crsc/samsi, may 22 2006 sava dediu solving the harmonic oscillator
TRANSCRIPT
CRSCSAMSI May 22 2006 Sava Dediu
Solving the Harmonic Oscillator
ContentsContents
1 Simple illustrative example Spring-mass system2 Free Vibrations Undamped3 Free Vibrations Damped4 Forced Vibrations Beats and Resonance5 Will this work for the beam6 Writing as a First Order System7 Summary amp References
1 Spring-mass system
What is a spring-mass system and why it is important
(Hookersquos Law)
1 Spring-mass system
Dynamic problem What is motion of the mass when acted by an external force or is initially displaced
1 Spring-mass system
Forces acting on the mass
Net force acting on the mass
1 Spring-mass system
Newtonrsquos Second Law of Motion
the acceleration of an object due to an applied force is in the direction of the force and given by
For our spring-mass system
2 Undamped Free Vibrations
no damping no external force
(general solution)
(particular solutions)
A and B are arbitary constants determined from initial conditions
2 Undamped Free Vibrations
Periodic simple harmonic motion of the mass
2 Undamped Free Vibrations
Period of motion
Natural frequency of the vibration
Amplitude (constant in time)
Phase or phase angle
3 Damped Free Vibrations
no external force
Assume an exponential solution
Then
and substituting in equation above we have
(characteristic equation)
3 Damped Free Vibrations
Solutions to characteristic equation
The solution y decays as t goes to infinity regardless the values of A and B
Damping gradually dissipates energy
overdamped
critically damped
underdamped
3 Damped Free Vibrations
The most interesting case is underdamping ie
3 Damped Free Vibrations Small Damping
4 Forced Vibrations
no damping
Periodic external force
Case 1
4 Forced Vibrations Beats
4 Forced Vibrations Beats
Rapidly oscillatingSlowly oscillating amplitude
4 Forced Vibrations Resonance
unbounded as
Case 2
5 Will this work for the beam
The beam seems to fit the harmonic conditions
bull Force is zero when displacement is zerobull Restoring force increases with displacementbull Vibration appears periodic
The key assumptions are
bull Restoring force is linear in displacementbull Friction is linear in velocity
6 Writing as a First Order System
Matlab does not work with second order equations
However we can always rewrite a second order ODE as a system of first order equations
1048710We can then have Matlab find a numerical solution to this system
6 Writing as a First Order System
Given the second order ODE
with the initial conditions
6 Writing as a First Order System
Now we can rewrite the equation in matrix-vector form
This is a format Matlab can handle
6 Constants are not independent
Notice that in all our solutions we never have c m or k alone We always have cm or km
1048710The solution for y(t) given (mck) is the same as y(t) given (αm αc αk)
important for the inverse problem
7 Summary
We can use Matlab to generate solutions to the harmonic oscillator
1048710At first glance it seems reasonable to model a vibrating beam
1048710We donrsquot know the values of m c or k
Need to solve the inverse problem
8 References
W Boyce and RC DiPrima Elementary Differential Equations and Boundary Value Problems
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
-
ContentsContents
1 Simple illustrative example Spring-mass system2 Free Vibrations Undamped3 Free Vibrations Damped4 Forced Vibrations Beats and Resonance5 Will this work for the beam6 Writing as a First Order System7 Summary amp References
1 Spring-mass system
What is a spring-mass system and why it is important
(Hookersquos Law)
1 Spring-mass system
Dynamic problem What is motion of the mass when acted by an external force or is initially displaced
1 Spring-mass system
Forces acting on the mass
Net force acting on the mass
1 Spring-mass system
Newtonrsquos Second Law of Motion
the acceleration of an object due to an applied force is in the direction of the force and given by
For our spring-mass system
2 Undamped Free Vibrations
no damping no external force
(general solution)
(particular solutions)
A and B are arbitary constants determined from initial conditions
2 Undamped Free Vibrations
Periodic simple harmonic motion of the mass
2 Undamped Free Vibrations
Period of motion
Natural frequency of the vibration
Amplitude (constant in time)
Phase or phase angle
3 Damped Free Vibrations
no external force
Assume an exponential solution
Then
and substituting in equation above we have
(characteristic equation)
3 Damped Free Vibrations
Solutions to characteristic equation
The solution y decays as t goes to infinity regardless the values of A and B
Damping gradually dissipates energy
overdamped
critically damped
underdamped
3 Damped Free Vibrations
The most interesting case is underdamping ie
3 Damped Free Vibrations Small Damping
4 Forced Vibrations
no damping
Periodic external force
Case 1
4 Forced Vibrations Beats
4 Forced Vibrations Beats
Rapidly oscillatingSlowly oscillating amplitude
4 Forced Vibrations Resonance
unbounded as
Case 2
5 Will this work for the beam
The beam seems to fit the harmonic conditions
bull Force is zero when displacement is zerobull Restoring force increases with displacementbull Vibration appears periodic
The key assumptions are
bull Restoring force is linear in displacementbull Friction is linear in velocity
6 Writing as a First Order System
Matlab does not work with second order equations
