Spring, Mass, and Viscous Friction

Figure 1-12 shows a system in which a spring is extended by applying a force F to the mass M starting from rest. Let v = the velocity at which the spring is extended, then

[1-57]

where

RF = constant of viscous friction, newton seconds per meter
S = stiffness coefficient of the spring, newtons per meter
M = mass in kilograms

Figure 1-12. Elasticity, mass, and viscous friction

Differentiation of Eq. 1-57 with respect to time yields

[1-58]

The relationship

[1-59]

is a solution that satisfies Eq. 1-58 and upon substitution of Eq. 1-59 in Eq. 1-58 there results

[1-60]

Equation 1-60 divided by ε^mt yields

[1-61]

or

where j = .

Since Eq. 1-57 is a second order differential equation, two constants of integration are involved in the general solution, and the velocity is expressed by

[1-62]

To evaluate the constants of integration A₁ and A₂ let t = 0, Now at t = 0, v = 0 as the mass is started from rest. Furthermore, if

must both be zero at t = 0. Therefore

So

[1-63]

[1-64]

[1-65]

From Eqs. 1-63 and 1-65 we have

[1-66]

Substitution of Eq. 1-66 in Eq. 1-62 yields

[1-67]

Frictional losses

The force required to overcome frictional resistance is

[1-68]

and the power converted into heat, i.e., heat expended in overcoming friction, is

[1-69]

The energy converted into heat through friction is

[1-70]

The same relationship can be derived in a simpler manner on the basis of Eq. 1-3. The gain in reversible energy is the energy stored in the spring, whereas the irreversible energy is in the form of heat, some of which raises the temperature of part of the system, thus representing a gain in irreversible energy; the remainder of the heat is dissipated to media surrounding the system. When Eq, 1-3 is applied to this case the following relationship is valid

[1-71]

Accordingly, for this situation Eq. 1-3 can be reduced to

[1-72]

where X = F/S so that the total energy input

The gain in reversible energy is the energy stored in the spring

Then from Eq. 1-72 the irreversible energy is

The final energy stored in the spring is also F²/2S. Hence the total energy input to the system is F²/S. One-half of this is converted into heat.