MODULE 3. KINEMATICS OF RIGID BODIES

This module tackled how the velocity and acceleration of the particles of a rigid body are related to the translational and rotational parts of its motion.

Various types of rigid-body motion can be conveniently grouped into following:

a.) Translation - A motion is said to be a translation if any straight line inside the body maintains the same orientation during the motion. In a translation, all of the particles forming the body move along parallel paths.

b.) Rotation About a Fixed Axis. In this motion, the particles forming the rigid body move in parallel planes along circles centered on the same fixed axis.

c.) General Plane Motion. Many other types of plane motion can occur, i.e., motions in which all the particles of the body move in a single plane. Any plane motion that is neither a rotation nor a translation is referred to as general plane motion.

d.)Motion About a Fixed Point. The three-dimensional motion of a rigid body attached at a fixed point O, such as the motion of a top on a rough floor.

translation and fixed axis rotation

rigid body in translation

Consider a rigid body in translation (either rectilinear or curvilinear translation), and let A and B be any two of its particles. Denoting the position vectors of A and B with respect to a fixed frame of reference by rA and rB. The relationship between velocity and accelerations A and B are shown in the figure.
When a rigid body is in translation, all the points of the body have the same velocity and the same acceleration at any given instant.

rigid body rotating about a fixed axis

The position of a rigid body rotating about a fixed axis is defined at any given instant by the angular position θ, which is usually measured in radians. Selecting the unit vector k along the fixed axis in such a way that the rotation of the body appears counterclockwise as seen from the tip of k, we define the angular velocity ω and the angular acceleration α of the body as shown in the left.
The velocity and acceleration of a point P of a body rotating about a fixed axis are shown in the left figure.
The vectors ω and α are both directed along the fixed axis of rotation and that their sense can be obtained by the right-hand rule.

rotation of a representative slab

In the case of the rotation of a representative slab in a plane perpendicular to the fixed axis. The z axis should be directed along the axis of rotation and point out of the page. Thus, the representative slab rotates in the xy plane about the origin O of the coordinate system. Problems under this type can be solve using the following steps:

a.) Draw a diagram of the representative slab showing its dimensions, its angular velocity and angular acceleration, and the vectors representing the velocities and accelerations of the points of the slab.

b.) Relate the rotation of the slab and the motion of points of the slab. Remember that the velocity v and the component at of the acceleration of a point P of the slab are tangent to the circular path described by P. The normal component an of the acceleration of P is always directed toward the axis of rotation.

equations defining rotation of rigid body

The motion of a rigid body rotating about a fixed axis is said to be known when we can express its angular coordinate θ as a known function of t. More often, the conditions of motion are specified by the angular acceleration of the body. For example, α may be given as a function of t, as a function of θ, or as a function of ω .
Two particular cases of rotation occur frequently:

a.) Uniform Rotation. This case is characterized by the fact that the angular acceleration is zero, therefore the angular velocity is constant.

b.) Uniformly Accelerated Rotation. In this case, the angular acceleration is constant.

general plane motion: velocity

A general plane motion can always be considered as the sum of a translation and a rotation.

analyzing general plane motion

Consider a wheel rolling on a straight track. Over some interval of time, two given points A and B will have moved, respectively, from A1 to A2 and from B1 to B2. However, we could obtain the same result through a translation that would bring A1 and B1 into A2 and B'1 (the line AB remaining vertical), followed by a rotation about A, bringing B into B2
In the general case of plane motion, we consider a small displacement that brings two particles A and B of a representative rigid body, respectively, from A1 and B1 into A2 and B2. We can divide this displacement into two parts: in one, the particles move into A2 and B'1 while the line AB maintains the same direction; in the other, B moves into B2 while A remains fixed. The first part of the motion is clearly a translation, and the second part is clearly a rotation about A.

absolute and relative velocity in plane motion

Any plane motion of a rigid body can be replaced by a translation of an arbitrary reference point A and a simultaneous rotation about A. The absolute velocity vB of a particle B of the rigid body from the relative velocity formula can be obtained as shown in the right figure
The angular velocity v of a rigid body in plane motion is independent of the reference point.