However we can always rewrite a second order ODE as a system of first order equations
1048710We can then have Matlab find a numerical solution to this system
6 Writing as a First Order System
Given the second order ODE
with the initial conditions
6 Writing as a First Order System
Now we can rewrite the equation in matrix-vector form
This is a format Matlab can handle
6 Constants are not independent
Notice that in all our solutions we never have c m or k alone We always have cm or km
1048710The solution for y(t) given (mck) is the same as y(t) given (αm αc αk)
important for the inverse problem
7 Summary
We can use Matlab to generate solutions to the harmonic oscillator
1048710At first glance it seems reasonable to model a vibrating beam
1048710We donrsquot know the values of m c or k
Need to solve the inverse problem
8 References
W Boyce and RC DiPrima Elementary Differential Equations and Boundary Value Problems
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
-
1 Spring-mass system
What is a spring-mass system and why it is important
(Hookersquos Law)
1 Spring-mass system
Dynamic problem What is motion of the mass when acted by an external force or is initially displaced
1 Spring-mass system
Forces acting on the mass
Net force acting on the mass
1 Spring-mass system
Newtonrsquos Second Law of Motion
the acceleration of an object due to an applied force is in the direction of the force and given by
For our spring-mass system
2 Undamped Free Vibrations
no damping no external force
(general solution)
(particular solutions)
A and B are arbitary constants determined from initial conditions
2 Undamped Free Vibrations
Periodic simple harmonic motion of the mass
2 Undamped Free Vibrations
Period of motion
Natural frequency of the vibration
Amplitude (constant in time)
Phase or phase angle
3 Damped Free Vibrations
no external force
Assume an exponential solution
Then
and substituting in equation above we have
(characteristic equation)
3 Damped Free Vibrations
Solutions to characteristic equation
The solution y decays as t goes to infinity regardless the values of A and B
Damping gradually dissipates energy
overdamped
critically damped
underdamped
3 Damped Free Vibrations
The most interesting case is underdamping ie
3 Damped Free Vibrations Small Damping
4 Forced Vibrations
no damping
Periodic external force
Case 1
4 Forced Vibrations Beats
4 Forced Vibrations Beats
Rapidly oscillatingSlowly oscillating amplitude
4 Forced Vibrations Resonance
unbounded as
Case 2
5 Will this work for the beam
The beam seems to fit the harmonic conditions
bull Force is zero when displacement is zerobull Restoring force increases with displacementbull Vibration appears periodic
The key assumptions are
bull Restoring force is linear in displacementbull Friction is linear in velocity
6 Writing as a First Order System
Matlab does not work with second order equations
However we can always rewrite a second order ODE as a system of first order equations
1048710We can then have Matlab find a numerical solution to this system
6 Writing as a First Order System
Given the second order ODE
with the initial conditions
6 Writing as a First Order System
Now we can rewrite the equation in matrix-vector form
This is a format Matlab can handle
6 Constants are not independent
Notice that in all our solutions we never have c m or k alone We always have cm or km
1048710The solution for y(t) given (mck) is the same as y(t) given (αm αc αk)
important for the inverse problem
7 Summary
We can use Matlab to generate solutions to the harmonic oscillator
1048710At first glance it seems reasonable to model a vibrating beam
1048710We donrsquot know the values of m c or k
Need to solve the inverse problem
8 References
W Boyce and RC DiPrima Elementary Differential Equations and Boundary Value Problems
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
-
1 Spring-mass system
Dynamic problem What is motion of the mass when acted by an external force or is initially displaced
1 Spring-mass system
Forces acting on the mass
Net force acting on the mass
1 Spring-mass system
Newtonrsquos Second Law of Motion
the acceleration of an object due to an applied force is in the direction of the force and given by
For our spring-mass system
2 Undamped Free Vibrations
no damping no external force
(general solution)
(particular solutions)
A and B are arbitary constants determined from initial conditions
2 Undamped Free Vibrations
Periodic simple harmonic motion of the mass
2 Undamped Free Vibrations
Period of motion
Natural frequency of the vibration
Amplitude (constant in time)
Phase or phase angle
3 Damped Free Vibrations
no external force
Assume an exponential solution
Then
and substituting in equation above we have
(characteristic equation)
3 Damped Free Vibrations
Solutions to characteristic equation
The solution y decays as t goes to infinity regardless the values of A and B
Damping gradually dissipates energy
overdamped
critically damped
underdamped
3 Damped Free Vibrations
The most interesting case is underdamping ie
3 Damped Free Vibrations Small Damping
4 Forced Vibrations
no damping
Periodic external force
Case 1
4 Forced Vibrations Beats
4 Forced Vibrations Beats
Rapidly oscillatingSlowly oscillating amplitude
4 Forced Vibrations Resonance
unbounded as
Case 2
5 Will this work for the beam
The beam seems to fit the harmonic conditions
