|ME 501A |
Call No. 12994
S. F. Felszeghy
|ADVANCED MECHANICS OF PARTICLES|
|1||Jan. 6||1-0 thru 1-1||1-3, 1-4|
|Jan. 8||1-2 thru 1-5||1-7, 1-8|
|2||Jan. 13||2-0 thru 2-5||2-1, 2-7|
|Jan. 15||2-6 thru 2-9||2-9, 2-13|
|3||Jan. 20||2-10 and 2-11||2-14, 2-17|
|Jan. 22||3-0 thru 3-2||3-1, 3-4|
|4||Jan. 27||3-3 thru 3-5||3-13, 3-19|
|Jan. 29||3-6, 3-9, 3-10||3-20, 3-24|
|Feb. 5||4-0 to 4-3||4-2, 4-3|
|6||Feb. 10||4-3 to 4-5||4-13, 4-14|
|Feb. 12||4-5 thru 4-7||4-16, 4-24|
|Feb. 19||6-0 to 6-4|
|8||Feb. 24||6-4 thru 6-5||6-1, 6-2|
|9||Mar. 2||Hamilton's Principle||6-3|
|10||Mar. 9||6-6 to 6-7||6-9, 6-12|
|Mar. 11||6-7 thru 6-8||6-13, 6-16|
Notebook containing solutions to problems is available in CSULA Library. Call No. is 133.
Concepts and Key Words
Classical mechanics deals with the response of physical bodies to the action of applied forces. By response we mean stresses in a body and the deformation and motion of a body.
Principal architect of classical mechanics was Sir Isaac Newton. Stated laws of motion for a particle and law of gravitation. Invented calculus. Contemporary of Newton, Leibniz, also invented calculus and originated analytical mechanics which is founded on the scalar quantities of kinetic energy and the work function. Subsequent contributors to analytical mechanics were Euler, Lagrange, and Hamilton.
Classical mechanics applies to relatively massive and slowly moving physical bodies compared to atomic sized particles and the speed of light. Exceptional cases fall within the realm of the theory of relativity (Einstein), quantum mechanics (Heisenberg, Schrö, Born), and relativistic quantum mechanics (Dirac).
Classical mechanics encompasses (a) mechanics of particles and rigid bodies: statics and dynamics, and (b) mechanics of deformable bodies: solids (strength of materials, elasticity, plasticity, viscoelasticity, etc.), and fluids (ideal, viscous, compressible).
Some basic concepts and definitions. Space - one dimensional, two dimensional, and three dimensional. Reference frame - inertial frame. Displacements, velocities, and accelerations measured in an inertial frame are called absolute. Time. Mass (inertial and gravitational - they are the same). Force (vector quantity). Particle. Rigid body.
Newton's laws of motion. First: A particle remains at rest or continues to move with a uniform velocity in a straight line if there is no unbalanced force acting on it. Second: The acceleration of a particle is proportional to the resultant force acting on it and is in the direction of this force. Third: The forces of action and reaction between interacting bodies are equal in magnitude, opposite in direction, and collinear.
Law of gravitation. Weight (W). Gravitational acceleration at surface of earth: g = 9.81 m/s^2 or 32.2 ft/sec^2.
Units. SI Units (Systè International). Fundamental quantities and units: length (m), time (s), mass (kg). Force (N) derived. USCU (U. S. Customary Units). Fundamental quantities and units: length (ft), time (sec), force (lb). Mass (slug) derived. W = mg in both systems (m is mass).
Quantities that can be represented by directed line segments and which combine according to the parallelogram law of addition (or triangle rule of addition) are called vectors, or geometric vectors to be precise.
Vector notation: underlined letters when handwritten. Vector algebra. Magnitude or length. Equality of two vectors. Sum or resultant of two vectors. Properties of vector addition. Multiplication of a vector by a scalar. Properties of this multiplication. Zero vector. Unit vector.
Base vectors. Resolution of a vector into vector components in the direction of three noncoplanar unit vectors. Expression of vector as sum of products of unit vectors (base vectors) and scalar components. Rectangular Cartesian coordinates and associated unit vectors. Rectangular scalar components of a vector. The dot product. Properties of dot product. The cross product. Properties of cross product. Continued products. Scalar triple product. Vector triple product.
