Viscous dampers

Viscous damping occurs in a mechanical system because of viscous friction that results from the contact of a system component and a viscous liquid. The damping force produced when a rigid body is in contact with a viscous liquid is usually proportional to the velocity of the body.

F= c v

Where c is called the damping coefficient and has dimensions of mass per time.

Viscous damping can occur naturally, as when a buoyant body oscillates on the surface of a lake or a column of liquid oscillates in a U-tube manometer. Viscous damping is often added to mechanical system as a means of vibration control. Viscous damping leads to an exponential decay in amplitude of free vibration control. Viscous reduction in amplitude in forced vibrations caused by a harmonic excitation. In addition, the presence of viscous damping gives rise to a linear term in the governing differential equation, and thus does not significantly complicate the mathematical modeling of the system. A mechanical device called a dashpot is added to mechanical system to provide viscous damping. A schematic of a dashpot in a one degree of freedom system is shown in next figure.

a)

b)

Figure 1 – a) schematic of one-degree-of freedom-mass-spring-dashpot system, b) Dashpot forces cx and opposes the direction of positive x’

A simple dashpot configuration is shown in next figure. The upper plate of the dashpot is connected to a rigid body. As the body moves, the plate slides over a reservoir of viscous liquid of dynamic viscosity. The area of the plate in contact with the liquid is A. The shear stress developed between the fluid and the plate creates resultant friction force acting on the plate. Assume the reservoir is stationary and the upper plate slides over the liquid with a velocity v. the reservoir depth h is small enough that the velocity profile in the liwuid can be approximated as linear, as illustrated in fig 2 b. If y is a coordinate measured upward from the bottom of the reservoir,

u(y)=v(y/h)

The shear stress developed on the plate is determined from Newton’s viscosity law

τ=µ(du/dy)= µ(v/h)

The viscous force acting on the plate is:

F= τA=(µA/h)v

Comparison of   shows that the damping coefficient for this dashpot is;

c=µA/h

Previous equation shows that a large damping force is achieve with a very viscous fluid, a small h, and a large A. A dashpot design with these parameters is often impractical and thus the devices of fig 2a is rarely used as a dashpot.

Figure 2 a) Simple dashpot model where plate is a fixed reservoir of viscous liquid b) Since h is small, a linear profile is assumed in the liquid

The analysis assumes the plate moves with a constant velocity. During the motion of a mechanical system the dashpot is connected to a particle which has a time dependent velocity. The changing velocity of the plate leads to unsteady effects in the liquid. If the reservoir depth h is small, the unsteady effects are small and can be neglected.