Engineering Mechanics

Equilibrium of Rigid Bodies:

Frames and Machines

Main Text• Major chunk of lectures based on:

• Vector Mechanics for Engineers, Beer, Johnston et al., 10th ed., McGraw-Hill.

• Referred to as BJ10. Indian Edition available.• Also, some problems from Beer and Johnston 3rd

and 8th editions, BJ3 and BJ8.• Dynamics will be exclusively taught from BJ10.

Many wonderful new resources available in BJ10. Become more clear as we proceed.

• Many slide contents in our lectures from BJ10Instructor resources.

• Attractive online features available for Instructorshttp://highered.mcgrawhill.com/sites/1259062910/information_center_view0/

• For purchase/other details kindly contact:• sagar.divekar@mheducation.com Sagar Divekar

santosh.joshi@mheducation.com Santosh Joshi

Secondary Text

• Many really interesting and challenging

problems from:

– Engineering Mechanics: Statics/Dynamics,

Meriam and Kraige, Eds. 2, 5, 7. (MK3,

MK5, MK7).

Online resources

• Nice demonstrations from Wolfram (look under

the mechanics section). Will show some of them.– http://demonstrations.wolfram.com/

• Beautiful lectures notes by Prof. Allan Bower at

Brown University– http://www.brown.edu/Departments/Engineering/Courses/En4/Notes/notes.html

• Nice general lectures on Dynamics on youtube:– http://www.youtube.com/user/mellenstei

• Nice animations to textbook problems:– http://wps.prenhall.com/wps/media/objects/3076/3149958/studypak/index_st.h

tml

Equilibrium

• System is in

equilibrium if

and only if the

sum of all the

forces and

moment (about

any point)

equals zero.

Supports and Equilibrium

• Any structure is made of many components.

• The components are the be connected by linkages.

• Other wise the structure will lose its integrity.

• Different component of structure talk to each other via linkages.

• The structure should be globally supported to prevent it from falling over.

Constraints and Reactions

• There is an intricate relationship between kinematics (motion) and reactions (forces).

• Always note that in the case of supports displacement (rotation) and force (torque) in any given direction are complementary.

• If a support rigidly constrains a given degree of freedom (DOF) for a rigid body then it gives rise to a reaction corresponding to that DOF.

• Similarly if a support freely allows motion of particular DOF then there is no reaction from the support in that direction.

Free Body Diagram (FBD)

• Single most important concept in

engineering mechanics.

• Zoom in on a given component of a structure.

• Means replace supports (connections) with the

corresponding reactions.

• Replace kinematic constraints with

corresponding reactions.

• Concepts will get more clear as we proceed

further.

Simple examples

• Copyright, Dr. Romberg

FBD

FBD

Simple examples

• Copyright, Dr. Romberg

FBD

FBD

Link: Two-Force Member

• Member with negligible weight and arbitrary shape connected to other members by pins

Two Force member

http://oli.cmu.edu

Hydraulic Cylinder

Show Mathematica demo

Equilibrium of a Three-Force

Body• Consider a rigid body subjected to forces

acting at only 3 points.

• Assuming that their lines of action intersect,

the moment of F1 and F2 about the point of

intersection represented by D is zero.

• Since the rigid body is in equilibrium, the sum

of the moments of F1, F2, and F3 about any axis

must be zero. It follows that the moment of F3

about D must be zero as well and that the line

of action of F3 must pass through D.

• The lines of action of the three forces must be

concurrent or parallel.

Slide from BJ10

Sample Problem 4.6 BJ-10

4 – 15 BJ10

A man raises a 10-kg joist,

of length 4 m, by pulling on

a rope.

Find the tension T in the

rope and the reaction at A.

SOLUTION:

• Create a free-body diagram of the

joist. Note that the joist is a 3 force

body acted upon by the rope, its

weight, and the reaction at A.

• The three forces must be concurrent

for static equilibrium. Therefore, the

reaction R must pass through the

intersection of the lines of action of

the weight and rope forces.

Determine the direction of the

reaction R.• Utilize a force triangle to determine

the magnitude of the reaction R.

Sample Problem 4.6

4 – 16 BJ10

• Create a free-body diagram of the

joist. • Determine the direction of the

reaction R.

63614141

3132tan

m2.313 m51508282

m515020tan m4141)2545(cot

m4141

m828245cosm445cos

ooo

21

oo

..

.

