sd lecture04 2d0ffreedom

7/28/2019 SD Lecture04 2D0fFreedom

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Soil Dynamics

Lecture 04

Systems with Two-Degrees of Freedom

Luis A. Prieto-Portar, August 2006.


2/18

The figure at right shows a simple

non-damped mass and spring system

with two-degrees of freedom.

This simple system can be excited

into vibration in two different ways:

(1) A sinusoidal force is applied to

the mass m1, thereby resulting

in a forced vibration of the

system, or

(2) The system is set to vibrate by

an impact force on the mass m2.


3/18

The calculation of the systems natural frequency. Consider the free-bodydiagram on the previous slide. The differential equations of motion are,

1 1 1 1 2 1 2

2 2 2 2 1

1 2

2

1 2 1 2

22 2 2

0

0

0

0

n n

n

n

m z k z k ( z z ) and

m z k ( z z )

Let z A sin t and z B sin t

Backsubstituting these solution s int o the basic differential equations,

A( k k m ) k B and

Ak ( k m )B

Since A and B are not zero, the n

+ + =

+ =

= =

+ =

+ =

2 2 2

1 2 1 2 2 2

4 21 1 2 2 2 1 1 2

1 2 1 2

0

n n

n n

on trivial solution is,

( k k m )( k m ) k

or ,

k m k m k m k k

m m m m

+ =

+ + + =


4/18

1 2

1 2 1 2

2

1

1 2

1 2 2

4 2 2 2 2 21 1 0

n

nl nl

n nl nl n nl nl

The equation for the natural frequency of the system can be simplified by

msettingm

k kand and

m m m

which yields,

( )( ) ( )( )( )

=

= =+

+ + + + =


5/18

Case 1: The amplitude of vibration for a

force on mass m1.

Consider the case when a vibration is induced

on the system through a force acting upon the

mass m1. The differential equations of motion

are now,

1 1 1 1 2 1 2 0

2 2 2 2 1

1 1 2 2

2

1 1 1 2 2 2 02

2 2 2 1 2

0

0

m z k z k ( z z ) Q sin t and

m z k ( z z )

Let z A sin t and z A sin t

Substituting back int o the d .e.s,

A ( m k k ) A k Q

A ( k m ) A k

+ + =

+ =

= =

+ + =

=


6/18

( ) ( )

( ) ( )( )

2 2

1 2 1 2

2

2 2 2

0 01 22 2

1 1

2 4 2 2 2 2 2

1

1 1

1 1

0

nl nl

nl nl nl nl

nl

From these two equations,we obta in t he two coefficients,

Q ( ) Q ( )A and Am m

where ( )( )( )

Notice that if A then

The is formemain system d by m and k , wherea

= =

= + + + +

= =

2

2 2

nl

s the is

formed by m and k .Thus ,the vibration of the main system can be reduced

or even e li minated

auxiliary syst

by the auxiliary system if its natural frequenc .

The auxiliary system is the

em

vibrationrefore absor era .b

=


7/18

Case 2: An impact force upon mass m2. This case can be modeled by assuming thatthe vibration is induced via an initial velocity v0 to the mass m2. Hence,

( )( )( )

( ) ( )

1 2

1 2

2 1 2 2 1 2

1 22 1 2

2 2 1 2

11 2

1 1 2

2 1 2

1 2 1 2 0

2 2 2 2

1 02 2 2

2 2 2

2 2 2

0 0 0

1

n n

n n

nl n nl n n n

n nnl n n

nl n n nl

nn n

z C sin t C sin t

z D sin t D sin t

At time t z z and z and z v

Substituting ,

sin t sin t z v and

sin t z

= +

= +

= = = = =

=

=

( )

( )( )( )

( )

( )

1 2

2

2 1 2 2

2 1 2 2

2 1

1 2 2

2

0

1 2 1 2

2 2 2 2

1 02 2 2

2 2

2 02 2

n n

n

nl n nl n

nl n n n

nl n

n n n

sin t v

therefore , the amplitudes Z and Z of the masses m and m are ,

Z v and

Z v

=

=


8/18

Example 1.

Calculate the natural frequencies of the system n1 and n2 shown below, ifW1 = 25 lb

and W2 = 5 lb, and the spring constants are k1 = 100 lb/in and k2 = 50 lb/in.


