Design a quick return four-bar Grashof crank-rocker for [tex]y = 90^\circ[/tex] of output rocker motion with a time ratio [tex]TR = 1/1.50[/tex].

1. Check the Grashof crank-and-rocker condition.
2. Explain your stepwise solution with appropriate construction lines.
3. Provide the final mechanism without the construction lines.

Problem-2: Designing a Quick Return Fourbar Grashof Crank RockerA quick-return mechanism is a mechanical system that has a motion sequence in which the return stroke is completed faster than the forward stroke. When designing a quick-return four-bar Grashof crank rocker for y=90° of output rocker motion with a time ratio TR=1/1.50, the following steps must be taken:

Step 1: Define the Grashof condition and determine the necessary dimensions.The Grashof condition, which is defined as (Lmin+Lmax)<(L1+L2), must be met in order to ensure that the crank rocker mechanism works. To begin, we must define the Grashof condition for the given mechanism as:

Lmin = L2 - L3Lmax = L2 + L3L1 = 2L2a = L1L2 = 3aL3 = 4aTR = 1/1.50y = 90°

Substituting the given values, we get:

Lmin = 2aLmax = 10aL1 = 6a

Step 2: Sketching the mechanism to meet the requirements of the problemStatementThe mechanism has to be designed so that it operates with y=90° of output rocker motion and a time ratio of TR=1/1.50.In this regard, the four-bar mechanism design is shown in the figure below:

Step 3: Checking Grashof's ConditionThe Grashof condition states that the sum of the shortest and longest links' lengths should be less than or equal to the sum of the remaining links. As a result, we must verify whether this criterion is met or not:

Therefore, Lmax+Lmin=10a+2a=12a, and L1+L2=6a+3a=9a.

The Grashof criterion is met because 12a>9a, as required. As a result, we can build the mechanism without fear of it failing to operate.

Step 4: Determining the Time Ratio TR and Coupler Point yWe can determine the time ratio TR and coupler point y using the following formulas:TR = 1/(1 + sinθ2/sinθ1) = 1/(1 + sin(180 - y)/sin(y))TR = 1/1.50Coupler Point:y = cos-1(((L1+L2 cosθ2-Lmin cosθ1)/(Lmax)))y = 90°

Step 5: Sketching the Final Mechanism without Construction LinesThe mechanism design without construction lines is shown in the figure below:

Therefore, the final mechanism for a quick-return four-bar Grashof crank rocker designed for y=90° of output rocker motion with a time ratio TR=1/1.50 is shown above.

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