AE

CE

..BDBFCE

..CDBD

.AFAECD

.ABBF

6.58

Sample Problem 4.6

4 – 17 BJ10

• Determine the magnitude of the

reaction R.

38.6sin

N 1.98

110sin4.31sin

RT

N 8.147

N9.81

R

T

Hydraulic Cylinder

show Mathematica demo on this

Multiple Rigid Bodies

Connected To Each Other

Pin Connections

• All figures from http://oli.web.cmu.edu

Modeling 3D Problem as 2D

Point Connections

• All figures from http://oli.web.cmu.edu

Contd..

• All figures from http://oli.web.cmu.edu

Contd..

Free Body Diagram at Pin-

Connection

• All figures from http://oli.web.cmu.edu

Summary

• Pin support

• Pin connection

• All figures from http://oli.web.cmu.edu

Roller

Slot Connection

Non-Symmetrical but bodies

connected by pin are very close to

each other

• All figures from http://oli.web.cmu.edu

Introduction to Structures

(BJ10)

6 - 30

• For the equilibrium of structures made of several

connected parts, the internal forces as well the external

forces are considered.

• In the interaction between connected parts, Newton’s

3rd Law states that the forces of action and reaction

between bodies in contact have the same magnitude,

same line of action, and opposite sense.

• Three categories of engineering structures are

considered:

a) Trusses: formed from two-force members, i.e.,

straight members with end point connections and

forces that act only at these end points.

b) Frames: contain at least one multi-force member,

i.e., member acted upon by 3 or more forces.

c) Machines: structures containing moving parts

designed to transmit and modify forces.

Analysis of a Frame

6 - 31

• Frames and machines are structures with at least one

multiforce (>2 forces) member. Frames are designed to

support loads and are usually stationary. Machines

contain moving parts and transmit and modify forces.

• A free body diagram of the complete frame is used to

determine the external forces acting on the frame.

• Internal forces are determined by dismembering the

frame and creating free-body diagrams for each

component.

• Forces between connected components are equal, have

the same line of action, and opposite sense.

• Forces on two force members have known lines of action

but unknown magnitude and sense.

• Forces on multiforce members have unknown

magnitude and line of action. They must be

represented with two unknown components.

Frames Which Cease To Be Rigid When Detached From Their

Supports

6 - 32

• Some frames may collapse if removed from

their supports. Such frames cannot be treated

as rigid bodies.

• A free-body diagram of the complete frame

indicates four unknown force components which

cannot be determined from the three

equilibrium conditions (statically

indeterminate).

• The frame must be considered as two distinct,

but related, rigid bodies.

• With equal and opposite reactions at the contact

point between members, the two free-body

diagrams show 6 unknown force components.

• Equilibrium requirements for the two rigid

bodies yield 6 independent equations. Thus,

taking the frame apart made the problem

solvable.(BJ10)

More examples

MK5 BJ8

Sample Problem 6.4 (BJ10)

6 - 34

Members ACE and BCD are connected

by a pin at C and by the link DE. For

the loading shown, determine the force

in link DE and the components of the

force exerted at C on member BCD.

SOLUTION:

1. Create a free body diagram for the

complete frame and solve for the support

reactions.

BJ10

Sample Problem 6.4

6 - 35

SOLUTION:

1. Create a free-body diagram for the complete frame

and solve for the support reactions.

N 4800 yy AF N 480yA

mm160 mm100N 4800 BMA

N 300B

N 300

0

x

xx

A

ABF

N 300xA

Sample Problem 6.4

6 - 36

SOLUTION (cont.):

2. Create a free body diagram for member BCD (since

the problem asked for forces on this body). Choose

the best FBD, then discuss your choice with a

neighbor. Justify your choice.

FDE,x

FDE,y

FDE

FDE

FDE,x

FDE,y

Sample Problem 6.4

6 - 37

N 561

mm 100N 480mm 06N 300mm 250sin0

DE

DEC

F

FM

CFDE N 561

• Sum of forces in the x and y directions may be used to find the force

components at C.

N 300cosN 561 0

N 300cos0

x

DExx

C

FCF

N 795xC

N 480sinN 5610

N 480sin0

y

DEyy

C

FCF

N 216yC

SOLUTION (cont.):

3. Using the best FBD for member BCD, what

is the one equilibrium equation that can

directly find FDE? Please discuss.

07.28tan150801

Sample Problem 6.4

6 - 38

• With member ACE as a free body with no

additional unknown forces, check the

solution by summing moments about A.

