Lagrange Equations of Motion for NonConservative Forces

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Lagrange Equations of Motion for NonConservative Forces
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1 Introduction to Lagrange Equations of Motion for NonConservative Forces[1]

The Lagrangian description of a mechanical system is different. For Lagrangian mechanics it is assumed that the position of the system at 2 instances of time (t1 and t2) are known (or knowable) and fixed. The system must move/behave between times t1 and t2 such that the system has the least "action". The "action" being defined by

LaTeX: \int_{t_{1}}^{t_{2}} L\left(q_{i}, \dot q_{i}\right) dt "Action"


where

LaTeX: L\left(q_{i}, \dot q_{i}\right) is called the Lagrange function or Lagrangian
LaTeX: q_{i} is a generalized coordinate which can represent the x, y, z of Cartesian coordinates or the θ, R of Polar coordinates, etc.

2 Lagrange Equation of Motion for Conversative Forces

The Lagrange Equation of Motion for Conservative Forces is below

LaTeX: \frac{d}{dt}\ \left(\frac{\partial K}{\partial \dot q_{1}}\right)-\frac{\partial K}{\partial q_{1}}=F_{q_{1}}. '


This results is based on D'Alembert's equation which starts by calculating differential work work done on a particle by calculating the change kinetic energy:

LaTeX: \delta W = m\left(\ddot x \delta x + \ddot y \delta y + \ddot z \delta z\right) = F_{x}\delta x + F_{y}\delta y + F_{z}\delta z '


This is work done by a force LaTeX: \mathbf{F}=[F_{x}, F_{y}, F_{z}]^{T} for a displacement LaTeX: \delta \mathbf{s}=[\delta x, \delta y, \delta x]^{T}. In terms of generalized coordinates we have

LaTeX: \begin{alignat}{2}
<p>\delta W & = & \left( F_{x} \frac{\partial x}{\partial q_{1}}+F_{y} \frac{\partial y}{\partial q_{1}}+F_{z} \frac{\partial z}{\partial q_{1}} \right) \delta q_{1} \\
& = & F_{q_{1}}\delta q_{1}\\
& = & \delta W_{q_{1}}
</p>
\end{alignat} LaTeX: \delta q_{2} = 0 for convenience


A generalized force LaTeX: F_{q_{r}} is of such nature that the product LaTeX: F_{q_{r}}\delta q_{r} is the work done by driving forces when LaTeX: q_{r} alone is changed by LaTeX: \delta q_{r}. We mention that a generalized force doe snot have to be a force in the usual sense. For example, if LaTeX: q_{r} is an angle, the LaTeX: F_{q_{r}} must be a torque in order that LaTeX: F_{q_{r}}\delta q_{r} be work.

3 Derivation of Lagrange Equation of Motion for NonConversative Forces[2]

The Lagrange equation above was derived for conservative forces only. A more general form would need to include nonconservative forces, such as friction, and that is what is derived in this section.


The Lagrange equation of motion for the q1 coordinate if some of the forces are nonconservative (such as friction) is The final result of this derivation is

LaTeX: \frac{d}{dt}\ \left(\frac{\partial L}{\partial \dot q_{1}}\right)-\frac{\partial L}{\partial q_{i}}=\tilde{F}_{q_{i}}. Lagrange Equation of Motion for All Forces


where:

  • LaTeX: L is the Lagrangian function,
  • LaTeX: q_{i} is the generalized coordinate, and
  • LaTeX: \tilde{F}_{q_{i}} are the conservative forces.

The kinetic energy of a particle is its ability to do work by virute of its motion. All forces, both conservative and nonconservative, acting on the particle contribute to the particle's motion. If we add a single nonconservative force, friction, to the conservative forces acting on the particle then

LaTeX: W_{f}+\sum W_{c}=\Delta K. '


where

  • LaTeX: \sum W_{c} is the total work done by all conservative forces,
  • LaTeX: W_{f} is the work done by friction, and
  • LaTeX: \Delta K is the change in kinetic (mechanical) energy.

