Of all possible time histories of displacement states that satisfy the compatibility equations and the constraints or the kinematic boundary conditions and that also satisfy the conditions at initial and final times t1 and t2, the history corresponding to the actual solution makes the lagrangian functional a minimum. Structural dynamics of linear elastic singledegreeof. Investigation of the benefits and effectiveness of hamilton s weak principle for multibody sytems analysis. Of all the possible paths along which a dynamical system may more from one point to. Close this message to accept cookies or find out how to manage your cookie settings. The total amplitude for a particle to go from one point in spacetime i. Hamiltons principle, axial and transverse vibration of beams, rayleighs method 3. The focus of this chapter is on continuum elastodynamics, and the related finite element developments are described via hamiltons principle as well as the hamiltons law of varying action. Fema 451b topic 3 notes slide 2 instructional material complementing fema 451, design examples sdof dynamics 3 2 structural dynamics equations of motion for sdof structures structural frequency and period of vibration behavior under dynamic load dynamic magnification and resonance effect of damping on behavior linear elastic response spectra. Civil engineers, mechanical engineers, aircraft engineers, ocean engineers, and engineering students encounter these problems every day, and it is up to them systematically to grasp the basic concepts, calculation principles and calculation methods of structural dynamics. We derived the eulerlagrange equation for a single variable, u, but we will now shift our attention to a system n particles of mass mi each. We will introduce the basic concepts using the singledegreeoffreedom springmass system. Advanced structural dynamics by eduardo kausel august 2017.
Bleich 1952 when all timerelated terms are dropped. Structural dynamics, and materials conference june 2012. Hamiltons principle isaac newton wasnt the only physicist to think about mechanics william r. Here, the equation of motion is obtained using hamiltons variational principle. The course aims fundamentals at imparting of vibration and structural dynamics to senior undergraduate and graduate students. Application of newtons laws and lagranges equations to lumpedparameter and continuous models. This same principle can be used to explain the law of reflection.
Introduction to dynamics of structures 7 washington university in st. Mathematical models for multipledegreeoffreedom mdof systems a. Ross is a registered professional engineer with the state of florida and has thirtyeight years of teaching and. Skip to main content accessibility help we use cookies to distinguish you from other users and to provide you with a better experience on our websites. Hamiltons the ory of r a y optics was a v ariational the ory. Lagranges equations, conservative and nonconservative forces, hamiltons principle. Hamiltons principle and hamiltons law of varying action. Substituting equations 18, 19 and 20 into dalemberts equation 12, rearranging the order of the summations, factoring out the common. Hamiltons principle an overview sciencedirect topics.
However, it has a critical weakness, termed endpoint. Newtonian mechanics and analytical mechanics referred to as variational principles in mechanics or energy methods in mechanics the principle of virtual displacements, hamiltons principle, and. Hamiltons principle hamilton, 1834, 1835 may be a theoretical base for dynamical systems by its nature of integral form in time with lagrangian density to account for continuous space. Hamiltons principle and noethers theorem michael fowler introduction. Hamiltons principle in analytical dynamics is certainly among the great intellectual. The course primarily consists of four modules, namely, single. Numerical methods for structural analysis in a previous work by the authors 45, it was shown how the vast amount of existing numerical methods can be grouped into three main sets according to the kind of physical phenomena they model and the type of differential equations they discretize. Of all possible paths between two points along which a dynamical system may move from one point to another within a given time interval from t0 to t1, the actual path followed by the system is the one which minimizes the line integral of. Allen ross is emeritus professor of the department of aerospace engineering, mechanics and engineering science at the university of florida, and is a faculty member at the graduate engineering research center, shalimar, florida. Hamiltons principle is stated in terms of the action s, which is a scalar quantity that shares all the invariances of the lagrangian l, and which is independent of any particular choice of generalised coordinates. Pdf hamiltons principle for the derivation of equations. For full treatment, see mechanics dynamics can be subdivided into kinematics, which describes motion, without regard to its causes, in terms of. Lecture notes on classical mechanics a work in progress. The study of dynamics may be subdivided into two main categories.
If a structure such as the one shown in the diagram below is in static equilibrium then from the principle of virtual work we can deduce the static equilibrium equation by computing. However, we can show that hamiltons principle implies that the trajectory which minimizes the action is the one that also. Hamilton set forth the law of varying action in papers concerning a general method in dynamics, published in 1834 and 1835. Across many disciplines of engineering, dynamic problems of structures are a primary concern. Structural dynamics of earthquake engineering sciencedirect. Extended framework of hamiltons principle for continuum. Energy methods were in use to solve statics problems before discovering them for dynamics. It offers a methodical manner of deriving equations motion for many systems, with the additional benefit that appropriate and correct boundary conditions are automatically. Structural dynamics introduction this chapter provides an elementary introduction to timedependent problems. It states that the dynamics of a physical system is determined by a variational problem for a functional based on a single function, the lagrangian, which contains all physical information concerning the system and the forces acting on it. Galileo and newton in the discussion of calculus of variations, we anticipated some basic dynamics, using the potential energy mgh for an element of the catenary, and conservation of energy. In physics, hamilton s principle is william rowan hamilton s formulation of the principle of stationary action. Lecture notes on classical mechanics a work in progress daniel arovas department of physics university of california, san diego may 8, 20. Mechanical vibrations 1741 hamiltons principle for conservative systems and lagrange equations hamiltons principle.
Numerical methods chapter 6 advanced structural dynamics. Generalized coordinates, lagranges equations, and constraints. The minimization approach to physics was formalized in detail by hamilton, and resulted in hamiltons principle which states. Gavin where q jt are generalized forces, collocated with the generalized coordinates, q jt.
Chapter 2 lagranges and hamiltons equations in this chapter, we consider two reformulations of newtonian mechanics, the lagrangian and the hamiltonian formalism. The rst is naturally associated with con guration space, extended by time, while the latter is. The eulerbernoulli beam model detailed derivations for the eulerbernoulli model can be found in text books bybenaroya 21, inman 22,meirovitch2325, rao 26 and thomson27. Investigation of the benefits and effectiveness of hamiltons weak principle for multibody sytems analysis. Hamiltons principle, which formed a subsection in the 1st edition, containing the basic equations, is discussed more briefly under the subheading, variational ap proach, in the 2nd edition. Mlf and great advantages to account for structural dynamics. However, while both editions contain a later chapter on variational formulations of the. Hamiltons principle is one of the great achievements of analytical mechanics. The fourier transform of a signal xt is defined by 36 and is related to the fourier transform of the derivatives of this function by 37 38.
Multibody system analysis based on hamiltons weak principle. We will include discussion of the stress analysis of the onedimensional bar, beam, truss, and plane frame. Hamiltons principle, hamiltons law 6 to the n power. Dynamics, branch of physical science and subdivision of mechanics that is concerned with the motion of material objects in relation to the physical factors that affect them. Abstractthree direct integration schemes for the matrix equations of motion of structural dynamicsthe newmark generalized acceleration operator, the wilson averaging variant of the linear acceleration operator and an averaging method based on a variational principle derived by gurtinare investigated for stability and approximation viscosity. It w as based on t he pr inc iple, due to f e rmat, that a ligh t ra y tra v e ling bet w een t w o point s will follow. Hamiltons principle, hamiltons law 6 to the n power correct formulations. Hamilton, in 1834 and 1835, came up with his own rule for how things move he begin by noting that, for a particle moving in a single dimension, the motion can be represented by a path in spacetime. Hamiltons principle is the variational principle for dynamical systems, and it has been widely used in mathematical physics and engineering. Example of multiple possible world lines of particle passing from event ato eventb.