11.4:
Virtual Work for a System of Connected Rigid Bodies
The principle of virtual work can be applied to machines with multiple connected rigid bodies under loads to solve problems involved in it.
Consider a toggle vise, where a force is applied at its midpoint, resulting in a reaction force exerted by the wooden block on the vise.
In the free-body diagram, friction is neglected, and a positive virtual displacement is considered.
Choose a system of coordinate axes, and define the positive and negative increments.
The reaction forces at the left point and the normal force do no work during this displacement.
As a result, only the virtual work done by the applied and reaction forces needs to be computed.
By expressing the coordinates in terms of the angle theta and differentiating, the total virtual work of both forces can be determined.
Setting the total virtual work to zero gives the equation for the reaction force exerted by the wooden block on the vise.
11.4:
Virtual Work for a System of Connected Rigid Bodies
Virtual work is a powerful method used to solve problems involving several connected rigid bodies. When the system is in equilibrium, virtual work is zero. This allows the calculation of the resulting forces when a system undergoes a virtual displacement. When attempting to analyze such a system, first, use a free-body diagram, where an independent coordinate represents the configuration of the links, and mark its deflected position resulting from the positive virtual displacement.
Next, measure the position coordinates from a fixed point on the free-body diagram and direct them toward the forces that do the work. This allows for each virtual displacement to be expressed in terms of the independent coordinate. Next step is to write down the equation of the virtual work, which helps in the determination of the resultant forces and moments. It is also important to note here that if a force or couple moment acts along the direction of positive virtual displacement, then their work is determined as positive virtual work; otherwise, it's negative virtual work.
The main advantage of using this method is that it can be used to solve problems that involve a system of several connected rigid bodies Each of these systems is said to have only one degree of freedom since the arrangement of the links can be completely specified using only one coordinate. This greatly simplifies complex calculations and helps better understand how different variables are related to each other within complex systems.
All in all, analytically evaluating equilibrium problems involving multiple connected rigid bodies can be done effectively with virtual work, enabling not just accurate answers but also clear insight into their internal workings as well.
Suggested Reading
- Hibbeler, R.C. (2016). Engineering Mechanics: Statics. Fourteenth Edition, New Jersey: Pearson. Section 11.1, Pp. 586-587.
- Beer, F.P., Johnston, E.R., Mazurek, D.F., Cornwell, P.J. and Self, B.P. (2016). Vector Mechanics For Engineers. Eleventh Edition, New York: McGraw-Hill Education. Pp. 578-579.