bull Force is zero when displacement is zerobull Restoring force increases with displacementbull Vibration appears periodic
The key assumptions are
bull Restoring force is linear in displacementbull Friction is linear in velocity
6 Writing as a First Order System
Matlab does not work with second order equations
However we can always rewrite a second order ODE as a system of first order equations
1048710We can then have Matlab find a numerical solution to this system
6 Writing as a First Order System
Given the second order ODE
with the initial conditions
6 Writing as a First Order System
Now we can rewrite the equation in matrix-vector form
This is a format Matlab can handle
6 Constants are not independent
Notice that in all our solutions we never have c m or k alone We always have cm or km
1048710The solution for y(t) given (mck) is the same as y(t) given (αm αc αk)
important for the inverse problem
7 Summary
We can use Matlab to generate solutions to the harmonic oscillator
1048710At first glance it seems reasonable to model a vibrating beam
1048710We donrsquot know the values of m c or k
Need to solve the inverse problem
8 References
W Boyce and RC DiPrima Elementary Differential Equations and Boundary Value Problems
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
-
1 Spring-mass system
Forces acting on the mass
Net force acting on the mass
1 Spring-mass system
Newtonrsquos Second Law of Motion
the acceleration of an object due to an applied force is in the direction of the force and given by
For our spring-mass system
2 Undamped Free Vibrations
no damping no external force
(general solution)
(particular solutions)
A and B are arbitary constants determined from initial conditions
2 Undamped Free Vibrations
Periodic simple harmonic motion of the mass
2 Undamped Free Vibrations
Period of motion
Natural frequency of the vibration
Amplitude (constant in time)
Phase or phase angle
3 Damped Free Vibrations
no external force
Assume an exponential solution
Then
and substituting in equation above we have
(characteristic equation)
3 Damped Free Vibrations
Solutions to characteristic equation
The solution y decays as t goes to infinity regardless the values of A and B
Damping gradually dissipates energy
overdamped
critically damped
underdamped
3 Damped Free Vibrations
The most interesting case is underdamping ie
3 Damped Free Vibrations Small Damping
4 Forced Vibrations
no damping
Periodic external force
Case 1
4 Forced Vibrations Beats
4 Forced Vibrations Beats
Rapidly oscillatingSlowly oscillating amplitude
4 Forced Vibrations Resonance
unbounded as
Case 2
5 Will this work for the beam
The beam seems to fit the harmonic conditions
bull Force is zero when displacement is zerobull Restoring force increases with displacementbull Vibration appears periodic
The key assumptions are
bull Restoring force is linear in displacementbull Friction is linear in velocity
6 Writing as a First Order System
Matlab does not work with second order equations
However we can always rewrite a second order ODE as a system of first order equations
1048710We can then have Matlab find a numerical solution to this system
6 Writing as a First Order System
Given the second order ODE
with the initial conditions
6 Writing as a First Order System
Now we can rewrite the equation in matrix-vector form
This is a format Matlab can handle
6 Constants are not independent
Notice that in all our solutions we never have c m or k alone We always have cm or km
1048710The solution for y(t) given (mck) is the same as y(t) given (αm αc αk)
important for the inverse problem
7 Summary
We can use Matlab to generate solutions to the harmonic oscillator
1048710At first glance it seems reasonable to model a vibrating beam
1048710We donrsquot know the values of m c or k
Need to solve the inverse problem
8 References
W Boyce and RC DiPrima Elementary Differential Equations and Boundary Value Problems
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
-
1 Spring-mass system
Newtonrsquos Second Law of Motion
the acceleration of an object due to an applied force is in the direction of the force and given by
For our spring-mass system
2 Undamped Free Vibrations
no damping no external force
(general solution)
(particular solutions)
A and B are arbitary constants determined from initial conditions
2 Undamped Free Vibrations
Periodic simple harmonic motion of the mass
2 Undamped Free Vibrations
Period of motion
Natural frequency of the vibration
Amplitude (constant in time)
Phase or phase angle
3 Damped Free Vibrations
no external force
Assume an exponential solution
Then
and substituting in equation above we have
(characteristic equation)
3 Damped Free Vibrations
Solutions to characteristic equation
The solution y decays as t goes to infinity regardless the values of A and B
Damping gradually dissipates energy
overdamped
critically damped
underdamped
3 Damped Free Vibrations
The most interesting case is underdamping ie
3 Damped Free Vibrations Small Damping
4 Forced Vibrations
no damping
Periodic external force
Case 1
4 Forced Vibrations Beats
4 Forced Vibrations Beats
Rapidly oscillatingSlowly oscillating amplitude
4 Forced Vibrations Resonance
unbounded as
Case 2
5 Will this work for the beam
The beam seems to fit the harmonic conditions
bull Force is zero when displacement is zerobull Restoring force increases with displacementbull Vibration appears periodic
