Generalized Single Degree of Freedom Systems
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About This Presentation
Generalized Single Degree of Freedom Systems
Slide Content
Generalized Single Degree of Freedom Systems
PVD, Generalized Parameters, Rayleigh Quotient
Giacomo Boffi
http://intranet.dica.polimi.it/people/boffi‐giacomo
Dipartimento di Ingegneria Civile Ambientale e Territoriale
Politecnico di Milano
March 24, 2020
PVD, Generalized Parameters, Rayleigh Quotient
Giacomo Boffi
http://intranet.dica.polimi.it/people/boffi‐giacomo
Dipartimento di Ingegneria Civile Ambientale e Territoriale
Politecnico di Milano
March 24, 2020
Generalized
SDOF
Giacomo Boffi
Outline
SDOF
Giacomo Boffi
Outline
Generalized
SDOF
Giacomo Boffi
Section 1
Introductory Remarks
SDOF
Giacomo Boffi
Section 1
Introductory Remarks
Generalized
SDOF
Giacomo Boffi
Introductory Remarks
Until now ourSDOF’s were described as composed by a single mass connected to a
fixed reference by means of a spring and a damper.
While the mass‐spring is a useful representation, many different, more complex
systems can be studied asSDOFsystems, either exactly or under some simplifying
assumption.
1SDOFrigid body assemblages, where the flexibility is concentrated in a number of
springs and dampers, can be studied, e.g., using the Principle of Virtual
Displacements and the D’Alembert Principle.
2simple structural systems can be studied, in an approximate manner, assuming a
fixed pattern of displacements, whose amplitude (the single degree of freedom)
varies with time.
SDOF
Giacomo Boffi
Introductory Remarks
Until now ourSDOF’s were described as composed by a single mass connected to a
fixed reference by means of a spring and a damper.
While the mass‐spring is a useful representation, many different, more complex
systems can be studied asSDOFsystems, either exactly or under some simplifying
assumption.
1SDOFrigid body assemblages, where the flexibility is concentrated in a number of
springs and dampers, can be studied, e.g., using the Principle of Virtual
Displacements and the D’Alembert Principle.
2simple structural systems can be studied, in an approximate manner, assuming a
fixed pattern of displacements, whose amplitude (the single degree of freedom)
varies with time.
Generalized
SDOF
Giacomo Boffi
Introductory Remarks
Until now ourSDOF’s were described as composed by a single mass connected to a
fixed reference by means of a spring and a damper.
While the mass‐spring is a useful representation, many different, more complex
systems can be studied asSDOFsystems, either exactly or under some simplifying
assumption.
1SDOFrigid body assemblages, where the flexibility is concentrated in a number of
springs and dampers, can be studied, e.g., using the Principle of Virtual
Displacements and the D’Alembert Principle.
2simple structural systems can be studied, in an approximate manner, assuming a
fixed pattern of displacements, whose amplitude (the single degree of freedom)
varies with time.
SDOF
Giacomo Boffi
Introductory Remarks
Until now ourSDOF’s were described as composed by a single mass connected to a
fixed reference by means of a spring and a damper.
While the mass‐spring is a useful representation, many different, more complex
systems can be studied asSDOFsystems, either exactly or under some simplifying
assumption.
1SDOFrigid body assemblages, where the flexibility is concentrated in a number of
springs and dampers, can be studied, e.g., using the Principle of Virtual
Displacements and the D’Alembert Principle.
2simple structural systems can be studied, in an approximate manner, assuming a
fixed pattern of displacements, whose amplitude (the single degree of freedom)
varies with time.
Generalized
SDOF
Giacomo Boffi
Further Remarks on Rigid Assemblages
Today we restrict our consideration to plane, 2‐D systems.
In rigid body assemblages the limitation to a single shape of displacement is a
consequence of the configuration of the system, i.e., the disposition of supports and
internal hinges.
When the equation of motion is written in terms of a single parameter and its time
derivatives, the terms that figure as coefficients in the equation of motion can be
regarded as thegeneralisedproperties of the assemblage: generalised mass, damping
and stiffness on left hand, generalised loading on right hand.
??????
⋆
̈?????? + �
⋆
̇?????? + ??????
⋆
?????? = ??????
⋆
(�)
SDOF
Giacomo Boffi
Further Remarks on Rigid Assemblages
Today we restrict our consideration to plane, 2‐D systems.
In rigid body assemblages the limitation to a single shape of displacement is a
consequence of the configuration of the system, i.e., the disposition of supports and
internal hinges.
When the equation of motion is written in terms of a single parameter and its time
derivatives, the terms that figure as coefficients in the equation of motion can be
regarded as thegeneralisedproperties of the assemblage: generalised mass, damping
and stiffness on left hand, generalised loading on right hand.
??????
⋆
̈?????? + �
⋆
̇?????? + ??????
⋆
?????? = ??????
⋆
(�)
Generalized
SDOF
Giacomo Boffi
Further Remarks on Continuous Systems
Continuous systems have an infinite variety of deformation patterns.
By restricting the deformation to a single shape of varying amplitude, we introduce an
infinity of internal contstraints that limit the infinite variety of deformation patterns,
but under this assumption the system configuration is mathematically described by a
single parameter, so that
ourmodelcan be analysed in exactly the same way as a strictSDOFsystem,
we can compute thegeneralisedmass, damping, stiffness properties of theSDOF
modelof the continuous system.
SDOF
Giacomo Boffi
Further Remarks on Continuous Systems
Continuous systems have an infinite variety of deformation patterns.
By restricting the deformation to a single shape of varying amplitude, we introduce an
infinity of internal contstraints that limit the infinite variety of deformation patterns,
but under this assumption the system configuration is mathematically described by a
single parameter, so that
ourmodelcan be analysed in exactly the same way as a strictSDOFsystem,
we can compute thegeneralisedmass, damping, stiffness properties of theSDOF
modelof the continuous system.
Generalized
SDOF
Giacomo Boffi
Final Remarks on GeneralisedSDOFSystems
From the previous comments, it should be apparent that everything we have seen
regarding the behaviour and the integration of the equation of motion of properSDOF
systems applies to rigid body assemblages and toSDOFmodels of flexible systems,
provided that we have the means for determining thegeneralisedproperties of the
dynamical systems under investigation.
SDOF
Giacomo Boffi
Final Remarks on GeneralisedSDOFSystems
From the previous comments, it should be apparent that everything we have seen
regarding the behaviour and the integration of the equation of motion of properSDOF
systems applies to rigid body assemblages and toSDOFmodels of flexible systems,
provided that we have the means for determining thegeneralisedproperties of the
dynamical systems under investigation.
Generalized
SDOF
Giacomo Boffi
Section 2
Assemblage of Rigid Bodies
SDOF
Giacomo Boffi
Section 2
Assemblage of Rigid Bodies
Generalized
SDOF
Giacomo Boffi
Assemblages of Rigid Bodies
planar, or bidimensional, rigid bodies, constrained to move in a plane,
the flexibility isconcentratedin discrete elements, springs and dampers,
rigid bodies are connected to a fixed reference and to each other by means of
springs, dampers and smooth, bilateral constraints (read hinges, double
pendulums and rollers),
inertial forces are distributed forces, acting on each material point of each rigid
body, their resultant can be described by
an inertial force applied to the centre of mass of the body, the product of the
acceleration vector of the centre of mass itself and the total mass of the rigid body,
� =∫d??????
an inertial couple, the product of the angular acceleration and the moment of
inertia??????of the rigid body,?????? =∫(??????
2
+ ??????
2
)d??????.
SDOF
Giacomo Boffi
Assemblages of Rigid Bodies
planar, or bidimensional, rigid bodies, constrained to move in a plane,
the flexibility isconcentratedin discrete elements, springs and dampers,
rigid bodies are connected to a fixed reference and to each other by means of
springs, dampers and smooth, bilateral constraints (read hinges, double
pendulums and rollers),
inertial forces are distributed forces, acting on each material point of each rigid
body, their resultant can be described by
an inertial force applied to the centre of mass of the body, the product of the
acceleration vector of the centre of mass itself and the total mass of the rigid body,
� =∫d??????
an inertial couple, the product of the angular acceleration and the moment of
inertia??????of the rigid body,?????? =∫(??????
2
+ ??????
2
)d??????.
Generalized
SDOF
Giacomo Boffi
Assemblages of Rigid Bodies
planar, or bidimensional, rigid bodies, constrained to move in a plane,
the flexibility isconcentratedin discrete elements, springs and dampers,
rigid bodies are connected to a fixed reference and to each other by means of
springs, dampers and smooth, bilateral constraints (read hinges, double
pendulums and rollers),
inertial forces are distributed forces, acting on each material point of each rigid
body, their resultant can be described by
an inertial force applied to the centre of mass of the body, the product of the
acceleration vector of the centre of mass itself and the total mass of the rigid body,
� =∫d??????
an inertial couple, the product of the angular acceleration and the moment of
inertia??????of the rigid body,?????? =∫(??????
2
+ ??????
2
)d??????.
SDOF
Giacomo Boffi
Assemblages of Rigid Bodies
planar, or bidimensional, rigid bodies, constrained to move in a plane,
the flexibility isconcentratedin discrete elements, springs and dampers,
rigid bodies are connected to a fixed reference and to each other by means of
springs, dampers and smooth, bilateral constraints (read hinges, double
pendulums and rollers),
inertial forces are distributed forces, acting on each material point of each rigid
body, their resultant can be described by
an inertial force applied to the centre of mass of the body, the product of the
acceleration vector of the centre of mass itself and the total mass of the rigid body,
� =∫d??????
an inertial couple, the product of the angular acceleration and the moment of
inertia??????of the rigid body,?????? =∫(??????
2
+ ??????
2
)d??????.
Generalized
SDOF
Giacomo Boffi
Assemblages of Rigid Bodies
planar, or bidimensional, rigid bodies, constrained to move in a plane,
the flexibility isconcentratedin discrete elements, springs and dampers,
rigid bodies are connected to a fixed reference and to each other by means of
springs, dampers and smooth, bilateral constraints (read hinges, double
pendulums and rollers),
inertial forces are distributed forces, acting on each material point of each rigid
body, their resultant can be described by
an inertial force applied to the centre of mass of the body, the product of the
acceleration vector of the centre of mass itself and the total mass of the rigid body,
� =∫d??????
an inertial couple, the product of the angular acceleration and the moment of
inertia??????of the rigid body,?????? =∫(??????
2
+ ??????
2
)d??????.
SDOF
Giacomo Boffi
Assemblages of Rigid Bodies
planar, or bidimensional, rigid bodies, constrained to move in a plane,
the flexibility isconcentratedin discrete elements, springs and dampers,
rigid bodies are connected to a fixed reference and to each other by means of
springs, dampers and smooth, bilateral constraints (read hinges, double
pendulums and rollers),
inertial forces are distributed forces, acting on each material point of each rigid
body, their resultant can be described by
an inertial force applied to the centre of mass of the body, the product of the
acceleration vector of the centre of mass itself and the total mass of the rigid body,
� =∫d??????
an inertial couple, the product of the angular acceleration and the moment of
inertia??????of the rigid body,?????? =∫(??????
2
+ ??????
2
)d??????.
