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Castigliano's method for calculating displacements is an application of his second theorem, which states: If the strain energy of a linearly elastic structure can be expressed as a function of generalised force Q i then the partial derivative of the strain energy with respect to generalised force gives the generalised displacement q i in the direction of Q i.
In classical mechanics, the Udwadia–Kalaba formulation is a method for deriving the equations of motion of a constrained mechanical system. [1] [2] The method was first described by Anatolii Fedorovich Vereshchagin [3] [4] for the particular case of robotic arms, and later generalized to all mechanical systems by Firdaus E. Udwadia and Robert E. Kalaba in 1992. [5]
In the application of the principle of virtual work it is often convenient to obtain virtual displacements from the velocities of the system. For the n particle system, let the velocity of each particle P i be V i, then the virtual displacement δr i can also be written in the form [2] = = ˙, =, …,.
The work W done by a constant force of magnitude F on a point that moves a displacement s in a straight line in the direction of the force is the product = For example, if a force of 10 newtons ( F = 10 N ) acts along a point that travels 2 metres ( s = 2 m ), then W = Fs = (10 N) (2 m) = 20 J .
After inserting the known value for each degree of freedom, the master stiffness equation is complete and ready to be evaluated. There are several different methods available for evaluating a matrix equation including but not limited to Cholesky decomposition and the brute force evaluation of systems of equations. If a structure isn’t ...
According to this formula, the graph of the applied force F s as a function of the displacement x will be a straight line passing through the origin, whose slope is k. Hooke's law for a spring is also stated under the convention that F s is the restoring force exerted by the spring on whatever is pulling its free end.
Torque-free precessions are non-trivial solution for the situation where the torque on the right hand side is zero. When I is not constant in the external reference frame (i.e. the body is moving and its inertia tensor is not constantly diagonal) then I cannot be pulled through the derivative operator acting on L .
Expressed in terms of components with respect to a rectangular Cartesian coordinate system, the governing equations of linear elasticity are: [1]. Equation of motion: , + = where the (), subscript is a shorthand for () / and indicates /, = is the Cauchy stress tensor, is the body force density, is the mass density, and is the displacement.