11.9:
Potential-Energy Criterion for Equilibrium
Consider a frictionless spring-mass system whose position is defined by one independent variable. Here, the potential energy is the sum of the gravitational and elastic potential energies, and the negative of its change equals the work done on the system.
If the system is in equilibrium and undergoes a virtual displacement, the work done can be replaced by virtual work.
From the principle of virtual work, the work is zero for all virtual displacements, and so the change in potential energy is also zero. For small virtual displacements, the potential energy is expressed in terms of its first derivative with the position coordinate.
Since the virtual displacement is not zero, the first derivative of the potential energy must be zero. So, a system is in equilibrium when the first derivative of its potential energy is zero.
Applying this criterion to the spring-mass system gives its equilibrium position.
If the potential energy depends on several independent variables, then the partial derivative of the potential energy for each each coordinate must be zero for equilibrium.
11.9:
Potential-Energy Criterion for Equilibrium
Potential energy or potential function plays an essential role in determining the stability of a mechanical system. If a system is subjected to both gravitational and elastic forces, the potential function of the system can be expressed as the algebraic sum of gravitational and elastic potential energy. If the system is in equilibrium and is displaced by a small amount, then the work done on the system equals the negative of the change in the system's potential energy from the initial to the final position. If the system undergoes a virtual displacement rather than an actual displacement, then the virtual work relation suggests that the change in potential energy of the system should be zero for all virtual displacements. This means that the equilibrium configuration of a mechanical system is one for which the total potential energy of the system has a stationary value. For a system of one degree of freedom where the potential energy and its derivatives are a continuous function of a single variable x, which describes the configuration, the equilibrium condition can be written as
The above equation states that a mechanical system is in equilibrium when the derivative of its total potential energy is zero. For systems with several degrees of freedom, the partial derivative of potential energy with respect to each coordinate must be zero for equilibrium.
The stability at the equilibrium configuration remains as long as its potential energy is minimum at that configuration. To ensure minimum potential energy, the second derivative of the potential function with respect to the displacement coordinates must be positive. This is the second essential condition for the stable equilibrium of a system.
Suggested Reading
- Hibbeler, R.C. (2016). Engineering Mechanics: Statics. Fourteenth Edition, New Jersey: Pearson. Pp. 600-602.
- Meriam, J.L., Kraige, L.G. and Bolten, J.N.(2016). Engineering Mechanics: Statics and Dynamics. Eighth Edition, Singapore: John Wiley & Sons. Pp. 415-416.