mechanics of rigid body

Mechanics of Rigid BodyKinematics, Kinetics and Static

1.- Introduction

2.- Kinematics. Types of Rigid Body Motion:

Translation, RotationGeneral Plane Motion

3.- Kinetics. Forces and Accelerations. Energy and Momentum Methods.

Angular Momentum and Moment of Inertia Fundamental Equations of Dynamics

4.- Statics. Equilibrium.

Mechanics of Rigid Body

System of Particles. Internal and External forces

Rigid Body is defined as a particular system of particles which does not deform. Let A and B be any two of its particles; then, in a rigid body the distance between A y B will remain without changes. Limits of this assumption: elasticity and break

Introduction. Rigid Body as a particular system of particles

ABA

B rrr −=

CrAB AB == /


1.- Introduction: Forces acting on a rigid body Forces acting of rigid bodies can be also separated in two groups: (a) The external forces, represent the action of other bodies on the rigid body under consideration; (b) The internal forces are the forces which hold together the particles forming the rigid body. Only external forces can impart to the rigid body a motion of translation or rotation or bothTransmissibility principle: The effect of an external force on a rigid body remains unchanged if that force is moved along its line of action. We will need the math concept of moment of a force (torque) to describe its principle.

Connected Rigid Bodies : Mechanisms. i.e. slider-crank, gear box

Free Body Diagrams: Showing all the forces acting on the body Reactions at supports and connections


Keeping Constant Speed over a incline with angle α

Exercise: Free-Body Diagram on the rod (beam) embedded in the wall

Exercise: Free-Body Diagram on the front wheel and on the rear wheel (a) constant speed (b) accelerating (c) Acting brakes

Exercise: Free-Body Diagram on the sliding stair with friction in supporting points

Restaurant


Introduction. Mass Center (Center of Gravity) of a System of Particles: Concept. Static and Dynamic propertiesThe center of mass of a system of particles (Rigid Body is a particular case) is the point of the space where the system of gravitational forces formed by all elementary gravitational forces acting on each elementary particle (dm g), is equivalent to one force (mg) placed there. The potential energy of a system of particles is simply mgy, where y is the height of center of mass. This concept provides a method to find the center of mass of a body.

The center of mass moves like a particle of mass m = Σmi under the influence of the external forcesacting on the system,

∑==∑i

iiCMCMext vmvmoramF

To compute the point where is placed the center of mass

∑∑∑∫∑

===

==

iiiCM

iiiCM

iiiCM

CMi

iiCM

zmzmymymxmxm

dmrrmorrmrm

;;

Find the center of mass (with “math” and without “math”)

m 2m

Fig. 15.1 a 15.4Mechanics of Rigid Body. Kinematics

1.- Kinematics. Types of motion: TRANSLATION, ROTATION about a fixed axis , GENERAL PLANE MOTION, MOTION about a fixed pint, GENERAL MOTION

TRANSLATION. A motion is said to be a translation if any straight line inside the body keeps the same direction during the movement.

All the particles forming the body move along parallel paths. If these paths are straight lines, the motion is said a rectilinear translation; if the paths are curved lines, the motion is a curvilinear motion

ROTATION about a fixed axis. The particles forming the rigid body move in parallel planes along circles centered on the same fixed axis. If this axis, called the axis of rotation intersects the rigid body, the particles located on the axis have zero velocity and zero acceleration

Exercise: Distinguish between curvilinear translation and rotation about a fixed axis

Mechanics of Rigid Body. Kinematics

GENERAL PLANE MOTION.Any plane motion which is neither a translation or a rotation is referred as a general plane motion. Plan motion is that in which all the particles of the body move in parallel planes. Translation and rotation are plane motions.

MOTION about a fixed point. The three-dimensional motion of a rigid body attached at a fixed point, for example, the motion of a top on a rough floor, is known as motion about a fixed point.

GENERAL MOTION Any motion of a rigid body which does not fall in any of the cathegories above described.