Generalized
SDOF
Giacomo Boffi
Assemblages of Rigid Bodies
planar, or bidimensional, rigid bodies, constrained to move in a plane,
the flexibility isconcentratedin discrete elements, springs and dampers,
rigid bodies are connected to a fixed reference and to each other by means of
springs, dampers and smooth, bilateral constraints (read hinges, double
pendulums and rollers),
inertial forces are distributed forces, acting on each material point of each rigid
body, their resultant can be described by
an inertial force applied to the centre of mass of the body, the product of the
acceleration vector of the centre of mass itself and the total mass of the rigid body,
� =∫d??????
an inertial couple, the product of the angular acceleration and the moment of
inertia??????of the rigid body,?????? =∫(??????
2
+ ??????
2
)d??????.
SDOF
Giacomo Boffi
Assemblages of Rigid Bodies
planar, or bidimensional, rigid bodies, constrained to move in a plane,
the flexibility isconcentratedin discrete elements, springs and dampers,
rigid bodies are connected to a fixed reference and to each other by means of
springs, dampers and smooth, bilateral constraints (read hinges, double
pendulums and rollers),
inertial forces are distributed forces, acting on each material point of each rigid
body, their resultant can be described by
an inertial force applied to the centre of mass of the body, the product of the
acceleration vector of the centre of mass itself and the total mass of the rigid body,
� =∫d??????
an inertial couple, the product of the angular acceleration and the moment of
inertia??????of the rigid body,?????? =∫(??????
2
+ ??????
2
)d??????.
Generalized
SDOF
Giacomo Boffi
Assemblages of Rigid Bodies
planar, or bidimensional, rigid bodies, constrained to move in a plane,
the flexibility isconcentratedin discrete elements, springs and dampers,
rigid bodies are connected to a fixed reference and to each other by means of
springs, dampers and smooth, bilateral constraints (read hinges, double
pendulums and rollers),
inertial forces are distributed forces, acting on each material point of each rigid
body, their resultant can be described by
an inertial force applied to the centre of mass of the body, the product of the
acceleration vector of the centre of mass itself and the total mass of the rigid body,
� =∫d??????
an inertial couple, the product of the angular acceleration and the moment of
inertia??????of the rigid body,?????? =∫(??????
2
+ ??????
2
)d??????.
SDOF
Giacomo Boffi
Assemblages of Rigid Bodies
planar, or bidimensional, rigid bodies, constrained to move in a plane,
the flexibility isconcentratedin discrete elements, springs and dampers,
rigid bodies are connected to a fixed reference and to each other by means of
springs, dampers and smooth, bilateral constraints (read hinges, double
pendulums and rollers),
inertial forces are distributed forces, acting on each material point of each rigid
body, their resultant can be described by
an inertial force applied to the centre of mass of the body, the product of the
acceleration vector of the centre of mass itself and the total mass of the rigid body,
� =∫d??????
an inertial couple, the product of the angular acceleration and the moment of
inertia??????of the rigid body,?????? =∫(??????
2
+ ??????
2
)d??????.
Generalized
SDOF
Giacomo Boffi
Rigid Barx
G
L
Unit mass ̄?????? =constant,
Length�,
Centre of Mass??????
??????= �/2,
Total Mass?????? = ̄??????�,
Moment of Inertia?????? = ??????
�
2
12
= ̄??????
�
3
12
SDOF
Giacomo Boffi
Rigid Barx
G
L
Unit mass ̄?????? =constant,
Length�,
Centre of Mass??????
??????= �/2,
Total Mass?????? = ̄??????�,
Moment of Inertia?????? = ??????
�
2
12
= ̄??????
�
3
12
Generalized
SDOF
Giacomo Boffi
Rigid RectangleG
y
a
b
Unit mass?????? =constant,
Sides�, �
Centre of Mass??????
??????= �/2, ??????
??????= �/2
Total Mass?????? = ??????��,
Moment of Inertia?????? = ??????
�
2
+ �
2
12
= ??????
�
3
� + ��
3
12
SDOF
Giacomo Boffi
Rigid RectangleG
y
a
b
Unit mass?????? =constant,
Sides�, �
Centre of Mass??????
??????= �/2, ??????
??????= �/2
Total Mass?????? = ??????��,
Moment of Inertia?????? = ??????
�
2
+ �
2
12
= ??????
�
3
� + ��
3
12
Generalized
SDOF
Giacomo Boffi
Rigid Triangle
For a right triangle.y
G
a
b
Unit mass?????? =constant,
Sides�, �
Centre of Mass??????
??????= �/3, ??????
??????= �/3
Total Mass?????? = ??????��/2,
Moment of Inertia?????? = ??????
�
2
+ �
2
18
= ??????
�
3
� + ��
3
36
SDOF
Giacomo Boffi
Rigid Triangle
For a right triangle.y
G
a
b
Unit mass?????? =constant,
Sides�, �
Centre of Mass??????
??????= �/3, ??????
??????= �/3
Total Mass?????? = ??????��/2,
Moment of Inertia?????? = ??????
�
2
+ �
2
18
= ??????
�
3
� + ��
3
36
Generalized
SDOF
Giacomo Boffi
Rigid Ovalx
y
a
b
Unit mass?????? =constant,
Axes�, �
Centre of Mass??????
??????= ??????
??????= 0
Total Mass?????? = ??????
??????��
4
,
Moment of Inertia?????? = ??????
�
2
+ �
2
16
When� = � = ?????? = 2??????the oval is a circle:
?????? = ?????? ????????????
2
, ?????? = ??????
??????
2
2
= ??????
????????????
4
2
.
SDOF
Giacomo Boffi
Rigid Ovalx
y
a
b
Unit mass?????? =constant,
Axes�, �
Centre of Mass??????
??????= ??????
??????= 0
Total Mass?????? = ??????
??????��
4
,
Moment of Inertia?????? = ??????
�
2
+ �
2
16
When� = � = ?????? = 2??????the oval is a circle:
?????? = ?????? ????????????
2
, ?????? = ??????
??????
2
2
= ??????
????????????
4
2
.
Generalized
SDOF
Giacomo Boffi
Rigid Ovalx
y
a
b
Unit mass?????? =constant,
Axes�, �
Centre of Mass??????
??????= ??????
??????= 0
Total Mass?????? = ??????
??????��
4
,
Moment of Inertia?????? = ??????
�
2
+ �
2
16
When� = � = ?????? = 2??????the oval is a circle:
?????? = ?????? ????????????
2
, ?????? = ??????
??????
2
2
= ??????
????????????
4
2
.
SDOF
Giacomo Boffi
Rigid Ovalx
y
a
b
Unit mass?????? =constant,
Axes�, �
Centre of Mass??????
??????= ??????
??????= 0
Total Mass?????? = ??????
??????��
4
,
Moment of Inertia?????? = ??????
�
2
+ �
2
16
When� = � = ?????? = 2??????the oval is a circle:
?????? = ?????? ????????????
2
, ?????? = ??????
??????
2
2
= ??????
????????????
4
2
.
Generalized
SDOF
Giacomo Boffi
trabacolo1c k c k
2 211
N
m , J
2 2
p(x,t) = P x/a f(t)
a 2 a a a a a
The mass of the left bar is??????
1= ̄?????? 4�and its moment of inertia is
??????
1= ??????
1
(4�)
2
12
= 4�
2
??????
1/3.
The maximum value of the external load is??????
max= ?????? 4�/� = 4??????and the resultant of
triangular load is?????? = 4?????? × 4�/2 = 8??????�
SDOF
Giacomo Boffi
trabacolo1c k c k
2 211
N
m , J
2 2
p(x,t) = P x/a f(t)
a 2 a a a a a
The mass of the left bar is??????
1= ̄?????? 4�and its moment of inertia is
??????
1= ??????
1
(4�)
2
12
= 4�
2
??????
1/3.
The maximum value of the external load is??????
max= ?????? 4�/� = 4??????and the resultant of
triangular load is?????? = 4?????? × 4�/2 = 8??????�
Generalized
SDOF
Giacomo Boffi
Forces and Virtual Displacements
�
1
̇�
4
??????
1
̈�
2
3??????
1�
4
�
2
̇�
2??????
2
̈�
3
??????�
3
�
�(�)
??????
2
̈�
3�
8??????� ??????(�)
??????
1
̈�
4�
??????�
4
??????�
2
3
??????�
4
??????�2
??????�
3
??????�
3
??????�
????????????
2= ??????�/(3�)????????????
1= ??????�/(4�)
� = 7� − 4�cos??????
1− 3�cos??????
2, ??????� = 4�sin??????
1????????????
1+ 3�sin??????
2????????????
2
????????????
1= ??????�/(4�), ????????????
2= ??????�/(3�)
sin??????
1≈ �/(4�),sin??????
2≈ �/(3�)
??????� =@
1
4??????
+
1
3??????
A� ??????� =
7
12??????
� ??????�
SDOF
Giacomo Boffi
Forces and Virtual Displacements
�
1
̇�
4
??????
1
̈�
2
3??????
1�
4
�
2
̇�
2??????
2
̈�
3
??????�
3
�
�(�)
??????
2
̈�
3�
8??????� ??????(�)
??????
1
̈�
4�
??????�
4
??????�
2
3
??????�
4
??????�2
??????�
3
??????�
3
??????�
????????????
2= ??????�/(3�)????????????
1= ??????�/(4�)
� = 7� − 4�cos??????
1− 3�cos??????
2, ??????� = 4�sin??????
1????????????
1+ 3�sin??????
2????????????
2
????????????
1= ??????�/(4�), ????????????
2= ??????�/(3�)
sin??????
1≈ �/(4�),sin??????
2≈ �/(3�)
??????� =@
1
4??????
+
1
3??????
A� ??????� =
7
12??????
� ??????�
Generalized
SDOF
Giacomo Boffi
Forces and Virtual Displacements
�
1
̇�
4
??????
1
̈�
2
3??????
1�
4
�
2
̇�
2??????
2
̈�
3
??????�
3
�
�(�)
??????
2
̈�
3�
8??????� ??????(�)
??????
1
̈�
4�
??????�
4
??????�
2
3
??????�
4
??????�2
??????�
3
??????�
3
??????�
????????????
2= ??????�/(3�)????????????
1= ??????�/(4�)
� = 7� − 4�cos??????
1− 3�cos??????
2, ??????� = 4�sin??????
1????????????
1+ 3�sin??????
2????????????
2
????????????
1= ??????�/(4�), ????????????
2= ??????�/(3�)
sin??????
1≈ �/(4�),sin??????
2≈ �/(3�)
??????� =@
1
4??????
+
1
3??????
A� ??????� =
7
12??????
� ??????�
SDOF
Giacomo Boffi
Forces and Virtual Displacements
�
1
̇�
4
??????
1
̈�
2
3??????
1�
4
�
2
̇�
2??????
2
̈�
3
??????�
3
�
�(�)
??????
2
̈�
3�
8??????� ??????(�)
??????
1
̈�
4�
??????�
4
??????�
2
3
??????�
4
??????�2
??????�
3
??????�
3
??????�
????????????
2= ??????�/(3�)????????????
1= ??????�/(4�)
� = 7� − 4�cos??????
1− 3�cos??????
2, ??????� = 4�sin??????
1????????????
1+ 3�sin??????
2????????????
2
????????????
1= ??????�/(4�), ????????????
2= ??????�/(3�)
sin??????
1≈ �/(4�),sin??????
2≈ �/(3�)
??????� =@
1
4??????
+
1
3??????
A� ??????� =
7
12??????
� ??????�
Generalized
SDOF
Giacomo Boffi
Principle of Virtual Displacements
�
1
̇�
4
??????
1
̈�
2
3??????
1�
4
�
2
̇�
2??????
2
̈�
3
??????�
3
�
�(�)
??????
2
̈�
3�
8??????� ??????(�)
??????
