ANALYTICAL MECHANICS Introduction and Basic …A reatiseT on the Analytical Dynamics of Particles...

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Transcript of ANALYTICAL MECHANICS Introduction and Basic …A reatiseT on the Analytical Dynamics of Particles...

ANALYTICAL MECHANICS

Introduction and Basic Concepts

Paweª FRITZKOWSKI, Ph.D. Eng.

Division of Technical MechanicsInstitute of Applied Mechanics

Faculty of Mechanical Engineering and Management

POZNAN UNIVERSITY OF TECHNOLOGY

Agenda

1 Introduction to the Course

2 Degrees of Freedom and Constraints

3 Generalized Quantities

4 Problems

5 Summary

6 Bibliography

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1. Introduction to the Course

Introduction to the Course

Analytical mechanics � What is it all about?

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Introduction to the Course

Analytical mechanics � What is it all about?

Analytical mechanics...

is a branch of classical mechanics

results from a reformulation of the classical Galileo'sand Newton's concepts

is an approach di�erent from the vector Newtonian mechanics:more advanced, sophisticated and mathematically-oriented

eliminates the need to analyze forces on isolated parts ofmechanical systems

is a more global way of thinking: allows one to treat a systemas a whole

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Introduction to the Course

Analytical mechanics � What is it all about? (cont.)

provides more powerful and easier ways to derive equations ofmotion, even for complex mechanical systems

is based on some scalar functions which describe an entire system

is a common tool for creating mathematical models for numericalsimulations

has spread far beyond the pure mechanics and in�uenced variousareas of physics

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Introduction to the Course

Analytical mechanics � What is it all about?

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Introduction to the Course

Analytical mechanics � Key contributors

Joseph Louis Lagrange (1736-1813),mathematician and astronomer, born in Italy,worked mainly in France

Sir William Rowan Hamilton (1805-1865),Irish mathematician, physicist and astronomer

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Introduction to the Course

Lecture contents

1 Introduction and Basic Concepts

2 Static Equilibrium

3 Lagrangian Dynamics

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Introduction to the Course

Objectives of the course

To enrich your knowledge on mechanics with some elements ofanalytical mechanics and vibration theory for discrete systems

To shape your skills in mathematical modelling and analyticaldescription of equilibrium and motion of mechanical systems

To develop your ability to analyze motion of mechanical systems

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Introduction to the Course

Related literature

1 Thornton S.T., Marion J.B., Classical Dynamics of Particles and

Systems. Brooks/Cole, 2004.

2 Török J.S., Analytical Mechanics with an Introduction to Dynamical

Systems. Wiley, 2000.

3 Hand L.N., Finch J.D., Analytical Mechanics. Cambridge UniversityPress, 1998.

4 Goldstein H., Poole Ch., Safko J., Classical Mechanics.Addison-Wesley, 2001.

5 Whittaker E.T., A Treatise on the Analytical Dynamics of Particles

and Rigid Bodies. Cambridge University Press, 1917.

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2. Degrees of Freedom and Constraints

Degrees of Freedom and Constraints

Mechanical system and its con�guration

Ci � the center of mass of the ith bodynb � the number of members (bodies) of the system

The radius vector of a body:

ri = ri(t) ,

where i = 1, 2, . . . , nb

In three dimensions:

xi = xi(t)yi = yi(t)zi = zi(t)

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Degrees of Freedom and Constraints

Degrees of freedom

Degrees of freedom (DOF) of a body � all the possible elementarymovements (translations or rotations) of the body

Degrees of freedom of a system � all the possible elementarymovements of all members within the system

Number of degrees of freedom (s) of a system � is equal tothe minimal number of variables that can completely describe thecon�guration of the system

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Degrees of Freedom and Constraints

Degrees of freedom � Free particle or rigid body

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Degrees of Freedom and Constraints

Constraints

Constraints � restrictions on motion of the system;conditions that limit motion of the system and reducethe number of degrees of freedom

Constraints result from:

• interconnections between various components of the system• attachments between the components and surroundings of the system

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Degrees of Freedom and Constraints

Constraints

Examples:

motion of a particle along a speci�ed curve

motion of a particle on a speci�edplane/surface

particles/rigid bodies connected bya rigid/�exible rod or a rope/cable

a rigid body attached to a wall/ground bya simple/roller support

two rigid bodies connected by a joint(pin/hinge)

a particle/rigid body impacting againsta wall/ground

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Degrees of Freedom and Constraints

Constraints

Valence (w) of a constraint � the number of degrees of freedom ofthe body/system reduced by the particular constraint type (support,connector, etc.)

