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MULTI-DEGREE OF FREEDOM LINEAR STRUCTURES
3. With K and M and the characteristic déterminant équation (8.61), calculate the consecutive frequencies uij,^,... ,cu/v, and order them from the
smallest to the largest.
4. Compute the components of the non-normalized modal vectors
using
N — 1 of the simultaneous équations (8.64). Note that
= 1, n = 1,2,... ,N.
5. Compute the N scalars en using équation (8.67); and then déterminé the
normalized modal vectors xn using équation (8.66).
6. Assemble the vectors xn to form the X matrix defined by équations (8.79)
and (8.80).
7. Fix two modal damping ratios Çk and s,n For instance, pick k — 1 and
m = 2, and Çj = Ç2 = 0.05; compute the Rayleigh coefficients aj and <12 from
équations (8.89); and compute the remaining N — 2 values of £n from équation
(8.88).
8. Using the results of wn from step 3 and £n from step 7, compute the N
damped frequencies a>dn from équation (8.97).
9. Compute yn(t) from équation (8.98) using numerical intégration. That
is, generate a table of yn(t) vs. t for each value of n.
10. Compute the modal coordinates
— £n(t) from équation (8.99). The
peak values of the coordinate components are useful in design.
It is advisable to implement this ten-step solution in a general computer
program. Numerical examples will be illustrated in the next chapter.
PROBLEMS
8.1. Consider a two degree of freedom lumped mass model of the jackup
rig shown in Fig. 2.17 and described in Example Problem 2.8. Choose the two
independent coordinates as v and 6, the displacement and rotation of the deck,
respectively. Let Pd and
be the arbitrary loadings corresponding to v and
6. For the deck, the mass moment of inertia about its mass center is Jd- Neglect
ail changes in élévation of the deck. Include damping.
(a) VA rite down the coordinate vector, the loading vector, and the damping
matrix.
(b) Define in words and symbols the two nonzero éléments of the mass matrix.
(< ) Each leg of the three-legged structure has a bending stiffness El and a
length l. Use classical beam theory to compute the éléments of the 2 x 2 stiffness
matrix.
(d) Show a free body sketch for each of the two masses.
(e) Use Newton s method to formulate the two équations of motion in terms
o. t and B. Include linear viscous damping in the form of équation (8.20).
8,2 ' For the structure described in Problem 8.1, formulate the expressions
r ■ inetic energy, the potential energy, and the Virtual work of the nonconservatixe onces. With these three scalar quantities, use Lagrange’s method to
denve the two équations of motion for the structure.
MULTI-DEGREE OF FREEDOM LINEAR STRUCTURES
3. With K and M and the characteristic déterminant équation (8.61), calculate the consecutive frequencies uij,^,... ,cu/v, and order them from the
smallest to the largest.
4. Compute the components of the non-normalized modal vectors
using
N — 1 of the simultaneous équations (8.64). Note that
= 1, n = 1,2,... ,N.
5. Compute the N scalars en using équation (8.67); and then déterminé the
normalized modal vectors xn using équation (8.66).
6. Assemble the vectors xn to form the X matrix defined by équations (8.79)
and (8.80).
7. Fix two modal damping ratios Çk and s,n For instance, pick k — 1 and
m = 2, and Çj = Ç2 = 0.05; compute the Rayleigh coefficients aj and <12 from
équations (8.89); and compute the remaining N — 2 values of £n from équation
(8.88).
8. Using the results of wn from step 3 and £n from step 7, compute the N
damped frequencies a>dn from équation (8.97).
9. Compute yn(t) from équation (8.98) using numerical intégration. That
is, generate a table of yn(t) vs. t for each value of n.
10. Compute the modal coordinates
— £n(t) from équation (8.99). The
peak values of the coordinate components are useful in design.
It is advisable to implement this ten-step solution in a general computer
program. Numerical examples will be illustrated in the next chapter.
PROBLEMS
8.1. Consider a two degree of freedom lumped mass model of the jackup
rig shown in Fig. 2.17 and described in Example Problem 2.8. Choose the two
independent coordinates as v and 6, the displacement and rotation of the deck,
respectively. Let Pd and
be the arbitrary loadings corresponding to v and
6. For the deck, the mass moment of inertia about its mass center is Jd- Neglect
ail changes in élévation of the deck. Include damping.
(a) VA rite down the coordinate vector, the loading vector, and the damping
matrix.
(b) Define in words and symbols the two nonzero éléments of the mass matrix.
(< ) Each leg of the three-legged structure has a bending stiffness El and a
length l. Use classical beam theory to compute the éléments of the 2 x 2 stiffness
matrix.
(d) Show a free body sketch for each of the two masses.
(e) Use Newton s method to formulate the two équations of motion in terms
o. t and B. Include linear viscous damping in the form of équation (8.20).
8,2 ' For the structure described in Problem 8.1, formulate the expressions
r ■ inetic energy, the potential energy, and the Virtual work of the nonconservatixe onces. With these three scalar quantities, use Lagrange’s method to
denve the two équations of motion for the structure.