Unformatted text preview: t be quantified. This is accomplished by finding the unit vectors in the direction of each
spring, shown below as ̂ and ̂ . ̂ ̂ Using the frame shown in the figure,
̂
̂ cos
cos ̂
̂ sin In vector form, the deflection of the mass ̅ is sin ̅ ̂ ̂ ̂ Therefore,
Δ ̂ ∙ ̅ cos ̂ ∙ ̅ Δ cos The potential energy is then simplified to be
cos
Lagrange’s equation is
0
21 cos 0 Thus, interesting enough, the system behaves as if it were restrained by a single spring of
magnitude
21 cos Example 2:
The example below is significantly more complicated than that discussed above. The Lshaped
bar is pinned at point , and is free to rotate about that point. Hence, the single degree of
freedom required for solution of the problem (obtaining the equation of motion) is , as shown in
the figure.
There are two springs, each of stiffness , and two dampers, each of magnitude . The mass of
the bar is , though you will see that it is not exactly relevant here. Relevant points on the body
are , which is the Lbeam weld, , where the force acts, and , the end of the bar. Both
springs are inclined an angle from the bottom portion of the beam. Relevant lengths are also
demonstrated in the figure.
The goal of the problem is to find the equation of motion of the system, using Lagrange’s
equation: where signifies Rayleigh’s dissipation function. The kinetic energy of the system is very easy to obtain, because the whole body purely rotates
about point :
1
2
For the sake of simplicity, the inertia term of the body about
knowledge of the mass is not entirely relevant). will be left as (hence, the Unit vectors are shown designating the direction of each spring and damper. The subscript on
the unit vector indicates the designation of the component. Therefore, the general forms of
and are
1
Δ
2 1
Δ
2 1
Δ
2 1
Δ
2 The symbol Δ designates a displacement (or stretching/compressing), whereas the symbol Δ
designates the velocity of one end of the compon...
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 Fall '12
 I.Green
 Energy, Force, Kinetic Energy, Strain

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