MDME: MANUFACTURING, DESIGN, MECHANICAL ENGINEERING 

MOMENTS


A MOMENT is the turning affect of a force, commonly known as Torque.  

Lecture Notes Moments.pdf    Moments.one

Videos: Moments can be Anywhere

1. Moment Definition

The Moment of a force is the turning effect about a pivot point. To develop a moment, the force must act upon the body to attempt to rotate it. A moment is can occur when forces are equal and opposite but not directly in line with each other.

 

The Moment of a force acting about a point or axis is found by multiplying the Force (F) by the perpendicular distance from the axis (d), called the lever arm. 

Moment = Force x Perpendicular Distance
M = F x d
(Nm) = (N)  x  (m)



Notes;
1. Clockwise is a positive moment.
2. Take care with units here - especially with mm. It is best to convert everything to m first.

2. Perpendicular Lever Arm

The force is not always perpendicular to the given lever arm. In this case, the correct perpendicular distance must be determined. (By the way, the perpendicular distance is also the SHORTEST distance between the force and the pivot point.)
Calculating the perpendicular (shortest) distance;

 

A pure moment has no force - just rotation. Example below shows a support stand to counter the vertical force so that only a pure moment (torque) is applied to the wheel nut.

www.tyreshopdirect.com

 

4. Moments Can be Placed Anywhere!

When a pure moment is applied to an object, it can be moved to different locations without changing anything - the reactions are exactly the same. This means the total moment can be calculated anywhere on the body and always giving the same answer. This is very handy - we can any location (the most convenient one) to calculate the total moment.

The most convenient place to take a moment is usually the busiest joint - like a pin joint with multiple unknown forces (more about this in the next chapter).

Example: Screwdriver in a long slot

The turning effect (torque or moment) of the screwdriver is exactly the same in each example above. The applied moment is independent of the location of the moment.

 

 

4. Moment vs Torque

There is no difference really, both are turning effects and both are measured in Nm.

A textbook definition: A turning effect is called Moment in static situations (with no motion). In dynamics applications (with movement) a turning effect is called Torque. So a spinning motor shaft has torque, but a lever has moment. This is the textbook definition, but in practice the two terms are often used interchangeably. E.g. "Torque wrench"

5. Equilibrium of Moments

When dealing with moments, equilibrium exists when the total moment is zero. (Otherwise it will accelerate in rotation, angular acceleration). Mathematically this is very simple - add the clockwise moments and subtract the anticlockwise ones.
For equilibrium of moments;


"Taking clockwise as positive, the sum of all moments about point A is zero"

These calculations are very simple. The most common mistake is not getting the perpendicular (shortest) distance between pivot and force. Another thing to watch is not keeping track of the signs (CW or CCW moments).

6. Force Couples

Two equal forces of opposite direction, with a distance d between them will cause a moment, where;
A special case of moments is a couple. A couple consists of two parallel forces that are equal in magnitude, opposite in sense and do not share a line of action. It does not produce any translation, only rotation. The resultant force of a couple is zero, but it produces a pure moment.


A tap wrench is an example of a couple. The two hand forces are equal but opposite direction.
Taking moments about the centre (both clockwise);
Moment = F * d + F * d = 2Fd 

The moment caused by a couple = The force * the distance between them.  

Example of a couple: Wheel brace using two hands pushing opposite directions.

www.hornbydrivertraining.co.uk 

7. Moments of components (Varignon’s Theorem)

In equilibrium, the sum of moments about any point is zero.

Principle of Moments, or Varignon’s Theorem

“The moment caused by the resultant force (of some system of forces) about some arbitrary point is equal to the sum of the moments due to all of the component forces of the system.”

Amazingly, you can choose ANY point and it still works! So we usually pick somewhere that is easy to calculate - like at the intersection of several forces so we don't have to include them in the calculation (because distance = 0, hence M=0)

In the tap-wrench example above, we can also take the moment about the left force:
Moment = F * 2d = 2Fd

It doesn't matter which point you pivot around, the moment is always zero. This is very handy when doing calculations.

For a body in equilibrium, the total moment about ANY point is always zero. So we usually pick a point that makes our calculations easier.



Demonstration of Varignon's Theorem

The Principle of Moments, also known as Varignon's Theorem, states that the moment of any force is equal to the algebraic sum of the moments of the components of that force. It is a very important principle that is often used in conjunction with the Principle of Transmissibility in order to solve systems of forces that are acting upon and/or within a structure. This concept will be illustrated by calculating the moment around the bolt caused by the 100 N force at points A, B, C, D, and E in the illustration.


