 |
|
|||
|
Accident reconstruction research
Research / Linear Momentum
 Text: Vector Mechanics for Engineers, (Statics and) Dynamics
Fifth Ed., F. P. Beer and E. R. Johnston, Jr., McGraw-Hill, 1989.
Concepts and Key WordsIntroduction Classical mechanics encompasses (a) mechanics of particles and rigid bodies: statics and dynamics, and (b) mechanics of deformable bodies. Statics deals with the action of forces on bodies at rest. Dynamics deals with the motion of bodies under the action of unbalanced forces. Some basic concepts and definitions. Space. Dimension of space: one, two, and three. 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 (Système 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). Kinematics of Particles 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. Rectilinear Motion Particle moving along a directed straight line. Position coordinate of particle. Also called: directed distance from an origin, or rectilinear displacement from an origin. Average velocity during an interval of time. Instantaneous velocity. Differential displacement equals instantaneous velocity times differential change in time. Average acceleration during an interval of time. Instantaneous acceleration. Differential change in instantaneous velocity equals instantaneous acceleration times differential change in time. Drop "instantaneous" qualifier. Case when velocity is a function of position coordinate. Differentiate with respect to time by chain rule. Acceleration times differential displacement equals velocity times differential change in velocity. Lecture 2 Four common cases in rectilinear motion: (a) acceleration is constant, (b) acceleration is a function of time, (c) acceleration is a function of velocity, (d) acceleration is a function of position coordinate. Formation of definite integrals with differential relations that incorporate initial and final conditions for an interval of elapsed time. Rectilinear motion of several particles. Dependent motions arising from particles being connected by inextensible cables. Relative position coordinate, relative velocity, relative acceleration. Angular motion of a line in a plane. Angular displacement (angular position coordinate, directed angle), angular velocity, angular acceleration. Analogy with rectilinear motion. Vectors 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. Unit vector. Zero vector. Lecture 3 Rectangular vector components. Rectangular base vectors. Rectangular scalar components. The dot (scalar, inner) product. Properties of dot product. The cross (vector) product. Properties of the cross product. Continued products: scalar triple product, vector triple product. Curvilinear Motion of a Particle. 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. Derivative of velocity is acceleration of particle. The velocity vector, the acceleration vector, and the local center of curvature of path lie in the osculating plane. Unit tangent vector. Velocity and acceleration in plane curvilinear motion are resolved three ways: in rectangular components, in normal and tangential components, and in radial and transverse components. Choice depends on problem. Rectangular components. Projectile motion example. Lecture 4 Motion relative to translating reference axes. Relative motion analysis. Absolute velocity (acceleration) of a particle B equals the absolute velocity (acceleration) of origin of translating frame, say A, plus vectorially the relative velocity (acceleration) of B with respect to A, that is, the translating frame. Normal and tangential components. Curvilinear motion of a particle in a plane. Tangential acceleration. Normal acceleration. Tangential unit vector, points in direction of increasing directed distance along curve. Normal unit vector, points towards local center of curvature. Radial and transverse components. Curvilinear motion of a particle in a plane. Polar coordinates. Radial velocity and acceleration. Transverse velocity and acceleration. Radial unit vector points in increasing direction of radius vector. Transverse unit vector points in direction of increasing polar angle. Kinetics of Particles Kinetics relates unbalanced forces with changes in motion. Basic relation is Newton's second law: sum of forces equals mass times acceleration. Holds in inertial frame. Application of second law gives equation of motion. Equation of motion establishes equivalence between a free-body diagram or FBD, that identifies all the forces acting on a particle, and kinetic diagram or KD, which shows graphically the mass times the acceleration. The FBD and KD representations and equivalence are among the most powerful problem solving devices in dynamics. Usual types of problems: (a) acceleration known and the value of an unknown force is desired, (b) resultant force known and resulting motion is wanted. Lecture 5 Example: Deceleration of a roller coaster car. Acceleration of two stacked blocks down an incline. 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. The work-energy equation (principle of work and energy). Example: the work-energy equation applied to a two-particle system connected by a massless, inextensible cable. The work done by the "internal" forces of action and reaction cancel when summed. The work done by the resultant force on a particle during a finite movement depends, in general, on the path followed. Lecture 6 Potential 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 a potential function or potential energy. In this case, force is minus the gradient of the potential function. Work done by such a force during a finite movement of its point of application is the negative change in value of the potential function 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 a potential function, the force is said to be conservative. Examples of conservative forces: gravitational force, linearly-elastic spring force. The work-energy equation extended in terms of: work done by nonconservative "work producing" forces, kinetic energy, and potential energy. Example: the extended work-energy equation applied to a two-particle system connected by a