1
̈�
4�
??????�
4
??????�
2
3
??????�
4
??????�2
??????�
3
??????�
3
??????�
????????????
2= ??????�/(3�)????????????
1= ??????�/(4�)
????????????
I= −??????
1
̈�
2
??????�
2
− ??????
1
̈�
4�
??????�
4�
− ??????
2
2̈�
3
2??????�
3
− ??????
2
̈�
3�
??????�
3�
= −F
??????
1
4
+ 4
??????
2
9
+
??????
1
16�
2
+
??????
2
9�
2
G̈� ??????�
????????????
D= −�
1
̇�
4
??????�
4
− −�
2� ??????� = −(�
2+ �
1/16)̇� ??????�
????????????
S= −??????
1
3�
4
3??????�
4
− ??????
2
�
3
??????�
3
= −F
9??????
1
16
+
??????
2
9
G� ??????�
????????????
Ext= 8??????� ??????(�)
2??????�
3
+ ??????
7
12�
� ??????�
SDOF
Giacomo Boffi
Principle of Virtual Displacements
�
1
̇�
4
??????
1
̈�
2
3??????
1�
4
�
2
̇�
2??????
2
̈�
3
??????�
3
�
�(�)
??????
2
̈�
3�
8??????� ??????(�)
??????
1
̈�
4�
??????�
4
??????�
2
3
??????�
4
??????�2
??????�
3
??????�
3
??????�
????????????
2= ??????�/(3�)????????????
1= ??????�/(4�)
????????????
I= −??????
1
̈�
2
??????�
2
− ??????
1
̈�
4�
??????�
4�
− ??????
2
2̈�
3
2??????�
3
− ??????
2
̈�
3�
??????�
3�
= −F
??????
1
4
+ 4
??????
2
9
+
??????
1
16�
2
+
??????
2
9�
2
G̈� ??????�
????????????
D= −�
1
̇�
4
??????�
4
− −�
2� ??????� = −(�
2+ �
1/16)̇� ??????�
????????????
S= −??????
1
3�
4
3??????�
4
− ??????
2
�
3
??????�
3
= −F
9??????
1
16
+
??????
2
9
G� ??????�
????????????
Ext= 8??????� ??????(�)
2??????�
3
+ ??????
7
12�
� ??????�
Generalized
SDOF
Giacomo Boffi
Principle of Virtual Displacements
�
1
̇�
4
??????
1
̈�
2
3??????
1�
4
�
2
̇�
2??????
2
̈�
3
??????�
3
�
�(�)
??????
2
̈�
3�
8??????� ??????(�)
??????
1
̈�
4�
??????�
4
??????�
2
3
??????�
4
??????�2
??????�
3
??????�
3
??????�
????????????
2= ??????�/(3�)????????????
1= ??????�/(4�)
????????????
I= −??????
1
̈�
2
??????�
2
− ??????
1
̈�
4�
??????�
4�
− ??????
2
2̈�
3
2??????�
3
− ??????
2
̈�
3�
??????�
3�
= −F
??????
1
4
+ 4
??????
2
9
+
??????
1
16�
2
+
??????
2
9�
2
G̈� ??????�
????????????
D= −�
1
̇�
4
??????�
4
− −�
2� ??????� = −(�
2+ �
1/16)̇� ??????�
????????????
S= −??????
1
3�
4
3??????�
4
− ??????
2
�
3
??????�
3
= −F
9??????
1
16
+
??????
2
9
G� ??????�
????????????
Ext= 8??????� ??????(�)
2??????�
3
+ ??????
7
12�
� ??????�
SDOF
Giacomo Boffi
Principle of Virtual Displacements
�
1
̇�
4
??????
1
̈�
2
3??????
1�
4
�
2
̇�
2??????
2
̈�
3
??????�
3
�
�(�)
??????
2
̈�
3�
8??????� ??????(�)
??????
1
̈�
4�
??????�
4
??????�
2
3
??????�
4
??????�2
??????�
3
??????�
3
??????�
????????????
2= ??????�/(3�)????????????
1= ??????�/(4�)
????????????
I= −??????
1
̈�
2
??????�
2
− ??????
1
̈�
4�
??????�
4�
− ??????
2
2̈�
3
2??????�
3
− ??????
2
̈�
3�
??????�
3�
= −F
??????
1
4
+ 4
??????
2
9
+
??????
1
16�
2
+
??????
2
9�
2
G̈� ??????�
????????????
D= −�
1
̇�
4
??????�
4
− −�
2� ??????� = −(�
2+ �
1/16)̇� ??????�
????????????
S= −??????
1
3�
4
3??????�
4
− ??????
2
�
3
??????�
3
= −F
9??????
1
16
+
??????
2
9
G� ??????�
????????????
Ext= 8??????� ??????(�)
2??????�
3
+ ??????
7
12�
� ??????�
Generalized
SDOF
Giacomo Boffi
Principle of Virtual Displacements
�
1
̇�
4
??????
1
̈�
2
3??????
1�
4
�
2
̇�
2??????
2
̈�
3
??????�
3
�
�(�)
??????
2
̈�
3�
8??????� ??????(�)
??????
1
̈�
4�
??????�
4
??????�
2
3
??????�
4
??????�2
??????�
3
??????�
3
??????�
????????????
2= ??????�/(3�)????????????
1= ??????�/(4�)
????????????
I= −??????
1
̈�
2
??????�
2
− ??????
1
̈�
4�
??????�
4�
− ??????
2
2̈�
3
2??????�
3
− ??????
2
̈�
3�
??????�
3�
= −F
??????
1
4
+ 4
??????
2
9
+
??????
1
16�
2
+
??????
2
9�
2
G̈� ??????�
????????????
D= −�
1
̇�
4
??????�
4
− −�
2� ??????� = −(�
2+ �
1/16)̇� ??????�
????????????
S= −??????
1
3�
4
3??????�
4
− ??????
2
�
3
??????�
3
= −F
9??????
1
16
+
??????
2
9
G� ??????�
????????????
Ext= 8??????� ??????(�)
2??????�
3
+ ??????
7
12�
� ??????�
SDOF
Giacomo Boffi
Principle of Virtual Displacements
�
1
̇�
4
??????
1
̈�
2
3??????
1�
4
�
2
̇�
2??????
2
̈�
3
??????�
3
�
�(�)
??????
2
̈�
3�
8??????� ??????(�)
??????
1
̈�
4�
??????�
4
??????�
2
3
??????�
4
??????�2
??????�
3
??????�
3
??????�
????????????
2= ??????�/(3�)????????????
1= ??????�/(4�)
????????????
I= −??????
1
̈�
2
??????�
2
− ??????
1
̈�
4�
??????�
4�
− ??????
2
2̈�
3
2??????�
3
− ??????
2
̈�
3�
??????�
3�
= −F
??????
1
4
+ 4
??????
2
9
+
??????
1
16�
2
+
??????
2
9�
2
G̈� ??????�
????????????
D= −�
1
̇�
4
??????�
4
− −�
2� ??????� = −(�
2+ �
1/16)̇� ??????�
????????????
S= −??????
1
3�
4
3??????�
4
− ??????
2
�
3
??????�
3
= −F
9??????
1
16
+
??????
2
9
G� ??????�
????????????
Ext= 8??????� ??????(�)
2??????�
3
+ ??????
7
12�
� ??????�
Generalized
SDOF
Giacomo Boffi
Principle of Virtual Displacements
�
1
̇�
4
??????
1
̈�
2
3??????
1�
4
�
2
̇�
2??????
2
̈�
3
??????�
3
�
�(�)
??????
2
̈�
3�
8??????� ??????(�)
??????
1
̈�
4�
??????�
4
??????�
2
3
??????�
4
??????�2
??????�
3
??????�
3
??????�
????????????
2= ??????�/(3�)????????????
1= ??????�/(4�)
????????????
I= −??????
1
̈�
2
??????�
2
− ??????
1
̈�
4�
??????�
4�
− ??????
2
2̈�
3
2??????�
3
− ??????
2
̈�
3�
??????�
3�
= −F
??????
1
4
+ 4
??????
2
9
+
??????
1
16�
2
+
??????
2
9�
2
G̈� ??????�
????????????
D= −�
1
̇�
4
??????�
4
− −�
2� ??????� = −(�
2+ �
1/16)̇� ??????�
????????????
S= −??????
1
3�
4
3??????�
4
− ??????
2
�
3
??????�
3
= −F
9??????
1
16
+
??????
2
9
G� ??????�
????????????
Ext= 8??????� ??????(�)
2??????�
3
+ ??????
7
12�
� ??????�
SDOF
Giacomo Boffi
Principle of Virtual Displacements
�
1
̇�
4
??????
1
̈�
2
3??????
1�
4
�
2
̇�
2??????
2
̈�
3
??????�
3
�
�(�)
??????
2
̈�
3�
8??????� ??????(�)
??????
1
̈�
4�
??????�
4
??????�
2
3
??????�
4
??????�2
??????�
3
??????�
3
??????�
????????????
2= ??????�/(3�)????????????
1= ??????�/(4�)
????????????
I= −??????
1
̈�
2
??????�
2
− ??????
1
̈�
4�
??????�
4�
− ??????
2
2̈�
3
2??????�
3
− ??????
2
̈�
3�
??????�
3�
= −F
??????
1
4
+ 4
??????
2
9
+
??????
1
16�
2
+
??????
2
9�
2
G̈� ??????�
????????????
D= −�
1
̇�
4
??????�
4
− −�
2� ??????� = −(�
2+ �
1/16)̇� ??????�
????????????
S= −??????
1
3�
4
3??????�
4
− ??????
2
�
3
??????�
3
= −F
9??????
1
16
+
??????
2
9
G� ??????�
????????????
Ext= 8??????� ??????(�)
2??????�
3
+ ??????
7
12�
� ??????�
Generalized
SDOF
Giacomo Boffi
Principle of Virtual Displacements
For a rigid body in condition of equilibrium the total virtual work must be equal to zero
????????????
I+ ????????????
D+ ????????????
S+ ????????????
Ext= 0
Substituting our expressions of the virtual work contributions and simplifying??????�, the
equation of equilibrium is
F
??????
1
4
+ 4
??????
2
9
+
??????
1
16�
2
+
??????
2
9�
2
G̈�+
+(�
2+ �
1/16)̇� +F
9??????
1
16
+
??????
2
9
G� =
8??????� ??????(�)
2
3
+ �
7
12�
�
SDOF
Giacomo Boffi
Principle of Virtual Displacements
For a rigid body in condition of equilibrium the total virtual work must be equal to zero
????????????
I+ ????????????
D+ ????????????
S+ ????????????
Ext= 0
Substituting our expressions of the virtual work contributions and simplifying??????�, the
equation of equilibrium is
F
??????
1
4
+ 4
??????
2
9
+
??????
1
16�
2
+
??????
2
9�
2
G̈�+
+(�
2+ �
1/16)̇� +F
9??????
1
16
+
??????
2
9
G� =
8??????� ??????(�)
2
3
+ �
7
12�
�
Generalized
SDOF
Giacomo Boffi
Principle of Virtual Displacements
Collecting�and its time derivatives give us
??????
⋆̈� + �
⋆̇� + ??????
⋆
� = ??????
⋆
??????(�)
introducing the so calledgeneralised properties, in our example it is
??????
⋆
=
1
4
??????
1+
4
9
9??????
2+
1
16�
2
??????
1+
1
9�
2
??????
2,
�
⋆
=
1
16
�
1+ �
2, ??????
⋆
=
9
16
??????
1+
1
9
??????
2−
7
12�
�, ??????
