Understanding the concepts on Degrees of Freedom

1.

AIM :

• To compute `underset(n)(L)` `underset (n)(M)` and `underset (n)(Gamma)`for each of the five modes from the given equation,
• To calculate effective model mass `underset (n)(M)` and the model mass participating ratio for the 5 modes.
• To obtain base shear of each of the five modes of the vibration and the combined base shear as per SRSS Rule.

PROCEDURE :

STEP 1:

`underset (n)(M)` = `underset(n)(phi)overset(T)` m `underset(n)(phi)`

Mode 1,

The first mode of vibration `underset(1)(T)` = 2 sec`

• `underset(1)(phi)` = `( (0.3344 ), (0.64) ,( 0.895) , (1.078) , (1.173) )`
• `underset (1)(M)` = `underset(1)(phi)overset(T)` m `underset(1)(phi)`

= `[[0.334],[0.64],[0.895],[1.078],[1.173]]` `[ [ m,0,0,0,0 ] , [ 0,m,0,0,0 ] , [ 0,0,m,0,0 ] , [0,0,0,m,0 ] , [ 0,0,0,0,m ] ]` `[0.334 , 0.64 , 0.895 , 1.078 , 1.173 ]`

By using the scientific calculater, we calculate `underset (1)(M)` = 3.681M `

Mode 2,

The first mode of vibration `underset(2)(T)` = 0.6852 seconds`

• `underset(2)(phi)` = `( (0.895 ), (1.173) ,( 0.641) , (0.334) , (1.078) )`
• `underset (2)(M)` = `underset(2)(phi)overset(T)` m `underset(2)(phi)`

`nbsp; = `[[0.895],[1.173],[0.641],[0.334],[1.078]]` `[ [ m,0,0,0,0 ] , [ 0,m,0,0,0 ] , [ 0,0,m,0,0 ] , [0,0,0,m,0 ] , [ 0,0,0,0,m ] ]` `[0.895 , 1.173 , 0.641 , 0.334, 1.078]`

`by using the scientific calculater, we calculate `underset (2)(M)` = 3.681M `

Mode 3,

The first mode of vibration `underset(3)(T)` = 0.4346 seconds`

• `underset(3)(phi)` = `( (1.173 ), (0.334) ,( 1.078) , (0.641) , (0.895) )`
• `underset (3)(M)` = `underset(3)(phi)overset(T)` m `underset(3)(phi)`

`nbsp; = `[1.173],[0.334],[1.078],[0.641],[0.895]]` `[ [ m,0,0,0,0 ] , [ 0,m,0,0,0 ] , [ 0,0,m,0,0 ] , [0,0,0,m,0 ] , [ 0,0,0,0,m ] ]` `[1.173 , 0,334 , 1.078 , 0.641, 0.895 ]`

`by using the scientific calculater, we calculate `underset (3)(M)` = 3.685M `

Mode 4,

The first mode of vibration `underset(4)(T)` = 0.3383 seconds`

• `underset(4)(phi)` = `( (1.078 ), (0.895) ,( 0.334) , (1.173) , (0.641) )`
• `underset (4)(M)` = `underset(4)(phi)overset(T)` m `underset(4)(phi)`

`nbsp; = `[[1.078],[0.895],[0.334],[1.173],[0.641]]` `[ [ m,0,0,0,0 ] , [ 0,m,0,0,0 ] , [ 0,0,m,0,0 ] , [0,0,0,m,0 ] , [ 0,0,0,0,m ] ]` `[1.078 , 0,895 , 0.334 , 1.173, 0.641 ]`

`by using the scientific calculater, we calculate `underset (4)(M)` = 3.861M `

Mode 5,

The first mode of vibration `underset(5)(T)` = 0.2966 seconds`

• `underset(5)(phi)` = `( (0.641 ), (1.078) ,( 1.173) , (0.895) , (0.334) )`
• `underset (5)(M)` = `underset(5)(phi)overset(T)` m `underset(5)(phi)`

`nbsp; = `[[0.641],[1.078],[1.173],[0.895],[0.334]]` `[ [ m,0,0,0,0 ] , [ 0,m,0,0,0 ] , [ 0,0,m,0,0 ] , [0,0,0,m,0 ] , [ 0,0,0,0,m ] ]` `[0.641 , 1.078 , 1.073 , 0.895 , 0.334 ]`

`by using the scientific calculater, we calculate `underset (5)(M)` = 3.861M `

STEP 2:

