines used every day, such as bicycles, cars, and even corkscrews, are systems that consist of and depend upon interacting simple machines.
PRINCIPAL TERMS
- efficiency: how effective a machine is at transforming or transferring energy, quantified as the ratio of the actual performance of the machine to an idealized, theoretical version of it.
- joule: the International System of Units (SI) unit of work and energy, equivalent to the work done by a force of one newton applied over a distance of one meter.
- mechanical advantage: the amplification of force provided by many machines, measured as the ratio of the output force to the input force.
- net force: the sum of all forces acting on an object.
- newton: the International System of Units (SI) unit of force; one newton is equal to the force required to accelerate a one-kilogram mass at one meter per second per second.
- power: the rate of work done (energy transfer) over time.
- system elements: the individual components that work together and make up an overall system used to complete a task.
- work: the force moving an object, or the successful transfer of energy.
Simple Machines
A simple machine is used as the easiest way to multiply a force to complete a task. These and other kinds of machines are examples of a system, which is a process that has a defined input and at least one output. The six classic simple machines are the lever, the wheel and axle, the pulley, the inclined plane, the wedge, and the screw. A lever, for example, is simply a rod or plank balanced over a fulcrum. A wheel functions like a circular lever as it revolves about its center. This full circular motion enables a wheel to transfer energy, power, or mass from one position to another.
All simple machines provide mechanical advantage by amplifying the input force to create a greater output force. A lever positioned so that the application of 1 newton of force as input enables the production of 3 newtons of force as output has a mechanical advantage of 3. The trade-off for mechanical advantage is efficiency, the measurement of the machine’s effectiveness at transferring energy. It can be thought of as the ratio of the output to the input. The greater the mechanical advantage of a machine and the more elements involved, the lower its efficiency. For example, if the input force of 3 newtons moves through a distance of 3 meters (9.84 feet) to move an output force of 1 newton through a distance of 6 meters (19.68 feet), the machine has an efficiency of 0.67, or 67 percent. The input work would be 9 joules, and the output work would be 6 joules. No machine can be 100 percent efficient. There are always forces that counteract the force being applied, particularly as friction. The actual force that can then be transferred by a machine to produce an output is the net force. This is the sum of the applied force and the opposing forces.
Simple machines can be combined to produce more complex systems that are defined by internal and external boundaries. In a linear system, one device affects one output. In a nonlinear (complex) system, a single component can affect other system elements and produce more than one output. A bicycle is a complex machine/system constructed from several simple machines working as a single unit. The pedals are levers, providing torque to the main gears. These are connected by a chain system to another set of gears that transfer the torque to the rear wheel. The derailleurs that shift the drive chain from one gear to another are based on the inclined plane for their function. The handlebars also act as levers, allowing the orientation of the front wheel to be changed. This lets the bicycle change direction as it moves. Tension is normally applied to the brake and shift cables by pulling on their attached levers.
Feedback and Control Systems
A machine is of no use without an effective means of controlling its actions and functioning. Control systems built with modern digital electronics allow for precise control of the movement and actions of machines. This is accomplished by incorporating feedback mechanisms. A feedback mechanism measures an output stream, such as voltage or pressure, and uses that information to adjust the input stream in a way that maintains the proper desired stream. Input from an external source is generally the primary input to a system. A common example is the power delivered by the engine of a car with an automatic transmission. Fluid pressure in the transmission of the car is used to control the power delivered by the engine, increasing it when going uphill and decreasing it when rolling on flat road surfaces or going downhill.
Sample Problem
Calculate the overall efficiency of a linear system made up of six components having individual efficiencies of 98 percent, 87 percent, 86 percent, 75 percent, 50 percent, and 66 percent. Calculate the efficiency of a nonlinear (complex) system consisting of the same six components arranged such that the 98 percent efficient component drives an output from the remaining components in three streams made up of the 87 percent and 86 percent, 75 percent and 66 percent, and 50 percent components.
Answer:
The linear system produces an output that reflects the sequential product of the individual efficiencies of the components in the system, as
0.98 × 0.87 × 0.86 × 0.75 × 0.50 × 0.66 = 0.1815, or 18.15 percent
The nonlinear system must be treated as three independent systems, since it produces three independent outputs. There will be one stream that has an efficiency of
0.98 × 0.87 × 0.86 = 0.7332, or 73.32 percent
a second stream having an efficiency of
0.98 × 0.75 × 0.66 = 0.4851, or 48.51 percent
and a third stream with an efficiency of
0.98 × 0.50 = 0.49, or 49 percent
Systems in Real Life
Linear and complex systems are the fundamental structures of modern technology. Every machine that is used is a combination of simple machine structures. Examination of almost every aspect of modern life will reveal an underlying linear or complex system structure involved.
Bibliography
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