Kinematics of a Particle
Kinematics deals with motion apart from considerations of force and mass. Serves as an introduction to kinetics which relates unbalanced forces with changes in motion.
Motion of a Particle in a Fixed Coordinate System
Position vector of a particle that is a function of time as particle moves along a curved path. Derivative of position vector with respect to time is the velocity of the particle. Velocity is tangent to path. Directed distance measured along path from a reference point to particle. Directed distance is a position coordinate and function of time. Derivative of directed distance is scalar component of velocity. Acceleration of particle. Differentiation of velocity produces a sum of two vectors, one that is tangent to path and one that is normal to path. These two components lie in the osculating plane that contains the local center of curvature of path. Radius of curvature. Unit tangent vector and principal normal. Tangential acceleration. Normal or centripetal acceleration.
Let the position vector of a particle moving along a circular path originate from a point that is on an axis perpendicular to the circular path center. This position vector has constant length. Define a vector that is the product of the derivative of the directed angle that the particle's radial position coordinate makes with a fixed radial axis, and a unit vector along the perpendicular axis, consistent with the right-hand rule. The derivative of the position vector with respect to time, that is, the velocity of the particle, can be expressed as a cross product of the earlier defined vector and the position vector. Call the defined vector angular velocity. Have to show that angular velocity behaves as a vector.
Examine total displacement in time dt of a position vector of constant length that rotates simultaneously with two noncollinear angular velocities. The commutativity of vector addition allows the reversal of the order of addition of the two displacements caused by each angular velocity. This leads to the conclusion that angular velocity, and also infinitesimal angular displacements, add as vectors.
Consider a position vector of constant length of a particle that rotates with an angular velocity having a fixed direction: velocity of particle can be expressed as cross product of angular velocity vector and position vector. Differentiation of velocity gives acceleration. Angular acceleration vector. Centripetal acceleration. Tangential acceleration. Generalize to case when direction of angular velocity changes with time. In this case direction of angular velocity represents instantaneous axis of rotation; angular velocity and angular acceleration are no longer necessarily parallel.
Derivative of a vector referenced to a rotating frame. Part I. Case when vector is constant (in length and direction) relative to a frame that rotates with angular velocity relative to an inertial frame. By analogy with rotating position vector above, derivative of such a vector as measured by an observer in the inertial frame is the cross product of the angular velocity vector and the given vector.
Motion of a Particle in Several Coordinate Systems.
Rectangular Cartesian coordinates: x, y, z. Unit vectors i, j, k. Position vector, velocity, and acceleration expressed in these unit vectors.
Cylindrical coordinates: r, , z (polar radius vector, polar angle, directed distance from polar plane). Unit vectors tangent to coordinate lines: base vectors. Angular velocity of unit vectors. Position vector, velocity, and acceleration expressed in unit vectors.
Spherical coordinates: r, , (radius vector, colatitude, longitude). Unit vectors tangent to coordinate lines: base vectors. Angular velocity of unit vectors. Position vector, velocity, and acceleration expressed in unit vectors.
Motion of a Particle Attached to a Moving Rigid Body
Rigid body has angular velocity and angular acceleration. Absolute velocity of a particle P equals absolute velocity of another particle A plus relative velocity of P with respect to a nonrotating frame with origin at A. Absolute acceleration of particle P equals absolute acceleration of other particle A plus relative acceleration of P with respect to nonrotating frame with origin at A. Plane motion. Example: velocity field in a wheel rolling without slipping.
Instantaneous center of rotation in plane motion of a rigid body.
Derivative of a vector referenced to a rotating frame. Part II. Case when vector varies with time relative to a frame that rotates with angular velocity relative to an inertial frame. The derivative of such a vector as measured by an observer in the inertial frame is the cross product of the angular velocity vector and the given vector plus vectorially the derivative of the vector as measured by an observer in the rotating frame.
