11.3 Principle of Virtual Work for a System of Connected Rigid Bodies Method of virtual work most suited for solving equilibrium problems involving a system of several connected rigid bodies Before applying the principle of virtual work to the systems, specify the number of degrees of freedom for the system and establish the coordinates that define the position of the system

11.3 Principle of Virtual Work for a System of Connected Rigid Bodies Degrees of Freedom A system of connected bodies takes on a unique shape that can be specified provided the position of a number of specific points on the system is known Positions are defined using independent coordinates q, measured from fixed reference points For every coordinate established, the system will have a degree of freedom for displacement along the coordinate axis such that it is consistent with the constraining action of supports

11.3 Principle of Virtual Work for a System of Connected Rigid Bodies Degrees of Freedom An n degree of freedom system requires n independent coordinates to specify all the locations of all its members Example Consider one degree of freedom system where independent coordinate q = θ is used to specify location of two connecting links and the block

11.3 Principle of Virtual Work for a System of Connected Rigid Bodies Degrees of Freedom Coordinate x could be used as the independent coordinate Since the block is constrained to move within the slot, x is not independent of θ; rather, it can be related to θ using the cosine law b 2 = a 2 + x 2 -2axcos θ

11.3 Principle of Virtual Work for a System of Connected Rigid Bodies Degrees of Freedom Consider double link arrangement as a 2 degrees of freedom system To specify the location of each link, the coordinate angles θ 1 and θ 2 must be known since a rotation of one link is independent of a rotation of the other

11.3 Principle of Virtual Work for a System of Connected Rigid Bodies Principles of Virtual Work A system of connected rigid bodies is in equilibrium provided that the virtual work done by all the external forces and couples acting on the system is zero for each independent displacement of the system δU = 0 For a system with n degrees of freedom, it takes n independent coordinates to completely specify the location of the system

11.3 Principle of Virtual Work for a System of Connected Rigid Bodies Principles of Virtual Work For the system, it is possible to write n independent virtual work equations, one for every virtual displacement taken along each of the independent coordinate axes while the remaining n-1 independent coordinates are held fixed

11.3 Principle of Virtual Work for a System of Connected Rigid Bodies Procedure for Analysis Free Body Diagram Draw the FBD of the entire system of connected bodies and sketch the independent coordinate q Sketch the deflected position of the system on the FBD when the system undergoes a positive virtual displacement δq

11.3 Principle of Virtual Work for a System of Connected Rigid Bodies Procedure for Analysis Virtual Displacements Indicate position coordinates s i, measured from a fixed point on the FBD to each i number of active forces and couples Each coordinate system should be parallel to line of action of the active force to which it is directed, to calculate the virtual work along the coordinate axis Relate each of the position coordinates s i to the independent coordinate q, then differentiate for virtual displacements δs i in terms of δq

11.3 Principle of Virtual Work for a System of Connected Rigid Bodies Procedure for Analysis Virtual Work Equation Write the virtual-work equation for the system assuming that all the position coordinates s i undergo positive virtual displacement δs i Using the relations for δs i, express the work of each active force and couple in the equation in terms of the single independent virtual displacement δq Factor out this common displacement from all the terms and solve for unknown force, couple, or equilibrium position, q

11.3 Principle of Virtual Work for a System of Connected Rigid Bodies Procedure for Analysis Virtual Work Equation If the system contains n degrees of freedom, n independent coordinates q n must be specified Follow the above procedure and let only one of the independent coordinate undergo a virtual displacement, while the remaining n – 1 coordinates are held fixed n virtual work equations can be written, one for each independent coordinate

11.3 Principle of Virtual Work for a System of Connected Rigid Bodies Example 11.1 Determine the angle θ for equilibrium of the two-member linkage. Each member has a mass of 10 kg.

11.3 Principle of Virtual Work for a System of Connected Rigid Bodies Solution FBD One degree of freedom since location of both links may be specified by a single independent coordinate θ undergoes a positive (CW) virtual rotation δθ, only the active forces, F and the N weights do work

11.3 Principle of Virtual Work for a System of Connected Rigid Bodies Solution Virtual Displacements Origin of the coordinates established at the fixed pin support D Location of F and W specified by position coordinates x B and y w To determine work, note these coordinates are parallel to lines of action of their associated forces Express position coordinates in terms of independent coordinate θ and taking derivatives

11.3 Principle of Virtual Work for a System of Connected Rigid Bodies Solution It can be seen by the signs of these equations that an increase in θ causes an increase in x B and an increase in y w

