118
3 Continuum Mechanics and Nonlinear Elasticity
k
m
f(t)
x(t)
(a)
(b)
m
f
cx
.
f s = kx
x
Fig. 3.20 Spring-mass system oscillating under water. (a) Schematic of system. (b) Free-body
diagram of mass
The internal energy consists of both mechanical and thermal components. The
mechanical component is stored in the deformed spring as a type of potential
energy called strain energy. The thermal component derives from heat generated by
internal friction in the deforming spring material, as well as by friction between the
spring-mass system and the water. This heat increases the vibrations of the particles
(molecules, etc.) making up the system, adding to the total kinetic energy. It also
can break bonds in the spring and, thereby, alter material properties. We assume
these thermal effects are relatively small. Thus, the internal energy is approximately
equal to the strain energy, which is equal to the work done by the internal spring
force f s = kx. This work is given by
f s dx, which yields
U =
1
2 kx
2 .
(3.165)
Mechanical power is the product of force and velocity. In the present problem,
the applied force f (t) and the viscous force f v from the water contribute to the
power input. The viscous force depends on velocity and is approximated by the socalled dashpot force f v = −c ˙
x, where c is a damping coefficient related to fluid
viscosity, and the minus sign is needed because f v opposes the motion. The total
power input is
P in = f (t) ˙
x − (c ˙
x) ˙
x.
(3.166)
As discussed above, thermal effects are ignored, so Q in = 0. Substituting
Eqs. (3.164)–(3.166) into (3.163) gives
m ˙
x ¨
x + kx ˙
x = f (t) ˙
x − c ˙
x
2
or, for arbitrary velocity ˙
x,
m ¨
x + c ˙
x + kx = f (t),
(3.167)
which is the classical equation of motion for a spring-mass-dashpot system.
Models like this one are called lumped parameter models, because multiple
system properties are lumped into a single parameter. In the present problem,
for example, all dissipative processes are lumped into the damping parameter c,
3 Continuum Mechanics and Nonlinear Elasticity
k
m
f(t)
x(t)
(a)
(b)
m
f
cx
.
f s = kx
x
Fig. 3.20 Spring-mass system oscillating under water. (a) Schematic of system. (b) Free-body
diagram of mass
The internal energy consists of both mechanical and thermal components. The
mechanical component is stored in the deformed spring as a type of potential
energy called strain energy. The thermal component derives from heat generated by
internal friction in the deforming spring material, as well as by friction between the
spring-mass system and the water. This heat increases the vibrations of the particles
(molecules, etc.) making up the system, adding to the total kinetic energy. It also
can break bonds in the spring and, thereby, alter material properties. We assume
these thermal effects are relatively small. Thus, the internal energy is approximately
equal to the strain energy, which is equal to the work done by the internal spring
force f s = kx. This work is given by
f s dx, which yields
U =
1
2 kx
2 .
(3.165)
Mechanical power is the product of force and velocity. In the present problem,
the applied force f (t) and the viscous force f v from the water contribute to the
power input. The viscous force depends on velocity and is approximated by the socalled dashpot force f v = −c ˙
x, where c is a damping coefficient related to fluid
viscosity, and the minus sign is needed because f v opposes the motion. The total
power input is
P in = f (t) ˙
x − (c ˙
x) ˙
x.
(3.166)
As discussed above, thermal effects are ignored, so Q in = 0. Substituting
Eqs. (3.164)–(3.166) into (3.163) gives
m ˙
x ¨
x + kx ˙
x = f (t) ˙
x − c ˙
x
2
or, for arbitrary velocity ˙
x,
m ¨
x + c ˙
x + kx = f (t),
(3.167)
which is the classical equation of motion for a spring-mass-dashpot system.
Models like this one are called lumped parameter models, because multiple
system properties are lumped into a single parameter. In the present problem,
for example, all dissipative processes are lumped into the damping parameter c,