Elastic Instability - Multiple Degrees of Freedom-systems

Multiple Degrees of Freedom-systems

By attaching another rigid beam to the original system by means of an angular spring a two degrees of freedom-system is obtained. Assume for simplicity that the beam lengths and angular springs are equal. The equilibrium conditions become

$F L ( \sin \theta_1 + \sin \theta_2 ) = k_\theta \theta_1$

$F L \sin \theta_2 = k_\theta ( \theta_2 - \theta_1 )$

where and are the angles of the two beams. Linearizing by assuming these angles are small yields

$\begin{pmatrix} F L - k_\theta & F L \\ k_\theta & F L - k_\theta \end{pmatrix} \begin{pmatrix} \theta_1 \\ \theta_2 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix}$

The non-trivial solutions to the system is obtained by finding the roots of the determinant of the system matrix, i.e. for

$\frac{F L}{k_\theta} = \frac{3}{2} \mp \frac{\sqrt{5}}{2} \approx \left\{\begin{matrix} 0.382\\2.618 \end{matrix}\right.$

Thus, for the two degrees of freedom-system there are two critical values for the applied force F. These correspond to two different modes of deformation which can be computed from the nullspace of the system matrix. Dividing the equations by yields

$\frac{\theta_2}{\theta_1} \Big|_{\theta_1 \ne 0} = \frac{k_\theta}{F L} - 1 \approx \left\{\begin{matrix} 1.618 & \text{for } F L/k_\theta \approx 0.382\\ -0.618 & \text{for } F L/k_\theta \approx 2.618 \end{matrix}\right.$

For the lower critical force the ratio is positive and the two beams deflect in the same direction while for the higher force they form a "banana" shape. These two states of deformation represent the buckling mode shapes of the system.

Read more about this topic: Elastic Instability

Famous quotes containing the words multiple and/or degrees:

“... the generation of the 20’s was truly secular in that it still knew its theology and its varieties of religious experience. We are post-secular, inventing new faiths, without any sense of organizing truths. The truths we accept are so multiple that honesty becomes little more than a strategy by which you manage your tendencies toward duplicity.”
—Ann Douglas (b. 1942)

“So that the life of a writer, whatever he might fancy to the contrary, was not so much a state of composition, as a state of warfare; and his probation in it, precisely that of any other man militant upon earth,—both depending alike, not half so much upon the degrees of his WIT—as his RESISTANCE.”
—Laurence Sterne (1713–1768)