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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 .
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]
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 ...
here is the mass matrix, is the damping matrix, and are internal force per unit displacement and external forces, respectively. Using the extended mean value theorem , the Newmark- β {\displaystyle \beta } method states that the first time derivative (velocity in the equation of motion ) can be solved as,
However, the solution for the displacement is not unique and depends on the frequency. These solutions are typically written as w ^ n = A 1 cosh ( β n x ) + A 2 sinh ( β n x ) + A 3 cos ( β n x ) + A 4 sin ( β n x ) with β n := ( μ ω n 2 E I ) 1 / 4 .
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] = = ˙, =, …,. This means that the generalized force, Q j , can also be determined as Q j = ∑ i = 1 n F i ⋅ ∂ V i ∂ q ˙ j , j = 1 , … , m . {\displaystyle Q_{j}=\sum _{i=1}^{n}\mathbf {F ...
The formula above for the principle of virtual work with applied torques yields the generalized force = = / = The mechanical advantage of the gear train is the ratio of the output torque T B to the input torque T A , and the above equation yields M A = T B T A = R . {\displaystyle MA={\frac {T_{B}}{T_{A}}}=R.}