⋆
=
16
3
??????�.
It is worth writing down the ex‐
pression of??????
⋆
:
??????
⋆
=
9??????
1
16
+
??????
2
9
−
7
12�
�
SDOF
Giacomo Boffi
Principle of Virtual Displacements
Collecting�and its time derivatives give us
??????
⋆̈� + �
⋆̇� + ??????
⋆
� = ??????
⋆
??????(�)
introducing the so calledgeneralised properties, in our example it is
??????
⋆
=
1
4
??????
1+
4
9
9??????
2+
1
16�
2
??????
1+
1
9�
2
??????
2,
�
⋆
=
1
16
�
1+ �
2, ??????
⋆
=
9
16
??????
1+
1
9
??????
2−
7
12�
�, ??????
⋆
=
16
3
??????�.
It is worth writing down the ex‐
pression of??????
⋆
:
??????
⋆
=
9??????
1
16
+
??????
2
9
−
7
12�
�
Generalized
SDOF
Giacomo Boffi
Principle of Virtual Displacements
Collecting�and its time derivatives give us
??????
⋆̈� + �
⋆̇� + ??????
⋆
� = ??????
⋆
??????(�)
introducing the so calledgeneralised properties, in our example it is
??????
⋆
=
1
4
??????
1+
4
9
9??????
2+
1
16�
2
??????
1+
1
9�
2
??????
2,
�
⋆
=
1
16
�
1+ �
2, ??????
⋆
=
9
16
??????
1+
1
9
??????
2−
7
12�
�, ??????
⋆
=
16
3
??????�.
It is worth writing down the ex‐
pression of??????
⋆
:
??????
⋆
=
9??????
1
16
+
??????
2
9
−
7
12�
�
SDOF
Giacomo Boffi
Principle of Virtual Displacements
Collecting�and its time derivatives give us
??????
⋆̈� + �
⋆̇� + ??????
⋆
� = ??????
⋆
??????(�)
introducing the so calledgeneralised properties, in our example it is
??????
⋆
=
1
4
??????
1+
4
9
9??????
2+
1
16�
2
??????
1+
1
9�
2
??????
2,
�
⋆
=
1
16
�
1+ �
2, ??????
⋆
=
9
16
??????
1+
1
9
??????
2−
7
12�
�, ??????
⋆
=
16
3
??????�.
It is worth writing down the ex‐
pression of??????
⋆
:
??????
⋆
=
9??????
1
16
+
??????
2
9
−
7
12�
�
Generalized
SDOF
Giacomo Boffi
Principle of Virtual Displacements
Collecting�and its time derivatives give us
??????
⋆̈� + �
⋆̇� + ??????
⋆
� = ??????
⋆
??????(�)
introducing the so calledgeneralised properties, in our example it is
??????
⋆
=
1
4
??????
1+
4
9
9??????
2+
1
16�
2
??????
1+
1
9�
2
??????
2,
�
⋆
=
1
16
�
1+ �
2, ??????
⋆
=
9
16
??????
1+
1
9
??????
2−
7
12�
�, ??????
⋆
=
16
3
??????�.
It is worth writing down the ex‐
pression of??????
⋆
:
??????
⋆
=
9??????
1
16
+
??????
2
9
−
7
12�
�
Geometrical stiffness
SDOF
Giacomo Boffi
Principle of Virtual Displacements
Collecting�and its time derivatives give us
??????
⋆̈� + �
⋆̇� + ??????
⋆
� = ??????
⋆
??????(�)
introducing the so calledgeneralised properties, in our example it is
??????
⋆
=
1
4
??????
1+
4
9
9??????
2+
1
16�
2
??????
1+
1
9�
2
??????
2,
�
⋆
=
1
16
�
1+ �
2, ??????
⋆
=
9
16
??????
1+
1
9
??????
2−
7
12�
�, ??????
⋆
=
16
3
??????�.
It is worth writing down the ex‐
pression of??????
⋆
:
??????
⋆
=
9??????
1
16
+
??????
2
9
−
7
12�
�
Geometrical stiffness
Generalized
SDOF
Giacomo Boffi
Section 3
Continuous Systems
SDOF
Giacomo Boffi
Section 3
Continuous Systems
Generalized
SDOF
Giacomo Boffi
Let’s start with an example...
Consider a cantilever, with varying
properties̄??????and????????????, subjected to a
dynamic load that is function of both
time�and position??????,
?????? = ??????(??????, �).
??????
̄?????? = ̄??????(??????)
�
???????????? = ????????????(??????)�(??????, �)
??????(??????, �)
??????
Even the transverse displacements�will be function of time and position,
� = �(??????, �)
and because the inertial forces depend on̈� =
??????
2
�/??????�
2
and the elastic forces on
�
″
=
??????
2
�/????????????
2
the equation of dynamic equilibrium must be written in terms of a
partial derivatives differential equation.
SDOF
Giacomo Boffi
Let’s start with an example...
Consider a cantilever, with varying
properties̄??????and????????????, subjected to a
dynamic load that is function of both
time�and position??????,
?????? = ??????(??????, �).
??????
̄?????? = ̄??????(??????)
�
???????????? = ????????????(??????)�(??????, �)
??????(??????, �)
??????
Even the transverse displacements�will be function of time and position,
� = �(??????, �)
and because the inertial forces depend on̈� =
??????
2
�/??????�
2
and the elastic forces on
�
″
=
??????
2
�/????????????
2
the equation of dynamic equilibrium must be written in terms of a
partial derivatives differential equation.
Generalized
SDOF
Giacomo Boffi
Let’s start with an example...
Consider a cantilever, with varying
properties̄??????and????????????, subjected to a
dynamic load that is function of both
time�and position??????,
?????? = ??????(??????, �).
??????
̄?????? = ̄??????(??????)
�
???????????? = ????????????(??????)�(??????, �)
??????(??????, �)
??????
Even the transverse displacements�will be function of time and position,
� = �(??????, �)
and because the inertial forces depend on̈� =
??????
2
�/??????�
2
and the elastic forces on
�
″
=
??????
2
�/????????????
2
the equation of dynamic equilibrium must be written in terms of a
partial derivatives differential equation.
SDOF
Giacomo Boffi
Let’s start with an example...
Consider a cantilever, with varying
properties̄??????and????????????, subjected to a
dynamic load that is function of both
time�and position??????,
?????? = ??????(??????, �).
??????
̄?????? = ̄??????(??????)
�
???????????? = ????????????(??????)�(??????, �)
??????(??????, �)
??????
Even the transverse displacements�will be function of time and position,
� = �(??????, �)
and because the inertial forces depend on̈� =
??????
2
�/??????�
2
and the elastic forces on
�
″
=
??????
2
�/????????????
2
the equation of dynamic equilibrium must be written in terms of a
partial derivatives differential equation.
Generalized
SDOF
Giacomo Boffi
... and an hypothesis
To study the previous problem, we introduce anapproximate modelby the following
hypothesis,
�(??????, �) = Ψ(??????) �(�),
that is, the hypothesis ofseparation of variables
Note thatΨ(??????), theshape function, is adimensional, while�(�)is dimensionally a
generalised displacement, usually chosen to characterise the structural behaviour.
In our example we can use the displacement of the tip of the chimney, thus implying
thatΨ(??????) = 1because
�(�) = �(??????, �)and
�(??????, �) = Ψ(??????) �(�)
SDOF
Giacomo Boffi
... and an hypothesis
To study the previous problem, we introduce anapproximate modelby the following
hypothesis,
�(??????, �) = Ψ(??????) �(�),
that is, the hypothesis ofseparation of variables
Note thatΨ(??????), theshape function, is adimensional, while�(�)is dimensionally a
generalised displacement, usually chosen to characterise the structural behaviour.
In our example we can use the displacement of the tip of the chimney, thus implying
thatΨ(??????) = 1because
�(�) = �(??????, �)and
�(??????, �) = Ψ(??????) �(�)
Generalized
SDOF
Giacomo Boffi
... and an hypothesis
To study the previous problem, we introduce anapproximate modelby the following
hypothesis,
�(??????, �) = Ψ(??????) �(�),
that is, the hypothesis ofseparation of variables
Note thatΨ(??????), theshape function, is adimensional, while�(�)is dimensionally a
generalised displacement, usually chosen to characterise the structural behaviour.
In our example we can use the displacement of the tip of the chimney, thus implying
thatΨ(??????) = 1because
�(�) = �(??????, �)and
�(??????, �) = Ψ(??????) �(�)
SDOF
Giacomo Boffi
... and an hypothesis
To study the previous problem, we introduce anapproximate modelby the following
hypothesis,
�(??????, �) = Ψ(??????) �(�),
that is, the hypothesis ofseparation of variables
Note thatΨ(??????), theshape function, is adimensional, while�(�)is dimensionally a
generalised displacement, usually chosen to characterise the structural behaviour.
In our example we can use the displacement of the tip of the chimney, thus implying
thatΨ(??????) = 1because
�(�) = �(??????, �)and
�(??????, �) = Ψ(??????) �(�)
Generalized
SDOF
Giacomo Boffi
... and an hypothesis
To study the previous problem, we introduce anapproximate modelby the following
hypothesis,
�(??????, �) = Ψ(??????) �(�),
that is, the hypothesis ofseparation of variables
Note thatΨ(??????), theshape function, is adimensional, while�(�)is dimensionally a
generalised displacement, usually chosen to characterise the structural behaviour.
In our example we can use the displacement of the tip of the chimney, thus implying
thatΨ(??????) = 1because
�(�) = �(??????, �)and
�(??????, �) = Ψ(??????) �(�)
SDOF
Giacomo Boffi
... and an hypothesis
To study the previous problem, we introduce anapproximate modelby the following
hypothesis,
�(??????, �) = Ψ(??????) �(�),
that is, the hypothesis ofseparation of variables
Note thatΨ(??????), theshape function, is adimensional, while�(�)is dimensionally a
generalised displacement, usually chosen to characterise the structural behaviour.
In our example we can use the displacement of the tip of the chimney, thus implying
thatΨ(??????) = 1because
�(�) = �(??????, �)and
�(??????, �) = Ψ(??????) �(�)
Generalized
SDOF
Giacomo Boffi
Principle of Virtual Displacements
For a flexible system, the PoVD states that, at equilibrium,
????????????
E= ????????????
I.
The virtual work of external forces can be easily computed, the virtual work of internal
forces is usually approximated by the virtual work done by bending moments, that is
????????????
I≈?� ????????????
where??????is the curvature and????????????the virtual increment of curvature.
SDOF
Giacomo Boffi
Principle of Virtual Displacements
For a flexible system, the PoVD states that, at equilibrium,
????????????
E= ????????????
I.
The virtual work of external forces can be easily computed, the virtual work of internal
forces is usually approximated by the virtual work done by bending moments, that is
????????????
I≈?� ????????????
where??????is the curvature and????????????the virtual increment of curvature.
Generalized
SDOF
Giacomo Boffi
????????????
E
The external forces are??????(??????, �),�and the forces of inertia??????
I; we have, by separation
of variables, that??????� = Ψ(??????)??????�and we can write
????????????
p=?
??????
0
??????(??????, �)??????�d?????? =H?
??????
0
??????(??????, �)Ψ(??????)d??????I??????� = ??????
⋆
(�) ??????�
????????????
Inertia=?
??????
0
− ̄??????(??????)̈�??????�d?????? =?
??????
0
− ̄??????(??????) kΨ(??????)̈�o(Ψ(??????) ??????�)d??????
=H?
??????
0
− ̄??????(??????)Ψ
2
(??????)d??????�(�) ??????� = ??????
