Home / Questions / Repeat Example 715 when ωr 20 and Jr 10 EXAMPLE 715 Natural frequencies and mode shapes...
Repeat Example 7.15 when ω_{r} = 2.0 and J_{r} = 1.0.
EXAMPLE 7.15
Natural frequencies and mode shapes of a pendulum attached to a translating mass Consider the system shown in Figure 7.17. We shall derive the governing equations of motion for small |θ| and then use these equations to determine the natural frequencies and mode shapes associated with this system. The equations of motion are obtained by using Lagrange’s equations. We will also illustrate how the expressions for the kinetic energy and the potential energy are appropriately truncated to obtain the linear equations of motion. Considering the translation of the point mass m, and the rotation of the rigid bar M, the system kinetic energy is
and the system potential energy is
where the datum for computing the potential energy due to gravity loading has been chosen at the center of mass of the pendulum. For “small” oscillations about θ = 0, Taylor-series expansions lead to
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