## 3. Forces

• WHAT?: We look at mechanical structures, and determine the distribution of forces and moments.

• WHY?: a fundamental subject for every form of Mechanical Engineering (and every other branch of engineering that has ever existed.

• The difference between statics and dynamics, in brief,

Statics: does nothing, just sits there

Dynamics: moving things

• Consider some of the applications of statics design techniques,

3.1 Some Basic Concepts

• Mass and force: A mass can exert forces through gravity and other effects. Forces can also be exerted by other phenomenon, such as magnetism.

• We can emphasize the relationship between mass as an absolute and gravity as a local. The effects of gravity are dealt with as forces in most statics problems.

• Force has magnitude and direction. Therefore it is well suited to vectors.

• many forces can also operate on the same object, we can replace these with equivalent forces, called RESULTANTS.

• We have both action and reaction forces as well. As we apply ACTION forces, there are forces that will resist, these are called REACTIONS.

• Some approximations,

we are pretending the forces are applied at points, but in reality a force must be distributed,

we generally assume there are no deflections. This is known as the RIGID BODY assumption.

we often use PARTICLE approximations that assume bodies have no size. This simplifies calculations significantly.

TRANSMISSIBILITY: a force can be moved along a line of action.

Parallelogram law: a method for adding two forces to get a resultant vector.

Forces that are the result of mechanical contact are called surface forces, and are transmitted by pressure on the surface of an object.

Body forces are the result of Physics (read magic) and will pull on all of the mass of the object, such as gravity, electrostatic, inertia, magnetic, etc.

3.2 Vector vs. Scalar Quantities

• definitions,

• Parallelogram Law: Recall that vectors can be added or subtracted. This is a variation of the triangle law. In both cases we are putting vectors head to tail. These methods favor drafting solutions to problems that are not really necessary with calculators, but they are still very useful for understanding.

3.3 Math Review

• Aside: recall the following terms when discussing vectors and algebra,

line of action

fixed/sliding/free vectors

colinear

coplanar

concurrent

resultant

commutative/associative

sine law (law of sines)

cosine law (law of cosines)

basic sin, cos, tan relationships: in triangle

orthogonal: Cartesian

Pythagorean formula

unit vectors

right/left handed coordinates: coordinate axes and positive rotations

cross products

dot products

identity matrix

matrix multiplication, addition, subtraction, division with matrices/scalars

matrix determinants

matrix inversion

matrix transpose

equation solutions in matrix form: Gauss-Jordan row reduction; Cramer’s rule

parametric equations

plug-and-chug solutions: trial and error, Newton-Raphson

inequalities: greater than, less than, etc.

differential calculus

integration single and double, and selecting elements rectangular, triangular, cylindrical

developing linear equations

3.4 Rectangular Form of Vectors

• Vectors can be added to get resultant forces in vector (rectangular component) form.

3.5 Polar Form of Vectors

• We can also represent the same forces as scalar magnitudes, and direction,

#### 3.5.1 Cartesian Vector Notation

• This is like the rectangular form, except that i, j & k are used as placeholders.

#### 3.5.2 Scalar Notation

• We can represent forces as simple scalar values (not vectors). But, we must still remember that theses are still vectors. For scalar values we need to take care to define their direction for the problem, or for specific values. One common way to do this is to define positive ‘x-y’ axes, and then refer to ‘x’ and ‘y’ components. From this a positive ‘x’ component implies one direction: a negative component implies the opposite direction.

• For example,

• Scalar notation is often made obvious by using ‘x’, and ‘y’, or similar subscripts.

• direction, location, signs, etc. are all defined by convention, and very compact mathematical methods can be used.

• These problems can also be solved using cosine and sine law force additions on force triangles. Considering the last example,

• Consider the large pendulum below as an example where a force triangle could be used to find the tensions in the cables.

• NOTE: all of the vectors added are position vectors. If we are to consider rotation vectors, they cannot be simply added. You must consider alternatives such as Euler angles, etc.

• Consider the example below,

#### 3.5.3 Unit Vector Representation

• The unit vector representation follows.

• The unit vector representation can be developed from basics.

#### 3.5.4 3D Vectors

• We will use right-handed coordinates

• Consider the following example,

• To emphasize the main relations,

• An example to illustrate this technique is,

Problem 3.1 ([Hibbeler, 1992], prob. 2-56, pg. 49) Find the resultant force for the two forces shown in vector projection (F1) and cosine notation (F2).

Problem 3.2 Given the system of vectors pictured, a) give the resultant force using Cartesian notation b) find the magnitude of the resultant force in metric units. c) Then then using cosine angles, and finally d) projected onto the x-y plane.

• Consider the case below, where we know positions, and forces, but we want to find the resultant force,

• As a practical example of where 3D vectors might be required, consider the power line pole. It uses a tension cable anchored in the ground to resist the forces exerted by the power lines.

#### 3.5.5 Dot (Scalar) Product

• The definition of the dot product is,

• Evaluating the dot product.

• We can use a dot product to find the angle between two vectors

• Projecting vectors.

• Consider the example below where we find the component of one vector that lies in the direction of the other vector.

• The use for the dot product will become obvious in later sections.

