MM203 Mechanics of Machines: Part 2

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MM203Mechanics of Machines Part 2
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Kinetics of systems of particles

• Extension of basic principles to general systems
  of particles
• Particles with light links
• Rigid bodies
• Rigid bodies with flexible links
• Non-rigid bodies
• Masses of fluid

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Newtons second law

• G centre of mass
• Fi external force, fi internal force
• ri position of mi relative to G

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Newtons second law

• By definition
• For particle i
• Adding equations for all particles

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Newtons second law

• Differentiating w.r.t. time
• gives
• Also
• so
• (principle of motion of the mass centre)

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Newtons second law

• Note that is the acceleration of the
  instantaneous mass centre which may vary over
  time if body not rigid.
• Note that the sum of forces is in the same
  direction as the acceleration of the mass centre
  but does not necessarily pass through the mass
  centre

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Example

• Three people (A, 60 kg, B, 90 kg, and C, 80 kg)
  are in a boat which glides through the water with
  negligible resistance with a speed of 1 knot. If
  the people change position as shown in the second
  figure, find the position of the boat relative to
  where it would be if they had not moved. Does the
  sequence or timing of the change in positions
  affect the final result? (Answer x 0.0947 m).
  (Problem 4/15, MK)

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Example

• The 1650 kg car has its mass centre at G.
  Calculate the normal forces at A and B between
  the road and the front and rear pairs of wheels
  under the conditions of maximum acceleration. The
  mass of the wheels is small compared with the
  total mass of the car. The coefficient of static
  friction between the road and the rear driving
  wheels is 0.8. (Answer NA 6.85 kN, NB 9.34
  kN). (Problem 6/5, MK)

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Work-energy

• Work-energy relationship for mass i is
• where (U1-2)i is the work done on mi during a
  period of motion by the external and internal
  forces acting on it.
• Kinetic energy of mass i is

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Work-energy

• For entire system

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Work-energy

• Note that no net work is done by internal forces.
• If changes in potential energy possible
  (gravitational and elastic) then
• as for single particle

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Work-energy

• For system
• Now
• and note that
• so

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Work-energy

• Since ri is measured from G,
• Now

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Work-energy

• Therefore
• i.e. energy is that of translation of mass-centre
  and that of translation of particles relative to
  mass-centre

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Example

• The two small spheres, each of mass m, are
  rigidly connected by a rod of negligible mass and
  are released in the position shown and slide down
  the smooth circular guide in the vertical plane.
  Determine their common velocity v as they reach
  the horizontal dashed position. Also find the
  force R between sphere 1 and the guide the
  instant before the sphere reaches position A.
  (Answer v 1.137(gr)½, R 2.29mg). (Problem
  4/9, MK)

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Rigid body

• Motion of particles relative to mass-centre can
  only be due to rotation of body
• Velocity of particles due to rotation depends on
  angular velocity and the distance to centre of
  rotation. Where is centre of rotation?
• Need to examine kinematics of rotation

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Plane kinematics of rigid bodies

• Rigid body
• distances between points remains unchanged
• position vectors, as measured relative to
  coordinate system fixed to body, remain constant
• Plane motion
• motion of all points is on parallel planes
• Plane of motion taken as plane containing mass
  centre
• Body treated as thin slab in plane of motion
  all points on body projected onto plane

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Kinematics of rigid bodies
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Translation

• All points move in parallel lines or along
  congruent curves.
• Motion is completely specified by motion of any
  point therefore can be treated as particle
• Analysis as developed for particle motion

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Kinematics of rigid bodies
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Rotation about fixed axis

• All particles move in circular paths about axis
  of rotation
• All lines on body (in plane of motion) rotate
  through the same angle in the same time
• Similar to circular motion of a particle
• where riO is distance to O, the centre of
  rotation, and IO is mass moment of inertia about O

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Mass moment of inertia

• Mass moment of inertia in rotation is equivalent
  to mass in translation
• Rotation and translation are analogous

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General plane motion

• Combination of translation and rotation
• Principles of relative motion used

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Rotation

• Angular positions of two lines on body are
  measured from any fixed reference direction

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Rotation

• All lines on a rigid body in its plane of motion
  have the same angular displacement, the same
  angular velocity, and the same angular
  acceleration
• Angular motion does not require the presence of a
  fixed axis about which the body rotates

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Angular motion relations

• Angular position, angular velocity, and angular
  acceleration
• Similar to relationships between s, v, and a.
• Also, combining relationships and cancelling out
  dt

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Angular motion relations

• If constant angular acceleration
• Direction of ve sense must be consistent
• Analogous to rectilinear motion with constant a
• Same procedures used in analysis