Fig. 15.1 a 15.4Mechanics of Rigid Body. Kinematics

Exercise: Identify different types of motion

Mechanics of Rigid Body. Kinematics

TRANSLATION. Motion equations Fig. 15.1 Fig. 15.7 pag 918

Conclusion: A rigid body in translation can be considered as a particle

ABAB rrr −=/

will be constant in magnitude (rigid body) and in direction (translation motion), then

zeroisrofderivativethe AB /

AB vv =

AB aa =

When a rigid body is in translation all the points of the body have the same velocity and the same acceleration.

Mechanics of Rigid Body. Kinematics. Rotation about a fixed axis

Angular velocity, ωAngular acceleration,α

dtdθω = dt

dωα =

Representative slab

PR

v

ROTATION about a fixed axis. Motion equations. Velocity

Rv

Rdtd

dtds

Rddsradiusxanglearc

ω

θθ

=

=

==

Where angle is in radians!!!

Basic relationships curvilinear motion

Angular velocity and angular acceleration are invariants. They are the same for all points of the solid. They are a

characteristic of the rotating motion of the solid

Exercise: A compact disk rotating at 500 rev/min is scanned by a laser that begins at the inner radius of about 2.4 cm and moves out the edge at 6.0 cm. Which is the linear (tangential) velocity of the disk where the laser beam strikes: (a) at the beginning of scanning and (b) at the end?. The same for acceleration


ROTATION about a fixed axis. Motion equations. Acceleration

RRdtd

dtRd

dtdvaT αωω

====)(

dtdωα =

aT

aN

P

RRR

RvaN

222 )( ωω===

R

Representative slab


ROTATION about a fixed axis. The vector motion equations

ABB rv /∧=ω)( // ABABB rra ∧∧+∧= ωωα

Vector expressions for velocity and acceleration in rotation about a fixed axis

α


ROTATION about a fixed axis. Mechanism…Rigid Body connected

The red arrow shows the angular velocity of the horizontal gear 1. Draw the angular velocity for the other gear, 2 and 3. Solve the problem with quantitative values:

ω1 = 500 rev/min; R1 = 2 cm

ω2 = ? rev/min; R2 = 5 cm; R´2=10 cm

ω3 = ? rev/min; R4 = 10 cm;

The bucket falls from the rest with a constant linear acceleration of 0.3 g. (a) Estimate the speed of the bucket after 5 seconds and the fallen distance. (b) Compute the angular acceleration of the pulley © How fast will it rotate after 5 s.

Gear 1 rotatesclockwise at angular velocity of12 rad/s. How fastwill gear 2 and 3 rotate. Data: R1:5 cm; R2:10 cm; R3:20 cm.

1

32

12

3

Mechanics of Rigid Body. Kinematics. General Plane Motion

GENERAL PLANE MOTION. Any general plane motion can be considered as a translation plus a rotation Euler´s

Theorem


GENERAL PLANE MOTION. Any general plane motion can be considered as a translation plus a rotation

Angular velocity and angular acceleration of rod are independent of the selected point to rotate

Sliding rod


Rolling without slipping.

Rolling with slipping.An object slides and rolls

.C

RaRv

Rs

C

Cαωθ

===

Rolling without slipping.

As the wheel of radius R rotates throughangle θ, the point of the contact betweenthe wheel and the plane moves a distance s that is related with θ by s= θ R.

If there is no sliding, the distance traveledby point C is exactly the same s.

.C.

RaRv

Rs

C

Cαωθ

≠≠≠

Rolling with slipping.


A bicycle travels with a speed of 40 km/h. How fast the cycle rider pedals in rev/min?. Data: Sprocket radius: 2.5 cm; Front gear radius: 10 cm; rear wheel radius: 40 cm

Find the angular velocity of sliding stair of length 3 m, when the velocity of contact point with the soil is 3 m/s. The angles between the stair and the floor is 45º

The slider-crank mechanism converts the rotational motion of crank in linear motion of slider. Find the relationship between the angular velocity of crank and the linear velocity of slider piston

mechanics of rigid body

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