The key assumptions are
bull Restoring force is linear in displacementbull Friction is linear in velocity
6 Writing as a First Order System
Matlab does not work with second order equations
However we can always rewrite a second order ODE as a system of first order equations
1048710We can then have Matlab find a numerical solution to this system
6 Writing as a First Order System
Given the second order ODE
with the initial conditions
6 Writing as a First Order System
Now we can rewrite the equation in matrix-vector form
This is a format Matlab can handle
6 Constants are not independent
Notice that in all our solutions we never have c m or k alone We always have cm or km
1048710The solution for y(t) given (mck) is the same as y(t) given (αm αc αk)
important for the inverse problem
7 Summary
We can use Matlab to generate solutions to the harmonic oscillator
1048710At first glance it seems reasonable to model a vibrating beam
1048710We donrsquot know the values of m c or k
Need to solve the inverse problem
8 References
W Boyce and RC DiPrima Elementary Differential Equations and Boundary Value Problems
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
-
2 Undamped Free Vibrations
no damping no external force
(general solution)
(particular solutions)
A and B are arbitary constants determined from initial conditions
2 Undamped Free Vibrations
Periodic simple harmonic motion of the mass
2 Undamped Free Vibrations
Period of motion
Natural frequency of the vibration
Amplitude (constant in time)
Phase or phase angle
3 Damped Free Vibrations
no external force
Assume an exponential solution
Then
and substituting in equation above we have
(characteristic equation)
3 Damped Free Vibrations
Solutions to characteristic equation
The solution y decays as t goes to infinity regardless the values of A and B
Damping gradually dissipates energy
overdamped
critically damped
underdamped
3 Damped Free Vibrations
The most interesting case is underdamping ie
3 Damped Free Vibrations Small Damping
4 Forced Vibrations
no damping
Periodic external force
Case 1
4 Forced Vibrations Beats
4 Forced Vibrations Beats
Rapidly oscillatingSlowly oscillating amplitude
4 Forced Vibrations Resonance
unbounded as
Case 2
5 Will this work for the beam
The beam seems to fit the harmonic conditions
bull Force is zero when displacement is zerobull Restoring force increases with displacementbull Vibration appears periodic
The key assumptions are
bull Restoring force is linear in displacementbull Friction is linear in velocity
6 Writing as a First Order System
Matlab does not work with second order equations
However we can always rewrite a second order ODE as a system of first order equations
1048710We can then have Matlab find a numerical solution to this system
6 Writing as a First Order System
Given the second order ODE
with the initial conditions
6 Writing as a First Order System
Now we can rewrite the equation in matrix-vector form
This is a format Matlab can handle
6 Constants are not independent
Notice that in all our solutions we never have c m or k alone We always have cm or km
1048710The solution for y(t) given (mck) is the same as y(t) given (αm αc αk)
important for the inverse problem
7 Summary
We can use Matlab to generate solutions to the harmonic oscillator
1048710At first glance it seems reasonable to model a vibrating beam
1048710We donrsquot know the values of m c or k
Need to solve the inverse problem
8 References
W Boyce and RC DiPrima Elementary Differential Equations and Boundary Value Problems
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
-
2 Undamped Free Vibrations
Periodic simple harmonic motion of the mass
2 Undamped Free Vibrations
Period of motion
Natural frequency of the vibration
Amplitude (constant in time)
Phase or phase angle
3 Damped Free Vibrations
no external force
Assume an exponential solution
Then
and substituting in equation above we have
(characteristic equation)
3 Damped Free Vibrations
Solutions to characteristic equation
The solution y decays as t goes to infinity regardless the values of A and B
Damping gradually dissipates energy
overdamped
critically damped
underdamped
3 Damped Free Vibrations
The most interesting case is underdamping ie
3 Damped Free Vibrations Small Damping
4 Forced Vibrations
no damping
Periodic external force
Case 1
4 Forced Vibrations Beats
4 Forced Vibrations Beats
Rapidly oscillatingSlowly oscillating amplitude
4 Forced Vibrations Resonance
unbounded as
Case 2
5 Will this work for the beam
The beam seems to fit the harmonic conditions
bull Force is zero when displacement is zerobull Restoring force increases with displacementbull Vibration appears periodic
The key assumptions are
bull Restoring force is linear in displacementbull Friction is linear in velocity
6 Writing as a First Order System
Matlab does not work with second order equations
However we can always rewrite a second order ODE as a system of first order equations
1048710We can then have Matlab find a numerical solution to this system
6 Writing as a First Order System
Given the second order ODE
with the initial conditions
6 Writing as a First Order System
Now we can rewrite the equation in matrix-vector form
This is a format Matlab can handle
6 Constants are not independent
Notice that in all our solutions we never have c m or k alone We always have cm or km
1048710The solution for y(t) given (mck) is the same as y(t) given (αm αc αk)
