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Forced Oscillations(Simple Spring-Mass System)
Recap of Section 1.3
In the previous section, we discussed how adding a damping component (e. g. a dashpot) to an
unforced, simple spring-mass system would affect the response of the system. In particular, we
learned that adding the dashpot to the system changed the natural frequency of the system from
to a new damped natural frequency , and how this change made the response of the system
change from a constant sinusoidal response to an exponentially-decaying sinusoid in which the
system either had an under-damped, over-damped, or critically-damped response.
In this section, we will digress a bit by going back to the simple (undamped) oscillator system of
the previous section, but this time, a constant force will be applied to this system, and we will
investigate this system's performance at low and high frequencies as well as at resonance. In
particular, this section will start by introducing the characteristics of the spring and mass
elements of a spring-mass system, introduce electrical analogs for both the spring and mass
elements, learn how these elements combine to form the mechanical impedance system, and
reveal how the impedance can describe the mechanical system's overall response characteristics.
Next, power dissipation of the forced, simple spring-mass system will be discussed in order to
corroborate our use of electrical circuit analogs for the forced, simple spring-mass system.
Finally, the characteristic responses of this system will be discussed, and a parameter called the
amplification ratio (AR) will be introduced that will help in plotting the resonance of the forced,
simple spring-mass system.
Mechanical Impedance of Spring-Mass System
As mentioned twice before, force is analogous to voltage and velocity is analogous to current.
Because of these relationships, this implies that the mechanical impedance for the forced, simple
spring-mass system can be expressed as follows:
In general, an undamped, spring-mass system can either be "spring-like" or "mass-like". "Springlike"
systems can be characterized as being "bouncy" and they tend to grossly overshoot their
target operating level(s) when an input is introduced to the system. These type of systems
relatively take a long time to reach steady-state status. Conversely, "mass-like" can be
characterized as being "lethargic" and they tend to not reach their desired operating level(s) .
Amplification Ratio
The amplification ratio is a useful parameter that allows us to plot the frequency of the springmass
system with the purports of revealing the resonant freq of the system solely based on the
force experienced by each, the spring and mass elements of the system. In particular, AR is the
magnitude of the ratio of the complex force experienced by the spring and the complex force
experienced by the mass.
Mechanical Resistance
For most systems, a simple oscillator is not a very accurate model. While a simple oscillator
involves a continuous transfer of energy between kinetic and potential form, with the sum of the
two remaining constant, real systems involve a loss, or dissipation, of some of this energy, which
is never recovered into kinetic nor potential energy. The mechanisms that cause this dissipation
are varied and depend on many factors. Some of these mechanisms include drag on bodies
moving through the air, thermal losses, and friction, but there are many others. Often, these
mechanisms are either difficult or impossible to model, and most are non-linear. However, a
simple, linear model that attempts to account for all of these losses in a system has been
developed.
Characterizing Damped Mechanical Systems
Characterizing the response of Damped Mechanical Oscillating system can be easily quantified
using two parameters. The system parameters are the resonance frequency ('''wresonance''' and
the damping of the system '''Q(qualityfactor)orB(TemporalAbsorption'''). In practice, finding
these parameters would allow for quantification of unkwnown systems and allow you to derive
other parameters within the system.