Motion of a particle in a moving frame. Moving frame has angular velocity and angular acceleration. Absolute velocity of a particle P equals absolute velocity of moving frame origin O' plus relative velocity of P' with respect to a nonrotating frame with origin at O', where P' is a point coincident with P but fixed in the moving frame, plus relative velocity of P with respect to the moving frame. Absolute acceleration of particle P equals absolute acceleration of origin O' plus relative acceleration of P' with respect to a nonrotating frame with origin at O', where P' is a point coincident with P but fixed in the moving frame, plus relative acceleration of P with respect to the moving frame plus Coriolis acceleration. Examples: Acceleration of a tack in a banked bicycle tire rolling without slipping. Example: Acceleration of and force on a particle at the end of an oscillating windshield wiper on a turning car.
Dynamics of a Particle
In principle, if resultant force is known as function of time, particle position vector, or particle velocity, the particle position vector can be found as a function of time by integrating the acceleration. Special cases: (a) resultant force is constant, (b) resultant force is a function of time, (c) resultant force is a function of particle velocity, (d) resultant force is a function of particle position. Example of case (a): Projectile motion. Equation of trajectory. Maximum horizontal range. Time of flight. Maximum slant range. Example of case (c): Projectile motion when particle is subject to a damping force in addition to the gravitational force. Particle position as function of time. Example of case (a): Path of particle that is projected on a moving belt that exerts a constant friction force on particle as long as particle slides on belt.
Work and energy
Calculate the cumulative effect of forming a line integral with resultant force along path of particle from an initial position to a final position. Work done by resultant force during a differential displacement along path equals dot product of resultant force and differential displacement. Work equals tangential component of force times differential directed distance along path. Work done by resultant force during a finite movement of particle along path. Units of work: J (joule), ft.lb.
Substitution of Newton's second law for resultant force in line integral gives change in kinetic energy between initial and final positions. Definition of kinetic energy. Units of kinetic energy. Principle of work and kinetic energy.
Case when the work done by a force during a differential displacement of its point of application is minus an exact differential of a scalar function of position, called potential energy. In this case, force is minus the gradient of the potential energy. Work done by such a force during a finite movement of its point of application is the negative change in value of the potential energy between the initial and final end points. The work done does not depend on the path followed between the initial and final end points. The work is independent of the path. When a force is the gradient of potential energy, the force is said to be conservative. Examples of conservative forces: gravitational force, linearly-elastic spring force. Sum of kinetic and potential energies called total mechanical energy. Principle of conservation of mechanical energy when nonconservative "work producing" forces are not present. Conservative system
Example: simple pendulum. Motion for small angular displacements. Motion for large angular displacements. Period expressed in trems of complete integral of the first kind.
Power: Rate of work done by a force equals the dot product of force and velocity of point of application. Rate of work done by a couple equals dot product of couple and angular velocity of body.
Linear impulse and linear momentum
Calculate the cumulative effect of integrating the resultant force acting on a particle with respect to time. Linear momentum is the product of mass and velocity. Integral of force with respect to time is linear impulse. Integration of Newton's second law gives: linear impulse equals corresponding change in linear momentum. This is the principle of linear impulse and momentum. Conservation of linear momentum when resultant force vanishes during an interval of time. Possibility that only a component is conserved.
Angular impulse and angular momentum
Moment of linear momentum or angular momentum about a fixed point O of a particle. Moment, about fixed point O, of resultant force acting on a particle equals time rate of change of angular momentum of particle about point O. Calculate the cumulative effect of integrating the moment of resultant force about point O with respect to time. Integral of moment of a force with respect to time is angular impulse. Integration gives: angular impulse equals corresponding change in angular momentum. Conservation of angular momentum when moment of resultant force about fixed point O vanishes during an interval of time. Possibility that only a component is conserved.
Example problems that apply conservation principles: particle that is launched on the inside smooth surface of an inverted right circular cone; a particle that slides down inside a vertical helical tube at constant speed; application of Stokes' theorem to proving that a force field is conservative.
Dynamics of a System of Particles
Extend principles learned from motion of a single particle to motion of a general system of particles. System of particles can be an arbitrary collection of particles. Only restriction is that particles are bounded in space by a real or imaginary closed surface or envelope. No particles enter or leave this expandable surface. System of particles can represent a rigid or nonrigid body.