important for the inverse problem
7 Summary
We can use Matlab to generate solutions to the harmonic oscillator
1048710At first glance it seems reasonable to model a vibrating beam
1048710We donrsquot know the values of m c or k
Need to solve the inverse problem
8 References
W Boyce and RC DiPrima Elementary Differential Equations and Boundary Value Problems
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
-
2 Undamped Free Vibrations
Period of motion
Natural frequency of the vibration
Amplitude (constant in time)
Phase or phase angle
3 Damped Free Vibrations
no external force
Assume an exponential solution
Then
and substituting in equation above we have
(characteristic equation)
3 Damped Free Vibrations
Solutions to characteristic equation
The solution y decays as t goes to infinity regardless the values of A and B
Damping gradually dissipates energy
overdamped
critically damped
underdamped
3 Damped Free Vibrations
The most interesting case is underdamping ie
3 Damped Free Vibrations Small Damping
4 Forced Vibrations
no damping
Periodic external force
Case 1
4 Forced Vibrations Beats
4 Forced Vibrations Beats
Rapidly oscillatingSlowly oscillating amplitude
4 Forced Vibrations Resonance
unbounded as
Case 2
5 Will this work for the beam
The beam seems to fit the harmonic conditions
bull Force is zero when displacement is zerobull Restoring force increases with displacementbull Vibration appears periodic
The key assumptions are
bull Restoring force is linear in displacementbull Friction is linear in velocity
6 Writing as a First Order System
Matlab does not work with second order equations
However we can always rewrite a second order ODE as a system of first order equations
1048710We can then have Matlab find a numerical solution to this system
6 Writing as a First Order System
Given the second order ODE
with the initial conditions
6 Writing as a First Order System
Now we can rewrite the equation in matrix-vector form
This is a format Matlab can handle
6 Constants are not independent
Notice that in all our solutions we never have c m or k alone We always have cm or km
1048710The solution for y(t) given (mck) is the same as y(t) given (αm αc αk)
important for the inverse problem
7 Summary
We can use Matlab to generate solutions to the harmonic oscillator
1048710At first glance it seems reasonable to model a vibrating beam
1048710We donrsquot know the values of m c or k
Need to solve the inverse problem
8 References
W Boyce and RC DiPrima Elementary Differential Equations and Boundary Value Problems
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
-
3 Damped Free Vibrations
no external force
Assume an exponential solution
Then
and substituting in equation above we have
(characteristic equation)
3 Damped Free Vibrations
Solutions to characteristic equation
The solution y decays as t goes to infinity regardless the values of A and B
Damping gradually dissipates energy
overdamped
critically damped
underdamped
3 Damped Free Vibrations
The most interesting case is underdamping ie
3 Damped Free Vibrations Small Damping
4 Forced Vibrations
no damping
Periodic external force
Case 1
4 Forced Vibrations Beats
4 Forced Vibrations Beats
Rapidly oscillatingSlowly oscillating amplitude
4 Forced Vibrations Resonance
unbounded as
Case 2
5 Will this work for the beam
The beam seems to fit the harmonic conditions
bull Force is zero when displacement is zerobull Restoring force increases with displacementbull Vibration appears periodic
The key assumptions are
bull Restoring force is linear in displacementbull Friction is linear in velocity
6 Writing as a First Order System
Matlab does not work with second order equations
However we can always rewrite a second order ODE as a system of first order equations
1048710We can then have Matlab find a numerical solution to this system
6 Writing as a First Order System
Given the second order ODE
with the initial conditions
6 Writing as a First Order System
Now we can rewrite the equation in matrix-vector form
This is a format Matlab can handle
6 Constants are not independent
Notice that in all our solutions we never have c m or k alone We always have cm or km
1048710The solution for y(t) given (mck) is the same as y(t) given (αm αc αk)
important for the inverse problem
7 Summary
We can use Matlab to generate solutions to the harmonic oscillator
1048710At first glance it seems reasonable to model a vibrating beam
1048710We donrsquot know the values of m c or k
Need to solve the inverse problem
8 References
W Boyce and RC DiPrima Elementary Differential Equations and Boundary Value Problems
- Slide 1
- Slide 2
- Slide 3
- Slide 4
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3 Damped Free Vibrations
Solutions to characteristic equation
The solution y decays as t goes to infinity regardless the values of A and B
Damping gradually dissipates energy
overdamped
critically damped
underdamped
3 Damped Free Vibrations
The most interesting case is underdamping ie
3 Damped Free Vibrations Small Damping
4 Forced Vibrations
no damping
Periodic external force
Case 1
4 Forced Vibrations Beats
4 Forced Vibrations Beats
Rapidly oscillatingSlowly oscillating amplitude
4 Forced Vibrations Resonance
unbounded as
Case 2
5 Will this work for the beam
The beam seems to fit the harmonic conditions
bull Force is zero when displacement is zerobull Restoring force increases with displacementbull Vibration appears periodic