Example

simple support: w = 2

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Degrees of Freedom and Constraints

Valence of typical supports in plane problems

Freely sliding guide: w = 1 Rod, cable, rope: w = 1

Pin connection, joint: w = 2 Simple support: w = 2

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Degrees of Freedom and Constraints

Number of degrees of freedom

Assumption:A mechanical system...

is composed of nb parts: np particles and nr rigid bodies (nb = np + nr)

is subjected to nc constraints with valences wk (k = 1, 2, . . . , nc)

The number of degrees of freedom of the system is given by

s = 2np + 3nr −nc

∑k=1

wk (2D problems)

s = 3np + 6nr −nc

∑k=1

wk (3D problems)

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Degrees of Freedom and Constraints

Constraints

Mathematically constraints can be expressed as functionalrelationships between certain coordinates and/or velocities of thebodies, e.g.

gj(t, r1, r2, . . . , rnb) = 0 , j = 1, 2, . . . , nc

or

gj(t, r1, r2, . . . , rnb , r1, r2, . . . , rnb) = 0 , j = 1, 2, . . . , nc ,

where nc � the number of constraints

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Degrees of Freedom and Constraints

Classi�cation of constraints [Thornton & Marion, 2004]

Constraints...

1 geometric or kinematic

2 bilateral or unilateral

3 scleronomic or rheonomic

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Degrees of Freedom and Constraints

Classi�cation of constraints [Thornton & Marion, 2004]

geometric constraint � the equation connects only coordinates:

gj(t, r1, r2, . . . , rnb) = 0

kinematic constraint � the equation connects both coordinates andvelocities:

gj(t, r1, r2, . . . , rnb , r1, r2, . . . , rnb) = 0

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Degrees of Freedom and Constraints

Classi�cation of constraints [Thornton & Marion, 2004] (cont.)

bilateral constraint � expressed by an equation:

gj(r1, r2, . . .) = 0

unilateral constraint � expressed by an inequality:

gj(t, r1, r2, . . .) > 0 or gj(t, r1, r2, . . .) ≥ 0

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Degrees of Freedom and Constraints

Classi�cation of constraints [Thornton & Marion, 2004] (cont.)

scleronomic (�xed) constraint � the equation does not contain timeexplicitly:

gj(r1, r2, . . .) = 0

rheonomic (moving) constraint � the equation involves time explicitly:

gj(t, r1, r2, . . .) = 0

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3. Generalized Quantities

Generalized Quantities

Generalized coordinates [Hand & Finch, 1998]

Generalized coordinates � a set of variables

q1, q2, . . . , qn

that completely specify the con�guration of a system

Two cases are possible:

• if n = s, the generalized coordinates are mutually independent

(the so called proper set of generalized coordinates)• if n > s, then only s generalized coordinates are mutually independent

(q1, q2, . . . , qs), while the rest (qs+1, qs+2, . . . , qn) are dependent onthe former ones

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Generalized Quantities

Generalized coordinates [Hand & Finch, 1998] (cont.)

Generalized coordinates qj can be of di�erent nature:they can describe translational or rotational motion

Transformation equations � relationships between the Cartesiancoordinates and the generalized coordinates:

ri = ri(t, q1, . . . , qn)

or

xi = xi(t, q1, . . . , qn)

yi = yi(t, q1, . . . , qn)

zi = zi(t, q1, . . . , qn)

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Generalized Quantities

Generalized velocities

Generalized velocities � time derivatives of the generalizedcoordinates:

q1, q2, . . . , qn

Transformation equations for velocities:

ri = ri(t, q1, . . . , qn, q1, . . . , qn)

or

xi = xi(t, q1, . . . , qn, q1, . . . , qn)

yi = yi(t, q1, . . . , qn, q1, . . . , qn)

zi = zi(t, q1, . . . , qn, q1, . . . , qn)

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Generalized Quantities

Generalized forces

Assumption:A set of external forces F1, F2, . . . , Fnb acts on a mechanical system, where Fi isa force applied to the ith member:

Fi = [Fxi, Fyi, Fzi]

Generalized force Qj associated with qj:

Qj =nb

∑i=1

Fi∂ri

∂qj=

nb

∑i=1

(Fxi

∂xi

∂qj+ Fyi

∂yi

∂qj+ Fzi

∂zi

∂qj

)

The nature of the quantity Qj is strictly related to the character of qj:"displacement � force", "angle � moment"

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Generalized Quantities

Generalized quantities

Quantity Symbol Translation Rotation

Coordinate qj x [m] ϕ [rad]

Velocity qj v = x [m/s] ω = ϕ [rad/s]

Acceleration qj a = x [m/s2] ε = ϕ [rad/s2]

Force Qj F [N] M [Nm]

Momentum pj p [kg m/s] k [kg m2/s]

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4. Problems

Problems

Determine the number of degrees of freedom of the given systems:

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Problems

Derive expressions for the generalized forces acting on the given systems:

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5. Summary

Summary

Conclusions and �nal remarks

Analytical mechanics...

can be regarded as a non-vector formulation of classical mechanics

involves scalar generalized quantities of di�erent nature which aretreated equally, based on the same rules

eliminates the need to take into account the internal forces(constraints forces)

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Bibliography

Hand L.N., Finch J.D., Analytical Mechanics. Cambridge University Press,1998.

Thornton S.T., Marion J.B., Classical Dynamics of Particles and Systems.Brooks/Cole, 2004.

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