1.2 Kinetic state functions 3 To=人p咖 (1.7) which,as (1.3),is independent of the velocity-momentum relation.Note, from Fig.1.1,that T(p)and T(v)are related by T -园 Tp T p T*(U）=pU-T(p】 (1.8) The total differential of the kinetic coenergy reads dT"pdv +vdp- p=pdw (1.9) if(1.4)is used.It follows that r p二d (1.10) Thus.the kinetic coenergy is a function of the instantaneous velocity v,with derivative equal to the instantaneous momentum.Equation(1.8) defines a Legendre transformation which allows us to change from one independent variable [p in T(p)]to the other v in T(v)]without loss of information on the constitutive behavior.For a Newtonian particle, combining (1.5)and(1.7),the kinetic coenergy reads
1.2 Kinetic state functions 3 T∗(v) = Z v 0 p dv (1.7) which, as (1.3), is independent of the velocity-momentum relation. Note, from Fig.1.1, that T(p) and T∗(v) are related by v dv dp T(p) T (p; v) p = mv ã(v) p v p T T mv p = c ã 1 à v2=c2 p Fig. 1.1. Velocity-momentum relation for (a) Newtonian mechanics (b) special relativity. T∗(v) = pv − T(p) (1.8) The total differential of the kinetic coenergy reads dT∗ = p dv + v dp − dT dp dp = p dv (1.9) if (1.4) is used. It follows that p = dT∗ dv (1.10) Thus, the kinetic coenergy is a function of the instantaneous velocity v, with derivative equal to the instantaneous momentum. Equation(1.8) defines a Legendre transformation which allows us to change from one independent variable [p in T(p)] to the other [v in T∗(v)] without loss of information on the constitutive behavior. For a Newtonian particle, combining (1.5) and (1.7), the kinetic coenergy reads
4 1 Lagrangian dynamics of mechanical systems T(c)= (1.11 This form is usually known as the kinetic energy in most engineering mechanics textbooks.Note,however that T(p)and T*(v)have different variables,even though they have identical values for a Newtonian particle. Since T and T are always identical in Newtonian mechanics,it has been a tradition not t make a n between them.This point en reinforced (based the irtual dis 1 to electromcchanical systems and the distinction between electrical and magnetic,energy and coenergy functions will become necessary.This is why we will use the kinetic coenergy T(v)instead of the classical notation of the kinetic energy T(v). To illustrate that T and T may have different values,it is interesting to mention that when going from Newtonian mechanics to special relativity, the constitutive equation (1.5)must be replaced by p=1-01厄 (1.12) where m is the rest mass and c is the speed of light.Equations (1.5)and (1.12）are almost 1.3 Generalized coordinates,kinematic constraints A kinematically admissible motion denotes a spatial configuration that is always compatible with the geometric boundary conditions.The gen are a s of coordina that allow a r geometri respect to a re Thi the double pendulu 1.2 tha ngles tod as rdin ates,while the absolute in the second case.Note that the generalized coordinates do not always have a simple physical meaning such as a displacement or an angle;they may also represent the amplitude of an assumed mode in a distributed system.as is done extensively in the analysis of fexible structures. like the kinetic vely on a displacement formulatio
4 1 Lagrangian dynamics of mechanical systems T∗(v) = 1 2 mv2 (1.11) This form is usually known as the kinetic energy in most engineering mechanics textbooks. Note, however that T(p) and T∗(v) have different variables, even though they have identical values for a Newtonian particle. Since T and T∗ are always identical in Newtonian mechanics, it has been a tradition not to make a distinction between them. This point of view has been reinforced by the fact that the variational methods in mechanics are almost exclusively displacement based (based on virtual displacements). However, in the following chapters, we will extend Hamilton’s principle to electromechanical systems and the distinction between electrical and magnetic, energy and coenergy functions will become necessary. This is why we will use the kinetic coenergy T∗(v) instead of the classical notation of the kinetic energy T(v). To illustrate that T and T∗ may have different values, it is interesting to mention that when going from Newtonian mechanics to special relativity, the constitutive equation (1.5) must be replaced by p = mv p1 − v2/c2 (1.12) where m is the rest mass and c is the speed of light. Equations (1.5) and (1.12) are almost identical at low speed, but they diverge considerably at high speeds (Fig.1.1.b), and T∗ and T are no longer identical. 1 1.3 Generalized coordinates, kinematic constraints A kinematically admissible motion denotes a spatial configuration that is always compatible with the geometric boundary conditions. The generalized coordinates are a set of coordinates that allow a full geometric description of the system with respect to a reference frame. This representation is not unique; Fig.1.2 shows two sets of generalized coordinates for the double pendulum in a plane; in the first case, the relative angles are adopted as generalized coordinates, while the absolute angles are taken in the second case. Note that the generalized coordinates do not always have a simple physical meaning such as a displacement or an angle; they may also represent the amplitude of an assumed mode in a distributed system, as is done extensively in the analysis of flexible structures. 1 unlike the kinetic coenergy T ∗, the potential coenergy V ∗ is often used in structural engineering; however, it will not be used in this text, because our variational approach will rely exclusively on a displacement formulation
13 Generalized coordinates,kinematic constraints (a) (b) Fig.1.2.Double pendulum inaplane (a)relative angles(b)absolute angl The number of degrees of freedom (d.o.f.)of a system is the minimum number of coordinates necessary to provide its full geometric descrip- tion.If the number of generalized coordinates is equal to the number of d.o.f.,they form a minimum set of generalized coordinates.The use of a minimum set of coordinate is not always possible,nor advisable;if their number r exce eotindependetand the gener tes g:can be mitienhematios f(q1,9n,t)=0 (1.13) they are called holonomic.If the time does not appear explicitly in the constraints,they are called scleronomic. (q1,9n)=0 (1.14) The algebraic constraints(113)or(1.14)can always be used to eliminate duce the coordiate ematic constraints ntegra (1.15) aidgi=0 (1.16)