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Example

• The angular velocity of a gear is controlled
  according to w 12 3t2 where w, in radians per
  second, is positive in the clockwise sense and
  where t is the time in seconds. Find the net
  angular displacement Dq from the time t 0 to t
  3 s. Also find the total number of revolutions
  N through which the gear turns during the 3
  seconds. (Answer Dq 9 rad, N 3.66 rev).
  (Problem 5/5, MK)

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Kinetic energy of rigid body

• If rotation about O

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Parallel axis theorem

• Now (P.A.T.)
• and so

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Radius of gyration

• Radius of gyration
• Mass moment of inertia of point mass m at radius
  of gyration is the same as that for body
• P.A.T.

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Work done on rigid body
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Work done by couple

• Couple is system of forces that causes rotation
  but no translation
• Moment about G
• Moment about O

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Work done by couple

• Moment vector is a free vector
• Forces have turning effect or torque
• Torque is force by perpendicular distance between
  forces
• Work done
• positive or negative

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Forces and couples

• Torque is
• Also unbalanced force

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Work-energy principle

• When applied to system of connected bodies only
  consider forces/moments of system ignore
  internal forces/moments.
• If there is significant friction between
  components then system must be dismembered

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Example

• A steady 22 N force is applied normal to the
  handle of the hand-operated grinder. The gear
  inside the housing with its shaft and attached
  handle have a combined mass of 1.8 kg and a
  radius of gyration about their axis of 72 mm. The
  grinding wheel with its attached shaft and pinion
  (inside housing) have a combined mass of 0.55 kg
  and a radius of gyration of 54 mm. If the gear
  ratio between gear and pinion is 41, calculate
  the speed of the grinding wheel after 6 complete
  revolutions of the handle starting from rest.
  (Answer N 3320 rev/min). (Problem 6/119, MK)

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Rotation about fixed axis

• Motion of point on rigid body

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Vector notation

• Angular velocity vector, w, for body has sense
  governed by right-hand rule
• free vector

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Vector notation

• Velocity vector of point A
• What are magnitude and direction of this vector?
• Note that

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Vector notation

• Acceleration of point

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Vector notation

• Vector equivalents
• Can be applied in 3D except then angular velocity
  can change direction and magnitude

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Example

• The T-shaped body rotates about a horizontal axis
  through O. At the instant represented, its
  angular velocity is w 3 rad/s and its angular
  acceleration is a 14 rad/s2. Determine the
  velocity and acceleration of (a) point A and (b)
  point B. Express your results in terms of
  components along the n- and t- axes shown.
  (Answer vA 1.2et m/s, aA -5.6et 3.6en
  m/s2, vB 1.2et 0.3en m/s, aB -6.5et
  2.2en m/s2). (Problem 5/2, MK)

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Example

• The two V-belt pulleys form an integral unit and
  rotate about the fixed axis at O. At a certain
  instant, point A on the belt of the smaller
  pulley has a velocity vA 1.5 m/s, and the point
  B on the belt of the larger pulley has an
  acceleration aB 45 m/s2 as shown. For this
  instant determine the magnitude of the
  acceleration aC of the point C and sketch the
  vector in your solution. (Answer aC 149.6
  m/s2). (Problem 5/16, MK)

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Linear impulse and momentum

• Returning to general system Linear momentum of
  mass i is
• For system (assuming m does not change with time)

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Linear impulse and momentum

• Differentiating w.r.t. time
• Same as for single particle only applies if
  mass constant
• Same for rigid body

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Example

• The 300 kg and 400 kg mine cars are rolling in
  opposite directions along a horizontal track with
  the speeds shown. Upon impact the cars become
  coupled together. Just prior to impact, a 100 kg
  boulder leaves the delivery chute and lands in
  the 300 kg car. Calculate the velocity v of the
  system after the boulder has come to a rest
  relative to the car. Would the final velocity be
  the same if the cars were coupled before the
  boulder dropped? (Answer v 0.205 m/s).
  (Problem 4/11, MK)

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Angular impulse and momentum

• Considered about a fixed point O and about the
  mass centre.

