part 02 - support motion - transmissibility

8102019 Part 02 - Support Motion - Transmissibility

httpslidepdfcomreaderfullpart-02-support-motion-transmissibility 114

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Response to Support Motion

1 Support Displacement

k

similar to forced vibration

It is more common to represent support motion in terms ofsupport acceleration

m

k(y-x) m

k

2 Support Acceleration

x and y are absolute coordinates

based on a fixed reference

system



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Free Body Diagram

FS = force on spring based on relative displacement

=

FD = force on damper based on relative velocity

= c

FI = Inertia force based on absolute acceleration

=

FS FD FI

Dynamic Equilibrium

Define ndash coordinate based on fixed support ie relative

coordinate

Note or

also

Hence c

or

(1)



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The system can be thought of as

983110983113983128983109983108

One special case of base excitation is that of harmonicsupport displacement How much of the support motion istransmitted to the mass

(due to Harmonic Excitation)

Response from equation (1)

983128

983161

If

TRANSMISSIBILITY





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Plot of Tr vs

Note All curves pass through the same point at 2

Damping tends to reduce the effectiveness of

vibration isolation for 2

Frequency Ratio β =ω

ω

Transmissibility vs frequency ratio for vibration isolation

Note λ = ζDamping ratio

In terms of relative motion the response is

subs in Eq (2a)

sin θ

1 2

x(t) = x0sint

m

u

FIXED

=



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1 2

or

Maximum Relative Displacement =

1 2

Example The frame shown is subjected to a sinusoidalground motion 2 Determinea Transmissibility of motion to girderb Maximum shearing force in columnsc Maximum stress in columns

SolutionDetermine

2

21

21

1 1

1 1

15rsquo W8X20

W = 15k

2



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a)

21 (Absolute)

b) Determine maximum relative displacement

1 2 2

Max shear in column =

2 22

c) Max moment 22 1 12

σ MS

V M

U

DEM

Steady State Response

θ

where

θ

The force transmitted to the foundation ue an

t k yt an tbull Since these two forces are 90 degrees out of phase with each other it is

evident that the amplitude of the total base reaction force is given by

t t

Force transmitted to the foundation (support)

y

F(t) = F0sint

F0

AT

FT



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sin θ

where 2

Define Transmissibility of force

- ratio of amplitude of force transmitted to the

support to the amplitude of applied force ie F0

From previous results

Where = maximum acceleration transmitted to the oscillator

= maximum acceleration of the support (ground)

= maximum displacement of the support (ground)

= maximum displacement transmitted to the oscillator



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983097983087983091983087983090983088



983097983087983091983087983090983088

Objective To measure vibration of support (displacement

or acceleration)

Seismic Instruments (Principle of Seismographs)



983097983087983091983087983090983088



983097983087983091983087983090983088



983097983087983091983087983090983088



983097983087983091983087983090983088

Free Body Diagram

FS = force on spring based on relative displacement

=

FD = force on damper based on relative velocity

= c

FI = Inertia force based on absolute acceleration

=

FS FD FI

Dynamic Equilibrium

Define ndash coordinate based on fixed support ie relative

coordinate

Note or

also

Hence c

or

(1)



983097983087983091983087983090983088


983110983113983128983109983108




983128

983161

If

TRANSMISSIBILITY





983097983087983091983087983090983088

Plot of Tr vs





ω




subs in Eq (2a)

sin θ

1 2

x(t) = x0sint

m

u

FIXED

=



983097983087983091983087983090983088

1 2

or


1 2


SolutionDetermine

2

21

21

1 1

1 1

15rsquo W8X20

W = 15k

2



983097983087983091983087983090983088

a)

21 (Absolute)


1 2 2


2 22


σ MS

V M

U

DEM


θ

where

θ




t t


y

F(t) = F0sint

F0

AT

FT



983097983087983091983087983090983088

sin θ

where 2











983097983087983091983087983090983088



983097983087983091983087983090983088



983097983087983091983087983090983088


or acceleration)




