Dynamics of Mechanisms

Quiz

Autumn Session 2019

Inverse and Forward Dynamics of Linkage Mechanisms

Directions to students             Submit your answer sheet as a single PDF file.

Show the derivation process of each equation. Do not just list the final answers.

The answers, with their derived steps, can be either typed or hand-written.

QUESTION 1: Inverse Dynamics of Mechanisms                                                                      (60 marks)

Figure 1(a) shows an inverted slider-crank mechanism driven by a motor torque Q on crank 2. Suppose that crank 2 is to be driven continuously with an instantaneous angular speed ω₂ = 30 rad/s counter clockwise and an instantaneous angular acceleration a₂ = 300 rad/s² counter clockwise at the shown instant. Symbols G₂, G₃, and G₄ represent the centres of mass of the crank, slider, and rocker, respectively. The geometric dimensions, masses (m_i), and mass moments of inertia (I_i) of the mechanism are listed in Table 1. For carrying out the kinematic and dynamic analyses, a vector loop and a fixed coordinate system (X, Y) are established as shown in Figure 1(b). Note that due to the kinematic constraint of the sliding motion, the angular motions of the rocker and the slider are identical.

Assume that the mechanism is working on a horizontal plane, and the friction effect is neglected.

For the shown instant (i.e., q₂ = 30°, ω₂ = 30 rad/s, and a₂ = 300 rad/s²), please answer the following equations:

1. Calculate the angular velocity and angular acceleration of the (5 marks)
2. Calculate the positions (p_x, p_y), velocities (v_x, v_y), and accelerations (a_x, a_y) of the mass centres of the crank, slider, and (15 marks)
3. Calculate the reaction forces at the ground pivots O and C, (30 marks)
4. Calculate the shaking force and shaking (10 marks)

• Mechanism sketch (b) Coordinate system and vector loop

Figure 1

Table 1 Given parameters of the mechanism in Fig. 1^*

^*Units: m for lengths, kg for masses, and kg-m² for moments of inertia.

QUESTION 2: Forward Dynamics of Mechanisms                                                                     (40 marks)

Now, let us consider the forward dynamics problem of the inverted slider-crank mechanism. All the geometric and mass parameters remain the same, and a motor torque Q = 500 N-m is applied on crank 2. Suppose that the mechanism initially stays at rest at q₂ = 30° (refer to Fig. 1(b)). Assume that the mechanism is working on a horizontal plane, and the friction effect is neglected.

Please derive the equation of motion at the initial position using the power-equation method and kinematic coefficients.

Hint:

Write the answer in the form as:

Q = (å A)q!! + (å B)q!2   with q

(0) = p

and q! (0) = 0

2

where the (SA) and (SB) are calculated.

2                       2                    6           2