11.1 Virtual Work Procedures and Strategies, page 1 of 2

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Arline Lyons
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1 11.1 Virtual Work

2 11.1 Virtual Work rocedures and Strategies, page 1 of 2 rocedures and Strategies for Solving roblems Involving Virtual Work 1. Identify a single coordinate, q, that will completely define the configuration (position of all parts) of the machine being analyzed (The method of virtual work is often the best method to use when you are analyzing a machine or mechanism, rather than a rigid structure). q = k 2. raw a free-body diagram of the machine. 3. On the free-body diagram, superimpose a dashed-line sketch of the position of the machine when the coordinate q is increased a small amount, q. y x x and y are not active forces because they do not move and thus do no work. 4. Identify the active forces, that is, the forces that do work when q is increased by an amount, q. 5. Introduce coordinates, x i, measured from a fixed point to the point of application of each active force. 6. Write the virtual work equation, W = 0. W is the total work done by each active force when its coordinate x i is increased a small amount. ssign a positive sign to the work if the force and incremental displacement are in the same direction, a minus sign if in opposite directions. mg x 1 x 1 F spring x 1 /2 /2 7. Relate each x i to q by using relations derived from the geometry of the mechanism. Often these relations between differentials can be derived by differentiating equations relating x i to q. x 2 x 2 x 2 = sin x 2 = cos

3 11.1 Virtual Work rocedures and Strategies, page 2 of 2 8. Substitute, in the virtual work equation, W = 0, the expression for each x i in terms of q, and then factor out q from each term in the equation. The coefficient of q must equal zero if W is to be zero for arbitrary values of q. This condition gives an equation from which you can find the unknown force (or unknown value of q, if that is what is requested in the problem statement). Notes: a) good way to account for the effect of the force F in a spring is to remove the spring and replace it by a pair of forces of magnitude F. The virtual work of each of these forces is then F x i, where x i is the virtual displacement of the points where the ends of the spring were attached. fter the virtual work equation has been derived, then you can replace F by ks, where s is the extension of the spring. Note that you do not apply a virtual displacement to s. b) If you find that it is very difficult to derive the relations between the virtual displacements x i and q, then probably you should analyze the machine by drawing free-body diagrams and writing equilibrium equations for the separate parts of the machine. 1 c) The above procedure applies to one degree-of-freedom machines, that is, to machines the position of which can be described by a single coordinate, q. For systems with more than a single degree of freedom, multiple coordinates q i must be defined. The steps described above can then be followed. q 1 = 1 q 2 = 2 2

4 11.1 Virtual Work roblem Statement for Example 1 1. etermine the force required to keep the two rods in equilibrium when the angle = 30 and weight W is 50 lb. The rods are each of length and of negligible weight. They are prevented from moving out of the plane of the figure by supports not shown. W Smooth surface

5 11.1 Virtual Work roblem Statement for Example 2 2. etermine the value of moment M required to maintain the mechanism in the position shown, if = 35 and W = 200 lb. M 2 ft W 2 ft

6 11.1 Virtual Work roblem Statement for Example 3 3. etermine the value of the weight W required to maintain the mechanism in the position shown, if = 50 N. 3 m 3 m 2 m 3 m E F W

7 11.1 Virtual Work roblem Statement for Example 4 4. etermine the force Q necessary to maintain equilibrium when force = 400 N. 300 mm 250 mm 150 mm 250 mm E Q 300 mm F 400 mm G

8 11.1 Virtual Work roblem Statement for Example 5 5. ink is connected to collar, which can slide with negligible friction on horizontal rod EF. etermine the value of force Q necessary to maintain equilibrium when = 50, = 300 mm, and = 100 N. E Q F /2 /2

9 11.1 Virtual Work roblem Statement for Example 6 6. Rotating the threaded rod of the automobile jack causes joints and to move closer together, thus raising the weight W. etermine the axial force in the rod, if = 30 and W = 2 kn. W 150 mm 150 mm 150 mm 150 mm

10 11.1 Virtual Work roblem Statement for Example 7 7. The original length of the spring is. etermine the angle for equilibrium if = 3 m and = 300 N. Spring constant, k = 200 N/m k E F

11 11.1 Virtual Work roblem Statement for Example 8 8. ollars and can slide freely on rods and E. etermine the values of x and y, given that forces = 900 N and Q = 800 N. The unstretched length of the spring is 0.2 m, and the weight of the collars is negligible. x y k 9 kn/m Q E

12 11.1 Virtual Work roblem Statement for Example 9 9. etermine the moment M applied to the crankshaft that will keep the piston motionless when a pressure p = 400 psi acts on the top of the piston and = 25. The diameter of the piston is 3 in., and the piston slides with negligible friction in the cylinder. p 9 in. 4 in. M

13 11.1 Virtual Work roblem Statement for Example in is rigidly attached to member and moves in the smooth quarter-circle slot EF. etermine the value of force Q necessary to keep the system in equilibrium, if = 30, = 400 mm, a = 120 mm, and = 200 N. /2 E /2 Q a F

14 11.1 Virtual Work roblem Statement for Example scissors lift is used to raise a weight W = 800 lb. etermine the force exerted on pin F by the hydraulic cylinder F when = 35. Each linkage member is 2-ft long and pin connected at its midpoint and endpoints. The lift consists of two identical linkages and cylinders the one shown and one directly behind it. W K I J H F G E

15 11.1 Virtual Work roblem Statement for Example The unstretched length of the spring is 1 m. etermine the value of for equilibrium when force = 2 kn. 3 m 1.5 m k 1.5 kn/m 2 m 4 m

16 11.1 Virtual Work roblem Statement for Example a) etermine the moment reaction at the wall F. b) etermine the force reaction at the roller. In both cases = 60 lb. Hinges E F 5 ft 5 ft 5 ft 5 ft 10 ft

17 11.1 Virtual Work roblem Statement for Example etermine the vertical reaction at support, if = 2 kn. 3 m 4 m 3 m

18 11.1 Virtual Work roblem Statement for Example etermine the vertical reaction at support I of the truss, if = 10 kip = Q. Q E F G H I 5 ft 5 ft 5 ft 5 ft 5 ft 5 ft 5 ft 5 ft

19 11.1 Virtual Work roblem Statement for Example etermine the tension in the cord. The pulleys are frictionless and m = 90 kg. m

20 11.1 Virtual Work roblem Statement for Example etermine the equilibrium values of and for the two-bar linkage. The couple moment M = 5 N m; each bar is uniform and has a mass m of 5 kg; the length = 400 mm; and the unstretched length of the spring is 250 mm. 500 mm k = 0.2 kn/m 2 1 M

5.2 Rigid Bodies and Two-Dimensional Force Systems

5.2 Rigid Bodies and Two-Dimensional Force Systems 5.2 Rigid odies and Two-Dimensional Force Systems 5.2 Rigid odies and Two-Dimensional Force Systems Procedures and Strategies, page 1 of 1 Procedures and Strategies for Solving Problems Involving Equilibrium