The key assumptions are
bull Restoring force is linear in displacementbull Friction is linear in velocity
6 Writing as a First Order System
Matlab does not work with second order equations
However we can always rewrite a second order ODE as a system of first order equations
1048710We can then have Matlab find a numerical solution to this system
6 Writing as a First Order System
Given the second order ODE
with the initial conditions
6 Writing as a First Order System
Now we can rewrite the equation in matrix-vector form
This is a format Matlab can handle
6 Constants are not independent
Notice that in all our solutions we never have c m or k alone We always have cm or km
1048710The solution for y(t) given (mck) is the same as y(t) given (αm αc αk)
important for the inverse problem
7 Summary
We can use Matlab to generate solutions to the harmonic oscillator
1048710At first glance it seems reasonable to model a vibrating beam
1048710We donrsquot know the values of m c or k
Need to solve the inverse problem
8 References
W Boyce and RC DiPrima Elementary Differential Equations and Boundary Value Problems
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3 Damped Free Vibrations
The most interesting case is underdamping ie
3 Damped Free Vibrations Small Damping
4 Forced Vibrations
no damping
Periodic external force
Case 1
4 Forced Vibrations Beats
4 Forced Vibrations Beats
Rapidly oscillatingSlowly oscillating amplitude
4 Forced Vibrations Resonance
unbounded as
Case 2
5 Will this work for the beam
The beam seems to fit the harmonic conditions
bull Force is zero when displacement is zerobull Restoring force increases with displacementbull Vibration appears periodic
The key assumptions are
bull Restoring force is linear in displacementbull Friction is linear in velocity
6 Writing as a First Order System
Matlab does not work with second order equations
However we can always rewrite a second order ODE as a system of first order equations
1048710We can then have Matlab find a numerical solution to this system
6 Writing as a First Order System
Given the second order ODE
with the initial conditions
6 Writing as a First Order System
Now we can rewrite the equation in matrix-vector form
This is a format Matlab can handle
6 Constants are not independent
Notice that in all our solutions we never have c m or k alone We always have cm or km
1048710The solution for y(t) given (mck) is the same as y(t) given (αm αc αk)
important for the inverse problem
7 Summary
We can use Matlab to generate solutions to the harmonic oscillator
1048710At first glance it seems reasonable to model a vibrating beam
1048710We donrsquot know the values of m c or k
Need to solve the inverse problem
8 References
W Boyce and RC DiPrima Elementary Differential Equations and Boundary Value Problems
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3 Damped Free Vibrations Small Damping
4 Forced Vibrations
no damping
Periodic external force
Case 1
4 Forced Vibrations Beats
4 Forced Vibrations Beats
Rapidly oscillatingSlowly oscillating amplitude
4 Forced Vibrations Resonance
unbounded as
Case 2
5 Will this work for the beam
The beam seems to fit the harmonic conditions
bull Force is zero when displacement is zerobull Restoring force increases with displacementbull Vibration appears periodic
The key assumptions are
bull Restoring force is linear in displacementbull Friction is linear in velocity
6 Writing as a First Order System
Matlab does not work with second order equations
However we can always rewrite a second order ODE as a system of first order equations
1048710We can then have Matlab find a numerical solution to this system
6 Writing as a First Order System
Given the second order ODE
with the initial conditions
6 Writing as a First Order System
Now we can rewrite the equation in matrix-vector form
This is a format Matlab can handle
6 Constants are not independent
Notice that in all our solutions we never have c m or k alone We always have cm or km
1048710The solution for y(t) given (mck) is the same as y(t) given (αm αc αk)
important for the inverse problem
7 Summary
We can use Matlab to generate solutions to the harmonic oscillator
1048710At first glance it seems reasonable to model a vibrating beam
1048710We donrsquot know the values of m c or k
Need to solve the inverse problem
8 References
W Boyce and RC DiPrima Elementary Differential Equations and Boundary Value Problems
- Slide 1
- Slide 2
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4 Forced Vibrations
no damping
Periodic external force
Case 1
4 Forced Vibrations Beats
4 Forced Vibrations Beats
Rapidly oscillatingSlowly oscillating amplitude
4 Forced Vibrations Resonance
unbounded as
Case 2
5 Will this work for the beam
The beam seems to fit the harmonic conditions
bull Force is zero when displacement is zerobull Restoring force increases with displacementbull Vibration appears periodic
The key assumptions are
bull Restoring force is linear in displacementbull Friction is linear in velocity
6 Writing as a First Order System
Matlab does not work with second order equations
However we can always rewrite a second order ODE as a system of first order equations
1048710We can then have Matlab find a numerical solution to this system
6 Writing as a First Order System
Given the second order ODE
with the initial conditions
6 Writing as a First Order System
