sign in sign up
<<
Home , Analytical dynamics

ANALYTICAL Introduction and Basic Concepts

Paweª FRITZKOWSKI, Ph.D. Eng.

Division of Technical Mechanics Institute of Faculty of 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

Paweª Fritzkowski Introduction: Basic Concepts 2 / 37 1. Introduction to the Course Introduction to the Course  What is it all about?

Paweª Fritzkowski Introduction: Basic Concepts 4 / 37 Introduction to the Course Analytical mechanics  What is it all about?

Analytical mechanics... is a branch of results from a reformulation of the classical Galileo's and Newton's concepts is an approach dierent from the vector Newtonian mechanics: more advanced, sophisticated and mathematically-oriented eliminates the need to analyze on isolated parts of mechanical systems is a more global way of thinking: allows one to treat a system as a whole

Paweª Fritzkowski Introduction: Basic Concepts 5 / 37 Introduction to the Course Analytical mechanics  What is it all about? (cont.)

provides more powerful and easier ways to derive equations of , 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 numerical simulations has spread far beyond the pure mechanics and inuenced various areas of

Paweª Fritzkowski Introduction: Basic Concepts 6 / 37 Introduction to the Course Analytical mechanics  What is it all about?

Paweª Fritzkowski Introduction: Basic Concepts 7 / 37 Introduction to the Course Analytical mechanics  Key contributors

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

Sir (1805-1865), Irish mathematician, and astronomer

Paweª Fritzkowski Introduction: Basic Concepts 8 / 37 Introduction to the Course Lecture contents

1 Introduction and Basic Concepts

2 Static Equilibrium

3 Lagrangian

Paweª Fritzkowski Introduction: Basic Concepts 9 / 37 Introduction to the Course Objectives of the course

To enrich your knowledge on mechanics with some elements of analytical mechanics and theory for discrete systems To shape your skills in mathematical modelling and analytical description of equilibrium and motion of mechanical systems To develop your ability to analyze motion of mechanical systems

Paweª Fritzkowski Introduction: Basic Concepts 10 / 37 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 University Press, 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.

Paweª Fritzkowski Introduction: Basic Concepts 11 / 37 2. Degrees of Freedom and Constraints Degrees of Freedom and Constraints Mechanical system and its conguration

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)  the center of of the th body Ci i  the number of members (bodies) of the system nb

Paweª Fritzkowski Introduction: Basic Concepts 13 / 37 Degrees of Freedom and Constraints Degrees of freedom

Degrees of freedom (DOF) of a body  all the possible elementary movements (translations or rotations) of the body

Degrees of freedom of a system  all the possible elementary movements of all members within the system

Number of degrees of freedom (s) of a system  is equal to the minimal number of variables that can completely describe the conguration of the system

Paweª Fritzkowski Introduction: Basic Concepts 14 / 37 Degrees of Freedom and Constraints Degrees of freedom  Free particle or

Paweª Fritzkowski Introduction: Basic Concepts 15 / 37 Degrees of Freedom and Constraints Constraints

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

Constraints result from: • interconnections between various components of the system • attachments between the components and surroundings of the system

Paweª Fritzkowski Introduction: Basic Concepts 16 / 37 Degrees of Freedom and Constraints Constraints

Examples: motion of a particle along a specied curve motion of a particle on a specied plane/surface particles/rigid bodies connected by a rigid/exible rod or a rope/cable a rigid body attached to a wall/ground by a simple/roller support two rigid bodies connected by a joint (pin/hinge) a particle/rigid body impacting against a wall/ground

Paweª Fritzkowski Introduction: Basic Concepts 17 / 37 Degrees of Freedom and Constraints Constraints

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

Example simple support: w = 2

Paweª Fritzkowski Introduction: Basic Concepts 18 / 37 Degrees of Freedom and Constraints Valence of typical supports in plane problems

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

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

Paweª Fritzkowski Introduction: Basic Concepts 19 / 37 Degrees of Freedom and Constraints Number of degrees of freedom

Assumption: A mechanical system... is composed of parts: particles and rigid bodies ( ) nb np nr nb = np + nr is subjected to constraints with valences ( ) nc wk k = 1, 2, . . . , nc

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

nc (2D problems) s = 2np + 3nr − ∑ wk k=1

nc (3D problems) s = 3np + 6nr − ∑ wk k=1

Paweª Fritzkowski Introduction: Basic Concepts 20 / 37 Degrees of Freedom and Constraints Constraints

Mathematically constraints can be expressed as functional relationships between certain coordinates and/or of the bodies, e.g.

