Revisica
Sign in
ENGR5004 / Week 5
ENGR216 2022 Dynamics Q A1Past paperOld spec ENGR2162:125 marks30 min

ENGR216 Summer 2022 Dynamics Q A1[VALID]

Figure A1-1 shows a single link, AB, rotating anticlockwise about a fixed pivot at A.

Figure A1-1: link AB pivoted at A at angle theta, angular velocity omega and angular acceleration alpha anticlockwise.
Figure A1-1: link AB pivoted at A at angle theta, angular velocity omega and angular acceleration alpha anticlockwise.
Figure A1-2: actuator mechanism; link AB pivoted at A, block C slides on AB, link CD in a horizontal slide at height y.
Figure A1-2: actuator mechanism; link AB pivoted at A, block C slides on AB, link CD in a horizontal slide at height y.
Formulas you may need
  • Fixed-axis rotation, vector form: vB=ω×rB/A\mathbf{v}_B = \boldsymbol\omega\times\mathbf{r}_{B/A}, aB=α×rB/A−ω2rB/A\mathbf{a}_B = \boldsymbol\alpha\times\mathbf{r}_{B/A} - \omega^2\mathbf{r}_{B/A} (learn this)
  • Polar coordinates: v=r˙ ur+rθ˙ uθ\mathbf{v} = \dot r\,\mathbf{u}_r + r\dot\theta\,\mathbf{u}_\theta, a=(r¨−rθ˙2) ur+(rθ¨+2r˙θ˙) uθ\mathbf{a} = (\ddot r - r\dot\theta^2)\,\mathbf{u}_r + (r\ddot\theta + 2\dot r\dot\theta)\,\mathbf{u}_\theta; Coriolis term 2r˙θ˙2\dot r\dot\theta (learn this)
  • Normal-tangential: v=v ut\mathbf{v} = v\,\mathbf{u}_t, at=v˙a_t = \dot v, an=v2/ρa_n = v^2/\rho toward the centre of curvature (learn this)
  • Constraint differentiation: x=ytan⁡θx = y\tan\theta, ddθtan⁡θ=sec⁡2θ\dfrac{d}{d\theta}\tan\theta = \sec^2\theta, ddθsec⁡2θ=2sec⁡2θtan⁡θ\dfrac{d}{d\theta}\sec^2\theta = 2\sec^2\theta\tan\theta, chain rule ddt=θ˙ddθ\dfrac{d}{dt} = \dot\theta\dfrac{d}{d\theta} (learn this)
  1. (a)
    Figure A1-1 shows a link connected to fixed pivot at A and having angular acceleration α\alpha and instantaneous angular velocity ω\omega. Define and show on a sketch the velocity and acceleration components of point B when described in i. Cartesian coordinate system components. ii. Polar coordinate system components. iii. Normal-tangential coordinate system components.
    [6]2:2
  2. (b)
    Which coordinate system would you choose for an in-plane kinematic analysis of Figure A1-1, and why?
    [2]Third
  3. (c)
    For the configuration shown in A1-1, would point B experience a Coriolis acceleration? Give your reasoning.
    [3]2:2
  4. (d)
    Figure A1-2 shows an actuator mechanism. Link CD moves in a horizontal slide and the end C carries a block which slides along AB. Show that the linear velocity of CD is x˙=(ysec⁡2θ) ω\dot{x} = (y\sec^2\theta)\,\omega and also that the linear acceleration of CD is x¨=ω2(2ysec⁡2θtan⁡θ)+αysec⁡2θ\ddot{x} = \omega^2 (2y\sec^2\theta\tan\theta) + \alpha y\sec^2\theta
    [6]
  5. (e)
    Calculate the linear acceleration of CD when θ=50∘\theta = 50^\circ, y=2y = 2 m, ω=1\omega = 1 rad/s clockwise and α=5 rad/s2\alpha = 5\ \mathrm{rad/s^2} clockwise.
    [4]2:2
  6. (f)
    If CD moves with constant linear velocity, and the angular velocity of AB is 1 rad/s clockwise, find the angular acceleration of AB when θ=50∘\theta = 50^\circ and y=2y = 2 m.
    [4]2:2