2 - sdof - free vibrations

Structural Analysis-

spring 2013

(single degree of freedom - free vibrations)

Fawad Muzaffar M.Sc. Structures (Stanford University)

Ph.D. Structures (Stanford University)

Civil Engineering

Department

1

• Free Vibration: When a structure is disturbed from its equilibrium position and allowed to vibrate without any external force.

• Components of Dynamic System:

i. Mass

ii. Elastic Properties (Stiffness)

iii. Energy Loss Mechanism (Damping)

iv. The External Force Acting on the System

• The First Derivation

Use d’Alembert’s Principle to express equilibrium of forces acting on the block

------Equation 1

Equation of Motion of The Basic Dynamic System

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Note: D’Alembert’s Principle (named after French Mathematician Jean le Rond d’Alembert) states that different of sum of forces acting on a system of mass particles and time derivative of momenta of the system is equal to zero.

From Newton’s Second Law:

Assuming Viscous Damping:

Finally, the Elastic Force can be obtained from

Plugging all of the above values into Equation 1, we have


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• The virtual work Derivation

If mass is given a virtual displacement 𝛿𝑣 , the total work done by

equilibrium of forces must be equal to zero.

The negative signs in the above equation indicates

that force act in the negative direction relative to the

direction of motion.

Plugging in all constituent values, we have

Since 𝛿𝑣 ≠ 0

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• Influence of Gravitational Forces

From Equilibrium of forces

Expressing

The component forces can be evaluated as

The equation of motion can then be written

Noting that , we have

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Since Δ𝑠𝑡 does not vary with time

The equation of motion can then be written as

Note: equation of motion expressed with reference to static-equilibrium position of dynamic system is not effected by gravity.

• Influence of Support Excitation Dynamic deflections can be induced by motion of support points e.g. i) Excitation of supports by E.Q. ii) Motions of base of equipment due

to vibration of foundation

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• Assumptions – The Girder is Rigid.

– The Columns are Weightless.

– The Columns are Inextensible in Axial Direction.

– Resistance to Lateral Displacement id Provided by Columns.

– The Damper Provides a Velocity Proportion Response to Displacement

From Equilibrium in Horizontal Direction.

The Inertial Force is given by

where 𝑣 𝑡 represents total displacement of mass from fixed reference axis.

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The total motion of mass, 𝑣𝑡 can be expressed as

Plugging this value of 𝑣𝑡 back into equation of motion results in

The above expression can then be manipulated to yield

Note: 𝑝𝑒𝑓𝑓(𝑡) is called effective support excitation

An alternate form of above equation can be obtained by using 𝑣𝑡 𝑡 instead of 𝑣(𝑡).

R.H.S. depends on velocity and displacement of earthquake motion.

L.H.S. depends on quantities depending on total displacement. 8 Fawad Muzaffar

Analysis of Free Degree of Freedom Structures – Math Preliminaries

• Math Preliminaries

– The Complex Number Concept • A Complex Number G has real and imaginary parts

• The Polar form of Complex Number in complex

plane.

where 𝐺 = |𝐺| is the length of the vector.

• Using Euler’s Identity, the polar form can be expressed as

• Note: Multiplying G by 𝑖 rotates it by 𝜋/2 e.g.

𝑮 × 𝒊 = 𝑮 𝒄𝒐𝒔𝜽 + 𝒊𝑮 𝒔𝒊𝒏𝜽 𝒊 = −𝑮 𝒔𝒊𝒏𝜽 + 𝑮 𝒄𝒐𝒔𝜽𝒊

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Analysis of Free Degree of Freedom Structures – Math Preliminaries

• Euler’s Identity 𝒆𝒊𝜽 = 𝒄𝒐𝒔 𝜽 + 𝒊𝒔𝒊𝒏 𝜽

Taylor’s Series

𝒇 𝒙 = 𝒇 𝒙𝟎 + 𝟏

𝒌!

𝒅𝒌𝒇

𝒅𝒙𝒌 𝒙=𝒙𝟎

𝒙 − 𝒙𝒐𝒌

∞

𝒌=𝟏

Using Taylor Expansion to express 𝑐𝑜𝑠 𝜃 and 𝑠𝑖𝑛 𝜃 with 𝑥 = 𝜃 and 𝑥0 = 0

𝒄𝒐𝒔 𝜽 = 𝟏 −𝟏

𝟐𝜽𝟐 +

𝟏

𝟒!𝜽𝟒 −

𝟏

𝟔!𝜽𝟔 +

𝟏

𝟖!𝜽𝟖 − ⋯

sin 𝜽 = 𝜽 −𝟏

𝟑!𝜽𝟑 +

𝟏

𝟓!𝜽𝟓 −

𝟏

𝟕!𝜽𝟕 +

𝟏

𝟗!𝜽𝟗 − ⋯

Expressing 𝑒𝑥 with 𝑥 = 𝑖𝜃 and 𝑥0 = 0

𝒆 𝒊𝜽 = 𝟏 + 𝜽. 𝒊 −𝟏

𝟐𝜽𝟐 −

𝒊

𝟑!. 𝜽𝟑 +

𝟏

𝟒!. 𝜽𝟒 +

𝒊

𝟓!. 𝜽𝟓 −

𝟏

𝟔!. 𝜽𝟔 −

𝒊

𝟕!. 𝜽𝟕 +

𝟏

𝟖!. 𝜽𝟖 +

𝒊

𝟗!. 𝜽𝟗 …

𝒆 𝒊𝜽 = 𝟏 −𝟏

𝟐𝜽𝟐 +

𝟏

𝟒!. 𝜽𝟒 −

𝟏

𝟔!. 𝜽𝟔 +

𝟏

𝟖!. 𝜽𝟖 − ⋯ + 𝒊. 𝜽 −

𝟏

𝟑!. 𝜽𝟑 +

𝟏

𝟓!. 𝜽𝟓 −

𝟏

𝟕!. 𝜽𝟕 +

𝟏

𝟗!. 𝜽𝟗 …

𝒆 𝒊𝜽 = 𝐜𝐨𝐬𝜽 + 𝒊. sin𝜽

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Analysis of Free Degree of Freedom Structures

• For a freely vibrating structure, the equation of motion becomes

• The solution of the above linear, 2nd order homogeneous differential

equation is

where G is a complex constant and s is a real constant.

• Plugging in the value of 𝑣 into the equation of motion results in

The equation of motion becomes

----Equation 2

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• Case 1 - Damping c=0:

– When c=0, the values of s becomes

– The general solution of the differential equation then comes out to be

----Equation 3

– The complex constants can be expressed as

– Also from Euler’s Identity

– Plugging all values in Equation 3 results in

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– The free vibration response HAS TO BE REAL. Terms inside square brackets should be zero. Only possible if 𝐺1 and 𝐺2 are complex conjugate pair.

– The solution equation becomes

– Using Euler Identity to expand 𝑒𝑖𝑤𝑡 in the above equation results in

where

– The value of A and B can be determined from initial conditions 𝑣(0) and 𝑣 0 , resulting in

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– The equation of motion becomes

– The Solution Represents SHM

– The time required to complete

1- Cycle, T is given by

– The frequency of motion is given

by

– T is measured in secs and f is measured in Hertz (Hz).

– The maximum value of displacement is given by

– The phase angle 𝜃 is given by

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– The equation of motion becomes

– The Solution Represents SHM

– The time required to complete

1- Cycle, T is given by

– The frequency of motion is given

by

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2 - sdof - free vibrations

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