⋆̈� ??????�.
The virtual work done by the axial force deserves a separate treatment...
SDOF
Giacomo Boffi
????????????
E
The external forces are??????(??????, �),�and the forces of inertia??????
I; we have, by separation
of variables, that??????� = Ψ(??????)??????�and we can write
????????????
p=?
??????
0
??????(??????, �)??????�d?????? =H?
??????
0
??????(??????, �)Ψ(??????)d??????I??????� = ??????
⋆
(�) ??????�
????????????
Inertia=?
??????
0
− ̄??????(??????)̈�??????�d?????? =?
??????
0
− ̄??????(??????) kΨ(??????)̈�o(Ψ(??????) ??????�)d??????
=H?
??????
0
− ̄??????(??????)Ψ
2
(??????)d??????�(�) ??????� = ??????
⋆̈� ??????�.
The virtual work done by the axial force deserves a separate treatment...
Generalized
SDOF
Giacomo Boffi
????????????
N
The virtual work of??????is????????????
N= ????????????�where??????�is the variation of the vertical displacement of the top of
the chimney.
We start computing the vertical displacement of the top of the chimney in terms of the rotation of the
axis line,?????? ≈ Ψ
′
(??????)�(�),
�(�) = ?????? −?
??????
0
cos??????d?????? =?
??????
0
(1 −cos??????)d??????,
substituting the well known approximation�??????�?????? ≈ 1 −
??????
2
2
in the above equation we have
�(�) =?
??????
0
??????
2
2
d?????? =?
??????
0
Ψ
′2
(??????)�
2
(�)
2
d?????? ⇒
⇒ ??????� =?
??????
0
Ψ
′2
(??????)�(�)??????�d?????? =?
??????
0
Ψ
′2
(??????)d?????? �??????�
and
????????????
N=H?
??????
0
Ψ
′2
(??????)d?????? ??????I� ??????� = ??????
⋆
??????� ??????�
SDOF
Giacomo Boffi
????????????
N
The virtual work of??????is????????????
N= ????????????�where??????�is the variation of the vertical displacement of the top of
the chimney.
We start computing the vertical displacement of the top of the chimney in terms of the rotation of the
axis line,?????? ≈ Ψ
′
(??????)�(�),
�(�) = ?????? −?
??????
0
cos??????d?????? =?
??????
0
(1 −cos??????)d??????,
substituting the well known approximation�??????�?????? ≈ 1 −
??????
2
2
in the above equation we have
�(�) =?
??????
0
??????
2
2
d?????? =?
??????
0
Ψ
′2
(??????)�
2
(�)
2
d?????? ⇒
⇒ ??????� =?
??????
0
Ψ
′2
(??????)�(�)??????�d?????? =?
??????
0
Ψ
′2
(??????)d?????? �??????�
and
????????????
N=H?
??????
0
Ψ
′2
(??????)d?????? ??????I� ??????� = ??????
⋆
??????� ??????�
Generalized
SDOF
Giacomo Boffi
????????????
Int
Approximating the internal work with the work done by bending moments, for an
infinitesimal slice of beam we write
d??????
Int=
1
2
��"(??????, �)d?????? =
1
2
�Ψ"(??????)�(�)d??????
with� = ????????????(??????)�"(??????)
??????(d??????
Int) = ????????????(??????)Ψ"
2
(??????)�(�)??????�d??????
integrating
????????????
Int=H?
??????
0
????????????(??????)Ψ"
2
(??????)d??????I�??????� = ??????
⋆
� ??????�
SDOF
Giacomo Boffi
????????????
Int
Approximating the internal work with the work done by bending moments, for an
infinitesimal slice of beam we write
d??????
Int=
1
2
��"(??????, �)d?????? =
1
2
�Ψ"(??????)�(�)d??????
with� = ????????????(??????)�"(??????)
??????(d??????
Int) = ????????????(??????)Ψ"
2
(??????)�(�)??????�d??????
integrating
????????????
Int=H?
??????
0
????????????(??????)Ψ"
2
(??????)d??????I�??????� = ??????
⋆
� ??????�
Generalized
SDOF
Giacomo Boffi
Remarks
the shape functionmustrespect the geometrical boundary conditions of the
problem, i.e., both
Ψ
1= ??????
2
and Ψ
2= 1 −cos
????????????
2??????
are accettable shape functions for our example, asΨ
1(0) = Ψ
2(0) = 0and
Ψ
′
1(0) = Ψ
′
2(0) = 0
better results are obtained when the second derivative of the shape function at
leastresemblesthe typical distribution of bending moments in our problem, so
that between
Ψ
′′
1=constant and Ψ
2" =
??????
2
4??????
2
cos
????????????
2??????
the second choice is preferable.
SDOF
Giacomo Boffi
Remarks
the shape functionmustrespect the geometrical boundary conditions of the
problem, i.e., both
Ψ
1= ??????
2
and Ψ
2= 1 −cos
????????????
2??????
are accettable shape functions for our example, asΨ
1(0) = Ψ
2(0) = 0and
Ψ
′
1(0) = Ψ
′
2(0) = 0
better results are obtained when the second derivative of the shape function at
leastresemblesthe typical distribution of bending moments in our problem, so
that between
Ψ
′′
1=constant and Ψ
2" =
??????
2
4??????
2
cos
????????????
2??????
the second choice is preferable.
Generalized
SDOF
Giacomo Boffi
Remarks 0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
vi"/Z(t)
v/Z(t)
x/H
f
1=1-cos(pi*x/2)
f
2=x
2
the shape functionmustrespect the geometrical boundary conditions of the
problem, i.e., both
Ψ
1= ??????
2
and Ψ
2= 1 −cos
????????????
2??????
are accettable shape functions for our example, asΨ
1(0) = Ψ
2(0) = 0and
Ψ
′
1(0) = Ψ
′
2(0) = 0
better results are obtained when the second derivative of the shape function at
leastresemblesthe typical distribution of bending moments in our problem, so
that between
Ψ
′′
1=constant and Ψ
2" =
??????
2
4??????
2
cos
????????????
2??????
the second choice is preferable.
SDOF
Giacomo Boffi
Remarks 0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
vi"/Z(t)
v/Z(t)
x/H
f
1=1-cos(pi*x/2)
f
2=x
2
the shape functionmustrespect the geometrical boundary conditions of the
problem, i.e., both
Ψ
1= ??????
2
and Ψ
2= 1 −cos
????????????
2??????
are accettable shape functions for our example, asΨ
1(0) = Ψ
2(0) = 0and
Ψ
′
1(0) = Ψ
′
2(0) = 0
better results are obtained when the second derivative of the shape function at
leastresemblesthe typical distribution of bending moments in our problem, so
that between
Ψ
′′
1=constant and Ψ
2" =
??????
2
4??????
2
cos
????????????
2??????
the second choice is preferable.
Generalized
SDOF
Giacomo Boffi
Remarks 0
0.5
1
1.5
2
2.5
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
vi"/Z(t)
v/Z(t)
x/H
f
1=1-cos(pi*x/2)
f
2=x
2
f
1"
f
2"
the shape functionmustrespect the geometrical boundary conditions of the
problem, i.e., both
Ψ
1= ??????
2
and Ψ
2= 1 −cos
????????????
2??????
are accettable shape functions for our example, asΨ
1(0) = Ψ
2(0) = 0and
Ψ
′
1(0) = Ψ
′
2(0) = 0
better results are obtained when the second derivative of the shape function at
leastresemblesthe typical distribution of bending moments in our problem, so
that between
Ψ
′′
1=constant and Ψ
2" =
??????
2
4??????
2
cos
????????????
2??????
the second choice is preferable.
SDOF
Giacomo Boffi
Remarks 0
0.5
1
1.5
2
2.5
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
vi"/Z(t)
v/Z(t)
x/H
f
1=1-cos(pi*x/2)
f
2=x
2
f
1"
f
2"
the shape functionmustrespect the geometrical boundary conditions of the
problem, i.e., both
Ψ
1= ??????
2
and Ψ
2= 1 −cos
????????????
2??????
are accettable shape functions for our example, asΨ
1(0) = Ψ
2(0) = 0and
Ψ
′
1(0) = Ψ
′
2(0) = 0
better results are obtained when the second derivative of the shape function at
leastresemblesthe typical distribution of bending moments in our problem, so
that between
Ψ
′′
1=constant and Ψ
2" =
??????
2
4??????
2
cos
????????????
2??????
the second choice is preferable.
Generalized
SDOF
Giacomo Boffi
Remarks
the shape functionmustrespect the geometrical boundary conditions of the
problem, i.e., both
Ψ
1= ??????
2
and Ψ
2= 1 −cos
????????????
2??????
are accettable shape functions for our example, asΨ
1(0) = Ψ
2(0) = 0and
Ψ
′
1(0) = Ψ
′
2(0) = 0
better results are obtained when the second derivative of the shape function at
leastresemblesthe typical distribution of bending moments in our problem, so
that between
Ψ
′′
1=constant and Ψ
2" =
??????
2
4??????
2
cos
????????????
2??????
the second choice is preferable.
SDOF
Giacomo Boffi
Remarks
the shape functionmustrespect the geometrical boundary conditions of the
problem, i.e., both
Ψ
1= ??????
2
and Ψ
2= 1 −cos
????????????
2??????
are accettable shape functions for our example, asΨ
1(0) = Ψ
2(0) = 0and
Ψ
′
1(0) = Ψ
′
2(0) = 0
better results are obtained when the second derivative of the shape function at
leastresemblesthe typical distribution of bending moments in our problem, so
that between
Ψ
′′
1=constant and Ψ
2" =
??????
2
4??????
2
cos
????????????
2??????
the second choice is preferable.
Generalized
SDOF
Giacomo Boffi
Example
UsingΨ(??????) = 1 −cos
????????????
2??????
, with̄?????? =constant and???????????? =constant, with a load
characteristic of seismic excitation,??????(�) = − ̄??????̈�
??????(�),
??????
⋆
= ̄???????
??????
0
(1 −cos
????????????
2??????
)
2
d?????? = ̄??????(
3
2
−
4
??????
)??????
??????
⋆
= ????????????
??????
4
16??????
4
?
??????
0
cos
2
????????????
2??????
d?????? =
??????
4
32
????????????
??????
3
??????
⋆
??????= �
??????
2
4??????
2
?
??????
0
sin
2
????????????
2??????
d?????? =
??????
2
8??????
�
??????
⋆
??????= − ̄??????̈�
??????(�)?
??????
0
1 −cos
????????????
2??????
d?????? = −F1 −
2
??????
Ḡ????????????̈�
??????(�)
SDOF
Giacomo Boffi
Example
UsingΨ(??????) = 1 −cos
????????????
2??????
, with̄?????? =constant and???????????? =constant, with a load
characteristic of seismic excitation,??????(�) = − ̄??????̈�
??????(�),
??????
⋆
= ̄???????
??????
0
(1 −cos
????????????
2??????
)
2
d?????? = ̄??????(
3
2
−
4
??????
)??????
??????
⋆
= ????????????
??????
4
16??????
4
?
??????
0
cos
2
????????????
2??????
d?????? =
??????
4
32
????????????
??????
3
??????
⋆
??????= �
??????
2
4??????
2
?
??????
0
sin
2
????????????
2??????
d?????? =
??????
2
8??????
�
??????
⋆
??????= − ̄??????̈�
??????(�)?
??????
0
1 −cos
????????????
2??????
d?????? = −F1 −
2
??????
