List of questions for MST: Advanced Dynamics (MCEN90041_2024_SM2)

List of questions for MST

You have below the system descriptions. The questions will be similar to the ones in Assignment 2 or past MST's.

System 1

Two identical hemispheres (blue) of mass m and radius R spin relative to the connecting shaft (grey) (Figure a). The centre of mass of the hemispheres are at a constant distance L. The hemispheres roll without slip on the inclined plane.

The whole system can move on top of an inclined plane (green). The angle of inclination of the plane (α) is a constant.

A force F (red) is applied at the point indicated in Figure 1 (a and b). The force isparallel to the inclined plane and makes an angle with the shaft.

Use the coordinates x and y of the centre of the shaft relative to the border of the incline as part of your coordinates. See (Figure b).

Assume the distance from the centre of the shaft to the centre of mass of each hemisphere is L/2, as indicated in the picture.

Hemisphere

For the frames indicated in the picture, assume Izz=Iyy=I1 and Iz= I2.

System 2

Gear B rolls over gear A, which is rolling over the rack on the ground.  The centre of mass of gear A is connected to the wall via a spring of constant k and unstretched length lo. The angle that the line connecting the centre of both gears makes with the horizontal is θ. 

Gears A and B are identical and have mass and radius R. Their inertia tensors can be assumed to be that of a thin disc:   when zb is perpendicular to the plane of the gear.

Use the set of generalized coordinates Q={z,θ,α, β}, where x is the linear displacement of the centre of mass of gear A, a is the angle of rotation of gear A and β is the angle of rotation of gear B.

System 3

Two rotors are mounted on identical massless shafts. The two rotors are coupled by alinear spring of constant k and a dashpot of constant c, which are attached as shown inthe figure. Consider the rotors to be thin discs of masses m1 and m2 and radii R1and R. The inertia tensor of a thin disc in its body-centered frame. b) is:

, when z is perpendicular to the plane of the disc.

Assume a force is applied to the left disc at an angle as indicated on the top-viewd picture (on the plane of the disk).

System 4

The system is presented in the figure below.

Two massless sliders slide on two fixed guide bars. The bars AB and BC are connected to the sliders via pin joints. The joint between the bars (B) is pushed by a vertical force F. A spring of constant k and unstretched length lo connects the centres of mass of the bars.

Bar AB has mass m and length a. Bar BC has mass m2 and length b. Assume both bars are homogeneous.

The angle ß is fixed. The masses of the guide bars can be assumed to be negligible. Assume no friction between sliders and bars.

The inertia tensor of a rod of length / about its own centre of mass is:

, when x is along the length of the rod, I.

Use the set of generalized coordinates Q=(s,x,lalpha,gammal) Where x is the distance from point O to slider A, s is the distance from point O to slider C and the angles are indicated in the picture.

System 5

Two uniform. sector gears of mass m and m and are rigidly attached to the ends of two uniform. thin rods (AB and CD) of mass mg and m and, respectively. The sector gears rotate about fixed points B and C and stay in contact with each other during the movement. The system contains two torsional springs (kt1, k2), two extensional springs (k1, k2) and a dashpot (c). A horizontal force F pushes the block (mass ms) at end A of bar AB.

The spring of constant k is connected to a massless and frictionless follower, keeping it always horizontal.

The inertia tensor of the sector gears about its own centre of mass (G) in the frame. (b) attached to it and depicted in the inset of the figure is:

, i=1,2 for each gear (g).

The longitudinal centreline of bar CD may be assumed to pass through point C. The longitudinal centreline of bar AB passes through points B and the centre of mass of section gear 1,G1.

The movement of the system can be described using the coordinates Q=fa,β} the rotation angles of the sections gears, shown in figure the. Assume all springs are undeformed when the angles (a, β) are zero.

System 6

The sketch of a crane is presented in figure. A rectangular container is rigidly attached at the end of a cable. The cable length (s) can vary, to be able to bring the container to the ground. The cable is connected to a massless slider.

Assume the cable is rigid and can not bend and it can rotate about the axis pointing out of the plane in the side view (a), and about the cable itself (ф). This last rotation causes the cable to twist about itself. Treat the twisting as a torsional spring of constant kr. Assume the change in length due to the twisting is negligible.

The inertia tensor of a rectangular prism with the frame. and dimensions shown inFigure 2 is:

System 7

Background: A solar tracker is a device that tracks the sun as it moves on its path through the sky during the day, exposing the PV cells to an increased amount of sunlight and hence producing more electricity. The angle of the sun in the sky changes throughout the year as the tilt of the earth relative to the sun alters. In winter the sun is lower in the sky and is around for a shorter time as it travels through a shorter arc.

The system presented below represents a possible design of a solar tracker with dualaxis of rotation that aims to adjust for the sun's height as well as east-to-west rotation. A linear actuator pushes the plate up by a force F changing its inclination. A moment M (see inset (a)) is applied to cause the plate to rotate, as indicated in the figure. Assume the rotation is around its central axis (b/2) and that the force F is applied at a fixed distance d along that central line. Assume the thickness of the plate and support are negligible, so that the point A, centre of mass and point of application of force are along the rotaton axis.

Assume that the centre of mass of the plate is at its geometrical centre. The distance from point A to the start of the plate is a constant (see inset (a)).The inertia tensor of the solar plate given about its centre of

Mass G in the frame. shown in inset (b) is: