lec-02 vibration of sdof system
TRANSCRIPT
CIVL7008 Seismic Analysis forBuilding Structures
Lec-02 Vibration of SDOF System
Lec-02 Vibration of SDOF System
Free Vibration of SDOF System
Part 1
Lec-02 Vibration of SDOF System
Lec-02 Vibration of SDOF SystemDynamic System
SDOF System
u
F(t)
dispK
MC
Stiffness, KDamping, CMass, M
StructureProperties
Dynamic Loading P(t)
K: Stiffness Spring Force, Elastic Resistance to displacement
M: Mass, Inertia Force
C: Damping, Energy Loss Mechanism
Undamped system
u
F(t)
dispK
M
Damped System
u
F(t)
dispK
MC
u(t)
time
u(t)
time
m
m
Equation of Motion
Fp=P(t) = Fi + Fd + Fs
Lec-02 Vibration of SDOF System
Free Vibration of SDOF (undamped System)
Basic Equations
0
22k
2
/
/ 2 /
T 2 /
1/
Critical DampingConstantsDamping Ratio
Circular frequency
Period
Frequency
T
Free Vibration
u
F(t)=0
dispK
M
uoK
M
Move to U0 Distance
time
U(t)
Lec-02 Vibration of SDOF System
General Solution
0 0
0
/
General Solution
Complex plane
Imaginary unit
Real
time
displacement
Lec-02 Vibration of SDOF System
Free Vibration of SDOF (undamped System)
Lec-02 Vibration of SDOF System
Free Vibration of SDOF (undamped System)
Lec-02 Vibration of SDOF System
Critical Damping Constant Ccr
0
→ 0
→ 2 0 ∶ ∙ ∙
∙ ∙ ∙ ∙ ∙ ∙
∙ ∙ ∙ 2 ∙ ∙ ∙ ∙ ∙ ∙ ∙ 02 0
, 1 ∙ 0
1.0 1.0 1.0
0.05 0.02
Lec-02 Vibration of SDOF System
Free Vibration(damped system)
0
1
0 ∗ 1 ∗ 0
0 ′
/
U
K A2KA1
R
I time
u(t)
Free Vibration(damped system)
+
Lec-02 Vibration of SDOF System
time
u(t)
u1 u2 u3
2
ln 2 2
12 2 ∗ ln
12 ∗ ln
Damping ratio test method
Lec-02 Vibration of SDOF System
SDOF SimulationFree Vibration
Harmonic Vibration of SDOF System
Part 2
Lec-02 Vibration of SDOF System
Harmonic Vibration (undamped system)u
M ∙ cos Ω
Equation of Motion
∙ cos Ω
Particular solution
∙ cos Ω
∙ Ω ∙ sin Ω
∙ Ω ∙ cos Ω
→ ∙ Ω ∙ cos Ω ∙ cos Ω ∙ cos Ω
→ Ω
Static deflection:
ΩFrequency ratio:
Ω Ω⇒ Ω
Ω ∙ Ω1
1
Lec-02 Vibration of SDOF System
Harmonic Vibration (undamped system)u
M ∙ cos Ω
Frequency-Response Equation
Ω1
1 →Ω
∙ cos Ω cos Ω
Ω ∙ ∙ cos Ω1
1 ∙ ∙ cos Ω
Steady Response
Forced Vibration
Solution = Particular + General
Vibration Response
Steady Response
Free Vibration Response
= +
⇒
1 ∙ cos Ω cos sin
0 0, 0 0 :Ω
1 ∙ sin Ω sin cos
∵ 0 cos 0
∴ 0
0 1 0
Lec-02 Vibration of SDOF System
Equation of Motion
∙ sin Ω
Particular solution
∙ sin Ω
∙ Ω ∙ cos Ω
∙ Ω ∙ sin Ω
→ ∙ Ω ∙ sin Ω ∙ sin Ω ∙ sin Ω
→ Ω
→ Ω
Static deflection:
Ω
Ω 1 Ω 1 Ω 1
Frequency-Response Equation
Ω1
1
1 Ω
1 ∙ sin Ω sin cos
Ω1 ∙ cos Ω cos sin
Lec-02 Vibration of SDOF System
0 0, 0 0 :
∵ 0 cos 0
∴ 0
∵ 0Ω
1 0
∴ 1
→ 1 ∙ sin Ω 1 sin
1 ∙ sin Ω sin cos
Ω1 ∙ cos Ω cos sin
→Ω
1 ∙ cos Ω 1 cos
→ 1 ∙ sin Ω sin
Frequency-Response Equation
Ω1
1
Ω
Lec-02 Vibration of SDOF System
Lec-02 Vibration of SDOF System
Lec-02 Vibration of SDOF System
Lec-02 Vibration of SDOF System
Lec-02 Vibration of SDOF System
SDOF Harmonic Vibration Damped System
∙ cos Ω
Particular solution∙ cos Ω
∙ Ω ∙ sin Ω
∙ Ω ∙ cos Ω
→ ∙ Ω ∙ cos Ω Ω ∙ sin Ω∙ cos Ω ∙ cos Ω
1
2
2
Ω Ω
→ Ω Ω
Static deflection:
→ Ω Ω
Ω
Ω Ω∙
1
1 Ω Ω1
1 2
Lec-02 Vibration of SDOF System
Ω
Ω
Ωt‐αα
Lec-02 Vibration of SDOF System
Program, SDOF systems due to harmonic Vibration
Lec-02 Vibration of SDOF System
Undamped System Damped System
Sympathetic Vibration Situation
Lec-02 Vibration of SDOF System
Sympathetic Vibration Situation