APL100

mechanics

dynamics

statucs

kinematics

Frame of reference

coordinate systems

time derivatives

notation dot use

click to edit

path coordinates

osculating plane

radius of curvature

unit vectors

formula in terms of veolcity acceleration

in terms of the equation of the plane curve

in terms of the normal acceleration

in term of the parametric form of the path

define it in terms of the centre of curvature

you need to understand that to solve a problem you need to choose a coordinate systems and knowing which coordinate system to use in which problem may help to solve the problem easily

HINT : USE THEM IF YOU KNOW THE PATH ESPECIALLY IF SOMETHING IS CONSTRIANT

DYNAMICS

THE INERTIA MATRIX ABOUT POINYT A

PRINCIPAL AXIS OF INERTIA always about a point

AN AXIS SUCH THAT THE UNIT VECTOR ALONG ITS DIRECTION IS AN EIGEN VECTOR OG=F THE INTERIA MATRIX

IMPORTANT POINTS

FINDING THIS AXIS

USE THE BASIC DEFINATION

GEOMETRICALLLY

USING THE PLAN EOF MASS SYMMETRY

IN CASE OF BODY OF REVOLUTION

EXISTENCE

HA vector if expressed with wrt to the bases vectors as the eigen vectors of the interia matrix(the vectors along the principal axes

only the three terms

use the basis for the inertia matrix as the eigen vectors

you understand what it menas to say that write the inertai matrix wrt to a basis set

can you connect the inertia matrix to an opertaor

the unit vectors along the axes as a means for teh matrix representation of an operator

doubt is inertia matrix operator a linaer operator

can ypu connect the change of bases for the matrix repe of an operator to the change of bases when inertia matrix

is inertia matrix an hermitain matrix

what is the consequences if it the case

if yes what are the vector spaces we are considering ???

kinetic energy and work energy relation for the rigid body

BELT FRICTION

MAX POWER TRANSMISSION

THREE CASES

NO SLIP

IMPENDING SLIP

SLIP

PULLEY

REST

MOVING

V belt vs flat belt

v belts more tanagetal force foor the same normal force

the slope leads to incerease in the effective coeffienct of friction

finding the relation between the tensions on the two sides

also find the pressure

more realistice picture of a element of the belt

average of the Force vector

average of the radius of curvature

average of the nornal force evctor

LIMITING

width goes to zero use of the def of a derivative

find two independent vectors perpendicular to the axis for whcih the product of inertia are zero is sufficient

normal to this plane at the point it intersects the the plane

about any point on the axis of revolution the axis of revolution and any line normal to it is also a PA

HA in direction of w if only if w along one of the principal axes at A

if you get two orthoganl planes of symmetry then about the point of there intersection the normal to these planes are the PA also the the line of intersection of the planes

STANDAR MOMENT OF INERTIAS

MI disc

MI ring

mr2

mr2 by 2

what is new about what you learned

the moemntum of inertia terms have increased

kinetic energy

special cases

if you have the instanatanous axis of rotation

when the angualr veoclity only about one axis

when you have the inertia matrix wrt to PA

work energy relations

without work due external moments

work including the work due to external moments

REMEMBER THE WORK DONE BY INTERNAL FORCES MAY NOT BE ZERO

solving techniques

read the question properly and making a rough FBD

then chhose / identify appropriate coordinate system

find the points where the EULERS 2ND AXIOM CAN BE APPLIED

APPLY THE 2ND AND THE FIRST AXIOM AND SOLVE GENERAL THEN PUT SPECIFIC VALUES

FIRST WRITE THE W AND FIND I MATRIX*W witjout expicitly finding I beacuse all the elements of the matrix may not be needed

remember if you have to apply the parallel axis theorem you myst use the axis through the COM

SYMMETRICAL CUT THE MI REMAINS SAME

BE CAREFUL DONT CHANGE THE MASS

FINDING THE INERTIA TENSOR

IF COMPLEX OBJECT AND ARBITARY AXIS

BETTER TO FIRST FIND THE INERTIA TENSOR WRT TO SOME STANDARD AXES FOR ALL THE PARTS THEN FIND THE TOTAL INERTIA TENSOR WRT TO THE STANDARD AXIS FOR THE COMPLEX OBJECT AND THEN APPLY THE TRANSFORMATION MATRIX TO THE FINAL TENSOR

PARALLEL AXIS THEOREM ONLY WHEN ONE AXIS COM

CUT OUT OBJECTS BE CAREFUL WITH THE MASS AND THE MOMENT OF INERTIA

DOUBT A IS ON THE RIGID BODY ??? OR THE MASSLESS EXTENSION???

THE COORDINATE SYSTEM USED TO FIND THE INERTIA MATRIX MUST BE THE SAME AS THE COORDINATE SYSTEM USED TO EXPRESS THE ANGULAR VELOCITY

