dynamics review 2013

7/30/2019 Dynamics Review 2013

1/18

FE Exam: Dynamics review

D. A. Lyn

School of Civil Engineering3 March 2013


2/18

Preliminaries

Units (relevant quantities:g, displacement, velocity,

acceleration, energy, momentum, etc.)

Notation (dot, vector)

Vectors (components and directions/signs, graphicaladdition and subtraction, dot and cross products,

vector polygons)

Coordinate systems (Cartesian and curvilinear, fixedand moving or relative, unit vectors)

Statics (free body diagram)


3/18

Classification of dynamics and problems Kinematics: description of motion without reference to

forces Particle (no rotation about itself, size unimportant) and rigid-

body

Coordinate systems (Cartesian, curvilinear, rotation)

Constraints on motion

Kinetics: inclusion of forces (mass, or momentum or energy) Types of forces: conservative (gravitational, spring, elastic

collisions) and non-conservative (friction, inelastic collisions)

Newtons 2nd law: linear and angular momentum Use of free body diagram to deal with external forces

Particles and rigid body (system of particles) Impulse (time involved) and momentum

still working with vectors (before and after)

Work (distances involved) and energy (velocities involved) working with scalars (usually easier) (before and after)


4/18

Particle kinematics

General relations between displacement (r

),velocity (u), and acceleration (a)

Given a formula for (or graph of) r as functionoft, take derivatives to find u and a

Given a formula for u or a as function oft,integrate to find r oru

Special case: constant (in magnitude and direction)acceleration, (initial conditions needed todetermine integration constants)

( ) , ( ) = ( , , ), ( , , ), ( , , )d d

t t x y z x y z x y z dt dt

r u

u r a u r r u a

0a a

20 0

1( ) ( 0) , ( ) ( 0) ( 0)

2t t t t t t t t

u u a r r u a


5/18

Sample problems The position of a particle moving horizontally is described by

, withs in m and tin s. At t= 2 s, what is itsacceleration?

Soln: Take derivatives ofs with respect to t, and evaluate at t=2s( ) so a(t=2s) = 4 m/s2.

Projectile problem: A projectile is launched with an initial speedofv0=100 ft/s at q=30to the horizontal, what is the horizontaldistance,L, covered by the projectile when it lands again?

Soln: constant acceleration (only gravitationalacceleration involved) problem, so apply

formulae in two directions

2

( ) 2 8 3s t t t

( ) ( ) 4 8, ( ) ( ) 4u t s t t a t s t

20 0

20 0

/ 2

/ 2

end x end x end

end y end y end

x x v t a t

y y v t a t

0 0Given : 0, , 100cos30 , 100sin30x y x ya a g v v

wish to findL=xend-x0, foryend-y0=0, so we solve

L=v0xtendand 0=v0ytend-g(tend2/2) for tendandL;

L

v0

q

x

y

tend=3.1 s and L=269.2 ft


6/18

Kinetics of a particle Linear momentum, L=mu (mass, m, i.e., measure of inertia)

Newtons 2nd law:

Forces determined from free body diagram (as in statics)

Types of forces: gravitational, frictional, spring, external

Angular momentum (about a point O) ,

Newtons 2nd law:

Impulse (used in impact and collision problems),

momentum conservation:

mini-problem: A golf ball of mass 50-g is hit with a club. If the initialvelocity of the ball is 20 m/s, what is the impulse imparted to the ball? If

the contact duration was 0.05 s, what was the average force on the ball?

m F a L

0 m H r u

0 0M H2

1

1 2

t

t

dt Imp F2

1

2 1 1 2

t

t

dt L L Imp F

1 2 1 2 2

1 20

0 Imp (0.05 kg)(20 m/s) = 1 Ns

Imp 1 Ns 1 Ns / 0.05 s 20 N

t

avg avg

L L mv

F dt F t F


7/18

Problem: kinetics of a particle (truck) A truck of weight W= 4000 lbf moves down a q=10

incline at an initial speed ofu0

= 20 ft/s. A constant

braking force ofFbrk=1200 lbf is experienced by the

truck from a time, t= 0. What is the distance

covered by the truck before it stops from the time

that the braking force is applied?

kinematics problem:

2sin ( ) sin 4.1 ft/sbrkbrk net s s FF W F ma W mg a gW

q q

0

2 20 0

( ) ( 0) / 4.9 s

( ) ( 0) ( / 2) / 2 48.8 ft

end s end end s

end end s end s

u t t u t a t t u a

s t t s t u t a t u a

Notes: forces involved kinetics problem, rectilinear (straight-line) motion: determine netforce on truck in direction of motion, apply Newtons 2nd law to evaluate distance covered

From free body diagram, sum of forces in direction of motion,

u0

q

W

Fbrk

Wsinq

mass

kinetics problem (force balance insdirn):


8/18

Curvilinear coordinates and motion Plane motion (motion on a surface, i.e., in only two dimensions)

Tangential (t) and normal (n) coordinates

is the radius of curvature of particlepath at the particle position(advantage: vn = 0, zero normal comp.)

Radial (r) and transverse (q) or polar coordinates

Special case: pure circular motion at an angular frequency,wt:

2( ) , ( ) ( ) ( 2 )r rt r r t r r r r q qq q q q v e e a e e

2( ) , ( ) /t n tt v t v v v e a e e

22 2

2

, 0, , [ , ]

0, ,

( ) , ( )

r n t

r

r v r r v

r r r v r r r r

a t r a t r

q

q

q w q w q w q w

w

e e e e

xx1

yy1

r

q

eren

eteq particlepath

particleat time t

r

v=

r

q

w

ar

r= w

2

ar

r

q=

w.

( is the angular acceleration)


9/18

Particle kinetics problem

Find the tension, T, in the string and the

angular acceleration, , at the instant shownif at the position shown the sphere of mass,m=10kg, has a tangential velocity ofv0=4m/s. R = 0.6 m, and q0=30.

Choose a radial-transverse coordinate system,perform free body analysis to determine sumof forces, and set equal to ma.

202

0 0 0

20 0

dir'n: - cos / cos 352 N

dir'n: sin sin / 8.2/s

r

vr T W ma mv R T m g

R

W ma mR W mRq

q q

q q q q q

W

T

r

q

Rq0

v0

m

2( ) , ( ) ( ) ( 2 )r rt r r t r r r r q qq q q q v e e a e e


10/18

Energy and work

Work of a force,F

, resulting in a change in position from state 1 tostate 2:

Constant force in rectilinear motion,Fxx2-x1)

Gravitational force, -Wy2-y1),y>0 upwards

Spring force, -k(x2

2-x1

2)/2, (x2


11/18

A problem solved using energy principles A 2-kg block (A) rests on a frictionless plane

inclined at an angle q=30. It is attached by an

inextensible cable to a 3-kg block (B) and to a

fixed support. Assume pulleys are frictionless

and weightless. If initially both blocks are

stationary, how far will the 2-kg block

Motion constraints: sB=sA/2 (and yA=-2yBsinq), and vB=vA/2

Frictionless system conservative gravitational forces only, only distances

and speeds explicitly involved apply energy equation

22 2 21 2Initially, 0; at end, / 2 / 2 / 2 1 / /A A B A B AA B

T T mv mv m v m m v v

1 2 2 1 2 1 21 2

22

22

11 1 1

2 2sin

1 2.24 m 0 (2 1 / 2 sin

A A B B A A B B A A B B

A A B B B B BA A A A

A A A A A

A B BA

B A A A

V V T T T W y W y W y W y W y W y U

m v m v W y W W y W y

m v W y W

v m vy

g W W m v

q

q

Block A rises)

/ sin 2 4.48 mA As y yq

3 kg3 kg

2kg

2kg

q=30 q=30

State 1 State 2

s yB= ByA

v

BB

A

A sA

travel before its speed is 4 m/s?


12/18

Rigid body (plane) dynamics rigid body: distance between any two arbitrary points

on body remain same types of motion

pure translation (can be treated as a particle)

pure rotation about a fixed axis convenient to characterize in terms of angular frequency, w, and

angular acceleration, general plane motion (combined translation and rotation

about a fixed axis)

rotation is distinguishing feature of rigid-bodydynamics moment equation: product ofmass moment of inertia and

angular acceleration analogous to ma

effective forces and moments and mass centers inforce and moment equations


13/18

Rigid body (plane) kinematics

general plane motion of

a rigid body is sum of a

translation of a point and

rotation about that point.

