Chemical Engineering - Apllied Mechanics

Dynamics of a point mass

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1. DYNAMIC ANALYSIS OF A POINT MASS A point mass is a geometric (0-dimensional) point that may be assigned a finite mass. Since a point has zero volume, the density of a point mass having a finite mass is infinite, so point masses do not exist in reality. However, it is often a useful simplification in real problems to consider bodies point masses, especially when the dimensions of the bodies are much less than the distances among them. Let us consider a point mass, m, that is supported on a frictionless surface (Figure 1.1), defined in a three dimensional space. Let us denote P the mass position. The considered point mass is subjected to a set of n 1 external forces, F j (with j = 1, 2, …, n 1). These forces can be both driving and resistance forces. They are expressed by means of 3-D vectors the resultant of which is vector F e (Figure 1.2). Let us assume that the z axis of the absolute Cartesian coordinate system coincides with the vertical direction and that the point mass is subjected to the gravitation field. Therefore, a further force, W = m g, where g is the gravity acceleration vector (- 9.81 m/s 2), is exerted on the point mass. In order to study the dynamic behaviour of the mass m it is necessary to isolate it, that is it is necessary to remove the supporting surface and show the corresponding reaction force, R vz, normal to the surface at point P. Let us denote by v the point mass velocity and t the unity vector that indicates the direction of v (Figure 1.3). Besides, let us denote by F x-y the resultant of the vector forces F ex and F ey : this vector is contained in the plane γ (Figure 1.3) The absolute acceleration a of the point mass can be null or not null. In the latter case, the acceleration vector a can be decomposed into the two projections, a n and a t, normal and tangent, respectively, to the supporting surface. The linear momentum or translational momentum, Q, (SI unit [kg m/s], or equivalently, [N s]) is the product of the mass and velocity of an object. Like velocity, linear momentum is a vector quantity, possessing a direction as well as a magnitude. m = Qv (1) Linear momentum is also a conserved quantity, meaning that if a closed system is not affected by external forces, its total linear momentum cannot change. In classical mechanics, conservation of linear momentum is implied by Newton’s laws. Figure 1.1 Point mass moving on a 3-D surface Figure 1.2 Resultant of the external forces Figure 1.3 Projections of the resultant F e of the external forces, along with weight and reaction force R vz Newton’s first law Newton’s first law states that if the net force (the vector sum of all forces acting on an object) is zero, then the velocity of the object is constant. Velocity is a vector quantity which expresses both the object speed and the direction of its motion. Therefore, the statement that the object velocity is constant is a statement that both its speed and the direction of its motion are constant. The first law can be stated mathematically as: 1 1 n ejv j ⎛⎞++ = ⎜⎟ ⎝⎠ ∑ FWR = ⇔ d dt = v0 (2) 0 That is: = F0 with: 1 1 n ejv j ⎛⎞ =++⎜⎟ ⎝⎠ ∑ FFW = R (3) Therefore: • an object that is at rest will stay at rest unless an external force acts upon it; • an object that is in motion will not change its velocity unless an external force acts upon it. This is known as uniform motion. An object continues to do whatever it happens to be doing unless a force is exerted upon it. If it is at rest, it continues in a state of rest (demonstrated when a tablecloth is skillfully whipped from under dishes on a tabletop and the dishes remain in their initial state of rest). If an object is moving, it continues to move without turning or changing its speed. This is evident in space probes that continually move in outer space. Changes in motion must be imposed against the tendency of an object to retain its state of motion. In the absence of net forces, a moving object tends to move along a straight line path indefinitely. Newton placed the first law of motion to establish frames of reference for which the other laws are applicable. The first law of motion postulates the existence of at least one frame of reference called a Newtonian or inertial reference frame, relative to which the motion of a particle not subjected to forces is a straight line at a constant speed. Newton’s first law is often referred to as the law of inertia. Thus, a condition necessary for the uniform motion of a particle relative to an inertial reference frame is that the total net force acting on it is zero. In this sense, the first law can be restated as: In every material universe, the motion of a particle in a preferential reference frame Φ is determined by the action of forces whose total vanished for all times when and only when the velocity of the particle is constant in Φ. That is, a particle initially at rest or in uniform motion in the preferential frame Φ continues in that state unless compelled by forces to change it Newton’s laws are valid only in an inertial reference frame. Any reference frame that is in uniform motion with respect to an inertial frame is also an inertial frame, i.e. Galileian