Equations of motion
Consider n particles. Typical particle m_i is subjected to resultant force F_i from sources external to the envelope, and forces f_ij from sources internal to the envelope, where f_ij is the force exerted on m_i by m_j. Write Newton's second law for each particle and sum equations for all particles. The sums involving the internal forces vanish if internal forces satisfy Newton's third law. Introduce concept of center of mass, labelled c.m. or C. Then, sum of all external forces equals total mass times absolute acceleration of center of mass. This is Newton's second law of motion for a mass system.
Work and energy
Write work-energy equation for each particle m_i and then sum all equations. Result: work done by all forces on all particles equals change in the total kinetic energy. The total kinetic energy can be written as the kinetic energy of the total mass moving with the velocity of the center of mass plus the kinetic energy of the particles due to motion relative to the center of mass.
Internal and external conservative forces can be separated from the work term as changes in potential energies. If all external and internal forces are conservative, and/or "workless" constraint forces, then total energy is conserved. Example: Vertically stacked masses, in gravitational field, separated by a spring. Minimum downward deflection of top mass required to cause top mass to pull bottom mass away from floor.
Linear impulse and momentum
Define total linear momentum as sum of all linear momenta of particles. Then, sum of external forces equals time rate of change of total linear momentum. Example: force exerted on floor by an impacting, free-falling vertical rope. Integral of resultant of all external forces with respect to time equals to the corresponding change in total linear momentum.
Angular impulse and momentum
Compute total angular momentum about a fixed point O. Then, time rate of change of angular momentum about fixed point O equals moment of all external forces about point O, assuming internal forces satisfy Newton's third law. Same result applies if moment center is center of mass O. Result: sum of moments of external forces about fixed point O equals time rate of change of angular momentum about point O. Same applies for C.
Integration with respect to time of moment-rate of change of angular momentum equation gives angular impulse-change in angular momentum equation, known as the principle of angular impulse and momentum. If the angular impulse, or a component, vanishes, then angular momentum, or the corresponding component, is conserved. Example: motion of two particles connected by a taught massless string when one particle is subjected to a linear impulse perpendicular to the string. Example: the steady precession of a spinning gyro ring, supported by a massless shaft, under action of gravity.
Impact refers to collision between bodies. Contact forces are large and act for a very short time. Two particles (smooth spheres) that collide in direct central impact. Construct t-n axes: t-axis is tangent at contact point; n-axis coincides with line of impact. Velocities and sphere centers on line of impact. Conservation of total linear momentum in n-direction. Conservation of kinetic energy gives another equation. Can show that for this case, velocity of separation equals velocity of approach. Impact called perfectly elastic. If particles stick together after impact, velocity of separation is zero, and impact called completely plastic or inelastic. All other impact cases fall between elastic and plastic cases. Define coefficient of restitution e, which can be shown to equal ratio of velocity of separation to velocity of approach. For elastic impact, e = 1; for plastic impact, e = 0. For all other cases, 0 <e <1. Oblique central impact. Construct t-n axes as in direct central impact. The only difference now is that velocities are not necessarily parallel to line of impact (n-axis). The hypothesis made in such cases is velocity components along line of impact (n-axis) follow same rules as in direct central impact. The velocity components in the t-direction are unchanged.
Examples: Two spherical masses, connected to separate dumbbells, collide inelastically. Two spherical masses, connected to separate dumbbells, collide elastically. Two spherical masses, connected to separate dumbbells, collide with e = 0.5. The solution of these problems is facilitated if diagrams are drawn of the dumbbells representing the relation: initial momenta + impulses = final momenta.
Free particle, in three dimensional space. Subject to resultant force. Motion governed by Newton's second law.
Constrained particle. Confined to move on a smooth surface (constraint) (expressed as a function of spatial coordinates equal to zero) while a given (applied) force acts on particle. Possible (differential) displacement in time dt governed by dot product of gradient of constraint and possible displacement. Reaction force due to surface called force of constraint. Force of constraint is parallel to constraint gradient. Hence, work done by constraint force during a possible displacement is zero. Note: forces are categorized now as "given" ( or "applied") forces and "constraint" forces. Compare this to the earlier "external" and "internal" categories.
