INTERNATIONAL JOURNAL OF RESEARCH AND SCIENTIFIC INNOVATION (IJRSI)
ISSN No. 2321-2705 | DOI: 10.51244/IJRSI |Volume XII Issue X October 2025
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Analysis of Structural Vibration and Damping Mechanisms for Optimal
Displacement/Deformation
Dr. Afolabi, Olusegun Adeleke
Department of Civil & Environmental Engineering, University of Lagos
DOI: https://doi.org/10.51244/IJRSI.2025.1210000066
Received: 20 October 2025; Accepted: 27 October 2025; Published: 03 November 2025
ABSTRACT:
Structures are defined usually with respect to tolerable/permissible limiting conditions of displacement,
deformation and stresses etc, which enables continuous load application and static equilibrium, otherwise the
equilibrium becomes dynamical resulting in unstable structural system. This study aims to evaluate, effect of
vibrations and damping on structural stability, and identified that vibration is mechanical phenomenon involving
action of impact forces that produces oscillatory motion on the structure and characterized with oscillations,
displacement, and frequency (f = w/2Л) about a static mean position of rest (ie, F = ma = 0). The research study
entails review of literatures and work on structural vibration, stability and damping mechanism to restraint effect
within permissible limits. Dampers are commonly used to constraint structures to infinitesimal displacement
during load application, also the paper identify that vibratory systems are means of storing potential energy
(mass), kinetic energy (spring) and means by which energy is gradually dissipated through oscillations (ie, F =
ma). The motion can be optimized using principle of minimum potential energy and virtual work” expressing
workdone on a system undergoing virtual displacement (W = Fδx = 0, because δx = 0). Vibration damping is an
influence upon a system that prevents or reduces its oscillation, and is implemented by processes that dissipate
energy stored in oscillations. The damping ratio describe the system parameters which varies from undamped
(ξ = 0), underdamped (ξ < 1), critically damped (ξ =1) and overdamped (ξ > 1). Static equilibrium requires
damping of structures between critically damped and overdamped to ensure minimal oscillatory amplitude as
expected for stability and functional performance. Similarly, structural evaluation is implemented using virtual
work method,analytically defined as total work done, W = F δs = 0 (ie, δs → 0 or negligible). In conclusion, the
paper identify that dynamical tendency is characterized with instability while structural performance is identified
with infinitesimal or limit state deformation and displacement; hence corresponding vibratory displacement and
oscillations must be minimal using appropriate damping mechanism to reduce the cumulative effect on structural
system and to provide stability and safety of structural systems.
Keywords. Structures, Vibration, Oscillation, Displacement, Deformation, Damping mechanism
INTRODUCTION:
Structural loads are forces, moments, or actions, applied to structural system and the effect of these loads includes
stress, strain, deformation and displacement. Excessive and unexpected load may cause instability since such
condition are not anticipated nor considered during the design analysis, by creating additional effect on the
structure (Thomson, 2003). Impact loads are suddenly applied with an effect greater than gently applied load,
likewise vibration will cause additional effect because of the oscillatory motion and displacement amplitudes,
which must not extend beyond maximum magnitude of displacement amplitudes in design codes and standards
Cyclic load on structures may lead to fatigue damage, cumulative damage and failure, and can be due to repeated
loading on a structure or due to vibration. Vibratory systems (Thomson, 2003) are means of storing potential
energy (eg, spring), kinetic energy (eg, mass) and means by which the energy is gradually lost in oscillation (eg,
dampers), that is, the alternate transfer of energy between its potential and kinetic forms. Vibration of structure
is undesirable and involves waste of energy because it distort the static equilibrium state expected for stability
and functional use of structural systems. Vibration is a mechanical phenomenon which create oscillatory motion
about a mean (or, equilibrium) position and may be periodic or random. Damped vibration is when the energy
of vibrating systems is gradually dissipated by friction and other resistances (Lazan and Garcia-Raffi, 2022),
INTERNATIONAL JOURNAL OF RESEARCH AND SCIENTIFIC INNOVATION (IJRSI)
ISSN No. 2321-2705 | DOI: 10.51244/IJRSI |Volume XII Issue X October 2025
Page 767
thus the vibrations gradually reduce or change frequency or cease and the system rest in its equilibrium position.
