SPRING (STIFFNESS) ELEMENT
Spring (Stiffness) Element
The constitutive equation (Hooke’s law) is
Note that we have differentiated the familiar force-deflection Hooke’s law, in order to be consistent with the response/state variable (velocity) that is used for the inertia element.
Now following the same steps as for the inertia element, the energy of a spring element may be expressed as
This is the well-known (elastic) potential energy. Also,
Ƒ(0 ) =ƒ (0- )
Unless an infinite velocity is applied to the spring element. In summary, we have
1. Force can represent the state of a stiffness (spring) element. This is justified because the force of a spring at any general time t may be completely determined with the knowledge of the initial force and the applied velocity, and also because the energy of a spring element can be represented in terms of ƒ alone.
2. Force through a stiffness element cannot change instantaneously unless an infinite velocity is applied to it.
3. Force ƒ is a natural output (state) variable and u is a natural input variable for a stiffness element.