6.4 Kinematic Analysis
Analysis of the pestle and mortar
The mobility of the four-bar mechanisms in the design can be quantified using the Greubler-Kutzbach Equation. From this calculation, our design has 1 degree of freedom. Furthermore, it meets the Grashof condition to be considered a Class I kinematic system. In other words, the crank can make a full rotation. The calculations shown below only account for the grinding wheel subsystems, but similar calculations can be done for the other two crank sliders present in the design that yield the same results.
To check the mobility, we use the Gruebler-Kutzbach Equation:
Gruebler-Kutzbach Equation:
𝑀 = 3(𝐿 − 1) − 2*J1 − J2
= 3(4- 1) - 2(4) - 0 = 1 DOF
To verify a Class I kinematic system, we use the Grashoff Condition equation, where S = shortest link, L = longest link, and P and Q = middle length links.
Grashof Condition:
𝑆 + 𝐿 ≤ 𝑃 + 𝑄
0 + 170 < 130 + 45
Class I KC - crank can make a full rotation
After ensuring motion with mobility and grashoff calculations, we now perform position, velocity, and acceleration analysis for the mechanism. As seen in the design iterations, since the motor is offset from the input link, the group utilized gears to transfer motion from the motor to the input “link” (which is actually a gear). This means that the angular speed of the input “link” will be our motor speed multiplied by the gear ratio, which is 15/20. This means that our input “link” is rotating at 75% the speed of our motor. And in turn, the torque of the input “link” is multiplied by 20/15, which can help give extra force that is needed for the crushing purposes of our pestle and mortar.
The since the speed the motor is controlled by a potentiometer, we can control the speed of input link 2, which is the variable ⍵2 - which is kept on average at 1 revolution per second.
Position Analysis
The desired motion for this system is for the grinding wheel to oscillate in a straight line, covering a linear distance of 90mm. This can be exactly achieved using a crank slider mentioned above with no error. A graph of the x-position of the grinding wheel versus the input angle of the crank slider is shown below.
Velocity Analysis
A graph of the velocity of the grinding wheel versus the input angle of the crank is shown below - We use an initial angular acceleration of Alpha = 1 rev/s = 6.28 rad/s
Acceleration Analysis
A graph of the acceleration of the grinding wheel versus the input angle of the crank is shown below. We use an initial angular acceleration of Alpha = 0 rad/s^2 (We make an assumption that the motor gets up to speed right away)
Force/Mechanical Advantage Analysis
The force output of the pestle and mortar is arguably the most important variable, as that will determine the utility of our mechanism and how hard of a pill we are able to crush.
To determine the approximate mechanical advantage of the system, we use the formula
MA=Fout/Fin=VFin/VFout=(in*rin) / Vslider
where in is the input angular velocity of link/gear 2 (2), rin is the radius of the gear connected to the motor (44mm), and Vslider is the velocity of the crusher (l1_dot). When we control the potentiometer, it is easier to visually see ourselves as controlling the input angular velocity, as opposed to the input torque. Since we are planning to set the motor to rotate at an angular velocity that we determine, it’s better to use input angular velocity in our mechanical advantage equation.
From velocity analysis, we see that the slider velocity is zero when the input angle is 0° or 180° degrees, in which the slider crank is completely extended or completely retracted, leading to a potentially infinite mechanical advantage.
Thus, we want most of the crushing to be at the opposite ends of our mortar, when the mechanism is completely extended and completely retracted, since that is where we will have the highest mechanical advantage of our mechanism. Naturally, the real mechanical advantage will be lower, since this analysis assumes the smartie and all links are rigid.
We are limited by the deformation of the smartie. Also, our mechanism will break if we exceed the yield strength of our links and bearing connections. Thus, we will leave enough room on the sides of the mortar to allow reasonable crushing, as well as ensure we can actually crush our object given our links are made of acrylic.
We also noticed during testing that the mechanism would lock up and stop if the links did not have enough force to crush the smartie. This is much better than the alternative where the links, or even worse, the motor, would break or snap. Although it would be nice to crush the smartie at the ends, where the mechanical advantage is near infinite, we actually need to crush the smartie slightly away from the edge, where the force is still enough to crush the smartie while still keeping the mechanism from locking up.