In relativity principle, a different action must be maximized or minimized as functional attain stable equilibrium,
therefore to minimized transverse displacement vibratory system, the action of the applied load on the structure
must be optimized in order to evaluate the extrema functions that will make the functional attains a maximum
or minimum value (ie, applied force, displacement, deformation etc). Study of vibration is concerned with
oscillatory motion of bodies and the forces associated with them. Oscillatory motion is usually characterized
with periodic (or cyclic) function, which is a wave-like function that repeats its values at regular intervals. All
bodies with mass and elasticity are capable of vibration, hence most engineering machines and structures
experience vibration to some degree (Arboleda Monsalvea et al, 2007), and their design requires the
consideration of oscillatory behaviors (ie, amplitude, frequency, motion etc) and to ascertain stability during
load application and service period.
Fig. 1: Vibration Oscillatory motion
1.1: Stability is an important factor that enables continuous load application and structural performance, which
indicates that stable structure will be in static equilibrium condition required for load application (Eriksson and
Nordmark, 2019), and sum of forces and moments acting equals to zero (ie, ∑F = 0, and ∑M =0). Stability is a
characteristic property of engineering systems and a function that describe time dependent of particle-points in
geometric space which are in static equilibrium under system of forces (Freitas et al, (2016). Instability is a
critical condition that occur when structure lacks the capacity to provide adequate support to applied load or
generally the situation of lower permissible strength of components/structure (eg, σa > σp) and when supports’
reactive forces are inadequate to ensure requirement of static equilibrium (eg, impending displacement and
motion). Equilibrium is said to be static, if small externally induced displacement from that state produced an
opposing force that returns the body or particle to the equilibrium state, similarly equilibrium is unstable if least
displacement produces forces that tend to increase the displacement and possibly motion. Dynamics, involve
application of force on rigid bodies, and according to Newton’s law, leads to motion (in the direction of the
force), with acceleration, velocity and displacement as function of time. Constraints are parameters that provide
resistance to continuous and or straight movement of rigid body system (Eriksson and Nordmark, 2019), hence
constraint motion are motion dictated by condition of the restraint. Typical examples include periodic motion,
circular motion, or static condition of rest during force application. If a system of particles moves parallel to a
fixed plane, the system is said to be constraint to planar movement, because of the resistance to motion in the
transverse direction. In this case, Newton’s laws for a rigid system of N particles, ie, Pi, I = 1, 2,..N, is considered
satisfied, because no movement (or, motion) is anticipated or expected in that direction. Therefore, the resultant
force and torque at a reference point R can be defined, as,
F = Σ m.a = 0 , ---- (1)
and, T = Σ (r-R) m.a = 0 , ---- (2)
where, ri, denotes the planar trajectory of each particle, mi, = mass of particles and ai = acceleration.
Dynamics of Structural Systems
Structural dynamics involve the behavior of structures subjected to dynamics which possess high tendency for
motion, and the dynamic analysis is used to determine the behavior such as, displacement history, time, and
modal analysis eg, frequency (Kuznetsov, 2008). The difference between the dynamic and static analysis is on
INTERNATIONAL JOURNAL OF RESEARCH AND SCIENTIFIC INNOVATION (IJRSI)
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the basis of whether the applied action produces sufficient acceleration compared to the structure’s natural
frequency, and if a load is applied sufficiently slowly, the inertia forces can be ignored and the analysis be
simplified as static, otherwise if it varies quickly (relative to the structures ability to respond), the response must
be determined with dynamic analysis to evaluate structures mode shapes and frequencies (Bazant, 2000).
2.1: Increase in the effect of dynamic load is expressed as the dynamic amplification factor (DAF) or dynamic
load factor (DLF), defined as
DAF = DLF = δmax/δstatic ---- (3)
where δ is the deflection of the structure due to the applied load
The static equilibrium equation used in the displacement method of analysis is of the form
F = k υ ----- (4)
where F is the applied force, k, the stiffness resistance and υ is the resulting displacement.
If statically applied force is replaced by a dynamic or time-varying force F(t), the equation of static equilibrium
becomes one of dynamic equilibrium (Bigoni et al, 2012), and has the following form,
F(t) = m ѷ(t) + c ύ(t) + k υ(t) ---- (5)
Where m ѷ(t) represent the accelerated force, c ύ(t) the motion constraint (or resistance to motion) and k υ(t)
is the force corresponding to static displacement.
The dynamic equation (Begoni et al, 2012) must be satisfied at each instant of time during the time interval under
consideration, and also the time dependence of the displacements provide two additional forces that resist the
applied force in the dynamic equation.
According to Newton’s second law of motion, which states that a particle acted on by force (tongue) moves so
that the time rate of change of its linear (angular) momentum is equal to the force (or torque).
F(t) =
��
����
(��
����
����
) = m ѷ(t) ----- (6)
2.2: Minimum Total Potential Energy suggest that a body (or, structure), shall deform or displace to a position
that locally minimizes the total “potential energy”, with the lost in potential energy being converted to kinetic
energy for possible motion and displacement Freitas et al, 2016), Potential energy is associated with forces which
act on a body, such that total work done by these forces on the body depend only on displacement, defines as
difference between initial and final position of the body in space. The total potential energy (π) is the sum of
elastic strain energy U, stored in the deformed body and the potential energy (PE) associated to the applied forces
π = U + PE ----- (7)
The principle of least displacement (ie, Δs ≈ 0 ), or more precisely the principle of minimal displacement action,
indicates that, displacement of a rigid body must be relatively minimal and negligible, for it to be assumed
stationery and/or at rest position (required for structural system and load application).
The displacement is at stationary position, when an infinitesimal variation from such position involves no change
in energy, (ie, conservation of energy principle)
Thus, Δ π = δU + δ(PE) = 0 -- (8)
2.3: Total potential energy and virtual work principle are necessary to minimized displacement and deformation
of rigid body system (Eriksson and Nordmark, 2019), since any displacement beyond permissible limit will
subject the structure to unstable equilibrium condition which may not be comfortable for functional load
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application.
External work-done by forces Fi on linear elastic solid that produces set of displacement Di along the force “line
of action” is defined as,
W = ½ΣFi Di = ½(F1D1 + F2D2 + … + FnDn) --- (9)
Virtual work principle states that a body subjected to force application and responses with negligible
displacement the work-done is zero.
Ie, W = F ΔD ≈ 0 --- (10)
since ΔD ≈ 0 and negligible that is a virtual displacement
Structural Vibration Analysis and Damping
Analysis of structural vibration is necessary to determine the natural frequency of structures, and response
expected excitation, and ascertain if the structure will fulfill its intended function (Adhikari, 2002), and thus, the
integrity and usefulness of a structure can be maximized. There are two factors that control the frequency and
amplitude of vibration in structures, which are (i) the excitation applied and (ii) the response of the structure to
that particular excitation. Therefore, changing either the excitation or the dynamic characteristics of the structure
will change the stimulated vibration. The excitation usually comes from external source, like earthquake, winds,
and sources internal to the structure including moving loads, reciprocating engines and machinery. The excitation
force and motion can be periodic or harmonic in time, due to shock or impulse loadings and may even be random
in nature. The level of vibration in a structure can be attenuated by reducing either the excitation or the response
of the structure or both (Lazam, 2022). Structural response can be altered by changing the mass or stiffness of
the structure, by moving the source of excitation to another location, or by increasing the damping available.
Systems where the restoring force on a body is directly proportional to its displacement like the dynamics of the
spring-mass system are described analytically as Simple harmonic oscillator (Thomson, 1996).
