Dynamics and Control of Mass-Spring-Damper System

Running head: DYNAMICS AND CONTROL
DYNAMICS AND CONTROL
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Table of Contents
Introduction:...............................................................................................................................2
Theoretical background:.............................................................................................................3
Experiment and Results:............................................................................................................7
Task 1:....................................................................................................................................7
Task 2:....................................................................................................................................7
Task 3:..................................................................................................................................10
Task 4:..................................................................................................................................11
Task 5:..................................................................................................................................13
Conclusion:..............................................................................................................................14
References:...............................................................................................................................15

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Introduction:
Vibration can be present in any system that contains mass or electricity. Finding the natural
frequency of vibration is mainly is of the primary interest as that determines the type of
oscillation of the system. A basic vibration model consists of a mass, spring and damper. If
the damping is moderate then it has very little effect on the natural frequency of the system
and those systems can be considered conservative. If there is no damper in the system then
the system is a free vibration system and considered to have one degree of freedom. The
mass-spring-damper is commonly represented by two models, which are fixed-base
configuration and base-excited configuration (Li et al. 2015). Both models are used for
studying many of the practical problems in engineering. The fixed-base configuration is used
for the buildings and mechanical structures, whereas, base-excited configuration are used for
seismic sensors, vehicle suspension. Additionally, these models do provide some insights in
the design of physical experiments and through the analysis of these models forced and
unforced response of the system can be obtained. The unforced systems are the systems
responding to initial conditions, where no forcing terms are present (f(t)=0, ̇y (t ) = 0)) and
transient and eventual rest condition is given for interest (Gómez-Aguilar et al. 2015). The
forced response consists of a forcing term traditionally modelled as step, harmonic, random
or impulse function or maybe combination of these for modelling irregular forces. A motion
or seismic sensors are the example of base-excited mass-spring-damper system. The damper
and the spring elements are mechanically parallel and ‘seismic mass’ is supported within the
case. Here, the base is the case which is excited by the base input motion y(t). The system can
be either un-damped, underdamped, critically damped or over-damped depending on its
damping factor and can be differentiated from its oscillatory behaviour (Aoki, Yamashita and
Tsubakino 2016). The exact solution of the differential equation can be calculated by using

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Particular Integral method, however, in practice numerical solution is used with computer
programming to find a near exact solution within short time.
Theoretical background:
The differential equation of the mass-spring damper system is derived from the equilibrium
of forces by newton’s laws. When the mass-spring-damper is subjected to a force then it is
known as forced mass-spring-damper system. The forced mass-spring-damper model is given
by the following block diagram.
The action and reaction forces acting in y-direction are given below.