invariance or the principle of Newtonian relativity. Newton’s second law The second law states that the net force on an object is equal to the rate of change (that is, the derivative) of its linear momentum Q in an inertial reference frame: ( ) d d dt dtm == v Q F (4) The second law can also be stated in terms of an object’s acceleration. Since Newton’s second law is only valid for constant-mass systems, mass can be taken outside the differentiation operator by the constant factor rule in differentiation. Thus: ( ) d dtm m == v Fa (5) where F is the net force applied, m is the mass of the body, and a is the body’s acceleration. Thus, the net force applied to a body produces a proportional acceleration. In other words, if a body is accelerating, then there is a force on it. Consistent with the first law, the time derivative of the momentum is non-zero when the momentum changes direction, even if there is no change in its magnitude; such is the case with uniform circular motion. The relationship also implies the conservation of momentum: when the net force on the body is zero, the momentum of the body is constant. Any net force is equal to the rate of change of the momentum. Any mass that is gained or lost by the system will cause a change in momentum that is not the result of an external force. A different equation is necessary for variable-mass systems. D’Alembert’s principle and generalized forces D’Alembert’s principle introduces the concept of virtual wok due to applied forces F i and inertial forces, acting on a three-dimensional accelerating system of n particles whose motion is consistent with its constraints. Mathematically the virtual work, δW, done on a particle of mass m i through a virtual displacement δr i (consistent with the constraints) is: () 1 0 n iii i i Wmδδ = =− =∑ Far (6) where a i are the accelerations of the particles in the system and i = 1, 2,..., n simply labels the particles. In terms of generalized coordinates: () 11 0 mni iii j jij Wm qδδ == ∂ =− ∂ ∑∑ r Fa q = (7) this expression suggests that the applied forces may be expressed as generalized forces, Q j. Dividing by δq j gives the definition of a generalized force: 1 n i ji i jj W Q qqδ δ = ∂ == ∂ ∑ r F (8) If the forces F i are conservative, there is a scalar potential field V in which the gradient of V is the force: 1 n i ij ijj V VQ V qq = ∂ ∂ =−∇ ⇒ =− ∇ =− ∂ ∂ ∑ r F (9) Being the gradient operator of a scalar function f (x ∇ 1, x 2, …, x n), that is: 1 1n n f f f xx ∂ ∂ ∇= + + ∂∂" ee (10) In the three-dimensional Cartesian coordinate system, this is given by: f ff f x yz ∂ ∂∂ ∇= + + ∂∂ ∂ij k (11) Therefore, generalized forces can be reduced to a pote ntial gradient in terms of generalized coordinates. The previous result may be easier to see by recognizing that V is a function of the r i , which are in turn functions of q j, and then applying the chain rule to the derivative of V with respect to q j. With reference to the point mass whose equation of motion is expressed by the eq.(5), if we denote by F in the inertia force given by: in m =− Fa (12) where a is the absolute acceleration of the mass m, it is possible to write: in + = FF 0 (13) That is: ( ) nt mm =− =− + Fa aa (14) Figure 1.4 Normal and tangential components of the point mass absolute acceleration Figure 1.5 Forces acting on the isolated point mass 2. DIRECT AND INVERSE PROBLEMS OF MECHANICAL SYSTEMS Most of the problems in dynamics of mechanical systems can be subdivided into two main classes: direct problems and inverse problems of mechanical systems. In direct problems the kinematic parameters of the system are known, or they can be evaluated by means of a common kinematic analysis, while the unknown of the problem may be a driving force or a resistance force (all the other forces are assumed to be known). In inverse problems all the driving and resistance forces are known, as well as the distribution of the velocity vectors of the system at the current time, while the unknown of the problem is the distribution of the acceleration vectors. Direct problems of mechanics need to solve a set of linear or non-linear equations of motion. This analysis can be performed at a given time or it can be repeated at several different instants. Conversely, inverse problems of mechanics need to integrate a set of differential equations of motion, in the time domain. This allows one to obtain the distribution of velocity and acceleration vectors of the system at the current time and at subsequent instants. In this case, the trend of the time-varying driving and resistance forces must be known. The integration of the differential equations of motion can be performed using different adequate numerical algorithms the choice of which mainly depends on the characteristics of the mechanical system and the harmonic content of the driving and resistance forces. A further important parameter that must be set for the numerical integration is the time interval used for the iterative computations. The solution of inverse problems of mechanical systems may be rather onerous and time-consuming. Besides, it requires the use of computer programs. Therefore, only direct problems of mechanics are shown in detail in the present course. However, a very simple example of the