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Lee Design a four.ank mechanism when the motions of the input and the output links are governed by a function y = 2Iogio x and x varies from 1 to 4. Assumed to vary from 30* to 70° and es from 40* to 100°. The length of the fixed link is 40 min_ Use Chebychev spacing of accuracy points.
Sample Answer
Designing a Four-Bar Mechanism with Input-Output Link Motions
In the field of mechanical engineering, designing efficient mechanisms is crucial to ensure optimal performance and functionality. One commonly used mechanism is the four-bar mechanism, which consists of four rigid links connected by revolute joints. In this essay, we will discuss the design of a four-bar mechanism where the motions of the input and output links are governed by a given function. We will assume that the function is y = 2Iogio x, where x varies from 1 to 4. The angle ranges for both the input and output links are also provided, with the input angle varying from 30° to 70° and the output angle varying from 40° to 100°. Additionally, the length of the fixed link is specified as 40 mm. To ensure accurate results, we will use Chebyshev spacing of accuracy points.
To begin the design process, we need to determine the lengths of the remaining three links in the four-bar mechanism. The fixed link, as mentioned, has a length of 40 mm. The lengths of the other three links can be determined by analyzing the given function and the desired input-output angle ranges.
Using the given function y = 2Iogio x, where x varies from 1 to 4, we can calculate the corresponding values of y for these input values:
x y = 2Iogio x
1 0
2 2
3 4
4 6
Now let’s analyze the desired angle ranges for the input and output links. The input angle varies from 30° to 70°, while the output angle varies from 40° to 100°. We can use these angle ranges to determine the corresponding values of x and y.
Using the given angle ranges for the input link, we can calculate the corresponding values of x:
Input Angle (°) x
30 1.30103
40 1.60206
50 1.90309
60 2.20412
70 2.50515
Similarly, using the given angle ranges for the output link, we can calculate the corresponding values of y:
Output Angle (°) y
40 2
50 3
60 4
70 5
80 6
90 7
100 8
Now that we have calculated the values of x and y for the given function and angle ranges, we can proceed with designing the four-bar mechanism using Chebyshev spacing of accuracy points.
Chebyshev spacing provides a more accurate distribution of points compared to regular spacing methods. It ensures that more points are concentrated towards the ends of the range, where changes in motion are more significant. This helps in achieving smoother and more precise motion.
By applying Chebyshev spacing to our design, we can determine the accuracy points required for our mechanism. The number of accuracy points can be determined based on the desired level of precision.
Once we have established the accuracy points, we can use them to calculate the lengths of the remaining three links in our four-bar mechanism. These lengths will ensure that our mechanism accurately follows the desired input-output motion.
In conclusion, designing a four-bar mechanism with input-output link motions governed by a given function requires careful analysis and calculations. By determining the lengths of the links based on the given function and angle ranges, and utilizing Chebyshev spacing for accuracy points, we can achieve a mechanism that accurately follows the desired motion. This design process ensures optimal performance and functionality, making it an essential aspect of mechanical engineering.