First consider the 100 N force.
Since the line of action of the force is not perpendicular to the wrench at A, the force is broken down into its orthogonal components. The 40mm horizontal and the 50mm diagonal measurement near point A should be recognized as belonging to a 3-4-5 triangle. Therefore, Fx = -4/5(100 N) or -80 N and Fy = -3/5(100 N) or -60 N.

Consider Point A.
The line of action of Fx at A passes through the handle of the wrench to the bolt (which is also the center of moments). This means that the magnitude of the moment arm is zero and therefore the moment due to FAx is zero. FAy at A has a moment arm of twenty mm and will tend to cause a positive moment.

FAy d = (60 N)(200mm) = 1200 Nmm 

The total moment caused by the 100 N force F at point A is 1200 Nmm (1.2Nm).

Consider Point B.
At this point the 100 N force is perpendicular to the wrench. Thus, the total moment due to the force can easily be found without breaking it into components.

FB d = (100 N)(120mm) = 1200 Nmm

The total moment caused by the 100 N force F at point B is again 1200 Nmm (1.2Nm).

Consider Point C.
The force must once again be decomposed into components. This time the vertical component passes through the center of moments. The horizontal component FCx causes the entire moment.

FCx d = (80 N)(150mm) = 1200 Nmm

Consider Point D.
The force must once again be decomposed into components. Both components will contribute to the total moment.

FDx d = (80 N)(210mm) = 1680 Nmm
FDy d = (60 N)(80mm) = -480 Nmm

Note that the y component in this case would create a counterclockwise or negative rotation. The total moment at D due to the 100 N force is determined by adding the two component moments. Not surprisingly, this yields 1200 Nm.

Consider Point E.
Following the same procedure as at point D.

FEx d = (80 N)(30mm) = -240 Nmm
FEy d = (60 N)(240mm) = 1440 Nmm

However, this time Fx tends to cause a negative moment. Once again the total moment is 1200 Nmm.

At each point, A, B, C, D and E the total moment around the bolt caused by the 100 N force equalled 1200 Nmm. In fact, the total moment would equal 1200 Nmm at ANY point along the line of action of the force. This is Varignon's Theorem.

Based on  http://darkwing.uoregon.edu/%7Estruct/courseware/461/461_lectures/461_lecture10/461_lecture10.html


8. Moment of a Resultant

It turns out that when we add up the moment of several forces we get the same answer as taking the moment of the resultant.

To obtain the total moment of a system of forces, we can either...
  1. Calculate each moment (from each force separately) and add them up, keeping in mind the CW and CCW sign convention.
  2. Calculate the moment caused by the resultant of the system of forces about that point.

9. Moment of Force Components

This means we can do the opposite too -we can break the force into components to easily find each moment. This is a common trick for solving complex moment problems because it usually makes it much easier to find the perpendicular distances to each force. (However, even this is pretty easy when the problem is drawn up using CAD)


Simple Worked Examples

1: Simple Moment

A mass of 30kg pushes on a lever (crank) 170mm long. What moment does it cause about the centre shaft?

M = F * d
Convert to correct units;
F = 30 * 9.81 = 294.3N
d = 0.17m

M =  294.3 * 0.17 =  50.03Nm

2: Moment Equlibrium

If the system is balanced (equilibrium),

 
+ (5 050) - ( F x 0.25) = 0

So force F = 250 Nm 025 m = 10 N

In order to balance the 5 N force acting at 05 m from the pivot, we require 10 N on the opposite side at 025m.

If in equilibrium, the anticlockwise turning effect of force F must equal the clockwise turning effect of the 5N load.

3: A couple

Hand forces F= 25N and distance d = 140mm. Find Moment M applied to the tapping cutter.

Moment = 2*F*d 
            = 2 * 25 * 0.14
            = 7Nm


4: Perpendicular distance

If these wheel nuts must be tightened to 85Nm, what is the force F?
Angle = 35 degrees, d = 420mm.

From M = F * d, then F = M / d

Perpendicular distance = d * cos (35)

F = 85 / (0.42 * cos(35))
   = 247.06 N



Whiteboard: 


Whiteboard

Questions:

Questions (Ivanoff old edition)

Homework Assignment:
Do all questions 6.1 to 6.9 (New edition) or 5:1 to 5:11 (Old edition).