MECHANICAL VIBRATIONS
MENG 473
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WEEK 11
MULTIDEGREE OF FREEDOM
SYSTEMS
LECTURE #1
MULTIDEGREE OF FREEDOM SYSTEMS
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Most engineering systems are continuous and have an infinite number
of degrees of freedom.
The vibration analysis of continuous systems requires the solution of
partial differential equations, which is quite difficult.
For many partial differential equations, in fact, analytical solutions do
not exist.
The analysis of a multidegreeof-freedom system, on the other hand,
requires the solution of a set of ordinary differential equations, which
is relatively simple.
Hence, for simplicity of analysis, continuous systems are often
approximated as multidegree-of-freedom systems.
MULTIDEGREE OF FREEDOM SYSTEMS
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All the concepts introduced in the preceding chapter can be directly
extended to the case of multidegree-of-freedom systems.
For example, there is one equation of motion for each degree of
freedom; if generalized coordinates are used, there is one
generalized coordinate for each degree of freedom.
The equations of motion can be obtained from Newton s second law
of motion or by using the influence coefficients.
However, it is often more convenient to derive the equations of motion
of a multidegree-of-freedom system by using Lagrange’s equations.
MULTIDEGREE OF FREEDOM SYSTEMS
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There are n natural frequencies, each associated with its own mode
shape, for a system having n degrees of freedom.
The method of determining the natural frequencies from the
characteristic equation obtained by equating the determinant to
zero also applies to these systems.
However, as the number of degrees of freedom increases, the
solution of the characteristic equation becomes more complex.
The mode shapes exhibit a property known as orthogonality, which
can be utilized for the solution of undamped forced-vibration
problems using a procedure known as Modal Analysis.
The solution of forced-vibration problems associated with viscously
damped systems can also be found conveniently by using a concept
called Proportional damping.
MULTIDEGREE OF FREEDOM SYSTEMS
Modeling of Continuous Systems as Multidegree-of-Freedom Systems
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Different methods can be used to approximate a continuous system as a
multidegree-offreedom system.
A simple method involves replacing the distributed mass or inertia of the
system by a finite number of lumped masses or rigid bodies.
The lumped masses are assumed to be connected by massless elastic and
damping members.
Linear (or angular) coordinates are used to describe the motion of the
lumped masses (or rigid bodies).
Such models are called lumped-parameter or lumped-mass or discretemass systems.
The minimum number of coordinates necessary to describe the motion of
the lumped masses and rigid bodies defines the number of degrees of
freedom of the system.
Naturally, the larger the number of lumped masses used in the model, the
higher the accuracy of the resulting analysis
MULTIDEGREE OF FREEDOM SYSTEMS
Modeling of Continuous Systems as Multidegree-of-Freedom Systems
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Some problems automatically
indicate the type of lumpedparameter model to be used. For
example, the three-story building
shown in Fig. 6.1(a) automatically
suggests using a three-lumped-mass
model, as indicated in Fig. 6.1(b).
In this model, the inertia of the
system is assumed to be
concentrated as three point masses
located at the floor levels, and the
elasticities of the columns are
replaced by the springs.
MULTIDEGREE OF FREEDOM SYSTEMS
Modeling of Continuous Systems as Multidegree-of-Freedom Systems
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Similarly, the radial drilling machine shown in Fig. 6.2(a) can be
modeled using four lumped masses and four spring elements
(elastic beams), as shown in Fig. 6.2(b).
MULTIDEGREE OF FREEDOM SYSTEMS
Modeling of Continuous Systems as Multidegree-of-Freedom Systems
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Another popular method of approximating a continuous
system as a multidegree-offreedom system involves
replacing the geometry of the system by a large
number of small elements.
By assuming a simple solution within each element, the
principles of compatibility and equilibrium are used to
find an approximate solution to the original system.
This method, known as the finite element method
MULTIDEGREE OF FREEDOM SYSTEMS
Using Newton s Second Law to Derive Equations of Motion
MULTIDEGREE OF FREEDOM SYSTEMS
Using Newton s Second Law to Derive Equations of Motion
MULTIDEGREE OF FREEDOM SYSTEMS
Using Newton s Second Law to Derive Equations of Motion
MULTIDEGREE OF FREEDOM SYSTEMS
Using Newton s Second Law to Derive Equations of Motion
MULTIDEGREE OF FREEDOM SYSTEMS
Using Newton s Second Law to Derive Equations of Motion
MULTIDEGREE OF FREEDOM SYSTEMS
Using Newton s Second Law to Derive Equations of Motion
MULTIDEGREE OF FREEDOM SYSTEMS
Using Newton s Second Law to Derive Equations of Motion
MULTIDEGREE OF FREEDOM SYSTEMS
Using Newton s Second Law to Derive Equations of Motion
if the mass matrix is not diagonal, the system is said
to have mass or inertia coupling.