/

/

/

/ /

( )

( )

2/ / /

( )

=

=

B A t

B A n

B A t

B A B A A B A

B A B A A B A B A

w

w

v

a

a

v v v v k r

a a a a k r r

choose point A to write equation for the motion of point B

direction ofvB/A, (aB/A)t, and

(aB/A)ncrucial to solution

directions of unknown w and

may be assumed

velocity problem: knowing vA anddirection ofvB and geometry, find

magnitude ofvB and w.

acceleration problem: knowing aA

and direction ofaB and geometry,

find magnitude ofaB and for plane problems, the direction of and

especially is known, usually key to solution./B Ak r

/B A

r


14/18

Problem: Kinematics of rigid body example The end A of rod AB of lengthL = 0.6 m moves at

velocity VA = 2 m/s and acceleration, aA = 0.2 m/s2,both to the left, at the instant shown, when q =

60. What is the velocity, VB ,and acceleration, aB ,

of end B at the same instant?

Pure kinematics problem: assuming clockwiserotation about A (may not be correct)

/

2 2/ /

222

cos , sin / ( sin ) 3.85/s, 1.16 m/s

0 sin cos cossin cos sin sin

B A B A B A

B A A B

B A B A B A B A

A A ABx A

By B

V V L

V L V L V L V

a a L L

a a Va a L L

L L L

a a

w w

w q w q w q

w w

w q w q q

q q q q

v v k r

a a k r r

22 2

cos sin 1 cos 11.7 m/ssin

A

A

V

L L a Lq w q q q

V , aA, A

A

BL

V ?Ba ?B

q

x

y

aA

aB

q q

L

w2

L

rB/A

VA

VB

qwL

rB/A

q

q

q


15/18

Kinetics of a system of particles (or rigid body)

For a system of particles (or a rigid body), analysis is performed interms of the mass center, G, located at radial vector, rG, and total

mass m Equations of motions:

where aG is the acceleration of the mass center, and HG is theangular momentum about the mass center

For a system with no external forces or moment acting, then linearmomentum, L, and angular momentum, H, is conserved, i.e., remainsconstant

May be more convenient to deal with effective forces and effectivemoments

For a system of particles (or a rigid body), and

where the mass moment of inertiaIis defined by(Standard formulae forI= mk2, where kis the radius of gyration, forstandard bodies are listed in tables; be careful about which axis I isdefined, whether centroidal axis or not, remember parallel axistheorem)

orG i i Gm m m dm r r r r= andG G Gm F a L M H

G G GI I H 2 2ori iI r m I r dm

G GIH


16/18


17/18

Problem: two-particle system A particle A of mass m and and a particle B, of

mass 2m are connected by rigid massless rod oflengthR. If mass B is suddenly given a verticalvelocity v perpendicular to the connecting rod,determine the location of the mass center, thevelocity of the mass center, the angularmomentum, and the angular velocity of the

system soon after the motion begins.

/ /

2 2/ /

23 2

3

2 23 2

3 3

2 1 20 23 3 3

A B G A A B B G A B G A B A

G A B G A A B B G A B G B

G A G A A B G B B

G G G A A G B B G

m m m m m m

m m m m m m v

m m m R m R v mvR

I I m r m r w

r r r r r r r r r r

L u u u u u u u u j

H r u r u k k k

H k

2 2

2

02

0

2 1 22

3 3 3

2 2

3 3

m R m R mR

v

mv R mR R

w w

w w

k k

k k

v

x

y j

rA rB

r rB A-

A B

G

m 2m

(2 /3)v j

A BG

m 2mw


18/18

Problem: rigid-body kinetics

What is the angular acceleration, , of the60-kg (cylindrical) pulley of radiusR = 0.2 mand the tension in the cable if a 30-kg block isattached to the end of the cable?

Analysis of block

Kinematic constraint (ablock=R)

OR

m=30-kg

mpulley=60-kg

( )y y yF ma T W ma m R

T m g R

Analysis ofpulley 2 20 0 0 pulley pulleywhere / 2 / 2M I I m R TR m R

2pulley pulley

2 1Solve for and : 147 N, 24.5

1 2 / s

mg TT T

m m m R

R

T T

W=mg

a Ry=

y

+

dynamics review 2013

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