FORCES IN SPACE

Mercedes Benz SLSDesign by AMG

3-D Non-Parallel ForcesIn this section, we will explore the solution of noncoplanar concurrent force problems. This method is similar to the solution of parallel force problems. In other words, one must project a three dimensional view into two two-dimensional views and solve simultaneously. The primary difference is handling non-parallel forces

To handle forces in space, we must be able to:

Break a force into its x, y, and z components

Combine x, y, and z components into a single force

Use these concepts to aid in finding the resultant of multiple forces in space

2

3

The magnitude is determined by using that form of the pythagorean theorem used for three dimensional geometry

NR

zyxR

9.55402530 222

222

=++=

++=

Forces in space (i.e.: three dimensions) have components along the x-, y-, and z-axis. Consider the example below. Find the magnitude and direction of force vector 'R' which is the result of force components: 30 N, 25 N, and 40 N

Resolving 3-D Forces in Space

The direction of vector 'R' can be stated by one of two ways: by stating the end coordinates of the vector or using the altitude and azimuth angles of the vector

To use coordinates, we simply determined the location of the tip of the vector. Vector 'R' can be stated as follows:

Although the former is a valid, the later is preferred since it reduces the coordinates by a least common denominator. However, most force vector coordinates are seldom integer values. If these coordinates are decimal value, it is valid to state them as such

4

R = 55.9 N 30, 25, 40

R = 55.9 N 6, 5, 8

Resolving 3-D Forces in Space

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Altitude and azimuth angles are determined using basic trigonometry. The use of altitude and azimuth angles tends to be used more frequently in non-engineering applications such as astronomy or navigation; although any engineering application where it is necessary to define the location of the sun often use them. It should be noted most disciplines measure azimuth from the 'North', whereas engineering applications may measure from the 'East'. To put that statement into perspective, we measure azimuth from the positive x-axis whereas other disciplines measure azimuth from the equivalent of our negative z-axis

Resolving 3-D Forces in Space

The azimuth angle is angle θ between the x-axis and the 'shadow' of vector 'R' in the 'xz' plane. This shadow vector is depicted by vector R' (R-prime). Angle θ is found by applying the arc tangent function to the components in the 'xz' plane. In other words:

= tan−140/30 = 53.1o

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Resolving 3-D Forces in Space

Altitude angle is the angle between this shadow vector, R' and the resultant 'R'. We already know the length of 'R' and the length of 'Ry', so we can apply the arcsin function to find angle φ

Thus the resultantcan be expressed as:

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= sin−1 25/55.9 = 26.6o

R = 55.9 N 36.8o az , 26.6o alt

Resolving 3-D Forces in Space

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In order to solve for reactions when dealing with 3-D non-parallel force systems, we will need to break a forces in space into its x, y, and z components. Doing so is similar to dealing with forces in two dimensions‑

In other words, the ratio of force components to total force magnitude is equal to the ratio of component length to total length. Let's clear this up with an example

Components of 3-D Forces

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Consider a 100 lb force vector with coordinates (4, 3, 5). We wish to determine its force components in the x, y, and z directions

Since force components are in the same ratio as the geometry of the vector, we need to determine the geometric length of 'R'

Components of 3-D Forces

07.7534 222 =++=R

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Components of 3-D ForcesUsing this information, we can find the components of the vector as follows:

R x = 3 /7.07⋅100 = 56.6 lb

R y = 4 /7.07⋅100 = 42.4 lb

R z = 5 /7.07⋅100 = 70.7 lb ↙

Now let's resolve the following system of concurrent forces to a single resultant. The coordinates of the 200 lb, 300 lb, and 500 lb forces are (4, 6, 3), (5, 1, -3), and (-8, -2, -1) respectively. To determine the force components of each vector, we must first determine the geometric length of each vector based upon its coordinates as follows:

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3.81283

9.53152

8.73641

222

222

222

=++=

=++=

=++=

L

L

L(200 lb)

(300 lb)

(500 lb)

Resolving Concurrent Forces

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Once the geometric lengths are found, we can determine force components. We'll do this by developing the table shown below. Note the use of our sign convention to indicate the direction of the components. As always, it is critical to maintain a proper sign convention to arrive at accurate results

X-component Y-component Z-component

200 lb Force 4/7.8 (200) = 102.5 6/7.8 (200) = 153.8 3/7.8 (200) = 76.9

300 lb Force 5/5.9 (300) = 254.2 1/5.9 (300) = 50.8 -3/5.9 (300) = -152.9

500 lb Force -8/8.3 (500) = -481.9 -2/8.3 (500) = -120.5 -1/8.3 (500) = -60.2

Resultant -125 +84 -136

Resolving Concurrent Forces

Now simply apply the Pythagorean theorem to determine the magnitude of the resultant 'R'

To determine the azimuth angle (q) and altitude (f) angles, we need to know the magnitude of the resultant as projected into the xz plane (shadow vector).

R = 1252 842 1362 = 203 lb

R xz = 1252 1362 = 185 lb = tan−1

−125/−136 = 42.5o

Resolving Concurrent Forces

The resultant is in the third quadrant of the xz plane. The azimuth relative to the positive x-axis is:

The altitude can be determined from the length of the y-component and the length of Rxz by using the sin function.

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= 90o 42.5o = 132.5o

= sin−184 /203 = 24.4o

Resolving Concurrent Forces

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Thus we can express the resultant vector as:

Resolving Concurrent Forces

R = 203 lb 24.4o Alt ,132.5o Az

R = 203 lb −125, 84,−136 +y

+x

+z

68

6742

209 lbs