Vibrations_Tutorial2

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Elementary Tutorial
Fundamentals of Linear Vibrations

• Prepared by Dr. An Tran
• in collaboration with Professor P. R. Heyliger
• Department of Civil Engineering
• Colorado State University
• Fort Collins, Colorado
• June 2003

Developed as part of the Research Experiences of
Undergraduates Program on Studies of Vibration
and Sound , sponsored by National Science
Foundation and Army Research Office (Award
EEC-0241979). This support is gratefully
acknowledged.
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Fundamentals of Linear Vibrations

1. Single Degree-of-Freedom Systems
2. Two Degree-of-Freedom Systems
3. Multi-DOF Systems
4. Continuous Systems

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Single Degree-of-Freedom Systems

• A spring-mass system
• General solution for any simple oscillator
• General approach
• Examples
• Equivalent springs
• Spring in series and in parallel
• Examples
• Energy Methods
• Strain energy kinetic energy
• Work-energy statement
• Conservation of energy and example

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A spring-mass system
Governing equation of motion

• General solution for any simple oscillator

where
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Any simple oscillator

• General approach
• Select coordinate system
• Apply small displacement
• Draw FBD
• Apply Newtons Laws

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Simple oscillator Example 1
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Simple oscillator Example 2
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Simple oscillator Example 3
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Simple oscillator Example 4
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Equivalent springs

• Springs in series
• same force - flexibilities add

• Springs in parallel
• same displacement - stiffnesses add

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Equivalent springs Example 1
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Equivalent springs Example 2
Consider ka2 gt Wl ?n2 is positive -
vibration is stable ka2 Wl statics - stays
in stable equilibrium ka2 lt Wl unstable -
collapses
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Equivalent springs Example 3
We cannot define ?n since we have sin? term If
? lt lt 1, sin? ? ?
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Energy methods

• Kinetic energy T

• Strain energy U
• energy in spring work done

Conservation of energy work done energy stored
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Work-Energy principles

• Work done Change in kinetic energy

• Conservation of energy for conservative systems

E total energy T U constant
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Energy methods Example
Work-energy principles have many uses, but one
of the most useful is to derive the equations of
motion.
Conservation of energy E const.
Same as vector mechanics
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Two Degree-of-Freedom Systems

• Model problem
• Matrix form of governing equation
• Special case Undamped free vibrations
• Examples
• Transformation of coordinates
• Inertially elastically coupled/uncoupled
• General approach Modal equations
• Example
• Response to harmonic forces
• Model equation
• Special case Undamped system

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Two-DOF model problem

• Matrix form of governing equation

where M mass matrix C damping
matrix K stiffness matrix P force
vector Note Matrices have positive diagonals and
are symmetric.
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Undamped free vibrations

• Zero damping matrix C and force vector P

• Assumed general solutions

• Characteristic equation

• Characteristic polynomial (for det 0)

• Eigenvalues (characteristic values)

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Undamped free vibrations

• Special case when k1k2k and m1m2m

• Eigenvalues and frequencies

• Two mode shapes (relative participation of each
  mass in the motion)

• The two eigenvectors are orthogonal

Eigenvector (1)
Eigenvector (2)
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Undamped free vibrations (UFV)
Single-DOF
For two-DOF

• For any set of initial conditions
• We know A(1) and A(2), ?1 and ?2
• Must find C1, C2, ?1, and ?2 Need 4 I.C.s

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UFV Example 1
Given
No phase angle since initial velocity is 0
From the initial displacement
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UFV Example 2
Now both modes are involved
From the given initial displacement
Solve for C1 and C2
Hence,
or
Note More contribution from mode 1
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Transformation of coordinates
UFV model problem
inertially uncoupled
elastically coupled

• Introduce a new pair of coordinates that
  represents spring stretch

z1(t) x1(t) stretch of spring 1
z2(t) x2(t) - x1(t) stretch of spring
2 or x1(t) z1(t) x2(t) z1(t)
z2(t)
Substituting maintains symmetry
inertially coupled
elastically uncoupled
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Transformation of coordinates

• We have found that we can select coordinates so
  that
• Inertially coupled, elastically uncoupled, or
• Inertially uncoupled, elastically coupled.
• Big question Can we select coordinates so that
  both are uncoupled?

• Notes in natural coordinates

• The eigenvectors are orthogonal w.r.t M
• The modal vectors are orthogonal w.r.t K
• Algebraic eigenvalue problem

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Transformation of coordinates

• General approach for solution

Governing equation
Let
or
We were calling A - Change to u to match
Meirovitch
Substitution
Modal equations
Known solutions
Solve for these using initial conditions then
substitute into ().
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Transformation - Example
Model problem with
1) Solve eigenvalue problem
2) Transformation
So
As we had before. More general procedure Modal
analysis do a bit later.
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Response to harmonic forces

• Model equation

F not function of time
M, C, and K are full but symmetric.
Assume
Substituting gives
Hence
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Special case Undamped system

• Zero damping matrix C

• Entries of impedance matrix Z

• Substituting for X1 and X2

• For our model problem (k1k2k and m1m2m), let
  F2 0

Notes 1) Denominator originally (-)(-) ().
As it passes through w1, changes sign. 2) The
plots give both amplitude and phase angle
(either 0o or 180o)
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Multi-DOF Systems

• Model Equation
• Notes on matrices
• Undamped free vibration the eigenvalue problem
• Normalization of modal matrix U
• General solution procedure
• Initial conditions
• Applied harmonic force

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Multi-DOF model equation
Multi-DOF systems are so similar to two-DOF.

