Dynamic Responses - Mechanics of Structures: Variational and Computational Methods - page 620

STEP 4. Obtain Uncoupled Equations of Motion
The uncoupled equations for the i th modal coordinate q i are

i q i

q i + 2 ζ ω q i ˙ + ω = P i ( t )/ M i (10.84)

i i

Note that modal damping has been included here. For undamped responses, set ζ = 0.

i

STEP 5. Express the Initial Conditions in Modal Coordinates
With the assistance of the orthogonality relationship for the mode shapes, the modal
initial conditions q i ( 0 ) and q i ˙ ( 0 ) can be computed in terms of the physical initial conditions

using

( )

MV 0

φ i ( ) =

q i 0

M i

(10.85)

V

( )

M 0

φ i

˙ ( ) =

q i 0

M i

STEP 6. Compute Modal Responses to Initial Conditions
The modal coordinate due to initial conditions is, from the complementary solution to
Eq. (10.84), for ζ < 1,

i

e − ζ i ω i t ω i 1 ω i 1

˙

q i ( 0 ) + q i ( 0 )ζ i ω i

2 2

i

q i I ( t ) =

i 1

sin − ζ t + q i ( 0 ) cos − ζ t (10.86)

i

i

ω − ζ

If the forced responses are started from zero initial conditions, Steps 5 and 6 can be bypassed.

STEP 7. Compute Modal Responses to Applied Loading
The modal coordinate due to an applied loading, which is the particular solution of
Eq. (10.84), is

t

ω i 1

1

e − ζ i ω i ( t − τ) sin

i

q i P ( t ) =

i 1

P i (τ ) − ζ ( t − τ) d τ (10.87)

2
i

M i ω − ζ

0

For the case of harmonic excitation, i.e., P i = P 0 sin ω t, the integration in Eq. (10.87) can be

carried out analytically, giving

i

( P 0 / M i )ω

q i P (

) = (ω − θ )

t sin t (10.88)

1

− ω i 2 2

i ω i 2 1 / 2

i

2

+ 4 ζ

with

tan − 1 2

ζ i ω i ω

θ i =

i

ω − ω

2

The total modal response is given by

( ) =

q i I (

) +

q i P (

)

q i t t t (10.89)