2. 4 OTHER SOURCES OF POTENTIAL ENERGY
2.5 VISCOUS DAMPING
Viscous damping occurs in a mechanical system when a component of the system is in con- tact with a viscous liquid. The damping forceis usually proportional to the velocity
(2.37) where cis called the viscous damping coefficientand has dimensions of (force)(time)/ (length).
Viscous damping is often added to mechanical systems as a means of vibration control.
Viscous damping leads to an exponential decay in amplitude of free vibrations and a reduc- tion 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 equa- tion, and thus does not significantly complicate the mathematical modeling of the system.
A mechanical device called a dashpot is added to mechanical systems to provide viscous damping. A schematic of a dashpot in a one degree-of-freedom system is shown in Figure 2.18(a). The free-body diagram of the rigid body, Figure 2.18(b), shows the viscous force in the opposite direction of the positive velocity.
A simple dashpot configuration is shown in Figure 2.19(a). 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 a resultant friction force acting on the plate. Assume the reservoir is stationary and the upper plate slides over the
m F =cv
r U1:2 = 1
2rgAx21 - 1 2rgAx22 FB = mg + rgAx
m c
k kx
x
(a) (b)
cx· FIGURE 2.18
(a) Schematic of SDOF mass-spring-dashpot system. (b) Dashpot force is and opposes the direction of positive velocity.
c x#
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liquid with a velocity v. The reservoir depth h is small enough that the velocity profile in the liquid can be approximated as linear, as illustrated in Figure 2.19(b). If yis a coordi- nate measured upward from the bottom of the reservoir,
(2.38) The shear stress developed on the plate is determined from Newton’s viscosity law
(2.39) The viscous force acting on the plate is
(2.40) Comparison of Equation (2.40) with Equation (2.37) shows that the damping coefficient for this dashpot is
(2.41) Equation (2.41) shows that a large damping force is achieved with a very viscous fluid, a small h, and a large A. A dashpot design with these parameters is often impractical and thus the device of Figure 2.19(a) is rarely actually used as a dashpot.
This 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 his small, the unsteady effects are small and can be neglected.
A more practical dashpot is a piston-cylinder arrangement, as shown in Figure 2.20.
The piston slides in a cylinder of viscous liquid. Because of the motion, a pressure difference c =
mA h F = tA =
mA h v t = mdu
dy = mv h u(y) = v y
h
υ
(a)
(b) h
υ
vy Plate of area A
Viscous fluid
u(y) =
ρ, µ h
FIGURE 2.19
(a) Simple dashpot model where plate slides over a fixed reservoir of a viscous liquid. (b) Sincehis small, a linear velocity profile is assumed in the liquid.
x·
FIGURE 2.20
A piston and cylinder device that serves as a viscous damper.
is formed across the head of the piston which is proportional to the velocity of the piston.
The pressure times the area of the head is the damping force.
A torsional viscous damper is illustrated in Figure 2.21. The shaft is rigidly connected to a point on a body undergoing torsional oscillations. As the disk rotates in a dish of vis- cous liquid, a net moment due to the shear stresses developed on the face of the disk acts about the axis of rotation. The moment is proportional to the angular velocity of the shaft
(2.42) where ct is the torsional viscous damping coefficient and has dimensions of force-length-time.
Any form of damping where the damping force is proportional to the velocity is referred to as viscous damping. Viscous damping can be produced by a body moving through a magnetic field, a body oscillating on the surface of a lake, or by the oscillations of a column of liquid in a U-tube manometer.
The schematic representation for viscous damping when present in mechanical systems is shown in Figure 2.22. The force developed in the dashpot is equal to and opposite of the force from the damper on the body. The force resists the motion of the system and is drawn to show it acting in the opposite direction of the velocity. The direction of the force takes care of itself. If the velocity is negative, the actual damping force is acting in the direction of positive velocity. However, it is drawn on the FBD in the direction of negative velocity and has a negative value, thus being in the positive direction.
The viscous damping force is the damping coefficient times the velocity of the point where the dashpot is attached acting in the opposite direction of the positive velocity of that point.
M = ctu
#
θ⋅
FIGURE 2.21
A disk rotates in a dish of a viscous liquid, producing a moment about the axis of the shaft and acting as a torsional viscous damper.
cx· > 0
x·
cx· < 0 x·
c
x
(a) (b) (c)
FIGURE 2.22
(a) Schematic of a viscous damper in a mechanical system. (b) The viscous damping force is always drawn as the opposite of the direction of positive velocity. (c) When velocity is nega- tive, the viscous clamping force is still drawn to the left, but since it is nega- tive, it goes toward the right.
Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).