Modelling a uniaxial inerter in a 2D or 3D environment: Implications of centripetal acceleration

The inerter completes the force-current analogy between mechanical and electrical components, providing the mechanical equivalent to the capacitor. As such, it is a two-terminal passive element that, when implemented ideally, is normally said to generate a force proportional to the relative acceleration between its two terminals. However, this is applicable only if the inerter does not rotate, so the only relative motion between the device’s terminals is axial. In many applications, this restriction is acceptable, such as in car suspension systems. However, in this paper, it is shown that the relationship between the terminal accelerations and the generated force is more complex if the inerter is used in a 2-dimensional (2D) or 3-dimensional (3D) environment, such as within a multi-bar mechanism (e.g., robotic arms or railway pantographs). Specifically, the inerter force is not given by simply the relative acceleration between the two terminals. The centripetal acceleration, resulting from the rotation of the inerter, needs to be accounted for to find the second derivative of the inerter length, which defines the generated force. Two case studies are presented to demonstrate the effects of this normally neglected centripetal acceleration term. It is shown that when an inerter is operating in a 2D or 3D environment, significant errors may occur in evaluating the inerter force and also the system response if the centripetal acceleration term is neglected. Equations are provided for both modelling the inerter in different coordinate systems and for incorporating the inerter in 2D and 3D multibody systems.