Nonlinear Fluid Viscous Damper

ComputeFVDamperElasticity material is used to simulate the hysteretic response of a nonlinear FVD element in MASTODON. Nonlinear fluid viscous dampers (FVDs) are seismic protective devices used to mitigate the effects of intense earthquake shaking on engineered structures such as nuclear power plants. Fluid viscous dampers operate on the principle of fluid flow through the orifice creating a differential pressure across the piston head and develops an internal resisting force in the damper. Reinhorn et al. (1995) proposed a simplified expression for the force in the damper as a function of the fractional power of relative velocity as shown in Eq. (1)

(1)

where, is the damping coefficient, is the velocity exponent, and is the instantaneous relative velocity in the damper.

The response of the FVD is characterized by two parameters, , and . For = 1, Eq. (1) models a linear damper and the typical range for is between 0.3 and 1.0 for seismic applications. Fluid viscous dampers are typically installed in the structure with an in-line brace, which provides stiffness to the damper assembly. Reinhorn et al. (1995) observed that such stiffness had to be addressed for response calculations and therefore modeled explicitly. The brace can be modeled as a spring in series with the nonlinear dashpot, and the combined behavior of spring and dashpot is best represented using the Maxwell formulation, as shown in Figure 1.

Figure 1: Mathematical model for a nonlinear FVD element (Akcelyan et al., 2018).

The Maxwell model of Figure 1 also includes the flexibility of the gussets, brackets and clevises. The overall stiffness of the damper assembly is calculated as

(2)

In the Maxwell model, the nonlinear dashpot and the spring are in series and therefore the force in the dashpot and spring are equal. This force is given by,

(3)

and the expressions for the total displacement and the total velocity of the damper are

(4)

(5)

where, and are the relative displacement and the relative velocity in the spring, and and are the relative displacement and the relative velocity in the dashpot.

From Eq. (3), the rate of change of force can be written as

(6)

Using Eq. (4) & Eq. (5), and rewriting the expression for the rate of change of force

(7)

Eq. (7) is a first-order differential equation in the form of an initial value problem (IVP)

(8)

Dormand and Prince (1980) proposed an iterative integration scheme known as Dormand Prince (DP54), to solve a generalized initial value problem of the form shown in Eq. (8) above. Iterative algorithms such as DP54 are computationally more efficient than traditional integration schemes for numerical solution of initial value problems. Akcelyan et al. (2018) proposed a framework to obtain the numerical solution of nonlinear FVD based on DP54 algorithm and that has been adopted here. For detailed information on the numerical implementation, specific to the FVD element, refer to Akcelyan et al. (2018).