8 1 Lagrangian dynamics of mechanical systems tothe surface,this simply state displa ong to th face.If we e that th eaction force ue that F.6z =FT6x =0 (1.19) the virtual work of the constraint forces on any virtual displacements is zero.We will accept this as a general statement for a reversible system (without frict e that stant tim 1.4 The principle of virtual work n of a mechanical system without friction.Consider a of N particles with nosition vectors1 N.Since the static equi- librium implies that the resultant R of the force applied to each particle i is zero,each dot product Ri.6ri=0,and for all virtual displacements r;compatible with the kinematic con- straints.Ri can be decomposed into the contribution of external forces applied F and the constraint(reaction)forcesF Ri=F+Fi and the previous equation becomes F.6z:+F'.6x=0 For s tes that the vi it follows that ∑F.6m:=0 (1.20)
8 1 Lagrangian dynamics of mechanical systems Since ∇f is parallel to the normal n to the surface, this simply states that the virtual displacements belong to the plane tangent to the surface. Let us now consider the reaction force F which constraints the particle to move along the surface. If we assume that the system is smooth and frictionless, the reaction force is also normal to the surface; it follows that F.δx = FT δx = 0 (1.19) the virtual work of the constraint forces on any virtual displacements is zero. We will accept this as a general statement for a reversible system (without friction); note that it remains true if the surface equation depends explicitly on t, because the virtual displacements are taken at constant time. 1.4 The principle of virtual work The principle of virtual work is a variational formulation of the static equilibrium of a mechanical system without friction. Consider a system of N particles with position vectors xi, i = 1, ., N. Since the static equilibrium implies that the resultant Ri of the force applied to each particle i is zero, each dot product Ri.δxi = 0, and X N i=1 Ri.δxi = 0 for all virtual displacements δxi compatible with the kinematic constraints. Ri can be decomposed into the contribution of external forces applied Fi and the constraint (reaction) forces F′ i Ri = Fi + F′ i and the previous equation becomes XFi.δxi + XF′ i.δxi = 0 For a reversible system (without friction), Equ.(1.19) states that the virtual work of the constraint forces is zero, so that the second term vanishes, it follows that XFi.δxi = 0 (1.20)
1.4 The principle of virtual work The virtual wo forceson displacements or t s,and (i)it can be written in ger eralized coordinates ∑Qk.6gk=0 (1.21) the force associated with the generalized coordi- te qk Fig.1.4.Motion amplification mechanism As an example of application,consider the one d.o.f.motion amplification mechanism of Fig.1.4.Its kinematics is governed by x=5asin It follows that 6x =5a cos060 y =-2asin060 The principle of virtual work reads f6x+wy=(f.5acos0-w.2asin0)60=0 for arbitrary 60,which implies that the static equilibrium forces f and w satisfy f=w号tan0
1.4 The principle of virtual work 9 The virtual work of the external applied forces on the virtual displacements compatible with the kinematics is zero. The strength of this result comes from the fact that (i) the reaction forces have been removed from the equilibrium equation, (ii) the static equilibrium problem is transformed into kinematics, and (iii) it can be written in generalized coordinates: XQk.δqk = 0 (1.21) where Qk is the generalized force associated with the generalized coordinate qk. f ò w y x a Fig. 1.4. Motion amplification mechanism. As an example of application, consider the one d.o.f. motion amplification mechanism of Fig.1.4. Its kinematics is governed by x = 5a sin θ y = 2a cos θ It follows that δx = 5a cos θ δθ δy = −2a sin θ δθ The principle of virtual work reads f δx + w δy = (f.5a cos θ − w.2a sin θ) δθ = 0 for arbitrary δθ, which implies that the static equilibrium forces f and w satisfy f = w 2 5 tan θ
10 I Lagrangian dynamics of mechanical systems 1.5 D'Alembert's principle D'Alembert's principle extends the principle of virtual work to dynamics. It states that a problem of dynamic equilibrium can be transformed into a problem of static equilibrium by adding the inertia forces-mi;to the externally applied forces F and constraints forces F. Indeed,Newton's law implies that,for every particle, R=+F-m4=0 account finds 之R-m动，=0 (1.22) on the virtual displacements compatible writh the constraints is zero.This principle is most general;unfortunately,it is difficult to apply,because it still refers to vector quantities expressed in an inertial frame and,unlike e the time doo in e co app ∑B.d-∑md=0 If the external forces can be expressed as the gradient of a potential v which does not depend explicitly on tEdr=-dv (if V depends explicitly on t,the total differential includes a partial derivative with respect to t).Such a force field is called conservative.The second term in the previous equation is the differential of the kinetic coenergy: m=(（∑m=r It follows that
