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where b is known as the damping coef cient, which has units of force/velocity such as lbf/(in/s) or N/(m/s) As with spring forces, the force exerted by a damper always acts to oppose the relative motion This form of damping is often called viscous friction since it captures the effects of damping due to viscosity effects in uids quite well For instance, given a moving plate sliding over a xed plate as in Fig 1119, the force exerted by the uid on the plate is F= mA d h (1161)

where m is the coef cient of absolute viscosity and A is the surface area (Chen, 1982) This clearly ts the form of Eq 1160 Viscous friction is a good model for damping in bearings and other lubricated parts of the cam-follower system Rotary versions of the damper in Fig 1118 can be de ned analogously by

See Figure 833 If we sum up the area contribution from each subinterval of the partition we obtain that the area of our surface of revolution is about

1 Fig 9-3

344 d

2 f (x j ) 1 + [ f (x j )]2

(1162)

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as illustrated in the example later in this section In this case, the units of the damping coef cient, br, take the form of torque/angular velocity, such as ft-lbf/(rad/s) or Nm/(rad/s) More general damping laws can also be used For instance, a power-law dependence upon position may be added, giving a damping force of F = bd nd (1163)

f (x) 1 + [ f (x)]2

as is commonly done for simulating damping due to impact (Hunt and Crossley, 1975; Herbert and McWhannell, 1977) The dissipated power, or the rate at which a linear viscous damper of the form in Eq 1160 dissipates energy, is Pviscous = Fd = bd 2 (1164)

The curve y = l/t and the graphical depiction of lnx as the area under the curve are shown in Fig 9-3 The area under the curve between x = l and x = 2 is ln2 This area can be determined by taking as many narrow trapezoids or rectangles approximating this area as necessary to achieve a desired precision

As the mesh of the partition gets finer, the sum ( ) more closely approximates what we think of as the area of the surface, but it also converges to the integral We conclude that the integral

From this expression, it is clear that power is dissipated regardless of the direction of motion since the damping force always acts to oppose the relative motion of the two moving parts in the damper No power is dissipated when the system is at rest Furthermore, the power dissipated depends only upon the velocity of the system and grows as the square of the velocity Damping factors are the most dif cult to pin down in analysis The choice of the value of a damping factor of 003 for structural damping and 004 to 008 for well-lubricated machine (closed lubrication) components is considered reasonable For open lubrication systems the damping factors are higher depending on the contaminants 1152 Coulomb or Dry Friction In contrast to viscous friction, where the damping force depends upon the relative velocity between the two elements, friction between dry surfaces depends only upon the direction of the velocity and not its magnitude The amplitude of the damping force is determined by the normal force at the interface and the coef cient of friction between the surfaces For the system in Fig 1120, therefore, the damping force can be represented by F = mFn sgn(d ) (1165)

where sgn(x) denotes the sign of x (positive or negative) since the force always acts to oppose motion The power dissipated by this model is Pcoulomb = Fd = mFn sgn(d )d (1166)

f (x) 1 + [ f (x)]2

As with the viscous friction model, the power dissipated depends upon the velocity Unlike the viscous friction model, this dependence is linear, not quadratic The power also scales

Let f ( x) = 2x 3 For 1 x 2 we rotate the graph of f about the x axis Calculate the resulting surface area

345 F

Look more closely at the piece of the curve between t = 1 and t = 11 (Fig 9-4) The area under this part of the curve is approximated by the area of the rectangle 010xO91=0091 and the (area of the) small triangle (1/2)(010)(009) = 00045

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