Applied Structural and Mechanical Vibrations Theory, Methods and Measuring Instrumentation 3 Analytical dynamics—an overview 3 1 Introduction In order to describe the motion of a physical system, it i.
3 Analytical dynamics—an overview 3.1 Introduction In order to describe the motion of a physical system, it is necessary to specify its position in space and time Strictly speaking, only relative motion is meaningful, because it is always implied that the description is made with respect to some observer or frame of reference In accordance with the knowledge of his time, Newton regarded the concepts of length and time interval as absolute, which is to say that these quantities are the same in all frames of reference Modern physics showed that Newton’s assumption is only an approximation but, nevertheless, an excellent one for most practical purposes In fact, Newtonian mechanics, vastly supported by experimental evidence, is the key to the explanation of the great majority of everyday facts involving force and motion If one introduces as a fundamental entity of mechanics the convenient concept of material particle—that is, a body whose position is completely defined by three Cartesian coordinates x, y, z and whose dimension can be neglected in the description of its motion—Newton’s second law reads (3.1) where F is the resultant (i.e the vector sum) of all the forces applied to the particle, is the particle acceleration and the quantity m characterizes the material particle and is called its mass Obviously, x is here the vector of components x, y, z Equation (3.1) must not be regarded as a simple identity, because it establishes a form of interaction between bodies and thereby describes a law of nature; this interaction is expressed in the form of a differential equation that includes only the second derivatives of the coordinates with respect to time However, eq (3.1) makes no sense if the frame of reference to which it is referred is not specified A difficulty then arises in stating the cause of acceleration: it may be either the interaction with other bodies or it may be due to some distinctive properties of the reference frame itself Taking a step further, we can consider a set of material particles and suppose that a frame of reference exists such that Copyright © 2003 Taylor & Francis Group LLC all accelerations of the particles are a result of their mutual interaction This can be verified if the forces satisfy Newton’s third law, that is they are equal in magnitude and opposite in sign for any given pair of particles Such a frame of reference is called inertial With respect to an inertial frame of reference a free particle moves uniformly in a straight line and every observer in uniform rectilinear motion with respect to an inertial frame of reference is an inertial observer himself 3.2 Systems of material particles Let us consider a system of N material particles and an inertial frame of reference Each particle is subjected to forces that can be classified either as: internal forces, due to the other particles of the system or external forces, due to causes that are external to the system itself We can write eq (3.1) for the kth particle as (3.2) where k=1, 2,…, N is the index of particle, and are the resultants of external and internal forces, respectively In addition, we can write the resultant of internal forces as (3.3) which is the vector sum of the forces due to all the other particles, Fkj being the force on the kth particle due to the jth particle Newton’s third law states that (3.4) hence (3.5) and eq (3.2), summing on the particle index k, leads to (3.6) These results are surely well known to the reader but, nevertheless, they are worth mentioning because they show the possibility of writing equations where internal forces not appear Copyright © 2003 Taylor & Francis Group LLC 3.2.1 Constrained systems Proceeding further in our discussion, we must account for the fact that, in many circumstances, the particles of a given system are not free to occupy any arbitrary position in space, the only limitation being their mutual influences In other words, we must consider constrained systems, where the positions and/or the velocities of the particles are connected by a certain number of relationships that limit their motion and express mathematically the equations of constraints A perfectly rigid body is the simplest example: the distance between any two points remains unchanged during the motion and 3N–6 equations (if and the points are not aligned) must be written to satisfy this condition In every case, a constraint implies the presence of a force which may be, a priori, undetermined both in magnitude and direction; these forces are called reaction forces and must be considered together with all other forces For the former, however, a precise law for their dependence on time, coordinates or velocities (of the point on which they act or of other points) is not given; when we want to determine the motion or the equilibrium of a given system, the information about them is supplied by the constraints equations Constraints, in turn, may be classified in many ways according to their characteristics and to the mathematical form of the equations expressing them, we give the following definitions: if the derivatives of the coordinates not appear in a constraint equation we speak of holonomic constraint (with the further subdivision in rheonomic and scleronomic), their general mathematical expression being of the type (3.7) where time t appears explicitly for a rheonomic constraint and does not appear for a scleronomic one In all other cases, the term nonholonomic is used For example, two points rigidly connected at a distance L must satisfy (3.8) A point moving in circle in the x–y plane must satisfy (3.9a) or, in parametric form, (3.9b) Copyright © 2003 Taylor & Francis Group LLC where the angle θ (the usual angle of polar coordinates) is the parameter and shows how this system has only one degree of freedom (the angle θ), a concept that will be defined soon Equations (3.8) and (3.9) are two