LINEARIZATION PROCESS
Actual springs are nonlinear and follow F=kx only up to a certain deformation
In many practical applications we assume that the deflections are small and make use of
the linear relation. Even, if the force-deflection relation of a spring is nonlinear, we
often approximate it as a linear one by using a linearization process
Let Static equilibrium load F cause
a deflection of x*,.
If an incremental force ∆F is
added to F, the spring
deflect
by
additional
quantity ∆x.
The new F+ ∆F can be expressed using
Taylor’s series expansion
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LINEARIZATION PROCESS
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EXAMPLE 1.2

Solution: Static equilibrium position

Keq at static equilibrium position can be determined as:
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MASS OR INERTIA ELEMENT






Mass or inertia element assume to be a rigid body
Can gain or lose kinetic energy whenever the velocity
of the body changes
Work is equal to the force multiplied by the
displacement in the direction of the force, and the
work done on a mass is stored in the form of the
mass’s kinetic energy
Mathematical model to represent the actual vibrating
system, and there are often several possible models
Once the model is chosen, the mass or inertia
elements of the system can be easily identified.
Consider Cantilever beam with an end mass?
Ignore mass and damping of the beam
 The tip mass m represents the mass element, and the
elasticity of the beam denotes the stiffness of the spring.

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MASS OR INERTIA ELEMENTS: COMBINATION
OF MASSES
Consider a multistory building subjected to an
earthquake.
 Assuming that the mass of the
frame is negligible compared to
the masses of the floors, the
building can be modeled as a
multi-degree-of-freedom system
 The masses at the various floor
levels represent the mass elements, and the
elasticities of the vertical members denote the
spring elements.

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MASS OR INERTIA ELEMENTS: COMBINATION
OF MASSES
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MASS OR INERTIA ELEMENTS: COMBINATION
OF MASSES
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MASS OR INERTIA ELEMENTS: COMBINATION
OF MASSES
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MASS OR INERTIA ELEMENTS: COMBINATION
OF MASSES
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EQUIVALENT MASS OF A SYSTEM
Find the equivalent mass of the system shown in Fig., where the rigid link 1 is
attached to the pulley and rotates with it.
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𝜗𝑝 = 𝜗1 =
𝑥
𝑟𝑝
𝑥
𝑥2 = 𝜗𝑝 𝑙1 = 𝑙1 .
𝑟𝑝
𝑥
𝑙1 . 𝑟
𝑥2
𝑝
𝜗𝑐 =
=
𝑟𝑐
𝑟𝑐
Assuming small displacement, 𝑚𝑒𝑞 can be determined using the
equivalence of KE
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EXAMPLE
𝑥
𝜗𝑟 =
𝑙1
𝑥
𝑥𝑣 = 𝜗𝑟 . 𝑙2 = . 𝑙2
𝑙1
𝑥
𝑥𝑟 = 𝜗𝑟 . 𝑙3 = . 𝑙3
𝑙1
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MASS OR INERTIA ELEMENTS: COMBINATION
OF MASSES
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