983097983087983091983087983090983088



983097983087983091983087983090983088



983097983087983091983087983090983088



983097983087983091983087983090983088


983110983113983128983109983108




983128

983161

If

TRANSMISSIBILITY





983097983087983091983087983090983088

Plot of Tr vs





ω




subs in Eq (2a)

sin θ

1 2

x(t) = x0sint

m

u

FIXED

=



983097983087983091983087983090983088

1 2

or


1 2


SolutionDetermine

2

21

21

1 1

1 1

15rsquo W8X20

W = 15k

2



983097983087983091983087983090983088

a)

21 (Absolute)


1 2 2


2 22


σ MS

V M

U

DEM


θ

where

θ




t t


y

F(t) = F0sint

F0

AT

FT



983097983087983091983087983090983088

sin θ

where 2











983097983087983091983087983090983088



983097983087983091983087983090983088



983097983087983091983087983090983088


or acceleration)




983097983087983091983087983090983088



983097983087983091983087983090983088



983097983087983091983087983090983088





983097983087983091983087983090983088

Plot of Tr vs





ω




subs in Eq (2a)

sin θ

1 2

x(t) = x0sint

m

u

FIXED

=



983097983087983091983087983090983088

1 2

or


1 2


SolutionDetermine

2

21

21

1 1

1 1

15rsquo W8X20

W = 15k

2



983097983087983091983087983090983088

a)

21 (Absolute)


1 2 2


2 22


σ MS

V M

U

DEM


θ

where

θ




t t


y

F(t) = F0sint

F0

AT

FT



983097983087983091983087983090983088

sin θ

where 2











983097983087983091983087983090983088



983097983087983091983087983090983088



983097983087983091983087983090983088


or acceleration)




983097983087983091983087983090983088



983097983087983091983087983090983088



983097983087983091983087983090983088



983097983087983091983087983090983088

1 2

or


1 2


SolutionDetermine

2

21

21

1 1

1 1

15rsquo W8X20

W = 15k

2



983097983087983091983087983090983088

a)

21 (Absolute)


1 2 2


2 22


σ MS

V M

U

DEM


θ

where

θ




t t


y

F(t) = F0sint

F0

AT

FT



983097983087983091983087983090983088

sin θ

where 2











983097983087983091983087983090983088



983097983087983091983087983090983088



983097983087983091983087983090983088


or acceleration)




983097983087983091983087983090983088



983097983087983091983087983090983088



983097983087983091983087983090983088



983097983087983091983087983090983088

a)

21 (Absolute)


1 2 2


2 22


σ MS

V M

U

DEM


θ

where

θ




t t


y

F(t) = F0sint

F0

AT

FT



983097983087983091983087983090983088

sin θ

where 2











983097983087983091983087983090983088



983097983087983091983087983090983088



983097983087983091983087983090983088


or acceleration)




983097983087983091983087983090983088



983097983087983091983087983090983088



983097983087983091983087983090983088



983097983087983091983087983090983088

sin θ

where 2











983097983087983091983087983090983088



983097983087983091983087983090983088



983097983087983091983087983090983088


or acceleration)




983097983087983091983087983090983088



983097983087983091983087983090983088



983097983087983091983087983090983088



983097983087983091983087983090983088



983097983087983091983087983090983088



983097983087983091983087983090983088


or acceleration)




983097983087983091983087983090983088



983097983087983091983087983090983088



983097983087983091983087983090983088



983097983087983091983087983090983088



983097983087983091983087983090983088


or acceleration)




983097983087983091983087983090983088



983097983087983091983087983090983088



983097983087983091983087983090983088



983097983087983091983087983090983088


or acceleration)




983097983087983091983087983090983088



983097983087983091983087983090983088



983097983087983091983087983090983088



983097983087983091983087983090983088



983097983087983091983087983090983088



983097983087983091983087983090983088



983097983087983091983087983090983088



983097983087983091983087983090983088



983097983087983091983087983090983088

part 02 - support motion - transmissibility

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