If the damping matrix is not diagonal, the system is
said to have damping or velocity coupling.
Finally, if the stiffness matrix is not diagonal, the
system is said to have elastic or static coupling.
Both mass and damping coupling are also known
as dynamic coupling
MULTIDEGREE OF FREEDOM SYSTEMS
Using Newton s Second Law to Derive Equations of Motion
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The differential equations of the spring-mass system considered in Example
6.1 (Fig. 6.3(a)) can be seen to be coupled; each equation involves more
than one coordinate.
This means that the equations cannot be solved individually one at a time;
they can only be solved simultaneously.
In addition, the system can be seen to be statically coupled, since stiffnesses
are coupled that is, the stiffness matrix has at least one nonzero offdiagonal term.
On the other hand, if the mass matrix has at least one off-diagonal term
nonzero, the system is said to be dynamically coupled.
Further, if both the stiffness and mass matrices have nonzero off-diagonal
terms, the system is said to be coupled both statically and dynamically
MULTIDEGREE OF FREEDOM SYSTEMS
Influence Coefficients
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The equations of motion of a multidegree-of-freedom system can also be
written in terms of influence coefficients, which are extensively used in
structural engineering.
Basically, one set of influence coefficients can be associated with each of
the matrices involved in the equations of motion.
The influence coefficients associated with the stiffness and mass matrices
are, respectively, known as the stiffness and inertia influence coefficients.
In some cases, it is more convenient to rewrite the equations of motion
using the inverse of the stiffness matrix (known as the flexibility matrix) or
the inverse of the mass matrix.
The influence coefficients corresponding to the inverse stiffness matrix are
called the flexibility influence coefficients, and those corresponding to the
inverse mass matrix are known as the inverse inertia coefficients.
MULTIDEGREE OF FREEDOM SYSTEMS
Stiffness Influence Coefficients
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Stiffness Influence Coefficients
MULTIDEGREE OF FREEDOM SYSTEMS
Stiffness Influence Coefficients
MULTIDEGREE OF FREEDOM SYSTEMS
Stiffness Influence Coefficients
MULTIDEGREE OF FREEDOM SYSTEMS
Stiffness Influence Coefficients
MULTIDEGREE OF FREEDOM SYSTEMS
Stiffness Influence Coefficients
MULTIDEGREE OF FREEDOM SYSTEMS
Stiffness Influence Coefficients
MULTIDEGREE OF FREEDOM SYSTEMS
Stiffness Influence Coefficients
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Stiffness Influence Coefficients
MULTIDEGREE OF FREEDOM SYSTEMS
Stiffness Influence Coefficients
MULTIDEGREE OF FREEDOM SYSTEMS
Flexibility Influence Coefficients
MULTIDEGREE OF FREEDOM SYSTEMS
Flexibility Influence Coefficients
MULTIDEGREE OF FREEDOM SYSTEMS
Flexibility Influence Coefficients
MULTIDEGREE OF FREEDOM SYSTEMS
Flexibility Influence Coefficients
MULTIDEGREE OF FREEDOM SYSTEMS
Flexibility Influence Coefficients
MULTIDEGREE OF FREEDOM SYSTEMS
Flexibility Influence Coefficients
MULTIDEGREE OF FREEDOM SYSTEMS
Flexibility Influence Coefficients
MULTIDEGREE OF FREEDOM SYSTEMS
Flexibility Influence Coefficients
MULTIDEGREE OF FREEDOM SYSTEMS
Flexibility Influence Coefficients
MULTIDEGREE OF FREEDOM SYSTEMS
Flexibility Influence Coefficients
MULTIDEGREE OF FREEDOM SYSTEMS
Flexibility Influence Coefficients
MULTIDEGREE OF FREEDOM SYSTEMS
Flexibility Influence Coefficients
MULTIDEGREE OF FREEDOM SYSTEMS
Flexibility Influence Coefficients
MULTIDEGREE OF FREEDOM SYSTEMS
Flexibility Influence Coefficients
MULTIDEGREE OF FREEDOM SYSTEMS
Inertia Influence Coefficients
MULTIDEGREE OF FREEDOM SYSTEMS
Inertia Influence Coefficients
MULTIDEGREE OF FREEDOM SYSTEMS
Inertia Influence Coefficients