Now we can rewrite the equation in matrix-vector form
This is a format Matlab can handle
6 Constants are not independent
Notice that in all our solutions we never have c m or k alone We always have cm or km
1048710The solution for y(t) given (mck) is the same as y(t) given (αm αc αk)
important for the inverse problem
7 Summary
We can use Matlab to generate solutions to the harmonic oscillator
1048710At first glance it seems reasonable to model a vibrating beam
1048710We donrsquot know the values of m c or k
Need to solve the inverse problem
8 References
W Boyce and RC DiPrima Elementary Differential Equations and Boundary Value Problems
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
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-
4 Forced Vibrations Beats
4 Forced Vibrations Beats
Rapidly oscillatingSlowly oscillating amplitude
4 Forced Vibrations Resonance
unbounded as
Case 2
5 Will this work for the beam
The beam seems to fit the harmonic conditions
bull Force is zero when displacement is zerobull Restoring force increases with displacementbull Vibration appears periodic
The key assumptions are
bull Restoring force is linear in displacementbull Friction is linear in velocity
6 Writing as a First Order System
Matlab does not work with second order equations
However we can always rewrite a second order ODE as a system of first order equations
1048710We can then have Matlab find a numerical solution to this system
6 Writing as a First Order System
Given the second order ODE
with the initial conditions
6 Writing as a First Order System
Now we can rewrite the equation in matrix-vector form
This is a format Matlab can handle
6 Constants are not independent
Notice that in all our solutions we never have c m or k alone We always have cm or km
1048710The solution for y(t) given (mck) is the same as y(t) given (αm αc αk)
important for the inverse problem
7 Summary
We can use Matlab to generate solutions to the harmonic oscillator
1048710At first glance it seems reasonable to model a vibrating beam
1048710We donrsquot know the values of m c or k
Need to solve the inverse problem
8 References
W Boyce and RC DiPrima Elementary Differential Equations and Boundary Value Problems
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
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-
4 Forced Vibrations Beats
Rapidly oscillatingSlowly oscillating amplitude
4 Forced Vibrations Resonance
unbounded as
Case 2
5 Will this work for the beam
The beam seems to fit the harmonic conditions
bull Force is zero when displacement is zerobull Restoring force increases with displacementbull Vibration appears periodic
The key assumptions are
bull Restoring force is linear in displacementbull Friction is linear in velocity
6 Writing as a First Order System
Matlab does not work with second order equations
However we can always rewrite a second order ODE as a system of first order equations
1048710We can then have Matlab find a numerical solution to this system
6 Writing as a First Order System
Given the second order ODE
with the initial conditions
6 Writing as a First Order System
Now we can rewrite the equation in matrix-vector form
This is a format Matlab can handle
6 Constants are not independent
Notice that in all our solutions we never have c m or k alone We always have cm or km
1048710The solution for y(t) given (mck) is the same as y(t) given (αm αc αk)
important for the inverse problem
7 Summary
We can use Matlab to generate solutions to the harmonic oscillator
1048710At first glance it seems reasonable to model a vibrating beam
1048710We donrsquot know the values of m c or k
Need to solve the inverse problem
8 References
W Boyce and RC DiPrima Elementary Differential Equations and Boundary Value Problems
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
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-
4 Forced Vibrations Resonance
unbounded as
Case 2
5 Will this work for the beam
The beam seems to fit the harmonic conditions
bull Force is zero when displacement is zerobull Restoring force increases with displacementbull Vibration appears periodic
The key assumptions are
bull Restoring force is linear in displacementbull Friction is linear in velocity
6 Writing as a First Order System
Matlab does not work with second order equations
However we can always rewrite a second order ODE as a system of first order equations
1048710We can then have Matlab find a numerical solution to this system
6 Writing as a First Order System
Given the second order ODE
with the initial conditions
6 Writing as a First Order System
Now we can rewrite the equation in matrix-vector form
This is a format Matlab can handle
6 Constants are not independent
Notice that in all our solutions we never have c m or k alone We always have cm or km
1048710The solution for y(t) given (mck) is the same as y(t) given (αm αc αk)
important for the inverse problem
7 Summary
We can use Matlab to generate solutions to the harmonic oscillator
1048710At first glance it seems reasonable to model a vibrating beam
1048710We donrsquot know the values of m c or k
Need to solve the inverse problem
8 References
W Boyce and RC DiPrima Elementary Differential Equations and Boundary Value Problems
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
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5 Will this work for the beam
