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Introduction and Basic Concepts ANALYTICAL MECHANICS Introduction and Basic Concepts Paweª FRITZKOWSKI, Ph.D. Eng. Division of Technical Mechanics Institute 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 Paweª Fritzkowski Introduction: Basic Concepts 2 / 37 1. Introduction to the Course Introduction to the Course Analytical mechanics 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 classical mechanics 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 forces 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 motion, 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 physics 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 William Rowan Hamilton (1805-1865), Irish mathematician, physicist 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 Dynamics 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 vibration 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 mass 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 rigid body 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 sliding 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 velocities 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 time 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 force 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 "displacement force", "angle moment" Paweª Fritzkowski Introduction: Basic Concepts 30 / 37 Generalized Quantities Generalized quantities Quantity Symbol Translation Rotation Coordinate [m] [rad] qj x j Velocity
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