Suppose particle is confined to move on a smooth surface which is a function of the spatial coordinates and time (constraint expressed as a function of spatial coordinates and time equal to zero). Possible (differential) displacement in time dt governed by dot product of four dimensional (space-time) gradient of constraint and vector consisting of possible displacement components augmented with time element dt. Force of constraint is parallel to spatial gradient of constraint. Hence, work done by constraint force during a possible displacement is not zero. This motivates the definition of another class of infinitesimal displacements called virtual displacements. These are the displacements for which the dot product between the displacement vector and the spatial gradient of the constraint vanishes. Hence, constraint force does zero work in a virtual displacement. Example: vertically rising elevator floor; virtual displacements lie in floor.
General first-order differential form of a constraint. If first-order differential form is not an exact differential, and no integrating factor exists, then constraint is not integrable. The constraint force is proportional to the vector formed by the coefficients of the spatial coordinate differentials in the first-order differential constraint equation. The proportionality is established with a function of time labelled "lambda." General equations that govern motion of constrained particle.
If a constraint in differential form is integrable, the constraint is said to be holonomic. If a holonomic constraint contains time as a variable, the constraint is said to be rheonomic. If it does not contain time, it is scleronomic. If the constraint in differential form is not integrable, the constraint is said to be nonholonomic. Inequality constraints are called nonholonomic.
The number of coordinates less the number of constraints is called the number of degrees of freedom. True also for systems of particles. Single particle subjected to two constraints in differential form. Contraint force is a linear combination of the vectors formed by the coefficients of the spatial coordinate differentials in the first-order differential constraint equations. Linear combination expressed hereafter with subscripted "lambda's." General equations that govern motion of constrained particle.
Systems of particles. Example: Two particles moving in common plane connected by a rod the length of which varies with time.
Consider N particles in three dimensional space. Number masses, rectangular position coordinates, and forces consecutively from 1 to 3N. Suppose there are l independent constraint equations in differential form. Then, the system has 3N - l degrees of freedom. The constraint force is a linear combination of the vectors formed by the coefficients of the spatial coordinate differentials in the first-order differential constraint equations.
Example: wedge, on horizontal frictionless surface, carrying sliding block on inclined top surface with friction. Equations governing motion formulated with constraint equations.
Types of constraint forces. (i) Particle on smooth surface; (ii) particle given prescribed motion as function of time; (iii) rigid body sliding on smooth surface; (iv) rigid body rolling without slipping on "perfectly rough" surface; (v) particles connected by a rod; rigid bodies connected by hinge; (vi) particle sliding on smooth surface of rigid body; (vii) smooth rigid bodies in contact; (viii) perfectly rough rigid bodies in contact. All constraint forces arising from these situations do no work in an arbitrary virtual displacement of system.
The fundamental equation. The work done during a virtual displacement by the difference between the particle masses times their accelerations and the given forces vanishes. The importance of this result lies in the absence of the constraint forces from this equation.
Virtual work . Work done by given (applied) forces in virtual displacements. Principle of virtual work: a system of particles is in equilibrium if and only if the virtual work of the given forces vanishes.
Conservative given forces. If the given forces are conservative, then the virtual work of the given forces in a virtual displacement can be expressed as the variation of a potential. The differential of a function of the position coordinates, velocities, and time is the dot product of the gradient of the function relative to these variables and the vector formed from the differentials of the variables. The variation of the function is the dot product of the gradient of the function relative to the position coordinates and velocities, and the vector formed by the virtual displacements and virtual velocities. The latter are time derivatives of the virtual displacements.
A digression from main topic. Hamilton's Principle.