10 1 Lagrangian dynamics of mechanical systems 1.5 D’Alembert’s principle D’Alembert’s principle extends the principle of virtual work to dynamics. It states that a problem of dynamic equilibrium can be transformed into a problem of static equilibrium by adding the inertia forces - mx¨i to the externally applied forces Fi and constraints forces F′ i. Indeed, Newton’s law implies that, for every particle, Ri = Fi + F′ i − mix¨i = 0 Following the same development as in the previous section, summing over all the particles and taking into account that the virtual work of the constraint forces is zero, one finds X N i=0 (Fi − mix¨i).δxi = 0 (1.22) The sum of the applied external forces and the inertia forces is sometimes called the effective force. Thus, the virtual work of the effective forces on the virtual displacements compatible with the constraints is zero. This principle is most general; unfortunately, it is difficult to apply, because it still refers to vector quantities expressed in an inertial frame and, unlike the principle of virtual work, it cannot be translated directly into generalized coordinates. This will be achieved with Hamilton’s principle in the next section. If the time does not appear explicitly in the constraints, the virtual displacements are possible, and Equ.(1.22) is also applicable for the actual displacements dxi = ˙xidt X i Fi.dxi − X i mix¨i.x˙idt = 0 If the external forces can be expressed as the gradient of a potential V which does not depend explicitly on t, P Fi.dxi = −dV (if V depends explicitly on t, the total differential includes a partial derivative with respect to t). Such a force field is called conservative. The second term in the previous equation is the differential of the kinetic coenergy: X i mix¨i.x˙idt = d dt à 1 2 X i mi x˙i.x˙i ! dt = dT∗ It follows that
1.6 Hamilton's principle 11 d(T*+V)=0 and T+V=Ct (1.23) This is the law of conservation of total energy.Note that it is restricted to systems where (i)the potential energy does not depend explicitly ont and (ii/the kinematical constraints are independent of time 1.6 Hamilton's principle D'Alembert's principle isa complete for the s not be for expresses the dynamic equilibrium in the form of the stationarity of a definite integral of a scalar energy function.Thus,Hamilton's principle be comes independent of the coordinate system.Consider again Equ.(1.22); the first contribution 8W=F1.6 virtua of th the den e second can d=在)-=)-62 where we have used the commutativity of 6 and ()It follows that of the sy cE) stem.Using this equation,we hand side elimi ated
1.6 Hamilton’s principle 11 d(T∗ + V ) = 0 and T∗ + V = Ct (1.23) This is the law of conservation of total energy. Note that it is restricted to systems where (i) the potential energy does not depend explicitly on t and (ii) the kinematical constraints are independent of time. 1.6 Hamilton’s principle D’Alembert’s principle is a complete formulation of the dynamic equilibrium; however, it uses the position coordinates of the various particles of the system, which are in general not independent; it cannot be formulated in generalized coordinates. On the contrary, Hamilton’s principle expresses the dynamic equilibrium in the form of the stationarity of a definite integral of a scalar energy function. Thus, Hamilton’s principle becomes independent of the coordinate system. Consider again Equ.(1.22); the first contribution δW = XFi.δxi represents the virtual work of the applied external forces. The second contribution to Equ.(1.22) can be transformed using the identity x¨i.δxi = d dt( ˙xi.δxi) − x˙i.δx˙i = d dt( ˙xi.δxi) − δ 1 2 ( ˙xi.x˙i) where we have used the commutativity of δ and (˙). It follows that X N i=1 mix¨i.δxi = X N i=1 mi d dt( ˙xi.δxi) − δT∗ where T∗ is the kinetic coenergy of the system. Using this equation, we transform d’Alembert’s principle (1.22) into δW + δT∗ = X N i=0 mi d dt( ˙xi.δxi) The left hand side consists of scalar work and energy functions. The right hand side consists of a total time derivative which can be eliminated by
12 1 Lagrangian dynamics of mechanical systems 6c(1)=(2)=0 (1.240 Taking this into account,one gets (aw+Ti=】 c胎=0 If some of the external forces are conservative. 6W =-6V +6Wn (1.25) where V is the potential and SWhe is the virtual work of the nonconser- vative forces.Thus,Hamilton's principle is expressed by the variational indicator (V.I): v1.=6(T-V)+6wldt=0 (1.26) or v1=6L+6mm=0 (1.27) where L=T-V (1.28) is the of the system.The statement of the dynamic equi- librium goes as follows:The actual path is that which cancels the value of the variational indicator (1.26)or (1.27)with respect to all arbitrary variations of the path between two instantst and t2,compatible with the kinematic constraints,and such that Sri(t)=ri(t2)=0. e stress that zi does not measure displacements on the true separation between the true path and a perturbed one at a ntial v does not de 211 s V to VV.6x,while dV =VV.dz+OV/Ot.dt). Hamilton's principle,that we derived here from d'Alembert's principle for a system of particles,is the most general statement of dynamic equi- librium,and it is,in many respects,more general than Newton's laws
12 1 Lagrangian dynamics of mechanical systems integrating over some interval [t1, t2], assuming that the system configuration is known at t1 and t2, so that δxi(t1) = δxi(t2) = 0 (1.24) Taking this into account, one gets Z t2 t1 (δW + δT∗)dt = X N i=1 mi[ ˙xi.δxi] t2 t1 = 0 If some of the external forces are conservative, δW = −δV + δWnc (1.25) where V is the potential and δWnc is the virtual work of the nonconservative forces. Thus, Hamilton’s principle is expressed by the variational indicator (V.I.): V.I. = Z t2 t1 [δ(T∗ − V ) + δWnc]dt = 0 (1.26) or V.I. = Z t2 t1 [δL + δWnc]dt = 0 (1.27) where L = T∗ − V (1.28) is the Lagrangian of the system. The statement of the dynamic equilibrium goes as follows: The actual path is that which cancels the value of the variational indicator (1.26) or (1.27) with respect to all arbitrary variations of the path between two instants t1 and t2, compatible with the kinematic constraints, and such that δxi(t1) = δxi(t2) = 0. Again, we stress that δxi does not measure displacements on the true path, but the separation between the true path and a perturbed one at a given time (Fig.1.5). Note that, unlike Equ.(1.23) which requires that the potential V does not depend explicitly on time, the virtual expression (1.25) allows V to depend on t, since the virtual variation is taken at constant time (δV = ∇V.δx, while dV = ∇V.dx + ∂V/∂t.dt). Hamilton’s principle, that we derived here from d’Alembert’s principle for a system of particles, is the most general statement of dynamic equilibrium, and it is, in many respects, more general than Newton’s laws