Ḡ????????????̈�
??????(�)
Generalized
SDOF
Giacomo Boffi
Section 4
Vibration Analysis by Rayleigh’s Method
SDOF
Giacomo Boffi
Section 4
Vibration Analysis by Rayleigh’s Method
Generalized
SDOF
Giacomo Boffi
Vibration Analysis
The process of estimating the vibration characteristics of a complex system is
known asvibration analysis.
We can use our previous results for flexible systems, based on theSDOFmodel,
to give an estimate of the natural frequency??????
2
= ??????
⋆
/??????
⋆
A different approach, proposed by Lord Rayleigh, starts from different premises
to give the same results but theRayleigh’s Quotientmethod is important
because it offers a better understanding of the vibrational behaviour, eventually
leading to successive refinements of the first estimate of??????
2
.
SDOF
Giacomo Boffi
Vibration Analysis
The process of estimating the vibration characteristics of a complex system is
known asvibration analysis.
We can use our previous results for flexible systems, based on theSDOFmodel,
to give an estimate of the natural frequency??????
2
= ??????
⋆
/??????
⋆
A different approach, proposed by Lord Rayleigh, starts from different premises
to give the same results but theRayleigh’s Quotientmethod is important
because it offers a better understanding of the vibrational behaviour, eventually
leading to successive refinements of the first estimate of??????
2
.
Generalized
SDOF
Giacomo Boffi
Vibration Analysis
The process of estimating the vibration characteristics of a complex system is
known asvibration analysis.
We can use our previous results for flexible systems, based on theSDOFmodel,
to give an estimate of the natural frequency??????
2
= ??????
⋆
/??????
⋆
A different approach, proposed by Lord Rayleigh, starts from different premises
to give the same results but theRayleigh’s Quotientmethod is important
because it offers a better understanding of the vibrational behaviour, eventually
leading to successive refinements of the first estimate of??????
2
.
SDOF
Giacomo Boffi
Vibration Analysis
The process of estimating the vibration characteristics of a complex system is
known asvibration analysis.
We can use our previous results for flexible systems, based on theSDOFmodel,
to give an estimate of the natural frequency??????
2
= ??????
⋆
/??????
⋆
A different approach, proposed by Lord Rayleigh, starts from different premises
to give the same results but theRayleigh’s Quotientmethod is important
because it offers a better understanding of the vibrational behaviour, eventually
leading to successive refinements of the first estimate of??????
2
.
Generalized
SDOF
Giacomo Boffi
Vibration Analysis
The process of estimating the vibration characteristics of a complex system is
known asvibration analysis.
We can use our previous results for flexible systems, based on theSDOFmodel,
to give an estimate of the natural frequency??????
2
= ??????
⋆
/??????
⋆
A different approach, proposed by Lord Rayleigh, starts from different premises
to give the same results but theRayleigh’s Quotientmethod is important
because it offers a better understanding of the vibrational behaviour, eventually
leading to successive refinements of the first estimate of??????
2
.
SDOF
Giacomo Boffi
Vibration Analysis
The process of estimating the vibration characteristics of a complex system is
known asvibration analysis.
We can use our previous results for flexible systems, based on theSDOFmodel,
to give an estimate of the natural frequency??????
2
= ??????
⋆
/??????
⋆
A different approach, proposed by Lord Rayleigh, starts from different premises
to give the same results but theRayleigh’s Quotientmethod is important
because it offers a better understanding of the vibrational behaviour, eventually
leading to successive refinements of the first estimate of??????
2
.
Generalized
SDOF
Giacomo Boffi
Rayleigh’s Quotient Method
Our focus will be on thefree vibrationof a flexible, undamped system.
inspired by the free vibrations of a properSDOFwe write
�(�) = �
0sin??????�and
�(??????, �) = �
0Ψ(??????)sin??????�,
̇�(??????, �) = ?????? �
0Ψ(??????)cos??????�.
the displacement and the velocity are in quadrature: when�is at its maximum
̇� = 0, hence?????? = ??????
max,?????? = 0and wheṅ�is at its maximum it is� = 0, hence
?????? = 0,?????? = ??????
max,
disregarding damping, the energy of the system is constant during free vibrations,
??????
max+ 0 = 0 + ??????
max⇒ ??????
max= ??????
max
SDOF
Giacomo Boffi
Rayleigh’s Quotient Method
Our focus will be on thefree vibrationof a flexible, undamped system.
inspired by the free vibrations of a properSDOFwe write
�(�) = �
0sin??????�and
�(??????, �) = �
0Ψ(??????)sin??????�,
̇�(??????, �) = ?????? �
0Ψ(??????)cos??????�.
the displacement and the velocity are in quadrature: when�is at its maximum
̇� = 0, hence?????? = ??????
max,?????? = 0and wheṅ�is at its maximum it is� = 0, hence
?????? = 0,?????? = ??????
max,
disregarding damping, the energy of the system is constant during free vibrations,
??????
max+ 0 = 0 + ??????
max⇒ ??????
max= ??????
max
Generalized
SDOF
Giacomo Boffi
Rayleigh’s Quotient Method
Our focus will be on thefree vibrationof a flexible, undamped system.
inspired by the free vibrations of a properSDOFwe write
�(�) = �
0sin??????�and
�(??????, �) = �
0Ψ(??????)sin??????�,
̇�(??????, �) = ?????? �
0Ψ(??????)cos??????�.
the displacement and the velocity are in quadrature: when�is at its maximum
̇� = 0, hence?????? = ??????
max,?????? = 0and wheṅ�is at its maximum it is� = 0, hence
?????? = 0,?????? = ??????
max,
disregarding damping, the energy of the system is constant during free vibrations,
??????
max+ 0 = 0 + ??????
max⇒ ??????
max= ??????
max
SDOF
Giacomo Boffi
Rayleigh’s Quotient Method
Our focus will be on thefree vibrationof a flexible, undamped system.
inspired by the free vibrations of a properSDOFwe write
�(�) = �
0sin??????�and
�(??????, �) = �
0Ψ(??????)sin??????�,
̇�(??????, �) = ?????? �
0Ψ(??????)cos??????�.
the displacement and the velocity are in quadrature: when�is at its maximum
̇� = 0, hence?????? = ??????
max,?????? = 0and wheṅ�is at its maximum it is� = 0, hence
?????? = 0,?????? = ??????
max,
disregarding damping, the energy of the system is constant during free vibrations,
??????
max+ 0 = 0 + ??????
max⇒ ??????
max= ??????
max
Generalized
SDOF
Giacomo Boffi
Rayleigh’s Quotient Method
Our focus will be on thefree vibrationof a flexible, undamped system.
inspired by the free vibrations of a properSDOFwe write
�(�) = �
0sin??????�and
�(??????, �) = �
0Ψ(??????)sin??????�,
̇�(??????, �) = ?????? �
0Ψ(??????)cos??????�.
the displacement and the velocity are in quadrature: when�is at its maximum
̇� = 0, hence?????? = ??????
max,?????? = 0and wheṅ�is at its maximum it is� = 0, hence
?????? = 0,?????? = ??????
max,
disregarding damping, the energy of the system is constant during free vibrations,
??????
max+ 0 = 0 + ??????
max⇒ ??????
max= ??????
max
SDOF
Giacomo Boffi
Rayleigh’s Quotient Method
Our focus will be on thefree vibrationof a flexible, undamped system.
inspired by the free vibrations of a properSDOFwe write
�(�) = �
0sin??????�and
�(??????, �) = �
0Ψ(??????)sin??????�,
̇�(??????, �) = ?????? �
0Ψ(??????)cos??????�.
the displacement and the velocity are in quadrature: when�is at its maximum
̇� = 0, hence?????? = ??????
max,?????? = 0and wheṅ�is at its maximum it is� = 0, hence
?????? = 0,?????? = ??????
max,
disregarding damping, the energy of the system is constant during free vibrations,
??????
max+ 0 = 0 + ??????
max⇒ ??????
max= ??????
max
Generalized
SDOF
Giacomo Boffi
Rayleigh’s Quotient Method
Our focus will be on thefree vibrationof a flexible, undamped system.
inspired by the free vibrations of a properSDOFwe write
�(�) = �
0sin??????�and
�(??????, �) = �
0Ψ(??????)sin??????�,
̇�(??????, �) = ?????? �
0Ψ(??????)cos??????�.
the displacement and the velocity are in quadrature: when�is at its maximum
̇� = 0, hence?????? = ??????
max,?????? = 0and wheṅ�is at its maximum it is� = 0, hence
?????? = 0,?????? = ??????
max,
disregarding damping, the energy of the system is constant during free vibrations,
??????
max+ 0 = 0 + ??????
max⇒ ??????
max= ??????
max
SDOF
Giacomo Boffi
Rayleigh’s Quotient Method
Our focus will be on thefree vibrationof a flexible, undamped system.
inspired by the free vibrations of a properSDOFwe write
�(�) = �
0sin??????�and
�(??????, �) = �
0Ψ(??????)sin??????�,
̇�(??????, �) = ?????? �
0Ψ(??????)cos??????�.
the displacement and the velocity are in quadrature: when�is at its maximum
̇� = 0, hence?????? = ??????
max,?????? = 0and wheṅ�is at its maximum it is� = 0, hence
?????? = 0,?????? = ??????
max,
disregarding damping, the energy of the system is constant during free vibrations,
??????
max+ 0 = 0 + ??????
max⇒ ??????
max= ??????
max
Generalized
SDOF
Giacomo Boffi
Rayleigh’ s Quotient Method
Now we write the expressions for??????
maxand??????
max,
??????
max=
1
2
�
2
0?
??????
????????????(??????)Ψ
′′2
(??????)d??????,
??????
max=
1
2
??????
2
�
2
0?
??????
̄??????(??????)Ψ
2
(??????)d??????,
equating the two expressions and solving for??????
2
we have
??????
2
=
∫
??????
????????????(??????)Ψ
′′2
(??????)d??????
∫
??????
̄??????(??????)Ψ
2
(??????)d??????
.
Recognizing the expressions we found for??????
⋆
and??????
⋆
we could question the utility of
Rayleigh’s Quotient...
SDOF
Giacomo Boffi
Rayleigh’ s Quotient Method
Now we write the expressions for??????
maxand??????
max,
??????
max=
1
2
�
2
0?
??????
????????????(??????)Ψ
′′2
(??????)d??????,
??????
max=
1
2
??????
2
�
2
0?
??????
̄??????(??????)Ψ
2
(??????)d??????,
equating the two expressions and solving for??????
2
we have
??????
2
=
∫
??????
????????????(??????)Ψ
′′2
(??????)d??????
∫
??????
̄??????(??????)Ψ
2
(??????)d??????
.
Recognizing the expressions we found for??????
⋆
and??????
⋆
we could question the utility of
Rayleigh’s Quotient...
Generalized
SDOF
Giacomo Boffi
Rayleigh’s Quotient Method
in Rayleigh’s method we know the specific time dependency of the inertial forces
??????
I= − ̄??????(??????)̈� = ̄??????(??????)??????
2
�
0Ψ(??????)sin??????�
??????
Ihas the sameshapewe use for displacements.
ifΨwere the real shape assumed by the structure in free vibrations, the
displacements�due to a loading??????
I= ??????
2
̄??????(??????)Ψ(??????)�
0should be proportional to
Ψ(??????)through a constant factor, with equilibrium respected in every point of the
structure during free vibrations.
starting from a shape functionΨ
0(??????), a new shape functionΨ
1can be
determined normalizing the displacements due to the inertial forces associated
withΨ
0(??????),??????
