Automated material handling systems, manufacturing assembly lines and autonomous industrial robotics are typically designed to perform...

 

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Automated material handling systems, manufacturing assembly lines and autonomous industrial robotics are typically designed to perform superposition motion in a vertical plane. The path of motion is called a trajectory and it is a parabola. In the horizontal direction of the plane, there exits a constant velocity motion and constant acceleration motion in the vertical direction. The constant acceleration motion is due to the action of the earth's gravitational force W = mg where m is the mass and g the gravitational acceleration. In the production of goods, higher throughput speeds are achievable with continuous motion than discontinuous motion where velocity and acceleration do not remain constant. Drag forces arising from the mixture of air and wind may change the velocity in both the horizontal and vertical directions of motion. The drag forces are known to be functions of either time, position or velocity. Whether a drag force is time, position and velocity dependent rests on the need for constant continuous motion. The motion of a projectile in a plane is the superposition of two independent motion under the influence of gravity and drag forces. A projectile of mass m is launched from the origin O(0, 0) with a velocity vo at angle lying between 0 < < 90. The reference frame for the motion of the projectile in the ry-plane is S = Ory, where O(ro, yo) its origin and (e.ej) = dij, i, j = 1,2 are the basis vectors satisfying the Kronecker delta dij = {0, if i j and 1, if i=j}. Assume that the launching and landing of the projectile are on the same level ground, influence of drag forces is neglected and the projectile is under the influence of gravity only. Also consider the projectile as a particle. (i) Determine the degrees of freedom for the motion of the projectile (ii) Sketch the free body diagram of the projectile motion in the vertical ry-plane, label the coordinates (x.y), the origin O(0,0), the unit vectors e, on z-axis and e2 on y-axis, the maximum height A(Th. Yh) and range from 0(0,0) to the target point B(TR. YR). (iii) Obtain the relevant kinematic equations of motion for the projectile under the influ- ence of gravity only. (iv) Determine the equation for the path followed by the projectile (v) Obtain expressions for the ratios of merit of the projectile motion of the maximum height h, range R and maximum range Rm, namely: (a) h/R, (b) h/Rmax and (c) R/Rmax Automated material handling systems, manufacturing assembly lines and autonomous industrial robotics are typically designed to perform superposition motion in a vertical plane. The path of motion is called a trajectory and it is a parabola. In the horizontal direction of the plane, there exits a constant velocity motion and constant acceleration motion in the vertical direction. The constant acceleration motion is due to the action of the earth's gravitational force W = mg where m is the mass and g the gravitational acceleration. In the production of goods, higher throughput speeds are achievable with continuous motion than discontinuous motion where velocity and acceleration do not remain constant. Drag forces arising from the mixture of air and wind may change the velocity in both the horizontal and vertical directions of motion. The drag forces are known to be functions of either time, position or velocity. Whether a drag force is time, position and velocity dependent rests on the need for constant continuous motion. The motion of a projectile in a plane is the superposition of two independent motion under the influence of gravity and drag forces. A projectile of mass m is launched from the origin O(0, 0) with a velocity vo at angle lying between 0 < < 90. The reference frame for the motion of the projectile in the ry-plane is S = Ory, where O(ro, yo) its origin and (e.ej) = dij, i, j = 1,2 are the basis vectors satisfying the Kronecker delta dij = {0, if i j and 1, if i=j}. Assume that the launching and landing of the projectile are on the same level ground, influence of drag forces is neglected and the projectile is under the influence of gravity only. Also consider the projectile as a particle. (i) Determine the degrees of freedom for the motion of the projectile (ii) Sketch the free body diagram of the projectile motion in the vertical ry-plane, label the coordinates (x.y), the origin O(0,0), the unit vectors e, on z-axis and e2 on y-axis, the maximum height A(Th. Yh) and range from 0(0,0) to the target point B(TR. YR). (iii) Obtain the relevant kinematic equations of motion for the projectile under the influ- ence of gravity only. (iv) Determine the equation for the path followed by the projectile (v) Obtain expressions for the ratios of merit of the projectile motion of the maximum height h, range R and maximum range Rm, namely: (a) h/R, (b) h/Rmax and (c) R/Rmax Automated material handling systems, manufacturing assembly lines and autonomous industrial robotics are typically designed to perform superposition motion in a vertical plane. The path of motion is called a trajectory and it is a parabola. In the horizontal direction of the plane, there exits a constant velocity motion and constant acceleration motion in the vertical direction. The constant acceleration motion is due to the action of the earth's gravitational force W = mg where m is the mass and g the gravitational acceleration. In the production of goods, higher throughput speeds are achievable with continuous motion than discontinuous motion where velocity and acceleration do not remain constant. Drag forces arising from the mixture of air and wind may change the velocity in both the horizontal and vertical directions of motion. The drag forces are known to be functions of either time, position or velocity. Whether a drag force is time, position and velocity dependent rests on the need for constant continuous motion. The motion of a projectile in a plane is the superposition of two independent motion under the influence of gravity and drag forces. A projectile of mass m is launched from the origin O(0, 0) with a velocity vo at angle lying between 0 < < 90. The reference frame for the motion of the projectile in the ry-plane is S = Ory, where O(ro, yo) its origin and (e.ej) = dij, i, j = 1,2 are the basis vectors satisfying the Kronecker delta dij = {0, if i j and 1, if i=j}. Assume that the launching and landing of the projectile are on the same level ground, influence of drag forces is neglected and the projectile is under the influence of gravity only. Also consider the projectile as a particle. (i) Determine the degrees of freedom for the motion of the projectile (ii) Sketch the free body diagram of the projectile motion in the vertical ry-plane, label the coordinates (x.y), the origin O(0,0), the unit vectors e, on z-axis and e2 on y-axis, the maximum height A(Th. Yh) and range from 0(0,0) to the target point B(TR. YR). (iii) Obtain the relevant kinematic equations of motion for the projectile under the influ- ence of gravity only. (iv) Determine the equation for the path followed by the projectile (v) Obtain expressions for the ratios of merit of the projectile motion of the maximum height h, range R and maximum range Rm, namely: (a) h/R, (b) h/Rmax and (c) R/Rmax