massless, inextensible cable. Sum of kinetic and potential energies called total mechanical energy. Conservation of (total mechanical) energy when nonconservative "work producing" forces are not present. Impulse and 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. Graphical representation: initial momentum plus impulse equals final momentum. Example: impulsive motion, impact and ricochet of a bullet off a steel plate. Lecture 7 Moment of linear momentum or angular momentum about a fixed point O of a particle. Sum of moments, about fixed point O, of forces acting on a particle equals time rate of change of angular momentum of particle about point O. Calculate the cumulative effect of integrating the sum of moments of forces about point O with respect to time. Integral of moment of forces with respect to time is angular impulse. Integration gives: angular impulse equals corresponding change in angular momentum. Interpretation in rectangular components. Conservation of linear momentum, when resultant force vanishes, and conservation of angular momentum, when sum of moments of forces about fixed point O vanishes, during an interval of time. Possibility that only a component is conserved. Example: conservation of angular momentum in central force motion. Conservation of total linear and angular momemtum for two particles that interact during an interval of time in such a way that forces of interaction satisfy Newton's third law. Example: suitcase thrown on stationary baggage carrier. Application of conservation principle to impact. 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 elastic. If particles stick together after impact, velocity of separation is zero, and impact called plastic. 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. Lecture 8 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. Good approach is to draw momentum-impulse diagrams, draw unknown velocity vector components in assumed directions, and use magnitude symbols for the vectors in the governing equations (as in Statics). Kinetics of Systems 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. 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 G. Then, sum of all external forces equals total mass times absolute acceleration of center of mass. This is generalized 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. If the internal forces are associated with massless, inextensible connections, then work done by all these forces cancel. Internal and external conservative forces can be separated from the work term as changes in elastic and gravitational energies. Thus, work done by nonconservative forces equals change in kinetic plus change in gravitational energy plus change in elastic potential energy. Linear and angular 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. 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 G. 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 G. Conservation of energy and momentum If only conservative forces (and possibly nonconservative "workless" forces) act on a system of particles during an interval of motion, then total energy is conserved. This is the law of conservation of energy. If the resultant external force is zero during an interval of time, then total linear momentum is conserved. This is the principle of conservation of linear momentum. If the resultant moment of all external forces about a fixed point O (or G) is zero during an interval of time, then angular momentum about O (or G) is conserved. This is the principle of conservation of angular momentum. Kinematics of Rigid Bodies A rigid body is a system of particles for which the distances between the particles remain unchanged. First look at motions of rigid bodies that take place in a single plane, called plane motion. Plane motion types: rectilinear translation, curvilinear translation, fixed-axis rotation, general plane motion. Lecture 9 All lines in a rigid body in plane motion have the same angular displacement, angular velocity, and angular acceleration. Angular rotation. The rotation of a body about a given axis can be characterized by the direction of the rotation axis and by the sense and magnitude of the rotation angle. One could expect that finite (large) angular rotations are vectors. But, they are not because two successive rotations do not give, in general, the same body orientation when the sequence of rotations is reversed. Infinitesimal rotations are vectors, as will be shown. Angular velocity vector Suppose a disk spins about an axis normal to the disk. Define the angular velocity vector consistent with the positive sense of the rotation angle of the disk and the right-hand rule. Let the position vector of a particle on the rim of the disk originate from a point that is on the axis of rotation of the disk. This position vector has constant length. 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 angular velocity vector and the position vector. 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. Absolute and relative velocities in plane motion Absolute velocities of two particles A and B in a common body in plane motion are related as follows: v_A = v_B + v_B/A. Velocity v_B/A is the velocity of B as seen by an observer in a frame translating with v_A. Lecture 10 v_B/A equals w x r_B/A where w is the angular velocity of the body and r_B/A is the relative position vector of B with respect to A. Plane motion velocity field can be resolved at any instant into a translation with velocity v_A and fixed axis rotation about A with angular velocity w. Example: velocity field in a wheel rolling without slipping. Velocity of particle in wheel at contact point with ground is zero. Example: four-bar linkage. Instantaneous center of rotation (I.C.R.). Point on or off body that has zero absolute velocity. Location is the intersection of lines constructed perpendicular to the lines of actions of velocities of two separate particles, at the particle locations, whose velocities are not parallel. Absolute and relative accelerations in plane motion Absolute accelerations of two particles A and B in a common body in plane motion