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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



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


The relationship


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


Equation 1-60 divided by εmt yields



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


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





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


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


Frictional losses

The force required to overcome frictional resistance is


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


The energy converted into heat through friction is


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


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


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.

Last Update: 2011-01-12