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Engineering Mechanics - Coggle Diagram
Engineering Mechanics
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3D Force Vectors
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Moment of a force in 3D
Earlier during Engineering Mechanics I we specified moment of a force about a point O as “force times shortest (perpendicular) distance from point to force vector” – or simply as force times moment arm
This was completely okay in 2D applications, but now when we move on to 3- dimensional structures, a vector definition makes a lot more sense. Moment of a force F about point O is the cross product of vectors r and F
Vector r is a position vector that goes from point O to any point which lies on the line of action of F
F is the forece of the vector
Notice: the i-component of the moment is not affected by Fx , the j-component of the moment is not affected by Fy etc. This is natural, because they produce no moment in the said direction; they are parallel.
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• The magnitude of the cross product equals the area of the parallelogram, which has original vectors as sides.
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MOMENT OF A COUPLE USING CROSS PRODUCT: Moment of a couple can be calculated using the cross product as a resultant of
𝑴𝑶 = 𝒓𝑨 × (−𝑭) + 𝒓𝑩 × F
Anyhow, if we think that the minus sign belongs to rA
, this can be written as
𝑴𝑶 = (𝒓𝑩 − 𝒓𝑨) × 𝑭
Using the triangle rule of vector addition, we get
𝑴𝑶 = 𝒓 × F
Example:
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MOMENT OF A FORCE ABOUT COORDINATE AXIS: If we calculate the moment of a force about a specific point, we will get a moment vector whose direction is perpendicular to the plane containing the force and the moment arm. This moment is not always the one that we want to use in calculations; we may
have to break it into parts
Different components of the moment have different effects on our structure:
➢ x-component of the moment is trying to turn our structure around x-axis
➢ y-component of the moment is trying to turn our structure around y-axis
➢ z-component of the moment is trying to turn our structure around z-axis
The structure usually has different “ability” to fight against the moment
depending on its direction
If we calculate the moment subject to point O, we will get a moment vector which acts on direction Ob; the magnitude of this moment has no clear physical meaning
The moment about y-axis does; it tries to turn the pipe around y-axis and hence creates torsion.
The moment about x-axis tries to turn the pipe around x-axis and hence creates bending
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MOMENT OF A FORCE ABOUT AN ARBITRARY AXIS: What if the component that we are interested in is not in the direction of a coordinate axis – i.e. what if the pipe in the previous example would stick out of point O to some arbitrary direction, say 𝒂 = 𝑎𝑥𝒊 + 𝑎𝑦𝒋 + 𝑎𝑧𝒌 (instead of 𝒋)? In this case we need to calculate the projection of the moment vector 𝑴𝑶 on to
vector 𝒂 – let’s call it 𝑴a or actually we just need the magnitude Ma
Luckily, this can be calculated easily using the dot product:
Here 𝒖𝒂 is the unit vector in the direction of 𝒂
TRIPLE SCALAR PRODUCT: Using rules of mathematics, we can simplify our calculation by merging the dot
product into our cross product that we need in calculating the 𝑴o
Hence, we only need to calculate one 3x3-determinant and we get the magnitude of our moment subject to axis in the direction of 𝒂
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