`underset(n)(L)` = `underset(n)(phi)overset(T)` mi`

Mode 1,

The first mode of vibration `underset(1)(T)` = 2 sec`

• `underset(1)(phi)` = `( (0.3344 ), (0.64) ,( 0.895) , (1.078) , (1.173) )`

`underset(1)(L)` = `underset(1)(phi)overset(T)` mi`

`` `[ [ m,0,0,0,0 ] , [ 0,m,0,0,0 ] , [ 0,0,m,0,0 ] , [0,0,0,m,0 ] , [ 0,0,0,0,m ] ]```[[1] , [1] , [1] ,[1] ,[1]]``[[ 0.334,0.64,0.895,1.078,1.173]]`

By using scientific calculator `underset(1)(L)` = 4.121M

Mode 2,

The first mode of vibration `underset(2)(T)``0.6852 sec`

• `underset(2)(phi)` = `( (0.895 ), (1.173) ,( 0.641) , (1.334) , (1.078) )`

`underset(2)(L)` = `underset(2)(phi)overset(T)` mi`

`nbsp; `` `[ [ m,0,0,0,0 ] , [ 0,m,0,0,0 ] , [ 0,0,m,0,0 ] , [0,0,0,m,0 ] , [ 0,0,0,0,m ] ]```[[1] , [1] , [1] ,[1] ,[1]]``[[ 0.895,1.173,0.641,0.334,1.078]]`

`y using scientific calculator `underset(2)(L)` = 1.298M`

Mode 3,

The first mode of vibration `underset(3)(T)``= 0.4346 sec`

• `underset(3)(phi)` = `( (1.173 ), (0.334) ,( 1.078) , (0.641) , (0.895) )`

`underset(3)(L)` = `underset(3)(phi)overset(T)` mi`

`nbsp; `` `[ [ m,0,0,0,0 ] , [ 0,m,0,0,0 ] , [ 0,0,m,0,0 ] , [0,0,0,m,0 ] , [ 0,0,0,0,m ] ]```[[1] , [1] , [1] ,[1] ,[1]]``[[ 1.173,0.334,1.078,0.641,0.895]]`

`y using scientific calculator `underset(3)(L)` = 0.687M`

Mode 4,

The first mode of vibration `underset(4)(T)``= 0.3383 sec`

• `underset(4)(phi)` = `( (1.078 ), (0.895) ,( 0.334) , (1.173) , (0.641) )`

`underset(4)(L)` = `underset(4)(phi)overset(T)` mi`

`nbsp; `` `[ [ m,0,0,0,0 ] , [ 0,m,0,0,0 ] , [ 0,0,m,0,0 ] , [0,0,0,m,0 ] , [ 0,0,0,0,m ] ]```[[1] , [1] , [1] ,[1] ,[1]]``[[ 1.078,0.895,0.334,1.173,0.641]]`

`y using scientific calculator `underset(4)(L)` = 0.381M`

Mode 5,

The first mode of vibration `underset(5)(T)``= 0.2966 sec`

• `underset(5)(phi)` = `( (0.641 ), (1.078) ,( 1.173) , (0.895) , (0.334) )`

`underset(5)(L)` = `underset(5)(phi)overset(T)` mi`

`nbsp; `` `[ [ m,0,0,0,0 ] , [ 0,m,0,0,0 ] , [ 0,0,m,0,0 ] , [0,0,0,m,0 ] , [ 0,0,0,0,m ] ]```[[1] , [1] , [1] ,[1] ,[1]]``[[ 0.641,1.078,1.073,0.895,0.334]]`

`y using scientific calculator `underset(5)(L)` = 0.175M`

STEP 3:

`underset(n)Gamma` = `underset(n)(L) // underset(n)(M)`

MODE 1 -

`underset(1)Gamma` = `underset(1)(L) // underset(1)(M)` ` = `4.121M//3.861M` = 1.067

MODE 2 -

`underset(2)`Gamma` = `underset(2)(L) // underset(2)(M)` ` = `1.298M//3.861M` = -0.33`

`MODE 3`

`underset(3)Gamma` = `underset(3)(L) // underset(3)(M)` ` = `0.687M//3.85M` = 0.178`

`MODE 4`

`underset(4)Gamma` = `underset(4)(L) // underset(4)(M)` ` = `0.381M//3.861M` = -0.09`

`MODE 5`

`underset(5)Gamma` = `underset(5)(L) // underset(5)(M)` ` = `0.175M//3.861M` = 0.04`

STEP 4:

MODE N

`underset(n)(M)overset(#) ` = `underset(n)(L)overset(2) // underset(n)(M)` `

MODE 1 -

`underset(1)(M)overset(#)` = `underset(1)(L)overset(2) // underset(1)(M)` ` = overset(2)(4.121M)//3.861M` = 4.39`

