4.13:
Equivalent Couples
Consider a valve subjected to two couples in the same plane having the same moments.
If the magnitude, direction, and rotation sense of their moments are the same, the couples are equivalent.
First, denote the points of intersection of the lines of action of the two couples.
Then, move the first couple's two forces to the second couple's points of intersection and resolve them into components.
One pair of components has the same magnitude, line of action and opposite sense and can be canceled.
So, the first couple reduces to a new couple formed by forces S and –S.
Its moment equals the moment of force S about a point lying on the line of action of –S.
Using Varignon's theorem, the first couple's moment equals the sum of the moments of its components, which in turn equals the new couple's moment.
So, the new forces are equal to the second couple's forces; implying that the two couples are equivalent.
Similarly, two couples contained in parallel planes having the same moments are also equivalent.
4.13:
Equivalent Couples
In mechanical engineering, the concept of equivalent couples plays a crucial role in understanding and analyzing various mechanical systems.
Two couples are considered to be equivalent if they produce the same rotational effect on a rigid body. In other words, the two couples have the same magnitude and act in the same direction, causing the same angular displacement or acceleration in the body.
For instance, consider two couples lying in the plane of the page, with one having a pair of equal and opposite forces of magnitude 30 N and a perpendicular distance of 0.4 m between them. The other couple has a pair of equal and opposite forces of magnitude 40 N separated by a perpendicular distance of 0.3 m. Now, each pair's moment has a magnitude of 12 N·m, and both are directed out of the plane of the page.
In the second couple, a larger force is needed to achieve the same rotational effect since the distance between the forces is less.
In many real-life scenarios, mechanical systems are subjected to multiple couples simultaneously. By identifying equivalent couples, engineers can reduce the complexity of the system and analyze it more efficiently. This simplification helps in designing, troubleshooting, and optimizing mechanical systems.
In automotive engineering, equivalent couples are used to analyze the forces acting on various vehicle components, such as suspension systems, drive shafts, and steering mechanisms. This information is vital for designing and optimizing vehicles that deliver optimal performance and safety.
Suggested Reading
- Hibbeler, R.C. (2016). Engineering Mechanics: Statics. Fourteenth Edition, New Jersey: Pearson. Pp. 155.
- Meriam, J.L., Kraige, L.G. and Bolten, J.N.(2016). Engineering Mechanics: Statics and Dynamics. Eighth Edition, Singapore: John Wiley & Sons. Pp. 50-51.
- Beer, F.P., Johnston, E.R., Mazurek, D.F., Cornwell, P.J. and Self, B.P. (2016). Vector Mechanics For Engineers. Eleventh Edition, New York: McGraw-Hill Education. Pp. 121-123.