typical examples of holonomic (scleronomic) constraints A point moving on a sphere whose radius increases linearly with time (R=at) is an example of rheonomic constraint, the constraint equation being now (3.10a) or, in parametric form (3.10b) where the usual angles for spherical coordinates have been used From the examples above, it can be seen that holonomic constraints reduce the number of independent coordinates necessary to describe the motion of a system to a number that defines the degrees of freedom of the system Thus, a perfectly rigid body of N material points can be described by only six independent coordinates; in fact, a total of 3N coordinates identify the N points in space but the constraints of rigidity are expressed by 3N–6 equations, leaving only six degrees of freedom Consider now a sphere of radius R that rolls without sliding on the x–y plane Let X and Y be the coordinates of the centre with respect to the fixed axes x and y and let ωx and ωy be the components of the sphere angular velocity along the same axes The constraint of rolling without sliding is expressed by the equations (the dot indicates the time derivative) (3.11a) which are the projections on the x and y axes of the vector equation where v and ω are the sphere linear and angular velocities, the symbol × indicates the vector product, C is the position vector of the point of contact of the sphere with the x–y plane and O is the position vector of the centre of the sphere Making use of the Euler angles φ, θ and (e.g Goldstein [1]), eqs (3.11a) become (3.11b) which express the nonholonomic constraint of zero velocity of the point C Copyright © 2003 Taylor & Francis Group LLC A disc rolling without sliding on a plane is another classical example— which can be found on most books of mechanics—of nonholonomic constraint These constraints not reduce the number of degrees of freedom but only limit the way in which a system can move in order to go from one given position to another In essence, they are constraints of ‘pure mobility’: they not restrict the possible configurations of the system, but how the system can reach them Obviously, a holonomic constraint of the kind (dropping the vector notation and using scalar quantities for simplicity) implies the relation on the time derivative but when we have a general relation of the kind which cannot be obtained by differentiation (i.e it is not an exact differential) the constraint is nonholonomic In the equation above, A and B are two functions of the variables x1 and x2 Incidentally, it may be interesting to note that the assumption of relativistic mechanics stating that the velocity of light in vacuum is an upper limit for the velocities of physical bodies is, as a matter of fact, a good example of nonholonomic constraint Thus, in the presence of constraints: The coordinates xk are no longer independent (being connected by the constraint equations) The reaction forces appear as unknowns of the problem; they can only be determined a posteriori, that is, they are part of the solution itself This ‘indetermination’ is somehow the predictable result of the fact that we omit a microscopical description of the molecular interactions involved in the problem and we make up for this lack of knowledge with information on the behaviour of constraints—the reaction forces— on a macroscopic scale So, unless we are specifically interested in the determination of reaction forces, it is evident the interest in writing, if possible, a set of equations where the reaction forces not appear For every particle of our system we must now write (3.12) Copyright © 2003 Taylor & Francis Group LLC where F is the resultant of active (internal and external) forces, Φ is the resultant of reactive (internal and external) forces and the incomplete knowledge on Φ is supplied by the equation(s) of constraint The nature of holonomic constraints itself allows us to tackle point (1) by introducing a set of generalized independent coordinates; in addition, we are led to equations where reaction forces disappear if we restrict our interest to reactive forces that, under certain circumstances of motion, no work These are the subjects of the next section 3.3 Generalized coordinates, virtual work and d’Alembert principles: Lagrange’s equations If m holonomic constraints exist between the 3N coordinates of a system of N material particles, the number of degrees of freedom is reduced to n=3N–m It is then possible to describe the system by means of n configuration parameters usually called generalized coordinates, which are related to the Cartesian coordinates by a transformation of the form (3.13) and time t does not appear explicitly if the constraints are not time dependent The advantage lies obviously in the possibility of choosing a convenient set of generalized coordinates for the particular problem at hand From eq (3.13) we note that the velocity of the kth particle is given by (3.14) Let us now define the kth (k=1, 2,…, N) virtual displacement δxk as an infinitesimal displacement of the kth particle compatible with the constraints In performing this displacement we assume both active and reactive forces to be ‘frozen’ in time at the instant t, that is to say that they not change as the system passes through this infinitesimal change of its configuration This justifies the term ‘virtual’, as opposed to a ‘real’ displacement dxk, which occurs in a time dt Similarly, we can define the kth virtual work done by active and reactive forces as and the total work on all of the N particles is Copyright © 2003 Taylor & Francis Group LLC (3.15) If the system is in equilibrium, eq (3.15) is zero, because each one of the N terms in parentheses is zero; if in addition we restrict our considerations to reactive forces whose virtual work is zero, eq (3.15) becomes (3.16) which expresses the principle of virtual work We point out that only active forces appear in eq (3.16) and, in general, because the virtual displacements are not all independent, being connected by the constraints equations The method leads to a number of equations