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