0mm 220795mm 100sin561mm 300cos561

mm 220mm 100sinmm 300cos

xDEDEA CFFM

(checks)

Machines

6 - 39

• Machines are structures designed to

transmit and modify forces. Typically they

transform input forces (P) into output forces

(Q).

• Given the magnitude of P, determine the

magnitude of Q.

• Create a free-body diagram of the complete

machine, including the reaction that the

wire exerts.

• The machine is a nonrigid structure. Use

one of the components as a free-body.

Discuss why the forces at A are such.

• Sum moments about A,

Pb

aQbQaPM A 0

Problem MK5

• The car hoist allows the car to be driven on to the platform, after which the rear wheel is raised. If the loading from the rear wheel is 3300kg, determine the force in the hydraulic cylinder AB. Neglect the weight of the platform itself. Member BCD is a right angle bell crank pinned to the ramp at C

Problem MK5

• For the paper punch shown in the figure find the punching force Q corresponding to a hand grip P.

Problem BJ8

• Knowing that P =

411 lb and Q = 0,

determine for the

frame and loading

shown (a) the

reaction at D, (b) the

force in member BF.

Questions?

Tutorial on 2D equilibrium

Problem 1

• Knowing that each pulley has a radius of 250mm, determine the components of reactions at D and E.

Problem 2 (BJ10)

• The shear shown is usedto cut and trimelectronic-circuit boardlaminates. For theposition shown,determine (a) the verticalcomponent of forceexerted on the shearingblade at D, and (b) thereaction at C. It is giventhat P = 400 N.

Problem 3 (MK5)• An adjustable tow bar connecting the tractor unit H with the landing

gear J of a large aircraft is shown in the figure. Adjusting the height ofthe hook F at the end of the tow bar is accomplished by the hydrauliccylinder CD is activated by a small hand-pump (not shown). For thenominal position shown of the triangular linkage ABC, calculate theforce P supplied by the cylinder to the pin C to position the tow bar. Therig has a total weight of 220kg and is supported by the tractor hitch E.

Problem 4

(MK2)

The figure shows a special rigfor erecting vertical sectionsof a construction tower. Theassembly A has a mass of1.5Mg and is elevated by theplatform B which itself has amass of 2Mg. The platform isguided up the fixed verticalcolumns by rollers and isactivated by the hydrauliccylinder CD and links EDFand FH. For the particularposition shown calculate theforce R exerted by thehydraulic cylinder at D.Neglect mass of cylinder andlinks.

Problem 5

(BJ10)

A hydraulic-lift table is used to raise a 1000 kg crate. It

consists of two identical linkages on which hydraulic cylinders

exert equal forces. Members EDB and CG are each of length

2a, and member AD is pinned to the midpoint of EDB. If the

crate is placed on the table, so that half of its weight is

supported by the system shown, determine the force exerted by

each cylinder in raising the crate for θ = 60 deg, a = 0.70m,

and L = 3.20m. Show that the result obtained is independent

of distance d.

Problem 6

(BJ3)

• The elevation of a platform is controlled by two identicalmechanisms only one of which is shown. A load of 5 kN isapplied to the mechanism shown. Knowing that the pin atC can transmit only a horizontal force, determine (a) theforce in link BE, (b) the components of the force exerted bythe hydraulic cylinder on pin H.

H

Problem 7

(Shames)

• A hydraulic lift platform for loading trucks supports aweight 2W. Only one side of the system has been shown;the other side is identical. What force F should thehydraulic cylinder provide to support the weight W.Assume that the hydraulic device is inclined at aclockwise of α with respect to the horizontal. Neglectfriction everywhere.

Problem 8,

MK2

• Obtain the clamping force Q developed for

the pliers when the handle force is P.

Problem 9

BJ8

• A 500-kg concrete slab is supported by a chainand sling attached to the bucket of the front-endloader shown. The action of the bucket iscontrolled by two identical mechanisms, onlyone of which is shown. Knowing that themechanism supports half of the 500-kg slab,determine the force (a) in the cylinder CD, (b) incylinder FH.

Problem 10

• In the toy folding chair shown, members ABEH and CFK are parallel. Determine the components of all forces acting on member ABEH when a 160 N weight is placed on the chair. Draw completely all free body diagrams required. It may be assumed that the floor is frictionless and that half the weight is carried by each side of the chair and is applied at point M as shown.