11.3 Principle of Virtual Work for a System of Connected Rigid Bodies Solution Virtual Work Equation If the virtual displacements δx B and δy w were both positive, then the forces W and F would do positive work since the forces and their corresponding displacements would have the same sense For virtual work equation for displacement δθ, δU = 0; Wδy w + Wδy w + Fδx B = 0 Relating virtual displacements to common δθ, 98.1(0.5cosθ δθ) (0.5cosθ δθ) + 25(-2sinθ δθ) = 0

11.3 Principle of Virtual Work for a System of Connected Rigid Bodies Solution Virtual Work Equation δθ ≠ 0, (98.1cosθ -50 sinθ) δθ = 0 θ = tan -1 (9.81/50) = 63.0° If problem solved using equations of equilibrium, dismember the links and apply 3 scalar equations to each link Principle of virtual work, by means of calculus, eliminated this task so that answer is obtained directly

11.3 Principle of Virtual Work for a System of Connected Rigid Bodies Example 11.2 Determine the angle θ required to maintain equilibrium of the mechanism. Neglect the weight of the links. The spring is un-sketched when θ = 0°, and it maintains a horizontal position due to the roller.

11.3 Principle of Virtual Work for a System of Connected Rigid Bodies Solution FBD One degree of freedom since location of both links may be specified by a single independent coordinate θ undergoes a positive (CW) virtual rotation δθ, links AB and EC rotates by the same amount since they have the same length and link BC only translates Since a couple moment does work only when it rotates, work done by M 2 = 0 Reactive forces at A and E does no work

11.3 Principle of Virtual Work for a System of Connected Rigid Bodies Solution FBD

11.3 Principle of Virtual Work for a System of Connected Rigid Bodies Solution Virtual Displacements Position coordinates x B and x D are parallel to lines of action of P and F s and these coordinates locate the forces with respect to fixed points A and E

11.3 Principle of Virtual Work for a System of Connected Rigid Bodies Solution Virtual Work Equation For positive virtual displacements F s is opposite to δx D and hence does negative work δU = 0; M 1 δθ + Pδx B - F s δx D = 0 Relating virtual displacements to common δθ, 0.5 δθ + 2(0.4cosθ δθ) - F s (0.2cosθ δθ) = 0 ( cosθ – 0.2F s cosθ) δθ = 0 For arbitrary angle θ, spring is sketched a distance of x D = (0.2sinθ)m

11.3 Principle of Virtual Work for a System of Connected Rigid Bodies Solution Virtual Work Equation Therefore, F s = 60N/m(0.2sinθ)m = (12sinθ)N δθ ≠ 0, cosθ – 0.2F s cosθ = 0 Since sin2θ = 2sinθcosθ 1 = 2.4sin2θ – 1.6cosθ By trial and Error, θ = 36.3°

11.3 Principle of Virtual Work for a System of Connected Rigid Bodies Example 11.3 Determine the horizontal force C x that the pin at C must exert on BC in order to hold the mechanism in equilibrium when θ = 45°. Neglect the weight of the members.

11.3 Principle of Virtual Work for a System of Connected Rigid Bodies Solution FBD C x obtained by releasing the pin constraint at C in the x direction and allowing the frame to be displaced in this direction One degree of freedom since location of both links may be specified by a single independent coordinate θ undergoes a positive virtual displacement δθ, only C x and the 200N force do work

11.3 Principle of Virtual Work for a System of Connected Rigid Bodies Solution FBD

11.3 Principle of Virtual Work for a System of Connected Rigid Bodies Solution Virtual Displacements Forces C x and 200N are located from the fixed origin A using position coordinates y B and x C Using cosine rule, Thus,

11.3 Principle of Virtual Work for a System of Connected Rigid Bodies Solution Virtual Work Equation y B and x C undergo positive virtual displacements δy B and δx C C s and 200N do negative work since they act in opposite sense to δy B and δx C δU = 0; -200δy B + C x δx C = 0 Relating virtual displacements to common δθ, -200(0.6cosθδθ) – C x [(1.2x C sinθ)/(1.2cosθ – 2x C )]δθ = 0 C x = [-120cosθ(1.2cosθ – 2x C )]/(1.2x C sinθ) At required equilibrium position, θ = 45°

11.3 Principle of Virtual Work for a System of Connected Rigid Bodies Solution (x C ) 2 – 1.2cos45°x C – 0.13 = 0 Solving for positive root, x C = 0.981m Thus, C x = 114N

11.3 Principle of Virtual Work for a System of Connected Rigid Bodies Example 11.4 Determine the equilibrium position of the two-bar linkage. Neglect the weight of the links.