In the spring-mass system, Hooke’s law state that the restoring force of a spring is,
F = -kυ --- (11)
and using Newton’s second law,
mѷ = -kυ --- (12)
and ѷ = -kυ/m = -w2υ --- (13)
The solution to the differential equation produces a sinusoidal position vector
. υ(t) = A Cos(wt – ϕ) ---- (14)
Where w is the frequency of the oscillation, A the amplitude and ϕ is the phase shift of the function, and
determined by the initial conditions of the system.
Two dimensions harmonic oscillators behave have similar behavior to one dimension oscillator, where the
restoring force is proportional to the displacement from equilibrium with the same restorative constant in all
directions.
. x(t) = Ax Cos(wt – ϕx) --- (15)
. y(t) = Ay Cos(wt – ϕy) --- (16)
3.21: Method of waves interference: Waves are propagating dynamics disturbance of one or more quantities in
a medium (Espinoza, 2017), and a periodic wave oscillates repeatedly about an equilibrium point at some
frequency. Principle of superposition of waves states that when two or more propagating waves of similar type
INTERNATIONAL JOURNAL OF RESEARCH AND SCIENTIFIC INNOVATION (IJRSI)
ISSN No. 2321-2705 | DOI: 10.51244/IJRSI |Volume XII Issue X October 2025
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are incident on the same point, the resultant amplitude at that point is equal to the vector sum of the amplitude
of individual waves. Thus, two coherent waves are combined by adding the intensities or displacements with
considerations of their phase difference and the resultant wave may have greater amplitude (ie constructive
interference) or lower amplitude (ie, destructive interference). Energy in an ideal medium is conserved at the
point of destructive interference such that when the waves amplitude cancel each other, the energy is
redistributed within the medium
Figure 3.1: interference of waves
Assuming that equation of sinusoidal wave amplitude traveling to the right along x-axis is expressed as,
W1 (x, t) = A Cos (kx – wt) ---- (13)
Where A is the peak amplitude, k = 2π/λ is the wave number and w = 2πf is the angular frequency (or, speed) of
the wave
Suppose a second wave of the same frequency and amplitude but with different phase is also travelling to the
right
W2 (x, t) = A Cos (kx – wt + φ) ---- (14)
Where φ is the phase difference between the waves, the phase of a wave or other periodic function F of some
real variable t is an angle-like quantity representing the function of the cycle covered upto t. It is expressed in
such a scale that varies by one full turn as the variable t goes through each period
Two waves will superpose and add as follows
W1 + W2 = A ( Cos(kx-wt) + Cos(kx-wt+φ) ) ---- (15)
Evaluating by using trigonometric identity sum of two cosines
W1 + W2 = 2A Cos(φ/2) Cos(kx – wt + φ/2) ---- (16)
The equation represents a wave at the original frequency traveling to the right like its components, whose
amplitude is proportional to the cosine of φ/2 (Espinoza, 2017 and Kuznetsev 2008), and further defined as
Constructive interference: of the phase difference is an even multiple of π, and φ = .., -4π, -2π, 0, 2π, 4π, ---. then
Cos φ/2 = 1.
W1 + W2 = 2A Cos(kx – wt) ---- (17)
That is, sum of two waves is a wave with twice the amplitude, with corresponding high structural response and
displacement, which occurs at resonant frequency of structural system. Resonant is a phenomenon that occurs
when an object or system is subjected to extreme force or vibration that matches the natural frequency that
generate maximum amplitude and structural response
Destructive interference: if the phase difference is an odd multiple of π and φ = .., -3π, -π, π, 3π, 5π, .., then
Cos(φ/2) = 0
INTERNATIONAL JOURNAL OF RESEARCH AND SCIENTIFIC INNOVATION (IJRSI)
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W1 + W2 = (0) 2A Cos(kx – wt) = 0 ---- (18)
That is, sum of the two waves is zero, hence according to Newton’s law, a body will remain in state if rest or
continue in uniform motion unless compelled by another force to change. Also structural performance is usually
specified with tolerable limits which are minimum criteria that enables the structure remain in static equilibrium,
eg, minimal displacement, deformation, fatigue etc beyond which the structure becomes unstable and unsafe for
load application (Freitas et al (2016).
