Page 216 - Mechatronics with Experiments
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202   MECHATRONICS
                              Assume the 300 ms motion time is equally divided between acceleration, run, and deceleration times,
                              t = t = t = 100 ms (Figure 3.13).
                              a
                                  r
                                     d
                                 1. Calculate the necessary torque (maximum and continuous rated) and maximum speed required
                                   at the motor shaft. Select an appropriate servo motor for this application.
                                 2. If an incremental position encoder is used on the motor shaft for control purposes, what is the
                                   required minimum resolution in order to provide a tool positioning accuracy of 40 μin.
                                 3. Select a proper flexible coupling for this application to be used between the motor shaft and
                                   the lead-screw. Include the inertia of the flexible coupling in the inertia calculations and motor
                                   sizing calculations. Repeat step 1.
                              5.  Repeat the same analysis and calculations of Problem 3 and 4, if a rack and pinion mechanism
                              was used to convert the rotary motion of motor to a translational motion of the tool.
                                 1. First, determine the rack and pinion gear that gives the same gear ratio (from rotary motion to
                                   translational motion) as the ball-screw mechanism.
                                 2. Discuss the differences between the ball-screw, rack and pinion, and belt and pulley (transla-
                                   tional version) mechanisms.

                              6.  Given a four-bar linkage (Figure 3.7), derive the geometric relationship between the following
                              motion variables of the mechanism. Let the link lengths be l = 1.0m, l = 2.0, l = 1.5m, l = 1.0m.
                                                                        1       2     3       4
                                 1. Input is the angular position (   (t)) of link 1, output is the angular position of link 3, (   (t)).
                                                         1                                        3
                                   Find    = f(   ). Plot    for one revolution of link 1 as function of    .
                                        3
                                                                                   1
                                                   3
                                             1
                                 2. Determine the x and y coordinates of the tip of the link 3 during the same motion cycle. Plot
                                   the results on the x-y plane (path of the tip of link 3 during one revolution of link 1).
                              7.  Consider the cam and follower mechanism shown in Figure 3.9. The follower arm is connected
                              to a spring. The follower is to make an up–down motion once per revolution of the cam. The travel
                              range of the follower is to be 2.0inintotal.
                                 1. Select a modified trapezoidal cam profile for this task.
                                 2. Assume the input shaft to the cam is driven at 1200 rpm constant speed. Calculate the maximum
                                   linear speed and linear accelerations experienced at the tool tip.
                                 3. Let the stiffness of the spring be k = 100 lb∕in and the mass of the follower and the tool it is
                                   connected to m = 10 lb. Assume the input shaft motion is not affected by the dynamics of
                                              f
                                   the follower and tool. The input shaft rotates at constant speed at 1200 rev∕min. Determine
                                   the net force function at the follower and tool assembly during one cycle of the motion and
                                   plot the result. Notice that

                                                            F(t) = m ̈ x(t) + k ⋅ x(t)          (3.335)
                                                                   f
                                   and x(t), ̈ x(t) are determined by the input shaft motion and cam function. What happens if the
                                   net force F(t) becomes negative? One way to assume that F(t) does not become negative is to
                                   use a preloaded spring. What is the preloading requirement to ensure F(t) is always positive
                                   during the planned motion cycle? The preload spring force can be taken into account in the
                                   above equation as follows,

                                                          F(t) = m ̈ x(t) + k ⋅ x(t) + F pre    (3.336)
                                                                f
                                   where F  = k ⋅ x is a constant force due to the preloading of the spring. This force can be
                                         pre    o
                                   set to a constant value by selection of spring constant and initial compression.
                              8.  Consider an electric servo motor and a load it drives through a gear reducer (Figure 3.14).
                                 1. What is the generally recommended relationship between the motor inertia and reflected load
                                   inertia?
                                 2. What is the optimal relationship in terms of minimizing the heating of the motor?
                                 3. Derive the relationship for the optimal relationship.
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