NEXT TIME YOU SOLVE A DYNAMICS PROBLEM BE AWARE OF THESE POINTS

IMPACT AND ENERGY METHODS

MISTAJKES

FORGETTING POTENTIAL ENERGY

FORGET TO TAKE INTO ACCOUNT SOME WORK DONE BY EXTERNAL FORCES

TOOK THE ACTION AND REACTION FORCES IN THE SAME DIRECTION

CONSERVE THE ANGULAR MOEMNTUM ABOUT A POINT WHICH IS NOT AT REST

ESPECIALLY WITH FRICTION AND NORMAL REACTION

WRITING THE ANGUALR MOENTUM ABOUT GENERAL POINT

KEEP IN MIND THE EXPERSSION INVIOLVES ANGUKLAR MOEMNTUM ABOUT CENTRE OF MASS AND THE CROSS PORODUCT TERM OF RCA AND VCA

IT IS NOT IW only

writing the kinetic energy then inertia about the centre of mass

smooth impact the tangetial velocur=ty of the centre of mass is same not the point of impact

tools and tips

work energy theorm

time derivative

integrated

restitution

angular momentum and angular impulse

moemtum and impulse moemntum

time derivative of cobstant angular moentum

be careful in case of angular veoclity and angular acceeration when more than one body is there

use the tnagential and normal coordinate system in impact

repeated exprsessions can be given a name

try to find general expression and put the values when required

soemtimes cylindrica; coordinate system also elps

identitfy the IAOR AND ICOR

write the kinetic enrgy easily

no slip cylinder and rod

write the cordinate system clealy

trick fr torisonal spring

smooth cenrtal impact of unconstrainted bodies

w is same

find w or theta dor from w dot or theta double dot genreal equation

dont confuse the com with the pooint of application of forces or other points

be very careful betwwen the concident points at rest and the ppints on the rigid bodes

NOTE THAT THE NAGULAR MOMENTUM ABOUT A POINT AT REST AND COINCIDENT WITH THE SOME POINT OF A RIGID BODY IS NOT SAME AS THE ANGULAR MOMENTUM ABOUT THE POINT ON THE RIGID BODY

FOR A POINT ON A RIGID BODY OR ITS EXTENSION YOU MAY WRITE H AS IW but not for the concident point look at the exercise questio on pagr 415

ADOPT A SINGLE CONEVENTION FOR VECTORS RCD IS A VECTORE FROM D TO C

FIXI NG A COORDINATE SYSTEM WRITING VECTORS IN THAT COORDINATE SYTEM AND HELPS A LOT

IF YOU HAVE A BIG EXPRESSION FOR FINDING IT IT IS BETTER TO FIND EACH TERM SEPERATELY AND THEN ADD THEM UP

CONNECTING THE QUANTITIES INVOVLED HEPS IN SIMPLIFYINGH THE EXPRESSIONS

EXAMPLE

LENGTH IS TWICE THE BREADTH

OR MASS OF ONE BODY IS TWICE THE OTHER

even for unknown quantities wirite them as vectors and seperatwe the scalar

keep things in fractions

sometimes the angular moementum about the com reamins fixed

becuase forces paas through the com

w is constant

STATICS

IDENTITFY ZERO FORCE MEMBERS

IN 3D TRUSS

GENEERAL IDEAS ABOUT FORCES

IF TWO FORCES ACT ON ONE POINT AND THERE SUM IS ZERO THEN TEHY MUST BE EQUAL AND OPPOSITE

FORCE PERPENDICULAR COMPONENT TO COPLANR FORCES

click to edit

ALL THE IDEAS IN 2D TRUSSES IF YOU FIND COPLANAR FORCES

SUM OF THREE NON COPLANR FORCES ARE ZERO

ALL THE FORCES HAVE TO BE ZERO A

TWO NON COLLINAER FORCES SUM ZERO MEANS BOTH MUST BE ZERO

METHOD OF SECTIONS

NOTE THAT A 4 MEMBER SECTION IS NOT ALWAYS USELESS

BASICALLY IN THIS CASE YOU CANNOT NOT BE ABLE TO FIND ALL THE 4 FORCES

BUT YOU MAY BE ABLE TO FIND SOME OF THE FORCES

ESPECIALLY IF THREE FORCE ARE CONCURRENT

IN EQUILIBRIUM YOU CAN TAKE MOMENT ABOUT ANY POINT TO BE ZERO

SO CAREFULLY CHOOSE THE POINTS

THEY NEED NOT ONLY BE THE POINTS FROM WHICH THE FORCES COMING IN THE FBD

MISTAKES TO AVOID

NOT FOLLOWING THE CONVENTION

FORCES AWAY FROM JOINTS

NOT TAKING ALL THE FORCES ON THE HALF PART AND FORGETTING THE FORCES OTHER THAN THE UNKNOWN ONES

TIPS

USEFULNESS OF THE EULERS 2ND AXIOM

FIND A POINT WHERE MANY FORCES ARE CONCURRENT

IN A SECTION FIND ONLY THE FORCES YOU WANT NOT ALL

MIX METHOD OF JOINTS AND SECTIONS

BE CAREFUL WITH THE ANGLES

KEEP IN MIND THE UNITS NEWTONS OR KILONEWTONS

YOU MAY NOT NEED TO FIND ALL THE FORCES AT A JOINT TO FIND A PARTICULAR FORCE AT THAT JOINT FIRST WRITE THE RELATION BETWEEN THE FORCES ON THE JOINTS

WRONGLY MARKING THE SHEAR AND THE NORMAL FORCE

NORMAL ALING THE LENGTH SHEAR IS PERPENDICUALR TO LENGTH

THE SIGN OF THE DISTRIVUTED FORCES OR MOMENTS

DONT FORGET THE JUMPS AND TAKE CARE OF SIGN THEIR

ALL THE FUNCTION WITH RESPECT TO SAME ORIGIN

EVEN THE N(X) ,Q(X), MEANS EVEN THE FUNCTION OF DISTRIBUTED FORCES OR MOMENTS

DONT FORGET TO SEE THE STARTING AND THE END JUMPS

USE CALCULUS AND

DONT JUST GO BY INTUITION

LIKE IF SOMETHING WAS POSITIVE AND THEN IT IS GOING TO ZERO THAT MEANS THE INITIAL POSITIVE VALUE WAS THE MAXIMUM

Sometimes you need to take section to find the reactions from dupport

Found forces in wrong member

In hurry ignore members and do mistakes in joints