solution of an inverse problem is shown below. A point mass m is moving on a curvilinear profile, q, contained in a vertical plane (Figure 2.1). Therefore, the curve q is the absolute trajectory of the point mass. The position of the point mass is given by the curvilinear coordinate s of the point P, that is: P = P (s). The point O is the centre of curvature of q, evaluated in P, while ρ in the corresponding curvature radius. The unity vectors n and t indicate the normal and tangential directions in P to the curvilinear profile. The angle α is formed by the vertical direction and the direction normal to the profile, in P. Therefore we have: PP()s= ()s ρ ρ= ()s =nn ()s =tt ()s α α= (15) That is, all the above mentioned parameters depend on the curvilinear coordinate s. The point mass m is subjected to a driving force F 1 and a resistance force F 2, acting in the plane of Figure 2.2, as well as to the weight W = m g , where g is the gravity acceleration. The forces F 1 and F 2 may depend on the curvilinear coordinate s: that is, F 1 = F 1 (s) and F 2 = F 2 (s). These functions are assumed to be known. Figure 2.1 Point mass trajectory Figure 2.2 External forces acting on the point mass Let us remove the profile q, in order to isolate the point mass m. The reaction force, normal to the profile in P, is N (with N = N (s) ). The point mass is assumed to move with an absolute velocity expressed by the vector v (Figure 2.2) Owing to the relative velocity and friction between the point mass and profile q, a friction force T (with T = T (s) ) is applied to the mass m in the tangential direction indicated by the versor t (Figure 2.3). It is well known that the force T is given by: T = f k N. Where f k is the kinetic friction coefficient between the point mass and profile q. The absolute acceleration, a, of the point mass is unknown. Anyhow it can be expressed by the sum of the its normal and tangential components, a n and a t, respectively. That is: aa ntn t =+= + aa a n t (16) Eq.(16) can be rewritten as: 2vdv dt ρ =+ant or: 2 v( ) dv( ) () () () () dtss ss s ρ =+ an s t (17) Figure 2.3 Reaction force, N, and friction force, T, acting on the point mass Figure 2.4 Normal and tangential components of the inertia force acting on the point mass The last right term, a t, of eq.(17) is unknown while the normal centripetal component on the absolute acceleration a is known. Let us assume that the right versus of vector a t coincides with the versus of the velocity vector v. The normal and tangential components of the inertia force F i are those illustrated in Figure 2.4. That is: aa iinitn mm =− = + =− − FaFF n t mt (18) All the forces applied to the point mass can be projected in the normal and tangential directions, as shown in Figure 2.5. Figure 2.5 Reaction force, N, and friction force, T, acting on the point mass Then, it is possible to write the equilibrium of the forces in these two directions: 2 12 12v FFWN dv FFWT 0 dt nn n tt t m m ρ ⎧−−−++ = ⎪ ⎪ ⎨ ⎪−− −− = ⎪ ⎩ 0 (19) The only unknown in the first of eqs.(19) is the reaction force N. This term is given by: 2 12v NFFgc nn mm os() α ρ =− + + + (20) The friction force T is given by: 2 kk12v TN FF gcos( nn ffm m )α ρ ⎛⎞ ==− +++⎜⎟ ⎝⎠ (21) Therefore, the scalar value of the tangential acceleration a t can be obtained as: () t12 dv 1 aFFW dt tt t m == −−− T (22) That is: 2 t12 r12 dv 1v aFFgsin() FFgco dt ttnn mfm m m ααρ ⎡⎤⎛⎞ == −− −− +++⎢⎥⎜⎟ ⎝⎠ ⎣⎦ s() t (23) Eq.(23) allows the tangential acceleration to be evaluated at a given time , at which the point mass velocity is and the corresponding position is . A prediction of the point mass velocity and the corresponding position is , at the time ta it iv is i+1v i+1s i+1 i itt = +Δ , can be obtained applying numerical integration techniques like, Runge-Kutta, Newmark, Newton-Raphson, etc. . Likely, the simplest integration method is Euler’s algorithm. The accuracy of the results provided by this methods often depends on the choice of the integration time-step tΔ . As an example, the prediction of the point mass velocity at the time can be obtained as: i+1t i+1 i t (i) ivv +at = Δ (24) While the prediction of the corresponding position of the point mass is given by: () i+1 ii+1 i i 1 +v v 2 ss =+ t Δ (25) Then, a new set of the parameters ρ (s), α (s), n (s), t (s), F 1 (s) and F 2 (s) can be evaluated for the new curvilinear coordinate . The velocity , given by eq.(24), can be substituted in eqs.(19 - 23) in order to obtain a prediction of the tangential acceleration . Then, this iterative procedure can be used to evaluate the kinematic parameters of the point mass for any subsequent time t (as said above, the forces F i+1s i+1v t(i+1)a 1 (s) and F 2 (s) are assumed to be known at any time t ). This is a very simple example of an inverse problem of mechanics, that is a problem in which the known data are: - the initial position of the system; - the initial velocity vectors of the system; - the trend of the external driving and resistance forces. Conversely, the problem unknowns are: - the system accelerations at the initial time; - the trend of the system displacements, velocities and accelerations at subsequent instants. This way, the motion of the system caused by a set of known external forces can be predicted.