# One-Degree-of-Freedom System

JoVE Core
Mechanical Engineering
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JoVE Core Mechanical Engineering
One-Degree-of-Freedom System

### Nächstes Video11.12: Stability of Equilibrium Configuration: Problem Solving

If a system's position is defined by one independent variable, it has one degree of freedom. Its potential energy function can then be plotted against that variable.

When the system reaches equilibrium, the slope of the function at that position is zero.

The stability of the equilibrium configuration can be investigated using the second derivative of the potential energy function with respect to the variable.

If the second derivative is positive, the potential energy is minimum, and the system is in a stable equilibrium configuration.

Conversely, if the second derivative is negative, the potential energy is maximum. This implies an unstable equilibrium configuration.

If the second derivative is zero, higher-order derivatives must be evaluated.

If the first non-zero derivative is positive and has an even order, the equilibrium configuration is stable.

Alternatively, if the first non-zero derivative is negative and has odd or even order, it is unstable.

If all the higher-order derivatives are zero, the system has achieved neutral equilibrium. Here, the potential-energy function remains constant around the equilibrium position.

## One-Degree-of-Freedom System

In mechanical engineering, one-degree-of-freedom systems form the basis of a wide range of electrical and mechanical components. Using these models, engineers can predict the behavior of various parts in a larger system, which gives them insight into how different forces interact with each other.

A one-degree-of-freedom system is defined by an independent variable that determines its state and behavior. One example of a one-degree-of-freedom system is a simple harmonic oscillator, such as a mass attached to a spring. The position of the mass along one axis determines its state and behavior, and the force exerted by the spring causes it to oscillate back and forth. The potential energy (V) in such systems can be plotted on a graph to show how much energy the object holds depending on its position relative to its resting point.

By understanding these one-degree-of-freedom systems, engineers can gain valuable knowledge when constructing complex designs. This allows them to accurately model and predict their functioning under different conditions with confidence before building the designs out in reality. Additionally, they can also gain insight into the basic equations linking forces and accelerations, which helps to understand physical properties like inertia.

For investigations into the stability of equilibrium of different configurations, engineers analyze the second derivative of the potential function to determine which equilibrium configurations are stable, unstable, or neutral. Stable equilibriums correspond to minimum values on the potential energy plot, while maximum values indicate instability, and constant values denote neutrality at those points. This helps give designers a better sense of what designs work best for given environments where outside factors may cause dynamic changes over time that could disrupt a machine's operation if not well predicted beforehand.

Overall knowledge about single-degree freedom systems represents an invaluable tool for engineers looking to construct robust machines or vehicles, from small alarm clocks to modern rockets heading toward outer space.