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MM203 Mechanics of Machines: Part 1

MM203 Mechanics of Machines: Part 1. Module. Lectures Tutorials Labs Why study dynamics? Problem solving. Vectors. Unit vectors - components. Direction cosines. l , m , and n – direction cosines between v and x -, y -, and z -axes. Calculate 3 direction cosines for.

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MM203 Mechanics of Machines: Part 1

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  1. MM203Mechanics of Machines: Part 1

  2. Module • Lectures • Tutorials • Labs • Why study dynamics? • Problem solving

  3. Vectors

  4. Unit vectors - components

  5. Direction cosines • l, m, and n – direction cosines between v and x-, y-, and z-axes • Calculate 3 direction cosines for

  6. Dot (or scalar) product • Component of Q in P direction

  7. Angle between vectors

  8. Dot product • Commutative and distributive

  9. Particle kinematics • What is kinematics? • What is a particle? • Rectilinear motion - review • Plane curvilinear motion - review • Relative motion • Space curvilinear motion

  10. Rectilinear motion • Combining gives • What do dv and ds represent?

  11. Example • The acceleration of a particle is a = 4t− 30 (where a is in m/s2 and t is in seconds). Determine the velocity and displacement in terms of time. (Problem 2/5, M&K)

  12. Vector calculus • Vectors can vary both in length and in direction

  13. Plane curvilinear motion • Choice of coordinate system (axes) • Depends on problem – how information is given and/or what simplifies solution • Practice

  14. Plane curvilinear motion • Rectangular coordinates • Position vector - r • e.g. projectile motion • ENSURE CONSISTENCY IN DIRECTIONS

  15. Plane curvilinear motion • Normal and tangential coordinates • Instantaneous radius of curvature – r • What is direction of v? Note that dr/dt can be ignored in this case – see M&K.

  16. Plane curvilinear motion

  17. Example • A test car starts from rest on a horizontal circular track of 80 m radius and increases its speed at a uniform rate to reach 100 km/h in 10 seconds. Determine the magnitude of the acceleration of the car 8 seconds after the start. (Answer: a = 6.77 m/s2). (Problem 2/97, M&K)

  18. Example • To simulate a condition of “weightlessness” in its cabin, an aircraft travelling at 800 km/h moves an a sustained curve as shown. At what rate in degrees per second should the pilot drop his longitudinal line of sight to effect the desired condition? Use g = 9.79 m/s2. (Answer: db/dt = 2.52 deg/s). (Problem 2/111, M&K)

  19. Example • A ball is thrown horizontally at 15 m/s from the top of a cliff as shown and lands at point C. The ball has a horizontal acceleration in the negative x-direction due to wind. Determine the radius of curvature of the path at B where its trajectory makes an angle of 45° with the horizontal. Neglect air resistance in the vertical direction. (Answer: r = 41.8 m). (Problem 2/125, M&K)

  20. Plane curvilinear motion • Polar coordinates

  21. Plane curvilinear motion

  22. Example • An aircraft flies over an observer with a constant speed in a straight line as shown. Determine the signs (i.e. +ve, -ve, or 0) for • for positions • A, B, and C. • (Problem 2/134, M&K)

  23. Example • At the bottom of a loop at point P as shown, an aircraft has a horizontal velocity of 600 km/h and no horizontal acceleration. The radius of curvature of the loop is 1200 m. For the radar tracking station shown, determine the recorded values of d2r/dt2 and d2q/dt2 for this instant. (Answer: d2r/dt2 = 12.5 m/s2, d2q/dt2 = 0.0365 rad/s2). (Problem 2/141, M&K)

  24. Relative motion • Absolute (fixed axes) • Relative (translating axes) • Used when measurements are taken from a moving observation point, or where use of moving axes simplifies solution of problem. • Motion of moving coordinate system may be specified w.r.t. fixed system.

  25. Relative motion • Set of translating axes (x-y) attached to particle B (arbitrarily). The position of A relative to the frame x-y (i.e. relative to B) is

  26. Relative motion • Absolute positions of points A and B (w.r.t. fixed axes X-Y) are related by

  27. Relative motion • Differentiating w.r.t. time gives • Coordinate systems may be rectangular, tangential and normal, polar, etc.

  28. Inertial systems • A translating reference system with no acceleration is known as an inertial system. If aB = 0 then • Replacing a fixed reference system with an inertial system does not affect calculations (or measurements) of accelerations (or forces).

