lecture 3 vector mechanics for engineers: dynamics mecn 3010 department of mechanical engineering...

LectureLecture

33Vector Mechanics for Engineers: Vector Mechanics for Engineers:

DynamicsDynamics MECN 3010 MECN 3010

Department of Mechanical EngineeringDepartment of Mechanical Engineering

Inter American University of Puerto RicoInter American University of Puerto Rico

Bayamon CampusBayamon Campus

Dr. Omar E. Meza CastilloDr. Omar E. Meza [email protected]

http://www.bc.inter.edu/facultad/omeza

Lecture 3Lecture 3MEC

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Tentative Lecture ScheduleTentative Lecture Schedule

TopicTopic LectureLecture

Kinematics of a ParticleKinematics of a Particle 11

Kinetics of a Particle: Force and AccelerationKinetics of a Particle: Force and Acceleration

Kinetics of a Particle: Work and EnergyKinetics of a Particle: Work and Energy

Kinetics of a Particle: Impulse and MomentumKinetics of a Particle: Impulse and Momentum

Planar Kinematics of a Rigid BodyPlanar Kinematics of a Rigid Body


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Introduction and Basic ConceptsIntroduction and Basic Concepts

Topic 1: Kinematics of a Topic 1: Kinematics of a ParticleParticle

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"Lo peor es educar por métodos basados en el temor, la fuerza, la autoridad, porque se destruye la sinceridad y la confianza, y sólo se consigue una falsa sumisión”

Einstein AlbertEinstein Albert


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Chapter ObjectivesChapter Objectives

To introduce the concepts of position, To introduce the concepts of position, displacement, velocity, and acceleration.displacement, velocity, and acceleration.

To study particle motion along a straight To study particle motion along a straight line and represent this motion graphically.line and represent this motion graphically.

To investigate particle motion along a To investigate particle motion along a curve path using different coordinate curve path using different coordinate systems.systems.

To present an analysis of dependent To present an analysis of dependent motion of two particles.motion of two particles.

To examine the principles of relative To examine the principles of relative motion of two particles using translating motion of two particles using translating axes.axes.

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12.7 Curvilinear Motion12.7 Curvilinear Motion: Normal and Tangential Components: Normal and Tangential Components

When the path along which a particle travels is known, then it is When the path along which a particle travels is known, then it is often convenient to describe the motion using n and t coordinate often convenient to describe the motion using n and t coordinate axes which act normal and tangent to the pathaxes which act normal and tangent to the path

a. Planar Motion: a. Planar Motion: The t axis is The t axis is tangenttangent to the curvature, and n to the curvature, and n is perpendicular to the t axis.is perpendicular to the t axis.

ρρ(rho) is the (rho) is the radius of curvature radius of curvature and O’ is the and O’ is the center of center of curvature.curvature.

The plane which contains the n The plane which contains the n and t axes is referred as the and t axes is referred as the embracing or embracing or osculating planeosculating plane..

Radius of curvature

ds


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b. Velocity:b. Velocity: Since the particle Since the particle moves, s is a function of time. moves, s is a function of time. The particle’s velocity The particle’s velocity vv has a has a direction that is always tangent direction that is always tangent to the path, and a magnitude to the path, and a magnitude that is determined by taking the that is determined by taking the time derivative of the path time derivative of the path function s=s(t), i.e., v= ds/dt. function s=s(t), i.e., v= ds/dt. HenceHence

vv=v=vuutt

wherewhere

sv

12.7 Curvilinear Motion12.7 Curvilinear Motion: Normal and Tangential Components: Normal and Tangential Components


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c. Acceleration: c. Acceleration: The acceleration of The acceleration of the particle is the time rate of the particle is the time rate of change of the velocity. Thuschange of the velocity. Thus

wherewhere

wherewhere

12.7 Curvilinear Motion12.7 Curvilinear Motion: : Normal and Tangential ComponentsNormal and Tangential Components


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12.8 Curvilinear Motion12.8 Curvilinear Motion: : Cylindrical ComponentsCylindrical Components

a. Polar Coordinates:a. Polar Coordinates: If motion is If motion is restricted to the plane, then restricted to the plane, then polar coordinates is used.polar coordinates is used.

We can specify the location of We can specify the location of the particle shown in figure the particle shown in figure using a using a radial coordinate radial coordinate r(r(uurr), ), which extend outward from the which extend outward from the fixed origin fixed origin OO to the particle, to the particle, and a transverse coordinate and a transverse coordinate θθ((uuθθ), which is the ), which is the counterclockwise angle between counterclockwise angle between a fixed reference line and the r a fixed reference line and the r axis.axis.

b. Position:b. Position: At any instant the At any instant the position of the particle, is position of the particle, is defined by the position vectordefined by the position vector

rr = r = ruurr


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c. Velocity:c. Velocity: The instantaneous The instantaneous velocity v is obtained by taking velocity v is obtained by taking the time derivative of r. Using a the time derivative of r. Using a dot to represent the time dot to represent the time derivative, we havederivative, we have

wherewhere

wherewhere


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d. Acceleration:d. Acceleration: Taking the time Taking the time derivatives of the last velocity derivatives of the last velocity equations, we obtain the equations, we obtain the particle’s instantaneous particle’s instantaneous accelerationacceleration

wherewhere

wherewhere


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e. Cylindrical Coordinates:e. Cylindrical Coordinates: IF the IF the particles moves along a space particles moves along a space curve as shown in figure, then curve as shown in figure, then its location may be specified by its location may be specified by the three cylindrical coordinates the three cylindrical coordinates r,r,θθ,z.,z.

The position, velocity, and The position, velocity, and acceleration of the particle can acceleration of the particle can be written in terms of its be written in terms of its cylindrical coordinates as cylindrical coordinates as follows:follows:


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1212


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Due, Due, Tuesday, February 06, 2012Tuesday, February 06, 2012

Omar E. Meza Castillo Ph.D.Omar E. Meza Castillo Ph.D.

Homework2 Homework2 WebPage WebPage

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lecture 3 vector mechanics for engineers: dynamics mecn 3010 department of mechanical engineering...

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