Consider the motion of a system of particles over an interval of time. Hamilton's Principle states that the time integral over any inerval of time of the sum of the virtual kinetic energy change and the virtual work of the given forces vanishes when the virtual displacements are made from configurations of the actual motion, and when the initial and final configurations are specified. That Hamilton's principle is a necessary condition for the motion follows from integrating the variation of the kinetic energy over the interval of time. The variation of the kinetic energy is the change in the kinetic energy from its value at any instant during the actual motion to its value at the same instant along the varied path. The varied path is the sequence of configurations obtained by displacing a system at any instant from its actual configuration by the virtual displacements. The virtual displacements are assumed to be twice continuously differentiable in time. The virtual work done by the given forces is added to the integrand formed by the variation of the kinetic energy, and the fundamental equation is then applied to show the vanishing of the resulting integral. The sufficiency of Hamilton's principle for the motion can also be shown. If the given forces are conservative, then the work done by the given forces can be replaced by the variation of a potential. The kinetic energy minus the potential is defined as the Lagrangian function L. Define the variation of the definite integral formed with L as the integrand as the change in the intergral when the varied path is substituted for the actual path in L. Then, if the constraints are holonomic, the variation of the definite integral equals the integral of the variation. Hamilton's principle states that the variation of the integral of L vanishes. This defines a problem in the calculus of variations. In this language, the integral of L is stationary along the actual path relative to all other paths having the same endpoints and differing from the actual path by virtual displacements lying in the neighborhood of the actual path. Hamilton's principle can be used as the departure point for establishing governing equations of motion.
Generalized coordinates (Lagrangian coordinates). The idea is to find n "generalized" coordinates that can represent uniquely the configuration of N particles with 3N rectangular position coordinates. The least number of n "generalized' coordinates needed is 3N minus the number of independent holonomic constraints that relate the rectangular coordinates and possibly time. This is possible by associating with each holonomic constraint a single new coordinate that remains constant during the motion.
Lagrange's equations. Start with the fundamental equation in rectangular coordinates x_i and transform it into the generalized coordinates q_i. In this process introduce generalized forces Q_i. If there are no nonholonomic constraints, then Lagrange's equations follow from the fundamental equation: for each q_i coordinate, the time derivative of the partial derivative of the kinetic energy with respect to the generalized velocity minus the partial derivative of the kinetic energy with respect to the generalized coordinate equals the generalized force. If nonholonomic constraints exist, transform these into the q_i coordinates first, and form the relations the virtual generalized displacements must satisfy. In view of the existence of these relations, Lagrange's equations are modified on the right-hand side by the addition of the constraint forces expressed as linear combinations of the constraint coefficients multiplying the virtual generalized displacements in the constraint equations. This is expressed with Langrange multipliers usually written as "lambda_i." The complete formulation consists of the Lagrange equations together with the constraint equations relating the generalized velocities. Remarks: Lagrange'e equations may be used even if n is not the minimum number, i.e., when holonomic constraints remain in the problem formulation. If all, or some, of the given forces are conservative, these can be expressed as potentials, and then it is convenient to express Lagrange's equations in terms of the Lagrangian function, L = T - V, or kinetic energy minus the potential. Example: Spherical pendulum. Three rectangular position coordinates can be expressed as functions of two "generalized" spherical angular coordinates. In this transformation, the constant cord length (a holonomic constraint) is a new coordinate that remains constant during the motion. Example: Equations of motion for a free particle in a gravitational field using spherical coordinates. Example: Equations of motion for a spherical pendulum formulated with the constraint force associated with the holonomic constraint. Example: Motion of a particle in a small circular tube when the tube rotates about a vertical diametrical axis in a gravitational field. Formulation of the equations of motion with two constraint forces, associated with two holonomic constraints, included.
Principle of virtual work revisited. If, in the formulation of the virtual work expression, some constraints are released, then the constraint forces are usually represented in the virtual work expression by the Lagrange multiplier method, and the corresponding virtual displacements become independent. The coefficient of such an independent virtual displacement in the virtual work expression must vanish, and the coefficient becomes thereby a statement of the equilibrium condition.
Lagrange's equations continued. Example: Motion of a particle in a small straight tube when the tube rotates about a vertical axis in a gravitational field. The tube is tilted 30 degrees upward from the ground surface, and the particle is projected towards the axis of rotation from an outward radial initial position. Formulation of the equations of motion with a single generalized radial position coordinate. Formulation of fictitious kinetic energy and potential energy expressions (in rotating coordinates) the sum of which is conserved. Interpretation of motion with these terms.
Example: Equations of motion for a double pendulum. Small angle approximation (linearization of equations). Example: Motion of a three-wheeled symmetrical cart, with equilateral triangular frame, a caster at one corner, parallel wheels at other corners, and equal masses at corners. The cart center of mass is given an initial velocity that is transverse to parallel wheels.
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