Crack Label Matrix 8.20
Mina Delahoussaye
As mentioned, these equations are coupled since both and appear in each of these equations. The implication with coupled equations is that they must be solved as a system. Neither of the equations can be solved individually.
In contrast, consider the same system as above but with the central spring removed as shown in Figure 8.8. This is still a TDOF system as two coordinates are still required to completely specify the positions of the masses. The two equations of motion for this system are
(8.21)
As can be seen, uncoupled systems are significantly easier to solve that coupled ones. Review the effort required to obtain the solution to equations (8.20) and compare that to the effort required to obtain the solution to those in (8.21). The difference is apparent, and will become more significant as systems with more degrees of freedom are considered. It is hopefully clear from this that it is preferable to deal with uncoupled equations if possible.
Note that in matrix form, equations (8.20) and (8.21) are
(8.23)
respectively. As can be seen, uncoupled equations correspond to diagonal matrices (both mass and stiffness matrices must be diagonal). Any off-diagonal term leads to a coupling of the equations. In equation (8.23) the off-diagonal terms occur in the stiffness matrix, which is referred to as static coupling. The situation in which the mass matrix contains off-diagonal is termed inertia coupling. It is possible for both type of coupling to be present at the same time.
Relationship to Mode Shapes
In the linear systems we have been discussing it is (almost) always possible to find a set of coordinates which uncouples the equations of motion. That is essentially what the mode shapes represent: linear combinations of the chosen coordinates that will uncouple the equations of motion:
The two degree of freedom system shown consists of a wheel (with mass and centroidal moment of inertia ) which rolls without slipping and a bar (with mass and centroidal moment of inertia ) connected by three springs each of stiffness .
(a) Using the coordinates (the horizontal position of the center of the wheel) and (the horizontal position of the lower end of the bar) show that if and , the equations of motion for this system become
(3) Updated paragraph 8.20.6.1.3 (8) with the approved Taxpayer Bill of Rights statement: The Taxpayer Bill of Rights (TBOR) lists rights that already existed in the tax code, putting them in simple language and grouping them into 10 fundamental rights. Employees are responsible for being familiar with and acting in accord with taxpayer rights. See IRC 7803(a)(3), Execution of Duties in Accord with Taxpayer Rights. For additional information about the TBOR, see -bill-of-rights.
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The Taxpayer Bill of Rights (TBOR) lists rights that already existed in the tax code, putting them in simple language and grouping them into 10 fundamental rights. Employees are responsible for being familiar with and acting in accord with taxpayer rights. See IRC 7803(a)(3), Execution of Duties in Accord with Taxpayer Rights. For additional information about the TBOR, see -bill-of-rights.
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