  29. Example • A yacht moving in the direction shown is tacking windward against a north wind. The log registers a hull speed of 6.5 knots. A “telltale” (a string tied to the rigging) indicates that the direction of the apparent wind is 35° from the centerline of the boat. What is the true wind velocity? (Answer: vw = 14.40 knots). (Problem 2/191, M&K)

  30. Example • To increase his speed, the water skier A cuts across the wake of the boat B which has a velocity of 60 km/h as shown. At the instant when q= 30°, the actual path of the skier makes an angle b= 50° with the tow rope. For this position, determine the velocity vA of the skier and the value of dq/dt.(Answer: vA = 80.8 km/h, dq/dt = 0.887 rad/s). (Problem 2/193, M&K)

  31. Example • Car A is travelling at a constant speed of 60 km/h as it rounds a circular curve of 300 m radius. At the instant shown it is at q = 45°. CarB is passing the centre of the circle at the same instant. Car A is located relative to B using polar coordinates with the pole moving with B. For this instant, determine vA/B and the values fo dq/dt and dr/dt as measured by an observer in car B.(Answer: vA/B = 36.0 m/s, dq/dt = 0.1079 rad/s, dr/dt = −15.71 m/s).(Problem 2/201, M&K)

  32. Space curvilinear motion • Rectangular coordinates (x, y, z) • Cylindrical coordinates (r, q, z) • Spherical coordinates (R, q, f) • Coordinate transformations – not covered • Tangential and normal system not used due to complexity involved.

  33. Space curvilinear motion • Rectangular coordinates (x, y, z) – similar to 2D

  34. Space curvilinear motion • Cylindrical coordinates (r, q, z)

  35. Space curvilinear motion • Spherical coordinates (R, q, f)

  36. Example • A section of a roller-coaster is a horizontal cylindrical helix. The velocity of the cars as they pass point A is 15 m/s. The effective radius of the cylindrical helix is 5 m and the helix angle is 40°. The tangential acceleration at A is gcosg. Compute the magnitude of the acceleration of the passengers as they pass A. (Answer: a = 27.5 m/s2). (Problem 2/171, M&K)

  37. Example • The robot shown rotates about a fixed vertical axis while its arm extends and elevates. At a given instant, f = 30°, df/dt = 10 deg/s = constant, l = 0.5 m, dl/dt = 0.2 m/s, d2l/dt2 = −0.3 m/s2, and W = 20 deg/s = constant. Determine the magnitudes of the velocity and acceleration of the gripped part P. (Answer: v = 0.480 m/s, a = 0.474 m/s2). (Problem 2/177, M&K)

  38. Particle kinetics • Newton’s laws • Applied and reactive forces must be considered – free body diagrams • Forces required to produce motion • Motion due to forces

  39. Particle kinetics • Constrained and unconstrained motion • Degrees of freedom • Rectilinear motion – covered • Curvilinear motion

  40. Rectilinear motion - example • The 10 Mg truck hauls a 20 Mg trailer. If the unit starts from rest on a level road with a tractive force of 20 kN between the driving wheels and the road, compute the tension T in the horizontal drawbar and the acceleration a of the rig. (Answer: T = 13.33 kN, a = 0.667 m/s2). (Problem 3/5, M&K)

  41. Example • The motorized drum turns at a constant speed causing the vertical cable to have a constant downwards velocity v. Determine the tension in the cable in terms of y. Neglect the diameter and mass of the small pulleys. (Problem 3/48, M&K) • Answer:

  42. Curvilinear motion • Rectangular coordinates • Normal and tangential coordinates • Polar coordinates

  43. Example • A pilot flies an airplane at a constant speed of 600 km/h in a vertical circle of radius 1000 m. Calculate the force exerted by the seat on the 90 kg pilot at point A and at point A. (Answer: RA = 3380 N, RB= 1617 N). (Problem 3/63, M&K)

  44. Example • The 30 Mg aircraft is climbing at an angle of 15° under a jet thrust T of 180 kN. At the instant shown, its speed is 300 km/h and is increasing at a rate of 1.96 m/s2. Also q is decreasing as the aircraft begins to level off. If the radius of curvature at this instant is 20 km, compute the lift L and the drag D. (Lift and drag are the aerodynamic forces normal to and opposite to the flight direction, respectively). (Answer: D = 45.0 kN, L = 274 kN). (Problem 3/69, M&K)

  45. Example • A child's slide has a quarter circle shape as shown. Assuming that friction is negligible, determine the velocity of the child at the end of the slide (q = 90°) in terms of the radius of curvature r and the initial angle q0. • Answer

  46. Slide • Does it matter what profile slide has? • What if friction added?

  47. Example • A flat circular discs rotates about a vertical axis through the centre point at a slowly increasing angular velocity w. With w = 0, the position of the two 0.5 kg sliders is x = 25 mm. Each spring has a stiffness of 400 N/m. Determine the value of x for w = 240 rev/min and the normal force exerted by the side of the slot on the block. Neglect any friction and the mass of the springs. (Answer: x = 118.8 mm, N = 25.3 N).(Problem 3/83, M&K)

  48. Work and energy • Work/energy analysis – don’t need to calculate accelerations • Work done by force F • Integration of F = ma w.r.t. displacement gives equations for work and energy

  49. Work and energy • Active forces and reactive forces (constraint forces that do no work) • Total work done by force • where Ft = tangential force component

  50. Work and energy • If displacement is in same direction as force then work is +ve (otherwise –ve) • Ignore reactive forces • Kinetic energy • Gravitational potential energy

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