CE 573: Structural Dynamics Characteristics of Structural Dynamic Systems
Restoring Forces
"Inertial Forces": Newton's 2nd Law and D'Alembert's Principle
Damping
Dynamic Forces
CE 573: Structural Dynamics Equivalent Viscous Damping
Energy Dissipation in General
ED =
cycle
FD du =
cycle
FD (t )
du dt ED = FD (t ) u (t ) dt dt cycle
Energy Dissipation for Viscous Damping
Fd (t ) = cx (t ) = cX cos( t )
FD
Problem 10.1:
Given: A uniform beam of flexural stiffness EI = 109 lb-in 2 and length 300 in has one end fixed and the
other simply supported.
Required: Determine the system stiffness matrix considering three beam segments and the nodal coordinates
CE 573: Structural Dynamics Example: Solving the Forced Vibration Problem
Given: A three story building is modeled using shear frames. The property matrices and modal information for this structure are: 0 0 0 0.1035 405.57 -349.63 0 kip-sec2 ;
CE 573: Structural Dynamics
Final Exam
Problem #1 (25 pts.)
K 18" jl
Sinai-we A 5+fuclure 5
Given: The structures shown are modeled with truss bar elements, which behave similarly to axial bar
elements except that they allow both axial and transvers
CE 573: Structural Dynamics Solutions to HW#3
Problem #1
m Y(t) y(t)
v = constant
Center of Mass
k
c
Frame
g
h yo y (x)
R
Wheel
x
Given: A car can be (crudely) modeled as a single degree of freedom system having the entire car's mass m lumpe
CE 573: Structural Dynamics Example: Rayleigh Quotient
Given: A six-story building is modeled as a shear-frame structure. Parameters are m = 66 lb-sec 2 / in , E = 30 106 psi, and I = 882 in 4 . Required: Estimate the fundamental natural frequency
CE 573: Structural Dynamics MDOF Earthquake Response Spectrum Example
Given: A three story building is modeled using shear frames. The mass and stiffness properties are as indicated. Assume = 0.02 for all modes. Each story has a height of 14 ft. Re
CE 573: Structural Dynamics HW#2 Supplemental Problems
Problem #1
Given: A viscously damped structure is set into free vibration with an initial velocity. The resulting
damped oscillations are shown above.
Required: (a) Determine the logarithmic de
CE 573: Structural Dynamics Numerical Evaluation of Dynamic Response Elastoplastic Behavior
Example
Given: Mass of frame m = 0.5 kip-sec2 / in , stiffness k = 20 kips/in, and = 0.02 . Tank starts at rest and is subjected to pulse F(t) as shown. A
CE 573: Structural Dynamics Example on Natural Frequencies and Modes
Given: A two-story building is modeled as a shear-frame structure. Parameters are m = 35 103 kg and k = 17.5 106 N/m. Required: Determine the natural frequencies and natural vibr
CE 573: Structural Dynamics Example: Inverse Vector Iteration for Higher Modes
Given: A six-story building is modeled as shown. Parameters are m = 66 lb-sec 2 / in , E = 30 106 psi, and I = 882 in 4 . Required: Use inverse vector iteration to find
CE 573: Structural Dynamics HW#5 Solutions
Given: Another common model used for buildings is the portal frame model, which tries to account for
flexibility of both horizontal and vertical members by allowing for rotations at connections. In this pro
CE 573: Structural Dynamics HW#4 Supplemental Problem
Given: A response spectrum is a plot of the maximum dynamic load factor (DLF) as a function of the
(nondimensional) duration of the loading. See Figures 4.4 and 4.5 in the text for examples. We w
CE 573: Structural Dynamics General Periodic Response
Theory
mu + cu + ku = F ( t ) ao + an cos ( n t ) + bn sin ( n t )
n =1 n =1 N N
Complementary Solution: uc (t ) = e - t [ A cos D t + B sin D t ]. Particular Solution:
For constant term: u
CE 573: Structural Dynamics HW#3 Supplemental Problem
v = constant
m Y(t) y(t)
Center of Mass
k
c
Frame
g
h yo y (x)
R
Wheel
x
Given: A car can be (crudely) modeled as a single degree of freedom system having the entire car's mass
m lumped a
CE 573: Structural Dynamics Damped Free Vibration
Case #1: Overdamped Vibration c > ccr > 1:
Case #2: Critically Damped Vibration c = ccr = 1:
Case #3: Underdamped Vibration c < ccr < 1:
CE 573: Structural Dynamics Example: Inverse Vector Iteration
Given: A two-story building is modeled as a shear-frame structure. Parameters are m = 35 103 kg and k = 17.5 106 N/m. mode shape {a}1 for this building. Solution: 1. Property matrices
CE 573: Structural Dynamics HW#7 Supplemental Problem
Problem #1
Given: Illustrative Example 13.2 in the textbook (pages 418 426) describes the evaluation of the dynamic response of a three dimensional space frame using SAP 2000. A careful reading o
CE 573: Structural Dynamics Generalized SDOF Systems
Theory
Assume dynamic equilibrium for displacement v( x, t ) :
beam
FI (t ) + FD (t ) + FS (t ) + FE (t ) = 0,
beam beam beam
with each force dependent on v( x, t ) except for the external
CE 573: Structural Dynamics Response to Arbitrary Dynamic Excitation
mu + ku = F (t ) u (t ) = 1 t F ( ) sin (t - ) d m 0
(In all the examples, assume that the system starts at rest.)
Example #1: Step Force
u (t ) = Fo t 1 t Fo sin (t - ) d
CE 573: Structural Dynamics HW#1 Supplemental Problems
Problem #1
Given: A concrete slab (mass = 20,000 kg) forms the floor of a manufacturing facility. To prevent
excessive vibrations from reaching the equipment placed on this floor, the slab is su
CE 573: Structural Dynamics HW#6 Supplemental Problems
Problem #1
Given: A popular method for limiting the effects of earthquakes on structures is to base isolate them; that is, to put in supports underneath the structure that isolate the main part
CE 573: Structural Dynamics HW#2 Solution
Problem #1
Given: A viscously damped structure is set into free vibration with an initial velocity. The resulting
damped oscillations are shown above.
Required: (a) Determine the logarithmic decrement, the
CE 573: Structural Dynamics Example: Response Spectrum Analysis using Modal Combinations
Given: A four story building is modeled using shear frames. The mass and stiffness properties are expressed in terms of m = 12 103 kg and k = 4 106 N/m; prope
CE 573: Structural Dynamics Example: Solving the Free Vibration Problem
Given: A three-story building is modeled as shown. The dead load per unit length for each horizontal girder varies with each floor; you may assume that the weight of each column
CE 573: Structural Dynamics Structural Control Devices
Supplemental Dampers
Idea: Add damping to the structural system by inserting dampers directly into the structure.
Advantages: Fairly cheap and easy to install good for seismic retrofitting. D