Method of virtual work III - solved examples

Engineering Mechanics

4 chapters7 takeaways9 key terms5 questions

Overview

This video demonstrates the method of virtual work as a powerful tool for solving equilibrium problems in mechanics, particularly for complex systems. It explains the core principle: the net virtual work done by external forces on a system undergoing a hypothetical, infinitesimal displacement is zero. The video walks through several solved examples, including a multi-link mechanism, a crane, and a screw jack, to illustrate how this method simplifies calculations compared to traditional force equilibrium equations. The key takeaway is that by imagining a virtual displacement, one can directly relate applied forces and torques to external loads without needing to analyze internal or constraint forces.

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Chapters

The method of virtual work simplifies equilibrium calculations for large or complex mechanical systems.
It relies on the principle that the total virtual work done by external forces during an imagined, infinitesimal displacement is zero.
This method bypasses the need to calculate internal or constraint forces, focusing directly on external forces and torques.
Mastering the method requires practice with numerous examples.

Understanding the method of virtual work provides a more efficient approach to solving statics problems, especially as the number of components in a system increases.

The video begins by extending a previous example of a multi-link mechanism, showing how applying virtual work yields the same equilibrium force (F = mg tan(theta/2)) as traditional methods, but much faster.

For a system with N links, each of mass m and length b, subjected to an external force F and gravity, we consider a virtual angular displacement (delta theta).
The virtual displacement of each joint causes a change in height (delta y) and a widening of gaps (delta x).
The virtual work done by the external force F is F times the horizontal displacement of its application point.
The virtual work done by gravity is the sum of the work done on each link's center of mass, considering the vertical displacement (delta y) and the gravitational force (mg).
Setting the total virtual work (work by F - work by gravity) to zero allows direct calculation of the required force F.

This example demonstrates how the method of virtual work scales effectively with system complexity, yielding the same result as traditional methods with significantly less effort.

In a system with N links, a virtual displacement leads to a total virtual work equation: (Nb cos(theta/2) * F * delta theta) - (N * 2mg * (b/2) * sin(theta/2) * delta theta) = 0, simplifying to F = mg tan(theta/2).

For a crane holding a weight W via tension T, the virtual displacement is described by a change in angle (delta theta) of a specific link.
The external forces to consider for virtual work are the weight W and the tension T; constraint forces from the shaft are ignored as their work is zero.
Calculate the virtual displacements (delta x and delta y) of the points where W and T act, based on the geometry and the virtual angular change.
The virtual work done by W is -W * delta y (since W acts downwards and delta y is upwards).
The virtual work done by T is T * (vertical component of displacement) + T * (horizontal component of displacement).
Equating the total virtual work to zero provides a direct relationship between T and W.

This example shows how to apply the method of virtual work to systems with multiple angled members and external forces like tension, requiring careful calculation of displacements.

For a crane with specific dimensions, a virtual displacement delta theta leads to work terms like -W * (5 cos(theta) delta theta) and T * (3/sqrt(73)) * (4 delta theta), which are then set to zero to solve for W in terms of T.

The screw jack problem involves balancing an external load W with an applied torque C.
The system has one degree of freedom: the rotation angle (theta) of the screw.
A virtual displacement is a small rotation of the screw (delta alpha), which causes a vertical lift of the platform.
The virtual work done by the torque C is C * delta alpha.
The virtual work done against the load W is -W * (vertical displacement of the platform).
The vertical displacement is related to the screw's pitch (L) and the angle of the supporting triangles (theta).

This example extends the method of virtual work to include torques and rotational motion, demonstrating its versatility in analyzing mechanisms like screw jacks.

For a screw jack, a rotation delta alpha causes a vertical lift of (L/pi) * cot(theta) * delta alpha. Setting the virtual work (C * delta alpha - W * (vertical lift)) to zero yields the required torque C = WL / (pi * cot(theta)).

Key takeaways

1The method of virtual work is a powerful shortcut for solving equilibrium problems in mechanics.
2It bypasses the need to analyze internal forces or constraint reactions by focusing only on external forces and torques.
3The core principle is that the total virtual work done by external forces during an infinitesimal, hypothetical displacement is zero.
4Virtual displacement must be consistent with the system's constraints.
5The method simplifies calculations significantly as the number of links or components in a mechanism increases.
6Careful calculation of the virtual displacements of force application points is crucial for accurate results.
7The method can be applied to systems involving both linear forces and applied torques.

Key terms

Method of Virtual WorkVirtual DisplacementVirtual WorkEquilibrium ConditionDegree of FreedomExternal ForcesConstraint ForcesTorquePitch of Screw

Test your understanding

1What is the fundamental principle behind the method of virtual work?
2How does the method of virtual work simplify the analysis of complex mechanical systems compared to traditional equilibrium equations?
3What constitutes a 'virtual displacement' in the context of this method?
4Why are constraint forces typically excluded when calculating virtual work?
5Describe the steps involved in applying the method of virtual work to solve an equilibrium problem.