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

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

where nc  the number of constraints

Paweª Fritzkowski Introduction: Basic Concepts 21 / 37 Degrees of Freedom and Constraints

Classication of constraints [Thornton & Marion, 2004]

Constraints...

1 geometric or kinematic

2 bilateral or unilateral

3 scleronomic or rheonomic

Paweª Fritzkowski Introduction: Basic Concepts 22 / 37 Degrees of Freedom and Constraints

Classication 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 and velocities:

gj(t, r1, r2,..., rnb , r˙1, r˙2,..., r˙nb ) = 0

Paweª Fritzkowski Introduction: Basic Concepts 23 / 37 Degrees of Freedom and Constraints

Classication of constraints [Thornton & Marion, 2004] (cont.)

bilateral constraint  expressed by an equation:

gj(r1, r2,...) = 0 unilateral constraint  expressed by an inequality:

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

Paweª Fritzkowski Introduction: Basic Concepts 24 / 37 Degrees of Freedom and Constraints

Classication of constraints [Thornton & Marion, 2004] (cont.)

scleronomic (xed) constraint  the equation does not contain explicitly:

gj(r1, r2,...) = 0 rheonomic (moving) constraint  the equation involves time explicitly:

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

Paweª Fritzkowski Introduction: Basic Concepts 25 / 37 3. Generalized Quantities Generalized Quantities

Generalized coordinates [Hand & Finch, 1998]

Generalized coordinates  a set of variables

q1, q2,..., qn that completely specify the conguration 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 ( ), while the rest ( ) are dependent on q1, q2,..., qs qs+1, qs+2,..., qn the former ones

Paweª Fritzkowski Introduction: Basic Concepts 27 / 37 Generalized Quantities

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

Generalized coordinates can be of dierent nature: qj they can describe translational or rotational motion

Transformation equations  relationships between the Cartesian coordinates 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)

Paweª Fritzkowski Introduction: Basic Concepts 28 / 37 Generalized Quantities Generalized velocities

Generalized velocities  time derivatives of the generalized coordinates:

q˙ 1, q˙ 2,..., q˙ n

Transformation equations for velocities:

r˙i = r˙i(t, q1,..., qn, q˙ 1,..., q˙ n) or

x˙ i = x˙ i(t, q1,..., qn, q˙ 1,..., q˙ n)

y˙ i = y˙ i(t, q1,..., qn, q˙ 1,..., q˙ n)

z˙ i = z˙ i(t, q1,..., qn, q˙ 1,..., q˙ n)

Paweª Fritzkowski Introduction: Basic Concepts 29 / 37 Generalized Quantities Generalized forces

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

Fi = [Fxi, Fyi, Fzi]

Generalized force associated with : Qj qj

nb nb   ∂ri ∂xi ∂yi ∂zi Qj = ∑ Fi = ∑ Fxi + Fyi + Fzi i=1 ∂qj i=1 ∂qj ∂qj ∂qj

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

Paweª Fritzkowski Introduction: Basic Concepts 30 / 37 Generalized Quantities Generalized quantities

Quantity Symbol Translation Rotation

Coordinate [m] [rad] qj x ϕ [m/s] [rad/s] q˙ j v = x˙ ω = ϕ˙ [m/s2] [rad/s2] q¨j a = x¨ ε = ϕ¨ Force [N] [Nm] Qj F M [kg m/s] [kg m2/s] pj p k

Paweª Fritzkowski Introduction: Basic Concepts 31 / 37 4. Problems Problems Determine the number of degrees of freedom of the given systems:

Paweª Fritzkowski Introduction: Basic Concepts 33 / 37 Problems

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

Paweª Fritzkowski Introduction: Basic Concepts 34 / 37 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 dierent nature which are treated equally, based on the same rules eliminates the need to take into account the internal forces (constraints forces)

Paweª Fritzkowski Introduction: Basic Concepts 36 / 37 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.

Paweª Fritzkowski Introduction: Basic Concepts 37 / 37