With those preliminaries out of the way, in the next video we move on to representing the orientation of a rigid body. If you align the thumb of your right hand with the axis of rotation, positive rotation is the direction that your fingers curl. Positive rotation about an axis is defined by the right-hand rule. Even if the body is moving, when we talk about the body frame, we mean the stationary frame coincident with the frame attached to the body at a particular instant in time. In this book, all frames are considered to be stationary. The configuration of the body is given by the position of the origin of the body frame and the directions of the coordinate axes of the body frame, expressed in the space-frame coordinates. If I want to represent the position and orientation of a body in space, I fix a frame to the body and fix a frame in space. You can create a right-handed frame using your right hand: your index finger is the x-axis, your middle finger is the y-axis, and your thumb is the z-axis. All frames are right-handed, which means that the cross product of the x and y axes creates the z-axis. A frame consists of an origin and orthogonal x, y, and z coordinate axes. Rigid-body configurations are represented using frames. This approach may be new to you if you haven't taken a course in three-dimensional kinematics before. In other words, our representation of a configuration will not use a minimum set of coordinates, and velocities will not be the time derivative of coordinates. As discussed in the last chapter, we'll use implicit representations of configurations, considering the C-space as a surface embedded in a higher-dimensional space. With clarity and precision, describe a sequence of rigid motions that would map figure ABC onto figure A'B'C'.In Chapter 3, we learn representations of configurations, velocities, and forces that we'll use throughout the rest of the book. Let two figures ABC and A'B'C' be given so that the length of curved segment AC = the length of curved segment A'C', |∠ B| = |∠ B'| = 80 ° and |AB| = |A'B'| = 5. Which basic rigid motion, or sequence of, would map one triangle onto the other?ĥ. Kinematic Constraints in Structural Analyses In a stationary or quasistatic structural analysis, we are looking for an equilibrium solution. Transparency and mapped onto triangle A'B'C'. Today, we showcase the Rigid Motion Suppression feature in the COMSOL Multiphysics software, which you can use to automatically figure out the constraints you need. In each pair, triangle ABC can be traced onto a In the following picture, we have two pairs of triangles. Which basic rigid motion, or sequence of, would map one triangle onto the other?Ĥ. In the following picture, triangle ABC can be traced onto a transparency and mapped onto triangle A'B'C'. So now we let E 3 be the image of E after the Translation 1(E)įollowed by the Rotation 2(E) followed by the Rotation 3(E)ġ - 3. Now the only transformation left is Reflection 3.Which transformation do we perform next?. Which transformation do we perform first, the translation, the reflection or the rotation? How do you know?.What is the Translation 1(E) followed Rotation 2(E) by the followed by the Reflection 3(E)? Let Translation 1 be the translation along the vector v from (1, 0) to (-1, 1), let Rotation 2 be the 90 degree rotation around (-1, 1), and let Reflection 3 be the reflection across line L joining (-3, 0) and (0, 3).Let E denote the ellipse in the coordinate plane as shown.Now that we know about rotation, we can move geometric figures around the plane by sequencing a combination of So far we have seen how to sequence translations, sequence reflections, and sequence translations and reflections. New York State Common Core Math Grade 8, Module 2, Lesson 10 Worksheets New York State Common Core Math Grade 8, Module 2, Lesson 10. Videos, examples, and solutions to help Grade 8 students describe a sequence of rigid motions to map one figure onto another.
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