11.3 Principle of Virtual Work for a System of Connected Rigid Bodies Solution FBD View Free Body Diagram

11.3 Principle of Virtual Work for a System of Connected Rigid Bodies Solution Two degrees of freedom since the independent coordinates θ 1 and θ 2 must be known to locate the position of both links Position vector x B, measured from the fixed point O, is used to specify the location of P If θ 1 is held fixed and θ 2 varies by an amount δ θ 2, for the virtual work equation, [δU = 0] θ1 ; P(δx B ) θ1 - M δθ 2 = 0 where P and M represent magnitudes of applied force and couple moment acting on link AB

11.3 Principle of Virtual Work for a System of Connected Rigid Bodies Solution Position coordinate x B related to independent coordinates θ 1 and θ 2 x B = lsinθ 1 +lsinθ 2 To obtain variation of δx B with respect to δθ 2 (δx B /δθ 2 ) = l cosθ 2 (δx B ) θ2 = l cosθ 2 δθ 2 Therefore, (Pl cosθ 2 – M) δθ 2 = 0 δθ 2 ≠ 0, θ 2 = cos -1 (M/Pl)

11.3 Principle of Virtual Work for a System of Connected Rigid Bodies Solution To obtain variation of δx B with respect to θ 1 (δx B /δθ 1 ) = l cosθ 1 (δx B ) θ1 = l cosθ 1 δθ 1 Therefore, (Pl cosθ 1 – M) δθ 1 = 0 δθ 1 ≠ 0, θ 1 = cos -1 (M/Pl)

11.4 Conservative Forces Work done by a force when it undergoes a differential displacement has been defined as dU = F cosθ ds If the force is displaced over a path that has finite length s, the work is determined by integrating over the path To evaluate the integral, obtain a relationship between F and the component of displacement ds cosθ

11.4 Conservative Forces In some instances, however, the work done by a force will be independent of its path and instead, will depend only on the initial and final locations of the force along the path As force with such a property is called a conservative force

11.4 Conservative Forces Weight Consider body initially at P’ If the body is moved down along arbitrary path a to second position, then for a given displacement ds along the path, the displacement component in the direction of W has a magnitude of dy = ds cos θ

11.4 Conservative Forces Weight Since both the force and displacement are in the same direction, the work is positive or Similarly, for work done by the weight when the body moves up a distance y back to P’, along arbitrary path A’,

11.4 Conservative Forces Weight Weight of a body is therefore a conservative force since the work done by the weight depends only on the body’s vertical displacement and is independent of the path along which the body moves

11.4 Conservative Forces Elastic Spring Force developed by an elastic spring (F s = ks) is also a conservative force If the spring attached to a body and the body is displaced along any path, such that it causes the spring to elongate or compress from position s 1 to s 2, the work will be negative since the spring exerts a force F s on the body that is opposite to the body’s displacement

11.4 Conservative Forces Elastic Spring For either extension or compression, work is independent of the path and is simply Friction Work done by a frictional force depends on the path; longer the path, the greater the work Frictional forces are non-conservative and work done is dissipated in the form of heat

11.5 Potential Energy Gravitational Potential Energy If a body is located a distance y above a horizontal reference or datum, weight of the body has positive potential energy V g since W has the capacity of doing positive work when the body is moved back down to the datum Likewise, if the body is located distance y below a horizontal reference or plane, V g is negative At the datum, V g = 0

11.5 Potential Energy Gravitational Potential Energy Measuring y as positive upwards, for gravitational potential energy of the body’s weight W, V g = W y

11.5 Potential Energy Elastic Potential Energy Elastic potential energy V e that a spring produces on an attached body, when spring is elongated or compressed from an undeformed position (s = 0) to a final position s, V e = ½ ks2 V e is always positive in the deformed position since the spring has capacity of doing positive work in returning back to undeformed position

11.5 Potential Energy Potential Function If a body is subjected to both gravitational and elastic forces, potential energy or potential function V of the body can be expressed as an algebraic sum V = V g + V e If a system of frictionless connected rigid bodies has a single degree of freedom such that its position from the datum is defined by the independent coordinate q, potential function for the system can be expressed as V = V (q)

11.5 Potential Energy Potential Function Work done by all the conservative forces acting on the system in moving it from q 1 to q 2 is measured by the difference in V U 1-2 = V(q 1 ) – V(q 2 ) Example Consider potential function for a system consisting of a black of weight W supported by a spring

11.5 Potential Energy Potential Function Expressed in terms of its independent coordinate (q =) y, measured from a fixed datum located at the un-stretched length of the spring, V = V g + V e = - Wy + ½ hy 2 Moving the black from y 1 to a further downwards position y 2, for work of W and F, U 1-2 = V(y 1 ) – V(y 2 ) = - W[y 1 – y 2 ] + ½ k(y 1 ) 2 – ½ k(y 2 ) 2