I= ̄??????(??????)Ψ
0(??????),
we are going to demonstrate that the new shape function is a better
approximation of the true mode shape
SDOF
Giacomo Boffi
Rayleigh’s Quotient Method
in Rayleigh’s method we know the specific time dependency of the inertial forces
??????
I= − ̄??????(??????)̈� = ̄??????(??????)??????
2
�
0Ψ(??????)sin??????�
??????
Ihas the sameshapewe use for displacements.
ifΨwere the real shape assumed by the structure in free vibrations, the
displacements�due to a loading??????
I= ??????
2
̄??????(??????)Ψ(??????)�
0should be proportional to
Ψ(??????)through a constant factor, with equilibrium respected in every point of the
structure during free vibrations.
starting from a shape functionΨ
0(??????), a new shape functionΨ
1can be
determined normalizing the displacements due to the inertial forces associated
withΨ
0(??????),??????
I= ̄??????(??????)Ψ
0(??????),
we are going to demonstrate that the new shape function is a better
approximation of the true mode shape
Generalized
SDOF
Giacomo Boffi
Rayleigh’s Quotient Method
in Rayleigh’s method we know the specific time dependency of the inertial forces
??????
I= − ̄??????(??????)̈� = ̄??????(??????)??????
2
�
0Ψ(??????)sin??????�
??????
Ihas the sameshapewe use for displacements.
ifΨwere the real shape assumed by the structure in free vibrations, the
displacements�due to a loading??????
I= ??????
2
̄??????(??????)Ψ(??????)�
0should be proportional to
Ψ(??????)through a constant factor, with equilibrium respected in every point of the
structure during free vibrations.
starting from a shape functionΨ
0(??????), a new shape functionΨ
1can be
determined normalizing the displacements due to the inertial forces associated
withΨ
0(??????),??????
I= ̄??????(??????)Ψ
0(??????),
we are going to demonstrate that the new shape function is a better
approximation of the true mode shape
SDOF
Giacomo Boffi
Rayleigh’s Quotient Method
in Rayleigh’s method we know the specific time dependency of the inertial forces
??????
I= − ̄??????(??????)̈� = ̄??????(??????)??????
2
�
0Ψ(??????)sin??????�
??????
Ihas the sameshapewe use for displacements.
ifΨwere the real shape assumed by the structure in free vibrations, the
displacements�due to a loading??????
I= ??????
2
̄??????(??????)Ψ(??????)�
0should be proportional to
Ψ(??????)through a constant factor, with equilibrium respected in every point of the
structure during free vibrations.
starting from a shape functionΨ
0(??????), a new shape functionΨ
1can be
determined normalizing the displacements due to the inertial forces associated
withΨ
0(??????),??????
I= ̄??????(??????)Ψ
0(??????),
we are going to demonstrate that the new shape function is a better
approximation of the true mode shape
Generalized
SDOF
Giacomo Boffi
Rayleigh’s Quotient Method
in Rayleigh’s method we know the specific time dependency of the inertial forces
??????
I= − ̄??????(??????)̈� = ̄??????(??????)??????
2
�
0Ψ(??????)sin??????�
??????
Ihas the sameshapewe use for displacements.
ifΨwere the real shape assumed by the structure in free vibrations, the
displacements�due to a loading??????
I= ??????
2
̄??????(??????)Ψ(??????)�
0should be proportional to
Ψ(??????)through a constant factor, with equilibrium respected in every point of the
structure during free vibrations.
starting from a shape functionΨ
0(??????), a new shape functionΨ
1can be
determined normalizing the displacements due to the inertial forces associated
withΨ
0(??????),??????
I= ̄??????(??????)Ψ
0(??????),
we are going to demonstrate that the new shape function is a better
approximation of the true mode shape
SDOF
Giacomo Boffi
Rayleigh’s Quotient Method
in Rayleigh’s method we know the specific time dependency of the inertial forces
??????
I= − ̄??????(??????)̈� = ̄??????(??????)??????
2
�
0Ψ(??????)sin??????�
??????
Ihas the sameshapewe use for displacements.
ifΨwere the real shape assumed by the structure in free vibrations, the
displacements�due to a loading??????
I= ??????
2
̄??????(??????)Ψ(??????)�
0should be proportional to
Ψ(??????)through a constant factor, with equilibrium respected in every point of the
structure during free vibrations.
starting from a shape functionΨ
0(??????), a new shape functionΨ
1can be
determined normalizing the displacements due to the inertial forces associated
withΨ
0(??????),??????
I= ̄??????(??????)Ψ
0(??????),
we are going to demonstrate that the new shape function is a better
approximation of the true mode shape
Generalized
SDOF
Giacomo Boffi
Selection of mode shapes
Given different shape functionsΨ
??????and considering the true shape of free vibrationΨ,
in the former cases equilibrium is not respected by the structure itself.
To keep inertia induced deformation proportional toΨ
??????we must consider the
presence of additional elastic constraints. This leads to the following considerations
the frequency of vibration of a structure with additional constraints is higher
than the true natural frequency,
the criterium to discriminate between different shape functions is: better shape
functions give lower estimates of the natural frequency, the true natural
frequency being a lower bound of all estimates.
SDOF
Giacomo Boffi
Selection of mode shapes
Given different shape functionsΨ
??????and considering the true shape of free vibrationΨ,
in the former cases equilibrium is not respected by the structure itself.
To keep inertia induced deformation proportional toΨ
??????we must consider the
presence of additional elastic constraints. This leads to the following considerations
the frequency of vibration of a structure with additional constraints is higher
than the true natural frequency,
the criterium to discriminate between different shape functions is: better shape
functions give lower estimates of the natural frequency, the true natural
frequency being a lower bound of all estimates.
Generalized
SDOF
Giacomo Boffi
Selection of mode shapes
Given different shape functionsΨ
??????and considering the true shape of free vibrationΨ,
in the former cases equilibrium is not respected by the structure itself.
To keep inertia induced deformation proportional toΨ
??????we must consider the
presence of additional elastic constraints. This leads to the following considerations
the frequency of vibration of a structure with additional constraints is higher
than the true natural frequency,
the criterium to discriminate between different shape functions is: better shape
functions give lower estimates of the natural frequency, the true natural
frequency being a lower bound of all estimates.
SDOF
Giacomo Boffi
Selection of mode shapes
Given different shape functionsΨ
??????and considering the true shape of free vibrationΨ,
in the former cases equilibrium is not respected by the structure itself.
To keep inertia induced deformation proportional toΨ
??????we must consider the
presence of additional elastic constraints. This leads to the following considerations
the frequency of vibration of a structure with additional constraints is higher
than the true natural frequency,
the criterium to discriminate between different shape functions is: better shape
functions give lower estimates of the natural frequency, the true natural
frequency being a lower bound of all estimates.
Generalized
SDOF
Giacomo Boffi
Selection of mode shapes
Given different shape functionsΨ
??????and considering the true shape of free vibrationΨ,
in the former cases equilibrium is not respected by the structure itself.
To keep inertia induced deformation proportional toΨ
??????we must consider the
presence of additional elastic constraints. This leads to the following considerations
the frequency of vibration of a structure with additional constraints is higher
than the true natural frequency,
the criterium to discriminate between different shape functions is: better shape
functions give lower estimates of the natural frequency, the true natural
frequency being a lower bound of all estimates.
SDOF
Giacomo Boffi
Selection of mode shapes
Given different shape functionsΨ
??????and considering the true shape of free vibrationΨ,
in the former cases equilibrium is not respected by the structure itself.
To keep inertia induced deformation proportional toΨ
??????we must consider the
presence of additional elastic constraints. This leads to the following considerations
the frequency of vibration of a structure with additional constraints is higher
than the true natural frequency,
the criterium to discriminate between different shape functions is: better shape
functions give lower estimates of the natural frequency, the true natural
frequency being a lower bound of all estimates.
Generalized
SDOF
Giacomo Boffi
Selection of mode shapes
Given different shape functionsΨ
??????and considering the true shape of free vibrationΨ,
in the former cases equilibrium is not respected by the structure itself.
To keep inertia induced deformation proportional toΨ
??????we must consider the
presence of additional elastic constraints. This leads to the following considerations
the frequency of vibration of a structure with additional constraints is higher
than the true natural frequency,
the criterium to discriminate between different shape functions is: better shape
functions give lower estimates of the natural frequency, the true natural
frequency being a lower bound of all estimates.
SDOF
Giacomo Boffi
Selection of mode shapes
Given different shape functionsΨ
??????and considering the true shape of free vibrationΨ,
in the former cases equilibrium is not respected by the structure itself.
To keep inertia induced deformation proportional toΨ
??????we must consider the
presence of additional elastic constraints. This leads to the following considerations
the frequency of vibration of a structure with additional constraints is higher
than the true natural frequency,
the criterium to discriminate between different shape functions is: better shape
functions give lower estimates of the natural frequency, the true natural
frequency being a lower bound of all estimates.
Generalized
SDOF
Giacomo Boffi
Selection of mode shapes 2
In general the selection of trial shapes goes through two steps,
1the analyst considers the flexibilities of different parts of the structure and the
presence of symmetries to devise an approximate shape,
2the structure is loaded with constant loads directed as the assumed
displacements, the displacements are computed and used as the shape function,
of course a little practice helps a lot in the the choice of a proper pattern of loading...
SDOF
Giacomo Boffi
Selection of mode shapes 2
In general the selection of trial shapes goes through two steps,
1the analyst considers the flexibilities of different parts of the structure and the
presence of symmetries to devise an approximate shape,
2the structure is loaded with constant loads directed as the assumed
displacements, the displacements are computed and used as the shape function,
of course a little practice helps a lot in the the choice of a proper pattern of loading...
Generalized
SDOF
Giacomo Boffi
Selection of mode shapes 3
?????? = ??????(??????)
?????? = �
?????? = ??????(??????)
?????? = ??????(??????)
?????? = ??????(??????)
(�)
(�) (�)
(�)
SDOF
Giacomo Boffi
Selection of mode shapes 3
?????? = ??????(??????)
?????? = �
?????? = ??????(??????)
?????? = ??????(??????)
?????? = ??????(??????)
(�)
(�) (�)
(�)
Generalized
SDOF
Giacomo Boffi
Refinement??????
00
Choose a trial functionΨ
(0)
(??????)and write
�
(0)
= Ψ
(0)
(??????)�
(0)
sin??????�
??????
max=
1
2
�
(0)2
?????????????Ψ
(0)′′2
d??????
??????
max=
1
2
??????
2
�
(0)2
?̄??????Ψ
(0)2
d??????
our first estimate??????
00of??????
2
is
??????
2
=
∫????????????Ψ
(0)′′2
d??????
∫̄??????Ψ
(0)2
d??????
.
SDOF
Giacomo Boffi
Refinement??????
00
Choose a trial functionΨ
(0)
(??????)and write
�
(0)
= Ψ
(0)
(??????)�
(0)
sin??????�
??????
max=
1
2
�
(0)2
?????????????Ψ
(0)′′2
d??????
??????
max=
1
2
??????
2
�
(0)2
?̄??????Ψ
(0)2
d??????
our first estimate??????
00of??????
2
is
??????
2
=
∫????????????Ψ
(0)′′2
d??????
∫̄??????Ψ
(0)2
d??????
.
Generalized
SDOF
Giacomo Boffi
Refinement??????
01
We try to give a better estimate of??????
maxcomputing the external work done by the
inertial forces,
??????
(0)
= ??????
2
̄??????(??????)�
(0)
= �
(0)
??????
2
Ψ
(0)
(??????)
the deflections due to??????