Resolving the given motion into a translation of B and a simultaneous rotation about B as shown in the pictorial representation of sliding rod below, we have the equation,

instantaneous center of rotation

At any given instant, the velocities of the various particles of the rigid body are the same as if the body were rotating about an axis perpendicular to the plane of the body, called the instantaneous axis of rotation. This axis intersects the plane of the rigid body at a point C, called the instantaneous center of rotation of the body or the instantaneous center of zero velocity.

Assuming that vA and ω are known and that they are both different from zero. We could obtain these velocities by letting the rigid body rotate with the angular velocity ω about a point C located on the perpendicular to vA at a distance r = vA/ω from A.
As far as the velocities are concerned, the rigid body seems to rotate about the instantaneous center C at the instant considered.

general plane motion: acceleration

absolute and relative acceleration in plane motion

The relative acceleration aB/A can be resolved into two components: a tangential component (aB/A)t perpendicular to the line AB and a normal component (aB/A)n directed toward A. We denote the position vector of B relative to A by rB/A and the angular velocity and angular acceleration of the rigid body with respect to axes of fixed orientation by ωk and αk, respectively.

As an example, there is a rod AB whose ends slide along a horizontal and a vertical track. Assuming that we know the velocity vA and the acceleration aA of A, we propose to determine the acceleration aB of B and the angular acceleration α of the rod. Choosing A as a reference point, the given motion is equivalent to a translation with A and a rotation about A. The absolute acceleration of B must be equal to the sum shown in the right figure, where (aB/A)n has magnitude lω^2 and is directed toward A, while (aB/A)t has the magnitude lα and is perpendicular to AB.
When a mechanism consists of several moving parts that are pinconnected, we can analyze the mechanism by considering each part to be a rigid body, keeping in mind that the points at which two parts are connected must have the same absolute acceleration.

analysis of plane motion in terms of a parameter

It is possible to express the coordinates x and y of all the significant points of the mechanism by means of simple analytic expressions containing a single parameter.
For example, considering again the rod AB whose ends slide, respectively, in a horizontal and a vertical track. The coordinates xA and yB of the ends of the rod can be expressed in terms of the angle θ that the rod forms with the vertical.
A positive sign for vA or aA indicates that the velocity vA or the acceleration aA is directed to the right; a positive sign for vB or aB indicates that vB or aB is directed upward

analyzing motion with respect to a rotating plane

Rate of change of a vector with respect to a fixed frame and with respect to a rotating frame

Denoting the rate of change of a vector Q with respect to a fixed frame OXYZ by (Q̇)OXYZ and its rate of change with respect to a rotating frame Oxyz by (Q̇)Oxyz, the fundamental relation can be obtained as shown in the left figure, where Ω is the angular velocity of the rotating frame.

plane motion of a particle relative to a rotating plane

Using the equation above and designating the rotating frame by ℱ ,the following expressions for the velocity and the acceleration of a particle P can be obtained using the given relation shown in the figure.
Take note of the following notations used which are as follows:

a.) The subscript P refers to the absolute motion of the particle P; that is, to its motion with respect to a fixed or newtonian frame of reference OXY.

b.) The subscript P' refers to the motion of the point P' of the rotating frame ℱ that coincides with P at the instant considered.

c.) The subscript P/ℱ refers to the motion of the particle P relative to the rotating frame ℱ .

d.) The term aC represents the Coriolis acceleration of point P. Its magnitude is 2ΩvP/ℱ , and its direction is found by rotating vP/ℱ through 90° in the sense of rotation of the frame ℱ.

motion of a rigid body in space

motion about a fixed point

The most general displacement of a rigid body with a fixed point O is equivalent to a rotation of the body about an axis through O (Euler's Theorem).
Listed below are the following steps on how to analyze the motion of a point B of a body rotating about a fixed point O:

1. Determine the position vector r connecting the fixed point O to point B.

2. Determine the angular velocity ω of the body with respect to a fixed frame of reference. You can often obtain the angular velocity ω by adding two component angular velocities ω1 and ω2 .

