Читать книгу Introduction to Mechanical Vibrations - Ronald J. Anderson - Страница 17

We consider again the bead on the semicircular wire (Figure 1.1) and derive the equation of motion using Lagrange's Equations.

Lagrange's Equation is:

(1.16)

where:

  = the total kinetic energy of the system

  = the total potential energy of the system

  = a generalized coordinate

  = the time derivative of

  = the generalized force corresponding to a variation of

We first determine the kinetic energy of the system. This requires that we have an expression for the absolute velocity of the mass. This was done previously and the result, from Equation 1.5, is

(1.17)

The kinetic energy of the system is then

(1.18)

which becomes, after substitution of Equation 1.17 and some simplification,

(1.19)

Alternatively, using the informal approach and referring to Figure 1.3, we can see that there will be a component of velocity equal to tangent to the wire and another component equal to perpendicular to the wire and into the page. These two components are mutually perpendicular so we can write, by applying Pythagoras' theorem,

(1.20)

After factoring out of the brackets, this becomes exactly the same expression we had in Equation 1.19.

The potential energy of the system is due to gravity only. If the datum for potential energy is taken to be at point , the potential energy, , of the system is determined simply by the vertical distance from to the bead. This distance is and, as the mass is below the datum, the potential energy is negative, leading to

(1.21)

Having expressions for and and a single degree of freedom, , we can apply Lagrange's Equation (Equation 1.16) and find

(1.22)

Substituting the expressions from Equation 1.22 into Lagrange's Equation (Equation 1.16) gives the desired equation of motion

(1.23)

where we note that this equation when divided throughout by yields the same result as Equations 1.11 and 1.15, the equation of motion derived using Newton's Laws.

Clearly, Equation 1.23 could be further simplified by factoring out the group but this would take away the ability to look at the individual terms and give a physical explanation for them. Whenever an equation is derived, the first test for correctness is to see if all of the terms have the same dimensions. In this case, the first term has dimensions of where is mass, is length, and is time. Note that angles such as are in radians, which are dimensionless since they are defined by an arc length divided by a radius. It follows that trigonometric functions such as and are also dimensionless. Angular velocities therefore have dimensions derived from angles divided by time, , and angular accelerations are expressed as . Using these conventions, it is easy to see that all three terms in Equation 1.23 have the same dimensions³.

The dimensions of force are or mass times acceleration. Taking this into account, we can see that the three terms in Equation 1.23 all have dimensions of or force times length. The terms are, in fact, all moments. The third term is the most obvious because it contains the gravity force multiplied by a moment arm of . The moment arm is simply the horizontal distance between the mass and point . Lagrange's Equation has produced an equation of motion based on a dynamic moment balance about the stationary point and it did so without requiring the derivation of acceleration expressions, the drawing of free body diagrams, or the production of force and moment balance relationships. This is the power of using Lagrange's Equation for deriving equations of motion.

Following are explanations of terms that arise when using Lagrange's Equations.