The beam seems to fit the harmonic conditions
bull Force is zero when displacement is zerobull Restoring force increases with displacementbull Vibration appears periodic
The key assumptions are
bull Restoring force is linear in displacementbull Friction is linear in velocity
6 Writing as a First Order System
Matlab does not work with second order equations
However we can always rewrite a second order ODE as a system of first order equations
1048710We can then have Matlab find a numerical solution to this system
6 Writing as a First Order System
Given the second order ODE
with the initial conditions
6 Writing as a First Order System
Now we can rewrite the equation in matrix-vector form
This is a format Matlab can handle
6 Constants are not independent
Notice that in all our solutions we never have c m or k alone We always have cm or km
1048710The solution for y(t) given (mck) is the same as y(t) given (αm αc αk)
important for the inverse problem
7 Summary
We can use Matlab to generate solutions to the harmonic oscillator
1048710At first glance it seems reasonable to model a vibrating beam
1048710We donrsquot know the values of m c or k
Need to solve the inverse problem
8 References
W Boyce and RC DiPrima Elementary Differential Equations and Boundary Value Problems
- Slide 1
- Slide 2
- Slide 3
- Slide 4
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6 Writing as a First Order System
Matlab does not work with second order equations
However we can always rewrite a second order ODE as a system of first order equations
1048710We can then have Matlab find a numerical solution to this system
6 Writing as a First Order System
Given the second order ODE
with the initial conditions
6 Writing as a First Order System
Now we can rewrite the equation in matrix-vector form
This is a format Matlab can handle
6 Constants are not independent
Notice that in all our solutions we never have c m or k alone We always have cm or km
1048710The solution for y(t) given (mck) is the same as y(t) given (αm αc αk)
important for the inverse problem
7 Summary
We can use Matlab to generate solutions to the harmonic oscillator
1048710At first glance it seems reasonable to model a vibrating beam
1048710We donrsquot know the values of m c or k
Need to solve the inverse problem
8 References
W Boyce and RC DiPrima Elementary Differential Equations and Boundary Value Problems
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
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-
6 Writing as a First Order System
Given the second order ODE
with the initial conditions
6 Writing as a First Order System
Now we can rewrite the equation in matrix-vector form
This is a format Matlab can handle
6 Constants are not independent
Notice that in all our solutions we never have c m or k alone We always have cm or km
1048710The solution for y(t) given (mck) is the same as y(t) given (αm αc αk)
important for the inverse problem
7 Summary
We can use Matlab to generate solutions to the harmonic oscillator
1048710At first glance it seems reasonable to model a vibrating beam
1048710We donrsquot know the values of m c or k
Need to solve the inverse problem
8 References
W Boyce and RC DiPrima Elementary Differential Equations and Boundary Value Problems
- Slide 1
- Slide 2
- Slide 3
- Slide 4
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6 Writing as a First Order System
Now we can rewrite the equation in matrix-vector form
This is a format Matlab can handle
6 Constants are not independent
Notice that in all our solutions we never have c m or k alone We always have cm or km
1048710The solution for y(t) given (mck) is the same as y(t) given (αm αc αk)
important for the inverse problem
7 Summary
We can use Matlab to generate solutions to the harmonic oscillator
1048710At first glance it seems reasonable to model a vibrating beam
1048710We donrsquot know the values of m c or k
Need to solve the inverse problem
8 References
W Boyce and RC DiPrima Elementary Differential Equations and Boundary Value Problems
- Slide 1
- Slide 2
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6 Constants are not independent
Notice that in all our solutions we never have c m or k alone We always have cm or km
1048710The solution for y(t) given (mck) is the same as y(t) given (αm αc αk)
important for the inverse problem
7 Summary
We can use Matlab to generate solutions to the harmonic oscillator
1048710At first glance it seems reasonable to model a vibrating beam
1048710We donrsquot know the values of m c or k
Need to solve the inverse problem
8 References
W Boyce and RC DiPrima Elementary Differential Equations and Boundary Value Problems
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
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- Slide 22
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7 Summary
We can use Matlab to generate solutions to the harmonic oscillator
1048710At first glance it seems reasonable to model a vibrating beam
1048710We donrsquot know the values of m c or k
Need to solve the inverse problem
8 References
W Boyce and RC DiPrima Elementary Differential Equations and Boundary Value Problems
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
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8 References
W Boyce and RC DiPrima Elementary Differential Equations and Boundary Value Problems
- Slide 1
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