Reduced-Order Network Models for Biological Scaffolding - Multiscale Modeling in Biomechanics and Mechanobiology - page 121

O(

2 , when 1

) = O( ) ˆ =

where t t

2 . Thus, inserting this into the equations

of equilibrium yields

) +

t
m i ψ

tot
i

) + O(

2

v i ( t + t ) = v i ( t ( t + ˆ t t ) . (6.34)

Note that adding a weighted sum of Eqs. ( 6.31 ) and ( 6.32 ) yields

2

v i ( t + ˆ t ) = ˆ v i ( t + t ) + ( 1 − ˆ) v i ( t ) + O( t ) , (6.35)

which will be useful shortly. Now expanding the position of the center of mass
in a Taylor series about t + ˆ t we obtain

d 2 r i
dt 2 | t + ˆ t (

d r i
dt | t + ˆ t (

1
2

2 2 3

r i ( t + t ) = r i ( t + ˆ t ) + 1 − ˆ) t + 1 − ˆ) ( t ) + O( t )

(6.36)

and

d 2 r i
dt 2 | t + ˆ t ˆ

d r i
dt | t + ˆ t ˆ

1
2

2 2 3

r i ( t ) = r i ( t + ˆ t ) − t + ( t ) + O( t ) . (6.37)

Subtracting the two expressions yields

r i ( t + t ) − r i ( t )

) + O( = v i ( t + ˆ t t ). (6.38)

t

Inserting Eq. ( 6.35 ) yields

O(

2

r i ( t + t ) = r i ( t ) + (ˆ v i ( t + t ) + ( 1 − ˆ) v i ( t )) t + t ) . (6.39)

and thus using Eq. ( 6.34 ) yields

2

+ ˆ(

t )

tot
i

) + O(

2

r i ( t + t ) = r i ( t ) + v i ( t ) t ψ ( t + ˆ t t ) . (6.40)

m i

tot
i

tot
i

tot
i

The term ψ ( t + ˆ t ) can be approximated by ψ ( t + ˆ t ) ≈ ˆ ψ ( r i ( t +

tot
i

. Thus, we have the following for the velocity 9

t )) + ( 1 − ˆ) ψ ( r i ( t ))

ˆ ψ

) ) +

t
m i

tot
i

tot
i

v i ( t + t ) = v i ( t ( t + t ) + ( 1 − ˆ) ψ ( t (6.41)

and for the position

9

In order to streamline the notation, we drop the cumbersome O ( t ) -type terms.