are related as follows: a_A = a_B + a_B/A. Acceleration a_B/A is the acceleration of B as seen by an observer in a frame translating with a_A. The term a_B/A equals w x (w x r_B/A) + dw/dt x r_B/A, where w is the angular velocity of the body, r_B/A is the relative position vector of B with respect to A, and dw/dt is the angular acceleration of the body. Plane motion acceleration field can be resolved at any instant into a translation with acceleration a_A, and fixed axis rotation about A with angular velocity w and angular acceleration dw/dt. Example: four-bar linkage. The derivative of a vector referenced to a rotating frame. Assume the scalar components of a vector referenced to rotating frame are functions of time. Then, time derivative of vector as seen by an observer in an inertial frame equals the cross product of the angular velocity of the rotating frame and the vector plus the time derivative of the vector as seen by an observer in the moving frame. Lecture 11 Plane motion of a particle relative to a rotating frame Suppose a particle P is in curvilinear motion relative to axes x-y, with origin A, which have angular velociy W and angular velocity dW/dt with respect to inertial axes X-Y. Then, the absolute velocity v_P equals the absolute velocity v_A plus W x r plus [dr/dt]_xy where r is the relative position vector of P with respect to A, referenced to x-y, and [dr/dt]_xy is the velocity of P measured in the x-y frame. The absolute acceleration a_P equals the absolute acceleration a_A plus dW/dt x r plus W x (W x r) plus 2W x [dr/dt]_xy plus [d^2r/dt^2], where [d^2r/dt^2]r is the acceleration of P measured in x-y. Example: Four-bar linkage containing a slider. Solution by semi-graphical and semi-analytical method using diagrams representing vector equations and trigonometry. Plane Kinetics of Rigid Bodies Kinetics establishes the relationships between external forces and translational and rotational motions of a rigid body. Need at least two translational coordinates and an angular coordinate to define uniquely position of rigid body in plane motion. Usually use position coordinates of G as translational coordinates. The governing equations are: sum of external forces equals total mass,m, times acceleration of center of mass, a^bar, and sum of moments of external forces about G equals time rate of change of angular momentum about G. We develop further the moment equation. Angular momemtum about G equals mass moment of inertia about an axis through G and perpendicular to plane of motion. Hence, sum of moments of external forces equals mass moment of inertia "about" G, I^bar, times absolute angular acceleration "about" G, dw/dt. Schematic representation of equations of motion: the left-hand side of the equations of motion is represented by the free-body diagram (FBD), the right-hand side is represented by the kinetic-diagram (KD) which shows a "force" ma^bar and a "couple" I^bar*dw/dt. The FBD and KD representations and equivalence are among the most powerful problem solving devices in dynamics. Note, once the FBD and KD have been established, any moment center may be used. This fact helps to minimize the number of unknowns in the moment equation. Lecture 12 Mass moment of inertia of a body about an axis. Mass moments of inertia of a thin uniform plate about rectangular axes centered in plate, with two axes parallel to plate face. Relation of these mass moments of inertia to area moments of inertia of plate face. Examples: thin uniform rectangular plate, thin uniform disk, both with centroidal axes. Radius of gyration. Parallel axis theorem: transfer of mass moment of inertia about a centroidal axis to a parallel noncentroidal axis. Computation of mass moment of inertia by integration using volume element of lowest order. Plane motion cases: translation, fixed-axis rotation, and general plane motion. Translation: sum of moments of forces about any point on line of action of acceleration of G is zero. Example: triangular plate in circular translation guided by parallel massless rods. Fixed-axis rotation: center of percussion is point about which sum of forces vanishes. Lecture 13 Example: compound pendulum consisting of uniform rod, hinged at top end, with an attached uniform cylinder at bottom end, subject to a tranverse force. Location of center of percussion. General plane motion. Example: spool lifted vertically through cable while spool rolls without slipping. Minimum coefficient of static friction required to maintain rolling without slipping. Variations of this problem when coefficients of static friction and kinetic friction are given. When it is not known if spool rolls without slipping, assume that it does, and check if friction force required to sustain rolling without slipping is less than or equal to maximum static friction force that can develop. Example: model for determining impact velocity of a driver's head against instrument panel when car experiences sudden stop. Lecture 14 Plane motion of rigid bodies. Work and energy methods. Work done on body by all external forces equals the change in the total kinetic energy of the body. If some external forces are conservative, the work done by these forces can be expressed as minus the change in potential energy. If only conservative forces, and other workless forces act on body, then the total mechanical energy is conserved. Example: uniform cylinder rolling without slipping down an incline. Kinetic energy for specific body motions. Translation: kinetic energy equals one half of total mass times velocity squared. Fixed-axis rotation: kinetic energy equals one half of mass moment of inertia about rotation axis times angular velocity squared. Example: ventilator door swinging shut under action of gravity and spring restraint. General plane motion: kinetic energy equals one half of total mass times velocity of center of mass squared plus one half of mass moment of inertia about center of mass times angular velocity squared. Work-energy equation can be extended to systems of connected bodies. Must account for work done by frictional forces and couples in connections