1.3 Generalized coordinates, kinematic constraints 5 ò1 ò2 l1 l2 O (a) ò1 ò2 l1 l2 O (b) Fig. 1.2. Double pendulum in a plane (a) relative angles (b) absolute angles. The number of degrees of freedom (d.o.f.) of a system is the minimum number of coordinates necessary to provide its full geometric description. If the number of generalized coordinates is equal to the number of d.o.f., they form a minimum set of generalized coordinates. The use of a minimum set of coordinates is not always possible, nor advisable; if their number exceeds the number of d.o.f., they are not independent and they are connected by kinematic constraints. If the constraint equations between the generalized coordinates qi can be written in the form f(q1, ., qn, t) = 0 (1.13) they are called holonomic. If the time does not appear explicitly in the constraints, they are called scleronomic. f(q1, ., qn) = 0 (1.14) The algebraic constraints (1.13) or (1.14) can always be used to eliminate the redundant set of generalized coordinates and reduce the coordinates to a minimum set. This is no longer possible if the kinematic constraints are defined by a (non integrable) differential relation X i aidqi + a0dt = 0 (1.15) or X i aidqi = 0 (1.16)
6 1 Lagrangian dynamics of mechanical system if the ti e is excl ded:non integrable constraints such as(1.15)and(1.16) are called non-he (E, Fig.1.3.Vertical disk rolling without slipping on an horizontal plane As an example of non-holonomic constraints,consider a vertical disk rolling without slipping on an horizontal plane (Fig.1.3).The system is fully characterized by four generalized coordinates,the location(.y)of the contact point in the plane,and the orientation of the disk,defined by (0,).The reader can check that,if the appropriate path is used,the fou fe。heasigauadart (.e.th can b ts e plan es are dent be the rolling conditions v=ro i=vcos0 =vsin0 combining these equations,we get the differential constraint equations dz-r cos0 do=0 dy-rsin0 do=0 which actually restrict the possible paths to go from one configuration to the other
6 1 Lagrangian dynamics of mechanical systems if the time is excluded; non integrable constraints such as (1.15) and (1.16) are called non-holonomic. (x; y) ò r v x y þ Fig. 1.3. Vertical disk rolling without slipping on an horizontal plane. As an example of non-holonomic constraints, consider a vertical disk rolling without slipping on an horizontal plane (Fig.1.3). The system is fully characterized by four generalized coordinates, the location (x, y) of the contact point in the plane, and the orientation of the disk, defined by (θ, φ). The reader can check that, if the appropriate path is used, the four generalized variables can be assigned arbitrary values (i.e. the disc can be moved to all points of the plane with an arbitrary orientation). However, the time derivatives of the coordinates are not independent, because they must satisfy the rolling conditions: v = rφ˙ x˙ = v cos θ y˙ = v sin θ combining these equations, we get the differential constraint equations: dx − r cos θ dφ = 0 dy − r sin θ dφ = 0 which actually restrict the possible paths to go from one configuration to the other
1.3 Generalized coordinates,kinematic constraints 1.3.1 Virtual displacements A virtual displacement,or more generally a virtual change of configura- tion,is an infinitesimal change of coordinates occurring at constant time and t with the ki constraints of rtual i 41a time i volved.It follows that,for a sys m with ge ordin by holonomic constraints (1.13)or (1.14),the admissible variations must satisfy (1.17) Note that the same form applies,whether t is explicitly involved in the constraints or not,because the virtual displacements are taken at con- stant time.For non-holonomic constraints (1.15)or (1.16),the virtual displacements must satisfy ∑ag:=0 (1.18) Cor nts lar t as it unfolds with time,while the virt al dis separation between two different trajectories at a given instant. Consider a single particle constrained to move on a smooth surface f(x,y,2)=0 The virtual displacements must satisfy the constraint equation r+y+=0 which is in fact the dot product of the gradient to the surface, gm=1=%器 and the vector of virtual displacement 6r=(6,y,6z)T: gradf.6x =(VS)T6r =0
1.3 Generalized coordinates, kinematic constraints 7 1.3.1 Virtual displacements A virtual displacement, or more generally a virtual change of configuration, is an infinitesimal change of coordinates occurring at constant time, and consistent with the kinematic constraints of the system (but otherwise arbitrary). The notation δ is used for the virtual changes of coordinates; they follow the same rules as the derivatives, except that time is not involved. It follows that, for a system with generalized coordinates qi related by holonomic constraints (1.13) or (1.14), the admissible variations must satisfy δf = X i ∂f ∂qi δqi = 0 (1.17) Note that the same form applies, whether t is explicitly involved in the constraints or not, because the virtual displacements are taken at constant time. For non-holonomic constraints (1.15) or (1.16), the virtual displacements must satisfy X i aiδqi = 0 (1.18) Comparing Equ.(1.15) and (1.18), we note that, if the time appears explicitly in the constraints, the virtual displacements are not possible displacements. The differential displacements dqi are along a particular trajectory as it unfolds with time, while the virtual displacements δqi measure the separation between two different trajectories at a given instant. Consider a single particle constrained to move on a smooth surface f(x, y, z) = 0 The virtual displacements must satisfy the constraint equation ∂f ∂xδx + ∂f ∂y δy + ∂f ∂z δz = 0 which is in fact the dot product of the gradient to the surface, gradf = ∇f = (∂f ∂x, ∂f ∂y , ∂f ∂z ) T and the vector of virtual displacement δx = (δx, δy, δz)T : gradf.δx = (∇f) T δx = 0