MULTIDEGREE OF FREEDOM SYSTEMS
Inertia Influence Coefficients
MULTIDEGREE OF FREEDOM SYSTEMS
Potential and Kinetic Energy Expressions in Matrix Form
MULTIDEGREE OF FREEDOM SYSTEMS
Potential and Kinetic Energy Expressions in Matrix Form
MULTIDEGREE OF FREEDOM SYSTEMS
Potential and Kinetic Energy Expressions in Matrix Form
MULTIDEGREE OF FREEDOM SYSTEMS
Potential and Kinetic Energy Expressions in Matrix Form
MULTIDEGREE OF FREEDOM SYSTEMS
Potential and Kinetic Energy Expressions in Matrix Form
MULTIDEGREE OF FREEDOM SYSTEMS
Potential and Kinetic Energy Expressions in Matrix Form
MULTIDEGREE OF FREEDOM SYSTEMS
Potential and Kinetic Energy Expressions in Matrix Form
MULTIDEGREE OF FREEDOM SYSTEMS
Potential and Kinetic Energy Expressions in Matrix Form
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It can be seen that the potential energy is a quadratic function of the
displacements, and the kinetic energy is a quadratic function of the velocities.
Hence they are said to be in quadratic form.
Since kinetic energy, by definition, cannot be negative and vanishes only when all
the velocities vanish, Eqs. (6.34) and (6.36) are called positive definite quadratic
forms and the mass matrix [m] is called a positive definite matrix.
On the other hand, the potential energy expression, Eq. (6.30), is a positive
definite quadratic form, but the matrix [k] is positive definite only if the system is
a stable one.
There are systems for which the potential energy is zero without the displacements
or coordinates being zero. In these cases the potential energy will be a positive
quadratic function rather than positive definite; correspondingly, the matrix [k] is
said to be positive.
A system for which [k] is positive and [m] is positive definite is called a
semidefinite system (see Section 6.12).
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MULTIDEGREE OF FREEDOM SYSTEMS
MULTIDEGREE OF FREEDOM SYSTEMS
MULTIDEGREE OF FREEDOM SYSTEMS
MULTIDEGREE OF FREEDOM SYSTEMS
MULTIDEGREE OF FREEDOM SYSTEMS
MULTIDEGREE OF FREEDOM SYSTEMS
MULTIDEGREE OF FREEDOM SYSTEMS
MULTIDEGREE OF FREEDOM SYSTEMS
MULTIDEGREE OF FREEDOM SYSTEMS
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MULTIDEGREE OF FREEDOM SYSTEMS
MULTIDEGREE OF FREEDOM SYSTEMS
MODAL ANALYSIS
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When external forces act on a multidegree-of-freedom system, the system undergoes
forced vibration.
For a system with n coordinates or degrees of freedom, the governing equations of
motion are a set of n coupled ordinary differential equations of second order.
The solution of these equations becomes more complex when the degree of freedom
of the system (n) is large and/or when the forcing functions are nonperiodic.
In such cases, a more convenient method known as modal analysis can be used to solve
the problem.
In this method, the expansion theorem is used, and the displacements of the masses
are expressed as a linear combination of the normal modes of the system.
This linear transformation uncouples the equations of motion so that we obtain a set
of n uncoupled differential equations of second order.
The solution of these equations, which is equivalent to the solution of the equations of
n single-degree-of-freedom systems, can be readily obtained.
We shall now consider the procedure of modal analysis.
MULTIDEGREE OF FREEDOM SYSTEMS
EXPANSION THEOREM
MULTIDEGREE OF FREEDOM SYSTEMS
MULTIDEGREE OF FREEDOM SYSTEMS
MULTIDEGREE OF FREEDOM SYSTEMS
MULTIDEGREE OF FREEDOM SYSTEMS
MULTIDEGREE OF FREEDOM SYSTEMS
MULTIDEGREE OF FREEDOM SYSTEMS
MULTIDEGREE OF FREEDOM SYSTEMS
MULTIDEGREE OF FREEDOM SYSTEMS
MULTIDEGREE OF FREEDOM SYSTEMS
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MULTIDEGREE OF FREEDOM SYSTEMS
EXAMPLE 6.24 (MATLAB)
MULTIDEGREE OF FREEDOM SYSTEMS
QUESTIONS
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