Dynamics and Vibrations Kinematics of Rigid Bodies Mohammad
Dynamics and Vibrations Kinematics of Rigid Bodies Mohammad I. Kilani
What is a Rigid Body? § A body is considered to be a rigid body when the distance between any two points on it remain unchanged as the body moves. § A result of the above definition is that the angle between any two lines on the body does not change as the body moves. This is because the angle between any two lines can be seen as an angle in a triangle containing the two lines. Since the lengths of the sides triangle do not change, the angles do not change either.
What is a Rigid Body? § Note that a rigid body contains an infinite number of particles. However, the above observations on rigid body motion allows determining the position of any point on the body if the position of one reference point and the angular orientation of one reference line on it are known. A B O § For example, the figure shows the location of all points on the body if reference point A is located at the origin and reference line AB is aligned with the x- axis (orientation = 0 degrees) A O B
What is a Rigid Body? § In the previous example, three scalar quantities were needed to fully define the location of a body in the plane. Those were the x- and y- coordinates of point A and the angular orientation of line AB. A B O § The degrees of freedom (DOF) for a system is the number of independent coordinates that need to be specified to completely describe the configuration of the system in space. A rigid body in plane motion has DOF = 3. A O B
What is a Rigid Body? § The independent coordinates for B determining the location of a rigid body is not unique. For example, one could choose the x- and ycoordinates of point B and the xcoordinates of point A. Another choice could be the x- coordinates of point A, the y-coordinates of point B, and the orientation of line AB. O A § There is an infinite number of possible choices for the coordinates needed to determine the location of the body. However, regardless of the choice, there number is always the same, and once they are given, the position of any point on the body can be determined. A O B
Analytical Determination of the location of points § Knowing the location of the reference point A, and the orientation of the reference line AB, the location of a point C on a body can be determined from the relative position equation below: r. C/A A γ r. A B O § In using the above equation, note that the angle γ and the magnitude of the vector rc/a does not change as the body moves. A O B
Example: Determination of the location of a point on a rectangle § The rectangle shown is 3 m long and 1 m wide. Determine the location of point C if point A is located at point (1, 1) and the line AB is oriented at 90 degrees from the x- axis r. C/A A r. A O C B A O C γ B
Example: Determination of the location of a point on a rectangle § The lengths of the sides AB and AC right angle triangle are 2 m and 1. 5 m, respectively. Determine the coordinates of the apex A if point B is located on the y- axis and point C is on the x-axis, and the line BC makes 45 degrees with the horizontal C A O B C B
Example: Determination of the location of a point on a rectangle § The lengths of the sides AB and AC right angle triangle are 2 m and 1. 5 m, respectively. Determine the coordinates of the apex A if point B is located on the x- axis and point C is on the y-axis, and the line AB makes 45 degrees with the horizontal C A O C B A A C O B
TYPES OF MOTION
Three Dimensional Motion q A rigid body free to move within a reference frame will, in the general case, have a simultaneous combination of rotation and translation. q In three-dimensional space, there may be rotation about any axis and translation that can be resolved into components along three axes.
Plane Motion q In a plane, or twodimensional space, rigid body motion becomes a combination of simultaneous rotation about one axis (perpendicular to the plane) and also translation resolved into components along two axes in the plane. q Planar motion of a body occurs when all the particles of a rigid body move along paths which are equidistant from a fixed plane
Translation q All points on the body describe parallel (curvilinear or rectilinear) paths. q A reference line drawn on a body in translation changes its linear position but does not change its angular orientation. Rectilinear Translation Curvilinear Translation
Fixed Axis Rotation q The body rotates about one axis that has no motion with respect to the “stationary” frame of reference. All other points on the body describe arcs about that axis. A reference line drawn on the body through the axis changes only its angular orientation. q When a rigid body rotates about a fixed axis, all the particles of the body, except those which lie on the axis of rotation, move along circular paths
General Plane Motion q When a body is subjected to general plane motion, it undergoes a combination of translation and rotation, The translation occurs within a reference plane, and the rotation occurs about an axis perpendicular to the reference plane.
DEGREES OF FREEDOM (DOF) OR MOBILITY
Definition of the DOF q The number of degrees of freedom (DOF) that a system possesses is equal to the number of independent parameters (measurements) that are needed to uniquely define its position in space at any instant of time. q Note that DOF is defined with respect to a selected frame of reference. x. A θB x. B y. A YB
DOF of a Rigid Body in a 2 D Plane q If we constrain the pencil to always remain in the plane of the paper, three parameters are required to completely define its position on the paper, two linear coordinates (x, y) to define the position of any one point on the pencil and one angular coordinate (θ) to define the angle of the pencil with respect to the axes. q The minimum number of measurements needed to define its position is shown in the figure as x, y, and θ. This system o has three DOF.
DOF of a Rigid Body in a 2 D Plane q Note that the particular parameters chosen to define the position of the pencil are not unique. A number of alternate set of three parameters could be used. q There is an infinity of sets of parameters possible, but in this case there must be three parameters per set, such as two lengths and an angle, to define the system’s position because a rigid body in plane motion always has three DOF.
DOF of a Rigid Body in 3 D Space q If the pencil is allowed to move in a three-dimensional space, six parameters will be needed to define its position. A possible set of parameters that could be used is three coordinates of a selected point, (x, y, z), plus three angles (θ, φ, ρ). q Any rigid body in a threedimensional space has six degrees of freedom. Note that a rigid body is defined as a body that is incapable of deformation. The distance between any two points on a rigid body does not change as the body moves. ρ θ ϕ
DOF of Mechanisms
Rotation about a Fixed Axis
Rotation about a Fixed Axis q Since a point is without dimension, it cannot have angular motion. Only lines or bodies undergo angular motion. For example, consider the body shown and the angular motion of a radial line r located within the shaded plane. q At the instant shown, the angular position of r is defined by the angle θ, measured from a fixed reference line to r
Rotation about a Fixed Axis
Rotation about a Fixed Axis
Relative Motion Analysis: Velocity
Relative Motion Analysis: Acceleration
Problem 16 -2
Problem 16 -2
Problem 16 -6
Problem 16 -6
Problem 16 -9
Problem 16 -9
Problem 16 -18
Problem 16 -18
Problem 16 -29
Problem 16 -29
Problem 16 -29
Problem 16 -29
Problem 16 -42
Problem 16 -42
Problem 16 -45
Problem 16 -45
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