3.2: Damping is an influence upon a system that can reduce or prevent its oscillation (Lazaro, 2019), also in
physical systems damping is produced by processes that dissipate the energy stored in oscillation. Damping ratio
is a dimensionless parameter that describes how oscillations in a system decay after a disturbance, since many
systems exhibit oscillatory behavior when disturbed from the position of static equilibrium (Thomson, 2003).
For example, a mass suspended from a spring, bounced up and down when subjected to a pull, and on each
bounce the system tends to return to its equilibrium position with loss in energy (eg, frictional drag), that damp
the system and which gradually decay the oscillation amplitude. Damping ratio provides a mathematical means
of expressing the level of damping in a system relative to critical damping requirement, and defined as the system
parameter (ξ) which varies from undamped (ie, ξ = 0), underdamped (ξ < 1), critically damped (ξ = 1) to
overdamped (ξ > 1). Structures are load bearing and due to their functional requirement are classified between
critically damped to overdamped (1 ≤ ξ ), and also only be exposed to minimal vibration in order not to change
damping criteria.
Fig. 3.2: Damping factor of oscillation
For the damped harmonic oscillator with mass m (Arboleda-Monsalvea, et al, 2007), damping coefficient c and
spring constant k, the ratio defines the damping coefficient in the system’s differential equation to critical
damping coefficient.
. ξ = e/ec = (actual damping)/critical damping) -- (17)
Where the system’s equation of motion is,
. m ѷ(t) + c ύ(t) + k υ(t) -- (18)
And the corresponding critical damping coefficient is,
Cc = 2(km)½ or Cc = 2m(k/m)½ = 2mwn --- (19)
Where wn = (k/m)½ -- (20), the natural frequency of the system.
A damped sine wave or damped sinusoid is a sinusoidal function whose amplitude approaches zero as time
increases, and corresponds to the underdamped case of damped second order systems or underdamped second
order differential equations. The most common form of damping which is usually assumed is the form found in
linear system, this form is exponential damping in which the outer envelope of successive peak is an exponential
decay curve. The general equation for an exponentials damped sinusoid may be expressed as,
. y(t) = Ae-λt Cos(wt – ϕ) ---- (21)
INTERNATIONAL JOURNAL OF RESEARCH AND SCIENTIFIC INNOVATION (IJRSI)
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Where, y(t) = instantaneous amplitude at time t
A = the initial amplitude of the envelope
λ = decay rate, in the reciprocal of time units of the independent variable t
ϕ = the phase angle at t = 0
w = angular frequency ( and f = w/2π)
Fig. a Fig. b (force diagram)
Fig. 3.3: Mass Spring Damper system, F is the applied force, K the spring force and D the damping constant
Assessment of Structural Vibration:
Structures vibrate in special shapes called mode shapes when excited at their resonant frequencies and under
normal operating conditions; also structures vibrate in complex combination of all the mode shapes (Freitas et
al, 2016). Mode shapes and resonant frequencies (ie, the modal response) of a structure can be predicted
analytically using finite element models (FEM), these models use data-points connected by elements with the
properties of the structure’s materials, and the applied forces/loads. Mode shape is deflection patterns related to
a particular natural frequency and represent the relative displacement of all parts of a structure for that particular
mode. Experimental modal analysis consists of exciting the structure with an impact hammer or vibrator, and
measuring the frequency response functions between the excitatory and many points on the structure, after which
software is used to analyze the mode shapes. Typically, the structure is divided into a grid pattern with sufficient
points to cover the whole structure, or atleast the area of interest and the size of the grids depend on accuracy
expected. A frequency response function measurement is made for every location on the structure. The number
of measurement points is determined by size and complexity of the structure and the highest resonant frequency
of interest. Each FRF identifies the resonant frequencies of the structure and modal amplitudes of the
measurement grid point associated with the frequency response functions. The modal amplitude defines the ratio
of vibration acceleration to the force input, and the mode shape is extracted by examining the vibration amplitude
of all the grid points. Specialized software applications, like the Engineering data management (EDM) Modal
use FRF data to visualize the mode analysis. Resonance is the phenomenon of increased amplitude that occurs
when the frequency of applied periodic force (or, fourier component) is equal or similar to a natural frequency
of the system on which it acts, and when an oscillatory force is applied at a resonant frequency of dynamic
system, the system will oscillate at higher amplitude than when the same force is applied at other non-resonant
frequencies. The frequency at which response amplitude is relatively maximum is also referred to as resonant
frequency of the system. Natural frequency or eigen-frequency is the frequency at which a system tends to
oscillate in the absence of any driving force. The motion pattern of a system oscillating at its natural frequency
is called the normal mode, if all the parts move sinusoidally with the same frequency.