9/18

( ) ( )( )( )

( ) ( )( )

( )

1 2 1 2

1

2

4 2 2 2 2 2

2 2

1 1

1

1 2

2

2

4

1 1 0

50 20

25

100 32 2 1236

25 5

50 32 2 12

625

1

n nl nl n nl nl

nl

nl

n

The equation for the natural frequencies is,

( )( ) ( )( )( )

where,

m W.

m W

.kradians / s

m m

.k

radians / sm

Therefore,

( )(

+ + + + =

= = = =

= = =+ +

= = =

+

( ) ( )

( ) ( )

1 2

2

1

1

2

1 2

2 2 2 2 2

2 24 2 2 2

4 2

2

2

1 0

1 0 2 36 62 1 0 2 36 62 0

6 140

34 70

5 900 000 0

6 140 6 140 4 5 900 000

2,

nl nl n nl nl

n n

n n

nn n

) ( )( )( )

( . ) ( . )( ) ( )

, , ,

, ,rad / s rad

, ,or and / s

+ + + =

+ + + + =

=

+ =

=

=


10/18

Example 2.

Calculate the amplitudes of vibrationZ1 andZ2 for the two masses ofExample 1, if avibratory force Q = 10 sint(lb) is applied to the mass m1 with a frequency =78.54

radians/s.

( ) ( )( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

1 2 1 2

1 1 2

2 4 2 2 2 2 2

4 2 2 2 2 2

1

1 1

78 54 1 0 2 36 62 78 54 1 0 2 36 62

6 000 000

nl nl nl nl

This is case # , where a vibration i s induced on m .We seek the values of A and A ,

( )( )( )

. . . .

, ,

From these two equations,we obta in t he coeff

= + + + +

= + + + +

=

( )

( ) ( ) ( )

( )

( )( )

( )

2

2

2 22 2

0

1 261

2 20

2 261

10 62 78 54

256 10

32 2

0 06

0

12

10 62

256 10

10

32 2 12

nl

nl

icients,

.Q ( )A

mx

. ( )

Q ( ) ( )A

mx

. inches

. inches

. ( )

= = =

= = =


11/18

A free vibration system of coupled translation and rotation.

The 2-D of freedom system shown below will experience both translation and rotation.

The two differential equations of motion of the mass m are,


12/18

1 1 2 2

2

1 1 1 2 2 2

2

2

2

0

0

mz k ( z l ) k ( z l )

mr l k ( z l ) l k ( z l )

where

is the angle of rotation of the mass m ,

ddt

r is the radius of gyration of the body about the center of gravity ,

mr J is the mass moment of inertia about the cen

+ + + =

+ + =

=

=

( )

( )

2 2

1 2 1 1 2 2

2 2 1 1

2

2 2 1 1

0

0

z

z

ter of gravity.

Let ,

k k k and l k l k k

Therefore,

mz k z l k l k

mr k l k l k z

+ = + =

+ + =

+ + =


13/18

1 1 2 2

2

2

00

z

Notice that if l k l k then the mz equation i s independent of .

Similarly ,the mr equation i s independent of z.

Therefore , the two motions, translation and rotation exist independent

of each other ,

mz k z

mr k

The

=

+ =+ =

2

nz

z

nz

n

n

natural circular frequency of translation is,

k

m

and the natural circular frequency of rotation is,

kmr

=

=


14/18

1 1 2 2

2 2 1 11 2 2

1 2

3 22 2

0

0

z

n

However , if l k l k the equations are coupled ,and can be solved ,

k kl k l k E and E and Em m m

So , the coupled differential equations are ,

z E z E

E E zr r

Let the solutions be ,

z Z cos t and

= = =

+ + =

+ + =

= =

( )

( )

1

1

2

2

1 1 2 2 1 1 2 2

2 2

31 2

2 2

1 1 2

2 2

32 2

2 2

2 1 2

n

n n n n

n

n

n

n

cos t

from whence the general equations of motion are ,

z Z cos t Z cos t and cos t cos t

where the amplitude ratios are,

E / rZ E andE E / r

E / rZ E

E E / r

= + = +

= =

= =


15/18

Homework Problem #1.

Determine the natural frequency of the undamped free vibration of the system shown

below.


16/18


Determine the natural frequency of the undamped free vibration of the system

shown below.


17/18


Determine the natural frequency and the period of the system shown below.

100 N/mm 200 N/mm

150 N/mm

100 N/mm 150 N/mm 100 kg

Q=50(N)sint where =47 rad/s


18/18

References.

Das, B., Principles of Soil Dynamics, PWS-Kent Publishing Co., Boston, 1993;

Richart F.E., Hall J.R., Woods R.D., Vibrations of Soils and Foundations, Prentice-

Hall Inc., New Jersey, 1970;

Humar J.L., Dynamics of Structures, Prentice-Hall, New Jersey, 1990;

Prakash S., Soil Dynamics, McGraw-Hill, New York, 1981;

sd lecture04 2d0ffreedom

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