if these forces and couples are active. Work done by a couple acting on a rigid body. Work done equals the product of the magnitude of the couple times the angular displacement of body. Result is positive if directions of couple and angular displacement are the same. Lecture 15 Plane motion of rigid bodies. Impulse and momentum methods. Linear momentum of body is total mass times velocity of center of mass. Resultant of external forces equals time rate of change of linear momentum. Integration of this equation with respect to time gives the linear impulse-change in linear momentum equation. Recall that sum of external forces about a fixed point, or center of mass, equals time rate of change of angular momentum about same point. Integration of this equation with respect to time gives the angular impulse-change in angular momentum equation. Specialized to plane motion: angular impulse equals mass moment of inertia times change in angular velocity. This relation applies to either about a fixed point or about the center of mass. Schematic representation of equations that state intial momenta plus impulses of external forces equals final momenta. Applications: Bullet impacting obliquely and passing through a vertically hanging rod. A block overturning about one corner and impacting plastically the ground at other corner. Kinetics of Rigid Bodies in Three Dimensions Kinematics. Translation. Every particle in body has same velocity and acceleration. True for rectilinear or curvilinear translation. Rotation about a fixed axis. Every particle in body travels on a circular path. Rotation about a fixed point. Vector equations for velocity and acceleration of particle same in form as those for fixed axis rotation except now angular velocity and angular acceleration of body are no longer necessarily collinear. Relations between the absolute and relative angular velocities and angular accelerations of a body rotating with respect to a rotating frame. Absolute angular velocity of body equals absolute angular velocity of frame plus relative angular velocity of body with respect to rotating frame. Absolute angular acceleration of body equals absolute angular acceleration of frame plus cross product of angular velocity of frame and relative angular velocity of body plus relative angular acceleration of body with respect to rotating frame. Lecture 16 Angular momentum. The angular momentum of a rigid body about a fixed point, or the center of mass, can be expressed by body integrals that are identical in form. Evaluation of the angular momentum body integral gives an angular momentum vector whose scalar components are the elements of a column vector that equals the product of a symmetric inertia matrix times a column matrix formed by the angular velocity scalar components. The diagonal elements of the inertia matrix are the mass moments of inertia about the rectangular axes while the off-diagonal elements are the products of inertia with respect to the rectangular axes. Rotational inertia properties. The mass moment of inertia about an arbitrary axis can be expressed as a quadratic form in the direction cosines of the axis, with coefficients that are the moments and products of inertia. Plotting a series of points that are each a distance one over the square root of the moment of inertia from the origin along the axis generates the ellipsoid of inertia. Inasmuch, every ellipsoid has at least one set of three orthogonal axes of symmetry, we can always orient a set of rectangular axes in the body so that the axes are aligned with the axes of symmetry of the ellipsoid of inertia. Relative to these axes, the inertia matrix has zero off-diagonal elements. The diagonal elements are the principal moments of inertia. The corresponding axes are the principal axes of inertia. For many bodies, the principal axes are evident from the symmetry of the bodies. Relative to the principal axes, the angular momentum scalar components are simply the products of the mass moment of inertia and angular velocity component about and along the same axis. Assuming that the reference axes are attached to the body, differentiation of the angular momentum vector with respect to time leads to Euler's equations. These relate the sums of the moments of the external forces about the rectangular axes to the angular acceleration scalar components of the body, products of the angular velocity scalar components, and the principal moments of inertia. Application: rotating shaft with a tilted disk at the shaft center. Lecture 17 Rotating shaft carrying tilted disk continued. The shaft is simply supported on bearings. Only one bearing takes axial thrust. Bearing reactions due to causes other than gravity, that is, the reactions caused by the rotation of the body alone, are called dynamic reactions. Rotation of a body about a fixed axis. Moment equation specialized to axes fixed in body with one axis being axis of rotation. Two products of inertia appear in rotational equation of motion, if tranvserse axes are not principal axes. Gyroscopic motion - steady precession - motion of a tilted top under action of gravity. Absolute angular velocity of top's symmetry axis in inertial axes called rate of precession. Relative angular velocity of top with respect to precessing axes called rate of spin. Simplified case when spin rate is much bigger than precession rate, and angular momentum vector is entirely due to spin. Moment equation of motion says precession rate is independent of tilt angle, and inversely proportional to rate of spin. Exact approach that includes precession rate in angular momentum. Moment equation of motion says two precession rates are possible for one spin rate. Approximation for large spin rate: fast precession rate (independent of g), slow precession rate (same as that given by simplified approach). Slow precession rate is the one usually observed. Demonstrations: the "rattleback;" the top; the "top" that inverts, or rights, itself; the "pillbox" puzzle that requires insertion of two beads into diametrically opposite pockets. THE END
|
|||
|
ARC NETWORK QUICK LINKS
|
||||
