CHAPTER 19. PARTIAL DIFFERENTIAL EQUATIONS OF MOTION
Evaluate Displacement and AxialForce Response:
CHAPTER 19. PARTIAL DIFFERENTIAL EQUATIONS OF MOTION
Multiplying by
and integrating yields
CHAPTER 19. PARTIAL DIFFERENTIAL EQUATIONS OF MOTION
CHAPTER 19. PARTIAL DIFFERENTIAL EQUATIONS OF MOTION
Example E192.
generalized loading leads to
stiffnessproportional damping zero, except for the term i = n
CHAPTER 19. PARTIAL DIFFERENTIAL EQUATIONS OF MOTION
Finally, introducing the damping ratio for the nth mode
(a) arrangement of beam and load; (b) applied step-function loading.
CHAPTER 19. PARTIAL DIFFERENTIAL EQUATIONS OF MOTION
Determine Mode Shapes and Frequencies: results were found in Example E181 to be
CHAPTER 19. PARTIAL DIFFERENTIAL EQUATIONS OF MOTION
Determine Mode Shapes and Frequencies(see Example E185):
Compute Generalized Mass and Loading:
Solve the GeneralizedCoordinate Response (see Example E192):
194 UNCOUPLED AXIAL EQUATIONS OF MOTION: UNDAMPED CASE
CHAPTER 19. PARTIAL DIFFERENTIAL EQUATIONS OF MOTION
Multiplying each term by
and applying the orthogonality relationships
discussions
velocity
CHAPTER 19. PARTIAL DIFFERENTIAL EQUATIONS OF MOTION
Response of pile to step-function loading.
CHAPTER 19. PARTIAL DIFFERENTIAL EQUATIONS OF MOTION
191 UNCOUPLED FLEXURAL EQUATIONS OF MOTION: UNDAMPED CASE
CHAPTER 19. PARTIAL DIFFERENTIAL EQUATIONS OF MOTION
When the two orthogonality relationships are applied to the rst two terms, it is evident that all terms in the series expansions, except the nth, vanish; thus,
CHAPTER 19. PARTIAL DIFFERENTIAL EQUATIONS OF MOTION
Example E194.
Pile subjected to end loading: (a) geometric configuration; (b) step-function loading.
Compute Generalized Mass and Loading:
CHAPTER 19. PARTIAL DIFFERENTIAL EQUATIONS OF MOTION
Solve the NormalCoordinate Response Equation: The Duhamel solution of this equation is
CHAPTER 19. PARTIAL DIFFERENTIAL EQUATIONS OF MOTION
Multiplying by
, integrating, and applying the two orthogonality
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relationships together with the denitions of generalized mass and
Evaluate Displacement, Moment and Shear Response:
CHAPTER 19. PARTIAL DIFFERENTIAL EQUATIONS OF MOTION
192 UNCOUPLED FLEXURAL EQUATIONS OF MOTION: DAMPED CASE