SITYCTHEORYMECHANICS1902FChapter 7 Basic motion of rigid body
THEORY MECHANICS Chapter 7 Basic motion of rigid body
1. Translation of a rigid body2. Rotation of a rigid body about a fixed axis3. The velocity and acceleration at eachparticle in the rotation of the rigid body4. The vector representation of angularvelocity and angular acceleration5. The cross product of velocity andacceleration
1. Translation of a rigid body 2. Rotation of a rigid body about a fixed axis 3. The velocity and acceleration at each particle in the rotation of the rigid body 4. The vector representation of angular velocity and angular acceleration 5. The cross product of velocity and acceleration
7.1 Translation of a rigid bodyDuring the motion of a rigid body, the direction of any straight line in the rigidbody is always the same, that is, its direction is always parallel to the originaldirection. The motion ofa rigid body with such a characteristic is called parallelmovement of the rigid body, or translation for short.Theorem: when the rigid body is translational, the trajectory shape ofeachparticle in the rigid body is the same, and each particle has the same velocityandaccelerationatthesameinstant.Therefore, the study of the translational motion of a rigid body can bereduced to the study of the motion of any particlein the rigid body
During the motion of a rigid body, the direction of any straight line in the rigid body is always the same, that is, its direction is always parallel to the original direction. The motion of a rigid body with such a characteristic is called parallel movement of the rigid body, or translation for short. Theorem: when the rigid body is translational, the trajectory shape of each particle in the rigid body is the same, and each particle has the same velocity and acceleration at the same instant. Therefore, the study of the translational motion of a rigid body can be reduced to the study of the motion of any particle in the rigid body. 7.1 Translation of a rigid body
7.2 Rotation of a rigid body about a fixed axis1.Rotation characteristics and rotation equationsIn the process of rigid body motion, if a straight line on or its extension isalways motionless,themotionofa rigid bodywith suchafeature is calledfixed axis rotationof the rigid body, or rotation for short. The fixed line iscalledtheaxisofrotationAs shown in the figure, angle is called position anglesAs the rigid body rotates, the angle is a single-valuedDcontinuousfunctionoftimetstaticplanedyhamΦ=p(t)planeThis is the equation of rotation for a rigid body
1. Rotation characteristics and rotation equations In the process of rigid body motion, if a straight line on or its extension is always motionless, the motion of a rigid body with such a feature is called fixed axis rotation of the rigid body, or rotation for short. The fixed line is called the axis of rotation. As the rigid body rotates, the angle is a single-valued continuous function of time t. =(t) This is the equation of rotation for a rigid body. As shown in the figure, angle is called position angles. 7.2 Rotation of a rigid body about a fixed axis O z static plane dynamic plane
7.2Rotation of arigid body about afixedaxis2.Angular velocity, Angular accelerationThe angularvelocityof a rigid body rotatingabout a fixed axisis equaltothefirstderivativeofits positionAngle withrespecttotimedpUnit: rad/ s0dtIn engineering, the speed n is often used to represent the speed of rigidbody rotation. The unit of n is revolution/minute( r/min ), and thetransformation relationship with n is2元元n0n6030Theangularaccelerationofarigid bodyrotatingabouta fixed axisis equaltothefirst derivative of its angular velocity withrespecttotime,expressedas,d?pdo6Qdt?dt
2. Angular velocity, Angular acceleration The angular velocity of a rigid body rotating about a fixed axis is equal to the first derivative of its position Angle with respect to time = = dt d rad s In engineering, the speed n is often used to represent the speed of rigid body rotation. The unit of n is revolution/minute ( ), and the transformation relationship with n is r min n n 60 30 2 = = The angular acceleration of a rigid body rotating about a fixed axis is equal to the first derivative of its angular velocity with respect to time, expressed as, 2 2 d d dt dt = = = = 7.2 Rotation of a rigid body about a fixed axis Unit：