Dimensional Synthesis of a Grinder Mechanism

By A. H. Haji

The grinder mechanism is a function generator mechanism. Knowing a minimum required energy for crushing the input stones, the driving motor power and the linkage dimensions should be so determined as to meet the minimum desired speed for link number 6 i.e. the crusher link. The energy sought at the output of this mechanism is due to relatively short periods of grinding process i.e. during a small rotation of the driving motor. Therefore a flywheel can be used to absorb energy at non – active period and release it at the active period.    

Fig. 1: A typical Grinder schematic

Selecting the precision points

Considering some operational positions as the precision points is usually based on the principal function of the mechanism i.e. the main goal for which the machine is made and to satisfy the kinematical and dynamical conditions due to the corresponding situations.

Therefore of the crucial importance, is to have an overall though intuitive insight about the general behavior of the mechanism and its dominant operational features.

Consequently, I initially run some arbitrary models in Working Model to be familiar with the nature of the mechanism. Presented in Fig.2 is a typical angular velocity diagram for link number 6. Some remarkable features of this diagram are as follows:

• There are 4 extremum points in the curve.
• The extremum points occur at nearly equal intervals. . If we neglect the fly wheel speed fluctuations, this equality  considering the relatively constant motor rpm means equal motor (or fly wheel) rotations in one complete revolution or 360 degrees.
• In the running process, energy is absorbed during   and released at  to.

Fig. 2: A typical angular velocity diagram for link number 6

Now considering the importance of the grinder link speed we can choose the points 1, 2, 3 and 4 as the decision points. Whence the angular velocity of link number 6 (grinder link) can be computed as:

 and  are the minimum and maximum angles of the link number 6 respectively. Also the angular velocity at points 2 and 3 are equal to zero. As we see the above set of selected precision points provides useful tools to satisfy the design requirements.

Analytical approach

The vector representation form for Fig. 3 is as follows:

The last equation states the rigidity of link number 3 and therefore can be replaced with the following equation:

Recalling that the angle of link number 4;  is not important, the equation  can be rewritten in a Freudenstain form as:

Fig. 3:  The vector representation of the mehanism

With the same reason the angle of link number 5 can be eliminated resulting in a Freudenstain form for equation  as: 

Having 4 precision points there are 23 unknowns in equations  and:

But the number of equations is 4*2 = 8, so we have 15 free choices 8 of which belongs to two angular positions of  and. Therefore 7 free choices are finally remained.

The free choices are:

And the unknowns are:

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