Problem 3.3 A force F acts at the end of a pipe. Determine the magnitudes of the components that act perpendicular to, and along the axis of the end of the pipe. (the pipe lies in the y-z plane)

#### 3.5.6 Summary

• the basics of statics as a topic were covered

• engineering units and calculations

• representations covered in this section were,

scalar values

vector values

rectangular

polar

Cartesian

direction cosines

vector projection

direction vectors

• The dot product was shown as a way to project one vector onto another, or final angles between them.

#### 3.5.7 Problems

Problem 3.4

Problem 3.5 Four forces act on bolt A as shown. Determine the resultant of the forces on the bolt.

Problem 3.6 Find the tension in cable A and B if the tension in cable C is 100N.

Problem 3.7 A disabled automobile is pulled by two ropes as shown below. If the resultant of the two forces must be 300lb, parallel to the forward roll of the car, find (a) the tension in each of the ropes, knowing that α = 30°, (b) the value of α such that the tension in rope 2 is minimum.

###### Answer 3.7 a) TA= , b) 70°

Problem 3.8 Convert between the representations given on the left, and the results requested on the right.

###### Answer 3.8 a) 7.1dN,7.1dN, b) 22.3lb, 243°, c) 4.0m,1.7m,2.5m

Problem 3.9 An F-117A stealth fighter is supposed top be flying N20°E, but a strong wind from East to West is pushing it off course. If the plane is pointed N20°E, but is actually moving N23°E, and its 22,000 lb engine is at full thrust, a) what force is the wind exerting on the plane? b) What is the answer in newtons?

#### 3.5.8 References

3.1 Hibbeler, R.C., Engineering Mechanics: Statics and Dynamics, 6th edition, MacMillan Publishing Co., New York, USA, 1992.

3.2 Soustas-Little, R.W. and Inman, D.J., Engineering Mechanics Statics, Prentice-Hall, 1997.

3.6 Equilibrium

• Put simply equilibrium describes the condition where all forces are balanced (no acceleration). Static equilibrium describes the state where all forces are balanced, and the object is not in motion.

• For an object to be in equilibrium, all the forces and moments must be balanced for,

each particle in a rigid body

each rigid body

each object made up of rigid bodies

3.7 The Basic Equations of Statics

• There are two basic balances that must exist in any statics problems.

• if the sum of the forces is not zero, then the system will undergo translation, and the problem cannot be solved with statics methods.

• if the sum of moments is not zero, then the system will undergo rotation, and the problem cannot be solved with statics methods.

• At least one of these two equations will appear in every statics problem.

• A simple example,

• Let’s consider a force balance problem, ([Hibbeler, 1992]prob 3-40, pg. )

#### 3.7.1 Problems

Problem 3.10 Two cables (AB and AC) and a force (P) act on the top of a flag pole (AD). Find the magnitude of force P required to keep the flag pole standing? Assume that cable AB is under 5 KN of tension.

Problem 3.11 The 500 lb pendulum below is hung from beam ABC with three cables (BD, DC and DE). The cables are joined at D with a ring. What is the tension in each of the cables? (25%)

Problem 3.12 F1 is a projected vector with a magnitude of 15KN, the magnitude of F2 is 10KN. F2 is represented with direction cosines, and lies in the negative ‘x’ direction. Find the resultant of these vectors in Cartesian vector notation. (50%)

#### 3.7.2 References

Hibbeler, R.C., Engineering Mechanics: Statics and Dynamics, 6th edition, MacMillan Publishing Co., New York, USA, 1992.

3.8 Free Body Diagrams (FBDs)

• Up to this point we assumed very simple forces acting on very small particles.

• In reality mechanical systems have many parts, and we draw an FBD for each part.

• We should divide forces on free body diagrams into two categories,

Internal: these forces act only within a free body, and cancel out, unless we are looking at a section of a free body.

External: these forces act on a free body, and they induce reaction forces. Examples are gravity, and other free bodies.

• An example of using free body diagrams for a system is given below with a system of masses, ropes, pulleys and anchors.

#### 3.8.1 Pulleys and Springs

• Pulleys are basically a wheeled roller that a rope can roll over freely,

• A simple example of a pulley used for lifting a mass is given below,

• Springs are a very important engineering tool,

• A sample problem that uses springs is given below, ([Hibbeler, 1992] prob 3-16, pg. )

Problem 3.13 Given the three masses below, connected by a cable through three pulleys, determine the final resting height (h2) for the center mass. Assume the pulleys are very small and static.

#### 3.8.2 Summary

• equilibrium of forces and moments

• free body diagrams (FBD’s)

pulleys

springs

anchors

cables

masses

rings

3.9 Problems

Problem 3.14 Determine the reactions at B, C and D.

###### Answer 3.14 Fd = 200N, Fb = Fc = 231N

Problem 3.15 Four masses are suspended by cables that are supported by pulleys. The frictionless pulleys are mounted on a flat ceiling. Each of the pulleys is a distance of 2m from the others. Determine the height of the center mass.

Problem 3.16 For the mass pulley system on the left, a) draw the force triangle on the right, with all angles and magnitudes indicated, then b) find the mass if the tension in BC is 70N.

b) Find M.

#### 3.9.1 References

3.3 Hibbeler, R.C., Engineering Mechanics: Statics and Dynamics, 6th edition, MacMillan Publishing Co., New York, USA, 1992.

3.4 Soustas-Little, R.W. and Inman, D.J., Engineering Mechanics Statics, Prentice-Hall, 1997.