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Since the constraint forces act perpendicular to the motion of each particle in the system to maintain the constraints, the total virtual work by the constraint forces acting on the system is zero: [23] [nb 3] = =, so that = =
The argument is as follows. The principle of virtual work states that in equilibrium the virtual work of the forces applied to a system is zero. Newton's laws state that at equilibrium the applied forces are equal and opposite to the reaction, or constraint forces. This means the virtual work of the constraint forces must be zero as well.
This also means the constraint forces do not add to the instantaneous power.) The time integral of this scalar equation yields work from the instantaneous power, and kinetic energy from the scalar product of acceleration with velocity. The fact that the work–energy principle eliminates the constraint forces underlies Lagrangian mechanics. [24]
Separating the total forces into applied forces, , and constraint forces, , yields [6] = + = If arbitrary virtual displacements are assumed to be in directions that are orthogonal to the constraint forces (which is not usually the case, so this derivation works only for special cases), the constraint forces don't do any work, ∑ i C i ⋅ δ r ...
The constraint force is the reaction force the wire exerts on the bead to keep it on the wire, and the non-constraint applied force is gravity acting on the bead. Suppose the wire changes its shape with time, by flexing. Then the constraint equation and position of the particle are respectively
Screw theory is the algebraic calculation of pairs of vectors, also known as dual vectors[1] – such as angular and linear velocity, or forces and moments – that arise in the kinematics and dynamics of rigid bodies. [2][3] Screw theory provides a mathematical formulation for the geometry of lines which is central to rigid body dynamics ...
The components of virtual displacement are related by a constraint equation. In analytical mechanics, a branch of applied mathematics and physics, a virtual displacement (or infinitesimal variation) shows how the mechanical system's trajectory can hypothetically (hence the term virtual) deviate very slightly from the actual trajectory of the ...
In analytical mechanics (particularly Lagrangian mechanics), generalized forces are conjugate to generalized coordinates. They are obtained from the applied forces Fi, i = 1, …, n, acting on a system that has its configuration defined in terms of generalized coordinates. In the formulation of virtual work, each generalized force is the ...