As a result of the nonconservative force, friction, the total kinetic energy is not constant. The change is equal to the amount of work done by the nonconservative force. The mechanical (kinetic) energy of the particle can be described as follows:

LaTeX: E-E_{0}=\Delta E=W_{f}. '


where:

  • LaTeX: E is the final mechanical energy of the particle and
  • LaTeX: E_{0} is the particle initial mechanical energy.

Since friction always removes kinetic energy (it is always negative) the final energy of the particle (LaTeX: E) is less than the intial energy (LaTeX: E_{0}). The difference between the final and initial energies captured as internal energy.

LaTeX: \Delta E + U_{int}=0 '


where:

  • LaTeX: U_{int} is the internal energy or "lost" mechanical energy.

Further generalizing we get an equation stating the principle of conservation of energy. In other words the total energy does not change, just the form of the energy from kinetic to other such as internal or heat.

LaTeX: \Delta K + \sum \Delta U + U_{int}+\mbox{ (change in other forms of energy)} = 0. '


Earlier we described a general force

LaTeX: F_{q_{i}}=F_{x}\frac{\partial x}{\partial q_{i}}+F_{y}\frac{\partial y}{\partial q_{i}}+F_{z}\frac{\partial z}{\partial q_{i}}. '


If all the forces are conservative then

LaTeX: F_{q_{i}}=-\left(\frac{\partial U}{\partial x}\frac{\partial x}{\partial q_{i}}+\frac{\partial U}{\partial y}\frac{\partial y}{\partial q_{i}}+\frac{\partial U}{\partial z}\frac{\partial z}{\partial q_{i}}\right) '


where:

  • LaTeX: U is potential energy and
  • LaTeX: \Delta U is the change in potential energy.

Then we have

LaTeX: F_{q_{i}}=-\frac{\partial U}{\partial q_{i}}. '


Therefore, the Lagrange equations of motion for a single particle for conservative forces can be represented as

LaTeX: \frac{d}{dt}\left(\frac{\partial K}{\partial \dot q_{i}}\right)-\frac{\partial K}{\partial q_{i}}=-\frac{\partial U}{\partial q_{i}} '


Rearranging the equation so that all non-zero components are on the left side of the equation we get

LaTeX: \frac{d}{dt}\left(\frac{\partial K}{\partial \dot q_{i}}\right)-\frac{\partial}{\partial q_{i}}\left(K-U\right)=0 1


Define the Lagrangian function as

LaTeX: L=K-U 2


The partial derivative with respect to LaTeX: \dot q_{i} is

LaTeX: \frac{\partial}{\partial \dot q_{i}}\left(L\right)=\frac{\partial}{\partial \dot q_{i}}\left(K-U\right) '


Since potential energy (LaTeX: U) is not a function of velocity (LaTeX: \dot q_{i}) then

LaTeX: \frac{\partial L}{\partial \dot q_{i}}=\frac{\partial K}{\partial \dot q_{i}}. 3


Substituting Eqn. (1) and (2) into (3), the Lagrange equations of motion for conservative forces becomes

LaTeX: \begin{array}{lcl}
<p>\frac{d}{dt}\left(\frac{\partial K}{\partial \dot q_{i}}\right)-\frac{\partial}{\partial q_{i}}\left(K-U\right) & = & 0 \\
\frac{d}{dt}\left(\frac{\partial K}{\partial \dot q_{i}}\right)-\frac{\partial}{\partial q_{i}}\left(L\right) & = & 0 \\
\frac{d}{dt}\left(\frac{\partial L}{\partial \dot q_{i}}\right)-\frac{\partial L}{\partial q_{i}} & = & 0
</p>
\end{array}. '


When there are nonconservative forces present the Lagrange equations of motion become

LaTeX: \frac{d}{dt}\ \left(\frac{\partial L}{\partial \dot q_{1}}\right)-\frac{\partial L}{\partial q_{i}}=\tilde{F}_{q_{1}} '


where:

  • LaTeX: \tilde{F}_{q_{i}} are the nonconservative forces acting on the particle.

4 Notes

4.1 References

  1. WikiBooks
  2. Zak, pp. 15-17