Vibration of structure is undesirable and usually evaluated to ascertain that it will not affect performance, which
is conducted with electronic sensors called accelerometers (Lazam and Garcia-Raffi, 2022). The sensors convert
acceleration signals to an electronic signal that can be measured, analyzed and recorded with electronic hardware.
The dynamic signal analyzer includes a calibration setting parameter for each transducer that allows the voltage
signal to be converted into the measurement of acceleration. It incorporates a source type of signal, which is
amplified and sent to the modal shaker to excite the structure under test. Signal analysis is generally divided into
time and frequency domains, each domain provides a different view and insight into the nature of the vibration.
Time domain analysis (Thomson 2003), starts by analyzing the signal as a function of time, an oscilloscope, data
acquisition device or dynamic analyzer can be used to acquire the signal, and the plot of vibration versus time
provides information that helps characterized the behavior of the structure, which can be characterized by
measuring the maximum vibration (or peak)level or finding the period (time between zero crossings), or
estimating the decay rate (ie, the amount of time for the envelope to decay to near zero). These characteristic
INTERNATIONAL JOURNAL OF RESEARCH AND SCIENTIFIC INNOVATION (IJRSI)
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parameters are the typical result of time domain analysis. Frequency analysis also provides valuable information
about structural vibration, and any time history signal can be transformed into frequency domain. The most
common mathematical technique for transforming time signals into frequency domain is called Fourier
transform, named after the French mathematician J B Fourier. Fourier transform theory states that any periodic
signal can be represented by a series of pure sine tones, and in structural analysis, usually time waveforms are
measured, and their Fourier transformed are computed.
CONCLUSION:
Structural performances are defined with condition of static equilibrium and stability which allows for
comfortable load application and functionality, as implemented during structural design process. Response to
structural loading, are guided by specified limits of stress, strain and deformation according to limit state
philosophy to ensure fitness during service period, and if this conditions are not feasible (nor, possible) it will
result in instability which leads to structural failure.
Impact load produces vibration and oscillatory motion of structural systems, involving high structural
displacement and amplitudes, This produces finite deformation and fast degradation if not adequately considered
during the design, since structures can only be subjected to negligible (infinite) deformation. Also the cumulative
effect of impact load and large deformation is structural failure if not controlled adequately.
Minimizing displacement of vibration requires that the action of applied forces on structures be optimized for
extrema function and conditions (ie, δs → 0), which can be achieved through structural damping to reduce or
prevent the oscillation using appropriate damping mechanism, which generate oscillation decay after an
excitation. For least structural action of structural system the damping ratio is within critically damped and
overdamped (ξ ≥ 1)
Vibration testing is an integral test performed on oscillatory system characterized with measurement of
maximum vibration level and the decay rate, it is used to predict maximum amplitude (ie, displacement),
deformation rate and fatigue level. Structural vibration is commonly measured by electronic sensors called
accelerometers, which converts acceleration signals to electronic voltage signal, and are evaluated, analyzed and
recorded with electronic hardware and compared with permissible magnitude for safety and stability as specified
in the design standards.
RECOMMENDATION
The study evaluates importance of damping mechanism in minimizing dynamical tendency and therefore
suggested that appropriate damping design is essential to maintain performance, functionality, safety and
serviceability of structures, to ensure static equilibrium state of structures that can be subjected to dynamics in
order to prevent structural failure or collapse
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