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MIT16_07F09_Lec02

MIT16_07F09_Lec02 - S Widnall 16.07 Dynamics Fall 2009...

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S. Widnall 16.07 Dynamics Fall 2009 Version 1.0 Lecture L2 - Degrees of Freedom and Constraints, Rectilinear Motion Degrees of Freedom Degrees of freedom refers to the number of independent spatial coordinates that must be specified to deter- mine the position of a body. If the body is a point mass, only three coordinates are required to determine its position. On the other hand, if the body is extended, such as an aircraft, three position coordinates and three angular coordinates are required to completely specify its position and orientation in space. Kinematic Constraints In many situations the number of independent coordinates will be reduced below this number, either because the number of spacial dimensions is reduced or because there are relationships specified among the spatial coordinates. When setting up problems for solution it is useful to think of these relationships as constraints. For example, if a point mass is constrained to move in a plane (two dimensions) the number of spatial coordinates necessary to describe its motion is two. If instead of being a point mass, this body has extended dimensions, such as a flat plate confined to a plane, it requires three coordinates to specify its position and orientation: two position coordinates and one angular coordinate. 1
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If a particle is confined to move on a curve in either two or three dimensions, such as a bead moving on a wire, the number of independent coordinates necessary to describe its motion is one. Another source of constraints on the motion of particles is connections between them. For example, the two particle connected by a cable passing over a pulley are constrained to move in equal and opposite directions. More complex arrangements are possible and can be analyzed using these ideas. Two gears in contact are constrain to move together according to their individual geometry. A cylinder rolling on a plane is constrained in two ways. Contact with the plane reduces the two-dimensional motion to one spatial coordinate along the plane, and the constraint of rolling provides a relationship be- tween the angular coordinates and the spatial position, resulting in a single degree of freedom system. 2
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Internal Force-Balance Constraints Another type of constraint occurs when we consider the of a system of particles and the necessary force balance that occurs between the parts. These constraints follow directly from Newton’s third law: the force of action and reaction between two bodies are equal in magnitude and opposite in direction. We will pursue these ideas in greater depth later in the course. For now, we will give a simple example to illustrate the principle. Consider the systems shown in a) and b) .
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