3. Compute the velocity of B from the equation v = ω x r.

4. Determine the angular acceleration α of the body. The angular acceleration α represents the rate of change (ὤ)OXYZ of the vector v with respect to a fixed frame of reference OXYZ and reflects both a change in magnitude and a change in direction of the angular velocity.

5. Compute the acceleration of B by using the equation shown in the left figure.

general motion of a rigid body

The general motion of a rigid body may be considered as the sum of a translation and a rotation.
In the translation part of the motion, all of the points of the body have the same velocity vA and the same acceleration aA as point A of the body that has been selected as the reference point. While in the rotation part of the motion, the same reference point A is treated as if it were a fixed point.
To determine the velocity of a point B of the rigid body when the velocity vA of the reference point A and the angular velocity ω of the body are known, simply add vA to the velocity vB/A = ω x rB/A of B in its rotation about A.
To determine the acceleration of a point B of the rigid body when the acceleration aA of the reference point A and the angular acceleration α of the body are known, simply add aA to the acceleration of B in its rotation about A.

motion relative to a moving reference frame

three-dimensional motion of a particle relative to a rotating frame

Consider the three-dimensional motion of a particle P relative to a rotating frame Oxyz constrained to have a fixed origin O. Let r be the position vector of P at a given instant, and let Ω be the angular velocity of the frame Oxyz with respect to the fixed frame OXYZ at the same instant. The absolute velocity vP of P (i.e., its velocity with respect to the fixed frame OXYZ) can be expressed as shown in the right figure., where (ṙ)Oxyz is the relative velocity of point P with respect to the rotating frame. The relation can also be denoted in alternative form.

The absolute acceleration aP of P can be expressed with its alternative form as shown in the left figure.

frame of reference in general motion

Consider a fixed frame of reference OXYZ and a frame Axyz that moves in a known, but arbitrary, fashion with respect to OXYZ. The position of P is defined at any instant by the vector rP in the fixed frame and by the vector rP/A in the moving frame. The velocity vP/A of P relative to AX'Y'Z' can be obtained using the relation shown in the figure (by substituting rP/A to r), where Ω is the angular velocity of the frame Axyz at the instant considered.
Similarly, the acceleration aP/A of P relative to frame AX'Y'Z' can be obtained using the relation (shown in the right figure).

classwork assignment #3

Classwork Assignment 3_BARTOLOME.pdf

coursework #3

BARTOLOME_FAUSTOCOURSEWORK-3-DORB001.pdf

reflection:

The module entails the study of the kinematics of rigid bodies, which started with the rigid body in translation, wherein all points of the body in motion have all the same velocity and same acceleration at any given instant. In this module, I have learned how to analyze a rigid body that is rotating about its fixed axis, wherein the velocity and angular acceleration of an object rotating can be solved using angle of rotation drawn from the rotation axis. I learned that when obtaining expressions for angular velocity and acceleration of a rigid body, two particular cases of rotation can be encountered. These are uniform rotation and uniformly accelerated rotation. In addition, I realized that the general plane motion of a body is just the sum of translation and rotation, which helped me to easily analyze problems and draw kinetic diagrams of a rigid body.

The latter part of the module tackled the motion of rigid body with fixed point, general motion of rigid body in space, 3d dimensional motion of a particle relative to rotating frame, and frame of reference in general motion. In these topics, I learned that the direction of angular velocity changes from one instant to the next, and the angular acceleration is not directed along the instantaneous axis of rotation. Lastly, I realized that in rotating frames of reference, a correct choice of rotating frame will help in solving three-dimensional motion of rigid bodies, because it will lead to much simpler analysis of motion of rigid body.

Page updated

Report abuse