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：
7.2Rotation of a rigid body about afixedaxistyExamplel Theblock moves horizontally ata constantvelocity.The bar OA can berotated around Oaxis,and thebarViskeptclosetothelateraledgeoftheblockasshownintheTA0figure. Given that the height of the block is H, try to find therotational equation, angular velocity and angular acceleration?hof the OA bar.xVotxSoluton: Create rectangularxtgphhcoordinates as shown in the figureβ = arctgTherefore,therotationequationofOAbarishVodp0angularvelocityh? +votdt-2hvitdo&angularaccelerationdt(h? +vit?)?
x y O A 0 v h x Example1 The block moves horizontally at a constant velocity. The bar OA can be rotated around O axis, and the bar is kept close to the lateral edge of the block, as shown in the figure. Given that the height of the block is H, try to find the rotational equation, angular velocity and angular acceleration of the OA bar. Soluton：Create rectangular coordinates as shown in the figure. h v t h x tg 0 = = Therefore, the rotation equation of OA bar is ( ) 0 h v t = arctg angular velocity 2 2 0 2 0 h v t hv dt d + = = angular acceleration 3 0 2 3 2 2 0 2 ( ) d hv t dt h v t − = = + 7.2 Rotation of a rigid body about a fixed axis
7.3Thevelocityaccelerationat each particleofafixed axis rotating rigid bodyWhen the rigid body rotates around a fixed axis, any particleM from the rotationaxis R moves in a circular motion with00O particle as the center and R as the radiuspeMo1.Equation of motion(+)s=r@Mav2. VolecitydsdpThedirectionisMagnitude:Vroshown in figuredtdtThat is, the magnitude of the velocity at any particle in the rotationof the rigidbody is equal to the productof the distance from the particle to the axis ofrotationand theangularvelocityoftherigid body
When the rigid body rotates around a fixed axis, any particle M from the rotation axis R moves in a circular motion with O particle as the center and R as the radius. 1. Equation of motion s = r 2. Volecity The direction is shown in figure. That is, the magnitude of the velocity at any particle in the rotation of the rigid body is equal to the product of the distance from the particle to the axis of rotation and the angular velocity of the rigid body. s (+) O M0 M r v a 7.3 The velocity acceleration at each particle of a fixed axis rotating rigid body Magnitude: r dt d r dt ds v = = =
7.3 The velocity accelerationat each particleof afixed axis rotating rigid body3.Acceleration0dvdoMoDirection asTangentiala.=rsaSdtshown in figuredtacceleration:MaVThat is, the magnitude ofthe normal acceleration at any particleof rotation is equal to the product of the distance from thisparticleto the axis of rotation and the square of the angularvelocityoftherigid bodyv2(ro)?NormalDirection asro2a.acceleration:shown in figurerpThat is, the tangential acceleration at any particle of rotation is equal to theproductof the distance from this particle to the axis of rotation and the angularaccelerationoftherigid body
Normal acceleration： 2 2 2 ( ) r r v r an = = = Direction as shown in figure That is, the magnitude of the normal acceleration at any particle of rotation is equal to the product of the distance from this particle to the axis of rotation and the square of the angular velocity of the rigid body. O M0 M s r (+) v a n a Tangential acceleration： r dt d r dt dv a = = = 3. Acceleration 7.3 The velocity acceleration at each particle of a fixed axis rotating rigid body That is, the tangential acceleration at any particle of rotation is equal to the product of the distance from this particle to the axis of rotation and the angular acceleration of the rigid body. Direction as shown in figure
7.3 The velocity acceleration at each particleof a fixed axis rotating rigid bodyTotal Acceleration0MMagnitude:a :-00gaAnd the deflection Angel of the radius:Ma.aaKα = arctga.0From the above, it can be seen that the velocity and acceleration at any particle inthe rotatingrigid body are directly proportional to the distance from thisparticleto the axis ofrotation,but the deflection Angleformed by the total addedvelocity and radius is independent of the radius of rotation
From the above, it can be seen that the velocity and acceleration at any particle in the rotating rigid body are directly proportional to the distance from this particle to the axis of rotation, but the deflection Angle formed by the total added velocity and radius is independent of the radius of rotation. Total Acceleration O M0 M a a an 7.3 The velocity acceleration at each particle of a fixed axis rotating rigid body Magnitude: 2 2 2 4 n a a a r = + = + And the deflection Angel of the radius: 2 n a arctg arctg a = =
7.3 The velocity accelerationat each particleofa fixed axis rotating rigid bodyDistribution diagram of velocity and acceleration of fixed axisrotating rigid body:00αα
O O Distribution diagram of velocity and acceleration of fixed axis rotating rigid body: 7.3 The velocity acceleration at each particle of a fixed axis rotating rigid body