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Angular impulse and momentum

• About O
• First term is zero since vi vi 0 so
• - sum of all external moments (net moment of
  internal forces is zero)

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Angular impulse and momentum

• About O same as for single particle. As before,
  does not apply if mass is changing.
• About G

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Example

• The two balls are attached to a light rod which
  is suspended by a cord from the support above it.
  If the balls and rod, initially at rest, are
  struck by a force F 60 N, calculate the
  corresponding acceleration a of the mass centre
  and the rate d2q/dt2 at which the angular
  velocity of the bar is changing. (Answer a 20
  m/s2, d2q/dt2 336 rad/s2). (Problem 4/17, MK)

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Rigid body

• Angular momentum

• Planar motion

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Rigid body

• Angular acceleration

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Kinetic diagrams
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Kinetic diagrams - translation

• Alternative moment equation for rectilinear
  translation

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Kinetic diagrams - translation

• Alternative moment equation for curvilinear
  translation

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Example

• The cart B moves to the right with acceleration a
  2g. If the steady-state angular deflection of
  the uniform slender rod of mass 3 m is observed
  to be 20, determine the value of the torsional
  spring constant K. The spring, which exerts a
  moment M Kq on the rod, is undeformed when the
  rod is vertical. The values of m and l are 0.5 kg
  and 0.6 m, respectively. Treat the small end
  sphere of mass m as a particle. (Answer K
  46.8 Nm/rad). (Problem 6/16, MK)

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Example

• The mass of gear A is 20 kg and its centroidal
  radius of gyration is 150 mm. The mass of gear B
  is 10 kg and its centroidal radius of gyration is
  100 mm. Calculate the angular acceleration of
  gear B when a torque of 12 Nm is applied to the
  shaft of gear A. Neglect friction. (Answer aB
  25.5 rad/s2 (CCW)). (Problem 6/46, MK)

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Example

• The 28 g bullet has a horizontal velocity of 500
  m/s when it strikes the 25 kg compound pendulum,
  which has a radius of gyration of kO 925 mm. If
  the distance h 1075 mm, calculate the angular
  velocity w of the pendulum with its embedded
  bullet immediately after the impact. (Answer w
  0.684 rad/s) (Problem 6/174, MK)

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Plane kinematics absolute motion

• Absolute motion analysis
• Get geometric relationships
• Get time derivatives to determine velocity and
  acceleration
• Straightforward if geometry is straightforward
• Must be consistent with signs

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Example

• Point A is given a constant acceleration a to the
  right starting from rest with x essentially 0.
  Determine the angular velocity w of link AB in
  terms of x and a. (Problem 5/24, MK)
• Answer

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Example

• The wheel of radius r rolls without slipping, and
  its centre O has a constant velocity vO to the
  right. Determine expressions for the velocity v
  and acceleration of point A on the rim by
  differentiating its x- and y-coordinates.
  Represent your result graphically as vectors on
  your sketch and show that v is the vector sum of
  two vO vectors. (Problem 5/25, MK)
• Answer

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Example

• One of the most common mechanisms is the
  slider-crank. Express the angular velocity wAB
  and the angular acceleration aAB of the
  connecting rod AB in terms of the crank angle q
  for a given constant crank speed w0. Take wAB and
  aAB to be positive counter-clockwise. (Problem
  5/54, MK)
• Answer

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Plane kinematics relative velocity

• Two points on same rigid body.
• Motion of one relative to the other must be
  circular since distance between them is constant.

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Plane kinematics relative velocity

• Relative linear velocity is always in direction
  perpendicular to line joining points

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Plane kinematics relative velocity
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Plane kinematics relative velocity

• Relative velocity principles may also be used in
  cases where there is constrained sliding contact
  between two links A and B may be on different
  links

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Example

• Determine the angular velocity of the telescoping
  link AB for the position shown where the driving
  links have the angular velocities indicated.
  (Answer wAB 0.96 rad/s). (Problem 5/61, MK)

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Example

• For an interval of its motion the piston rod of
  the hydraulic cylinder has a velocity VA 1.2
  m/s as shown. At a certain instant q b 60.
  For this instant determine the angular velocity
  wBC of the link BC. (Answer wBC 15.56 rad/s).
  (Problem 5/66, MK)

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Example

• The mechanism is designed to convert from one
  rotation to another. Rotation of link BC is
  controlled by the rotation of the curved slotted
  arm OA which engages pin P. For the instant
  represented q 30 and b, the angle between the
  tangent to the curve at P and the horizontal, is
  40. If the angular velocity of OA is as shown,
  determine the velocity of the point C. (Answer
  vC 4.33 m/s). (Problem 5/85, MK)

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Relative acceleration

• May need to know velocities first

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Example

• If OA has a CCW angular velocity w0 10 rad/s
  (giving wBC 5.83 rad/s and wAB 2.5 rad/s),
  calculate the angular acceleration of link AB for
  the position where the coordinates of A are x
  -60 mm and y 80 mm. Link BC is vertical for
  this position. (Answer aAB 2.5 rad/s2).
  (Problem 5/137, MK)