• Model equation

1. Vector mechanics (Newton or D Alembert)
2. Hamilton's principles
3. Lagrange's equations

We derive using

• Notes on matrices

• They are square and symmetric.
• M is positive definite (since T is always
  positive)
• K is positive semi-definite
• all positive eigenvalues, except for some
  potentially 0-eigenvalues which occur during a
  rigid-body motion.
• If restrained/tied down ? positive-definite. All
  positive.

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UFV the eigenvalue problem
Equation of motion
in terms of the generalized D.O.F. qi
Substitution of
leads to

• Matrix eigenvalue problem

• For more than 2x2, we usually solve using
  computational techniques.
• Total motion for any problem is a linear
  combination of the natural modes contained in u
  (i.e. the eigenvectors).

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Normalization of modal matrix U

• We know that

? Let the 1st entry be 1

• So far, we pick our
• eigenvectors to look like

• Instead, let us try to pick
• so that

• Do this a row at a time to form U.

• Then

and

• This is a common technique
• for us to use after we have solved
• the eigenvalue problem.

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General solution procedure

• For all 3 problems
• Form Ku w2 Mu (nxn system)
• Solve for all w2 and u ? U.
• Normalize the eigenvectors w.r.t. mass matrix
  (optional).

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Initial conditions
General solution for any D.O.F.

• 2n constants that we need to determine by 2n
  conditions

Alternative modal analysis

• Displacement vectors

• UFV model equation

• n modal equations

? Need initial conditions on h, not q.
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Initial conditions - Modal analysis

• Using displacement vectors

• As a result, initial conditions

is

• Since the solution of

hence we can easily solve for

• And then solve

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Applied harmonic force

• Driving force Q Qocos(wt)

Equation of motion
Substitution of
leads to
Hence,
then
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Continuous Systems

• The axial bar
• Displacement field
• Energy approach
• Equation of motion
• Examples
• General solution - Free vibration
• Initial conditions
• Applied force
• Motion of the base
• Ritz method Free vibration
• Approximate solution
• One-term Ritz approximation
• Two-term Ritz approximation

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The axial bar

• Main objectives
• Use Hamiltons Principle to derive the equations
  of motion.
• Use HP to construct variational methods of
  solution.

A cross-sectional area uniform E modulus of
elasticity (MOE) u axial displacement r mass
per volume
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Energy approach

• For the axial bar

• Hamiltons principle

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Axial bar - Equation of motion

• Hamiltons principle leads to

• If area A constant ?

Since x and t are independent, must have both
sides equal to a constant.

• Separation of variables

Hence
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Fixed-free bar General solution
Free vibration
wave speed
EBC
NBC
General solution

• EBC ?

• NBC ?

For any time dependent problem
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Fixed-free bar Free vibration
For free vibration
General solution
Hence
are the frequencies (eigenvalues)
are the eigenfunctions
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Fixed-free bar Initial conditions
Give entire bar an initial stretch. Release and
compute u(x, t).
Initial conditions

• Initial velocity

• Initial displacement

or
Hence
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Fixed-free bar Applied force
Now, B.Cs
From
we assume

• Substituting

• B.C. at x 0

• B.C. at x L

or
Hence
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Fixed-free bar Motion of the base
From
Using our approach from before

• B.C. at x 0

• B.C. at x L

Hence
Resonance at
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Ritz method Free vibration

• Start with Hamiltons principle after I.B.P. in
  time

• Seek an approximate solution to u(x, t)
• In time harmonic function ? cos(wt)
  (w wn)
• In space X(x) a1f1(x)
• where a1 constant to be determined
• f1(x) known function of position

• f1(x) must satisfy the following
• Satisfy the homogeneous form of the EBC.
• u(0) 0 in this case.
• Be sufficiently differentiable as required by HP.

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One-term Ritz approximation 1
Substituting
Hence

• Ritz estimate is higher than the exact
• Only get one frequency
• If we pick a different basis/trial/approximation
  function f1, we would get a different result.

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One-term Ritz approximation 2
Substituting
Hence

• Both mode shape and natural frequency are exact.
• But all other functions we pick will never give
  us a frequency lower than the exact.

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Two-term Ritz approximation
In matrix form
where
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Two-term Ritz approximation (cont.)

• Substitution of

leads to

• Solving characteristic polynomial (for det 0)
  yields 2 frequencies

Let a1 1

• Mode 1

• Mode 2