`MODE 2`

`underset(2)(M)overset(#)` = `underset(2)(L)overset(2) // underset(2)(M)` ` = overset(2)(-0.298M)//3.861M` = 0.43`

`MODE 3`

`underset(3)(M)overset(#)` = `underset(3)(L)overset(2) // underset(3)(M)` ` = overset(2)(0.687M)//3.85M` = 0.12`

`MODE 4`

`underset(4)(M)overset(#)` = `underset(4)(L)overset(2) // underset(4)(M)` ` = overset(2)(-0.381M)//3.861M` = 0.03`

`MODE 5`

`underset(5)(M)overset(#)` = `underset(5)(L)overset(2) // underset(5)(M)` ` = overset(2)(0.175M)//3.861M` = 0.007`

STEP 5 :

MODEL mass participating ratio = `underset(n)(M)overset(#) //j`

Effective model mass = `underset(n)(M)overset(#)`

`underset (j)(M)` = Total mass = M+M+M+M+M = 5M

MODEL 1:

`underset(1)(M)overset(#) //underset(j)(M)` = `4.39//5` = 0.88 = 88%

`MODE 2`

`nbsp;`underset(2)(M)overset(#) //underset(j)(M)` = `0.43//5` = 0.086 = 8.6%`

`MODE 3`

`nbsp;`underset(3)(M)overset(#) //underset(j)(M)` = `0.12//5` = 0.024 = 2.4%`

`MODE 4`

`nbsp;`underset(4)(M)overset(#) //underset(j)(M)` = `0.03//5` = 0.006 = 0.6%`

`MODE 5`

`nbsp;`underset(5)(M)overset(#) //underset(j)(M)` = `0.007//5` = 0.001 = 0.1%`

STEP 6:

Now we need to obtain the base shear for each of the 5 modes of vibration as shown below,

`underset(bn)(V)` = `underset(n)(L) overset(2) //underset(n)(M)` * `underset (n)(A)`

• `underset(1)(T)` = 2 sec , `underset(1)(A)` = 0.27g

• `nbsp; `underset(2)(T)` = 0.6852 sec , `underset(2)(A)` = 0.76g`

• `nbsp; `underset(3)(T)` = 0.4346 sec , `underset(3)(A)` = 1.03g`

• `nbsp; `underset(4)(T)` = 0.3383 sec , `underset(4)(A)` = 1.03g`

• `nbsp; `underset(5)(T)` = 0.2966 sec , `underset(5)(A)` = 0.76g`

MODEL 1 :

`underset(b1)(V)` = `underset(1)(L) overset(2) //underset(1)(M)` * `underset (1)(A)` ` = 4.39M * 0.27 g = 2.619 kip`

`MODEL 2`

`underset(b2)(V)` = `underset(2)(L) overset(2) //underset(2)(M)` * `underset (2)(A)` ` = 0.43M * 0.76 g = 0.719 kip`

`MODEL 3`

`underset(b3)(V)` = `underset(3)(L) overset(2) //underset(3)(M)` * `underset (3)(A)` ` = 1.03M * 0.12 g = 0.272 kips`

`MODEL 4`

`underset(b4)(V)` = `underset(4)(L) overset(2) //underset(4)(M)` * `underset (4)(A)` ` = 0.03M * 0.12 g = 0.067 kips`

`MODEL 5`

`underset(b5)(V)` = `underset(5)(L) overset(2) //underset(5)(M)` * `underset (5)(A)` ` = 0.011M * 0.76 g = 0.011 kips`

STEP 7:

Combined base shear as per SRSS Rule,

`underset(b)V` = `sqrt underset(b1)(V)overset(2) + sqrt underset(b2)(V)overset(2) + sqrt underset(b3)(V)overset(2) + sqrt underset(b4)(V)overset(2) + sqrt underset(b5)(V)overset(2)`

= `sqrt overset(2)(2.613)+overset(2)(0.719)+overset(2)(0.272)+overset(2)(0.067)+overset(2)(0.0011)`

= 3.20 kips

STEP 8:

• The two modes are to be considered such that a model mass participating ratio of 90% is obtained
• `underset(1)(M)overset(#)//underset(j)(M)` = 88% + 8.6% = 96.6%

RESULT :

• Calculated `underset(n)(L)` `underset(n)(M)` and `underset(n)(Gamma)` for each of the five modes from the given equation
• calculated the effective Model mass `underset(n)(M)overset(#)` and the model mass participating ratio for the five modes
• the base shear for each of the five modes of vibration are obtained
• as per SRSS Rule the combined base shear is obtained
• The two modes are to be considered such that a model mass participating ratio of 90% is obatined.

M