(0)
are
�
(1)
= ??????
2
�
(1)
??????
2
= ??????
2
Ψ
(1)
�
(1)
??????
2
= ??????
2
Ψ
(1)̄�
(1)
,
where we writē�
(1)
because we need to keep the unknown??????
2
in evidence.
The maximum strain energy is
??????
max=
1
2
???????
(0)
�
(1)
d?????? =
1
2
??????
4
�
(0)̄�
(1)
?̄??????(??????)Ψ
(0)
Ψ
(1)
d??????
Equating to our previus estimate of??????
maxwe find the??????
01estimate
??????
2
=
�
(0)
̄�
(1)
∫̄??????(??????)Ψ
(0)
Ψ
(0)
d??????
∫̄??????(??????)Ψ
(0)
Ψ
(1)
d??????
SDOF
Giacomo Boffi
Refinement??????
01
We try to give a better estimate of??????
maxcomputing the external work done by the
inertial forces,
??????
(0)
= ??????
2
̄??????(??????)�
(0)
= �
(0)
??????
2
Ψ
(0)
(??????)
the deflections due to??????
(0)
are
�
(1)
= ??????
2
�
(1)
??????
2
= ??????
2
Ψ
(1)
�
(1)
??????
2
= ??????
2
Ψ
(1)̄�
(1)
,
where we writē�
(1)
because we need to keep the unknown??????
2
in evidence.
The maximum strain energy is
??????
max=
1
2
???????
(0)
�
(1)
d?????? =
1
2
??????
4
�
(0)̄�
(1)
?̄??????(??????)Ψ
(0)
Ψ
(1)
d??????
Equating to our previus estimate of??????
maxwe find the??????
01estimate
??????
2
=
�
(0)
̄�
(1)
∫̄??????(??????)Ψ
(0)
Ψ
(0)
d??????
∫̄??????(??????)Ψ
(0)
Ψ
(1)
d??????
Generalized
SDOF
Giacomo Boffi
Refinement??????
11
With little additional effort it is possible to compute??????
maxfrom�
(1)
:
??????
max=
1
2
??????
2
?̄??????(??????)�
(1)2
d?????? =
1
2
??????
6̄�
(1)2
?̄??????(??????)Ψ
(1)2
d??????
equating to our last approximation for??????
maxwe have the??????
11approximation to the
frequency of vibration,
??????
2
=
�
(0)
̄�
(1)
∫̄??????(??????)Ψ
(0)
Ψ
(1)
d??????
∫̄??????(??????)Ψ
(1)
Ψ
(1)
d??????
.
Of course the procedure can be extended to compute better and better estimates of
??????
2
but usually the refinements are not extended beyond??????
11, because it would be
contradictory with the quick estimate nature of the Rayleigh’s Quotient methodand
also because??????
11estimates are usually very good ones.
Nevertheless, we recognize the possibility of itereatively computing better and better
estimates opens a world of new opportunities.
SDOF
Giacomo Boffi
Refinement??????
11
With little additional effort it is possible to compute??????
maxfrom�
(1)
:
??????
max=
1
2
??????
2
?̄??????(??????)�
(1)2
d?????? =
1
2
??????
6̄�
(1)2
?̄??????(??????)Ψ
(1)2
d??????
equating to our last approximation for??????
maxwe have the??????
11approximation to the
frequency of vibration,
??????
2
=
�
(0)
̄�
(1)
∫̄??????(??????)Ψ
(0)
Ψ
(1)
d??????
∫̄??????(??????)Ψ
(1)
Ψ
(1)
d??????
.
Of course the procedure can be extended to compute better and better estimates of
??????
2
but usually the refinements are not extended beyond??????
11, because it would be
contradictory with the quick estimate nature of the Rayleigh’s Quotient methodand
also because??????
11estimates are usually very good ones.
Nevertheless, we recognize the possibility of itereatively computing better and better
estimates opens a world of new opportunities.
Generalized
SDOF
Giacomo Boffi
Refinement??????
11
With little additional effort it is possible to compute??????
maxfrom�
(1)
:
??????
max=
1
2
??????
2
?̄??????(??????)�
(1)2
d?????? =
1
2
??????
6̄�
(1)2
?̄??????(??????)Ψ
(1)2
d??????
equating to our last approximation for??????
maxwe have the??????
11approximation to the
frequency of vibration,
??????
2
=
�
(0)
̄�
(1)
∫̄??????(??????)Ψ
(0)
Ψ
(1)
d??????
∫̄??????(??????)Ψ
(1)
Ψ
(1)
d??????
.
Of course the procedure can be extended to compute better and better estimates of
??????
2
but usually the refinements are not extended beyond??????
11, because it would be
contradictory with the quick estimate nature of the Rayleigh’s Quotient method
and
also because??????
11estimates are usually very good ones.
Nevertheless, we recognize the possibility of itereatively computing better and better
estimates opens a world of new opportunities.
SDOF
Giacomo Boffi
Refinement??????
11
With little additional effort it is possible to compute??????
maxfrom�
(1)
:
??????
max=
1
2
??????
2
?̄??????(??????)�
(1)2
d?????? =
1
2
??????
6̄�
(1)2
?̄??????(??????)Ψ
(1)2
d??????
equating to our last approximation for??????
maxwe have the??????
11approximation to the
frequency of vibration,
??????
2
=
�
(0)
̄�
(1)
∫̄??????(??????)Ψ
(0)
Ψ
(1)
d??????
∫̄??????(??????)Ψ
(1)
Ψ
(1)
d??????
.
Of course the procedure can be extended to compute better and better estimates of
??????
2
but usually the refinements are not extended beyond??????
11, because it would be
contradictory with the quick estimate nature of the Rayleigh’s Quotient method
and
also because??????
11estimates are usually very good ones.
Nevertheless, we recognize the possibility of itereatively computing better and better
estimates opens a world of new opportunities.
Generalized
SDOF
Giacomo Boffi
Refinement??????
11
With little additional effort it is possible to compute??????
maxfrom�
(1)
:
??????
max=
1
2
??????
2
?̄??????(??????)�
(1)2
d?????? =
1
2
??????
6̄�
(1)2
?̄??????(??????)Ψ
(1)2
d??????
equating to our last approximation for??????
maxwe have the??????
11approximation to the
frequency of vibration,
??????
2
=
�
(0)
̄�
(1)
∫̄??????(??????)Ψ
(0)
Ψ
(1)
d??????
∫̄??????(??????)Ψ
(1)
Ψ
(1)
d??????
.
Of course the procedure can be extended to compute better and better estimates of
??????
2
but usually the refinements are not extended beyond??????
11, because it would be
contradictory with the quick estimate nature of the Rayleigh’s Quotient methodand
also because??????
11estimates are usually very good ones.
Nevertheless, we recognize the possibility of itereatively computing better and better
estimates opens a world of new opportunities.
SDOF
Giacomo Boffi
Refinement??????
11
With little additional effort it is possible to compute??????
maxfrom�
(1)
:
??????
max=
1
2
??????
2
?̄??????(??????)�
(1)2
d?????? =
1
2
??????
6̄�
(1)2
?̄??????(??????)Ψ
(1)2
d??????
equating to our last approximation for??????
maxwe have the??????
11approximation to the
frequency of vibration,
??????
2
=
�
(0)
̄�
(1)
∫̄??????(??????)Ψ
(0)
Ψ
(1)
d??????
∫̄??????(??????)Ψ
(1)
Ψ
(1)
d??????
.
Of course the procedure can be extended to compute better and better estimates of
??????
2
but usually the refinements are not extended beyond??????
11, because it would be
contradictory with the quick estimate nature of the Rayleigh’s Quotient methodand
also because??????
11estimates are usually very good ones.
Nevertheless, we recognize the possibility of itereatively computing better and better
estimates opens a world of new opportunities.
Generalized
SDOF
Giacomo Boffi
Refinement??????
11
With little additional effort it is possible to compute??????
maxfrom�
(1)
:
??????
max=
1
2
??????
2
?̄??????(??????)�
(1)2
d?????? =
1
2
??????
6̄�
(1)2
?̄??????(??????)Ψ
(1)2
d??????
equating to our last approximation for??????
maxwe have the??????
11approximation to the
frequency of vibration,
??????
2
=
�
(0)
̄�
(1)
∫̄??????(??????)Ψ
(0)
Ψ
(1)
d??????
∫̄??????(??????)Ψ
(1)
Ψ
(1)
d??????
.
Of course the procedure can be extended to compute better and better estimates of
??????
2
but usually the refinements are not extended beyond??????
11, because it would be
contradictory with the quick estimate nature of the Rayleigh’s Quotient methodand
also because??????
11estimates are usually very good ones.
Nevertheless, we recognize the possibility of itereatively computing better and better
estimates opens a world of new opportunities.
SDOF
Giacomo Boffi
Refinement??????
11
With little additional effort it is possible to compute??????
maxfrom�
(1)
:
??????
max=
1
2
??????
2
?̄??????(??????)�
(1)2
d?????? =
1
2
??????
6̄�
(1)2
?̄??????(??????)Ψ
(1)2
d??????
equating to our last approximation for??????
maxwe have the??????
11approximation to the
frequency of vibration,
??????
2
=
�
(0)
̄�
(1)
∫̄??????(??????)Ψ
(0)
Ψ
(1)
d??????
∫̄??????(??????)Ψ
(1)
Ψ
(1)
d??????
.
Of course the procedure can be extended to compute better and better estimates of
??????
2
but usually the refinements are not extended beyond??????
11, because it would be
contradictory with the quick estimate nature of the Rayleigh’s Quotient methodand
also because??????
11estimates are usually very good ones.
Nevertheless, we recognize the possibility of itereatively computing better and better
estimates opens a world of new opportunities.
Generalized
SDOF
Giacomo Boffi
Refinement Example
??????
1.5??????
2??????
??????
2??????
3??????
Ψ
(0)
1
11/15
6/15
1
1
1
1
1.5
2
Ψ
(1)
??????
(0)
??????
2
??????
?????? =
1
2
??????
2
× 4.5 × ?????? �
2
0
?????? =
1
2
× 1 × 3?????? �
2
0
??????
2
=
3
9/2
??????
??????
=
2
3
??????
??????
�
(1)
=
15
4
??????
??????
??????
2
Ψ
(1)
̄�
(1)
=
15
4
??????
??????
??????
(1)
=
1
2
??????
15
4
??????
??????
??????
4
(1 + 33/30 + 4/5)
=
1
2
??????
15
4
??????
??????
??????
4
87
30
??????
2
=
9
2
??????
??????
87
8
??????
??????
=
12
29
??????
??????
= 0.4138
??????
??????
SDOF
Giacomo Boffi
Refinement Example
??????
1.5??????
2??????
??????
2??????
3??????
Ψ
(0)
1
11/15
6/15
1
1
1
1
1.5
2
Ψ
(1)
??????
(0)
??????
2
??????
?????? =
1
2
??????
2
× 4.5 × ?????? �
2
0
?????? =
1
2
× 1 × 3?????? �
2
0
??????
2
=
3
9/2
??????
??????
=
2
3
??????
??????
�
(1)
=
15
4
??????
??????
??????
2
Ψ
(1)
̄�
(1)
=
15
4
??????
??????
??????
(1)
=
1
2
??????
15
4
??????
??????
??????
4
(1 + 33/30 + 4/5)
=
1
2
??????
15
4
??????
??????
??????
4
87
30
??????
2
=
9
2
??????
??????
87
8
??????
??????
=
12
29
??????
??????
= 0.4138
??????
??????
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