Domain

Explanation

Dynamics of structures

• Forces & displacements being static or time dependent
• Models: SDOF, 2DOF, MDOF, continuous
• Equation of motion: Newton’s 2^nd law
• D’Alembert’s law: Inertia forces = Resultant forces
• Damping forces: viscous, velocity dependent, Newtonian fluid
• Spring forces: stiffness, restoring, displacement dependent
• Frequency: natural, forcing, resonance, physical vibration wave transmission
• Exciting forces: general time-varying forces, periodic, harmonic, random (ensemble, probabilistic, ergodic)
• Linear analysis: POS of uncoupled effects
• Nonlinear analysis: forces (nonlinear spring & dashpots) & displacements (large amplitude, >0.01)

SDOF systems

• Derivation of (governing) equation of motion:

1. Equilibrium method: force or moment equilibrium
2. Energy method: Rayleigh’s Principle T_max=U_max
3. Conservation of energy:
4. Principle of Virtual Work

• Response x(t) = transient x_c(t) + steady-state x_p(t)
• I.C.: constants A, B
• External forces: x_p(t), f
• Energy dissipated by damping: in one cycle
• External forces:

1. Step force P₀
2. Ramp pt
3. Blast P₀e^-at

• For linear systems, use POS of responses of independent time-limited forces of the above types
• POS of free vibration responses of impulsive strips
• Duhamel integral: x(t) = (unit impulse)*(impulse) =
• Fourier series: for periodic loads represented as a sum of harmonic forces

MDOF systems

• Multi-mass, multi-dof systems where M, C, K & P(t) are matrices
• Giving multiple natural frequencies , damping ratios z _n
• Derivation of equation of motion:

1. FBD equilibrium
2. Lagrange’s equation & extended Hamilton principle
3. Stiffness method
4. Flexibility method

• FBD equilibrium: inertia forces = resultant forces = P(t) – restoring forces
• Lagrange’s equation: where p_i are the generalized coordinates
• Stiffness method: modeling & assembly for K
• Flexibility method: for a=K^-1 or displacement per unit force
• Modes: number (=dof), frequencies and relative shapes of peaks (mode-shapes)
• Mode-shapes f :

1. where (exact to model)
2. 

where [D]=[a][M] (approximate estimate

• 1^st mode: important to civil engineering due to

1. Large inherent stiffness: long period, low w_n, critical displacement & velocity
2. Excitations (moving loads – human / traffic, wind, deterioration & even earthquakes) are of low frequencies w

• Approximate estimate of w_n:

1. Matrix iteration: given initial , iterate numerically to convergence, [D]=[a][M]; for higher frequencies, add in orthogonality relationship
2. Dunkerley’s method:

, with 10~15% error [needs scaling: ()*1.15]

(Exact)

• Properties:

1. Suitable for uniform sections, difficult for varying
2. Prelim. estimation of large structures
3. More versatile for MDOF: adjust M_n, K_n
4. Exact model relations possible
5. Infinite frequencies & modes

• Types: [responses derived fr. governing equation of motion]

1. Axial vibration
2. Torsional vibration: tall buildings
3. Flexural vibration: tall buildings
4. Others: rotary inertia (deep beams, high modes); shear effect (short, deep beams)

• Critical:

1. B.C. & I.C.
2. Derivation of frequency equation & modeshapes
3. Response

• Typical procedure:

1. Set coordinate system: directions
2. Derive general equation for type of vibration
3. Set B.C. (displacement, slope, shear, moment, compatibility, forces)
4. Set I.C.
5. Solve for unknown constants using B.C. & I.C.
6. Derive frequency equation & modeshapes
7. Responses, if necessary

• Axial vibration: where , needs 2 B.C. & 2 I.C.
• Torsional vibration: replace the above u(x,t) with q (x,t) and , needs 2 B.C. & 2 I.C.
• Governing equations for flexural vibrations:
• : assuming slender beam (l/d ratio), ignored: rotary inertia & shear; sections normal to N.A. after bending
• : rotary inertia included; for deep beams, higher modes (higher slopes)
• : shear effect included; for short, deep beams
• 

• Flexural vibrations: => , , needs 4 B.C. & 2 I.C.
• Note that different coordinate systems would require different number of B.C. for solution

(Approximate)

• Approximate methods for (i) estimate 1^st or 1^st few modes, (ii) include effects of springs, lumped mass at discrete points
• Rayleigh’s principle: T_max=U_max
• Rayleigh’s Method:

1. Assume 1^st mode shape function
2. Derive static deflection curves
3. Derive the mode displacement at discrete M & K
4. Subst. Into appropriate frequency equation below

• Estimated
• Flexure beam: cantilever with tip mass
• Axial rod:
• Torsional systems:
• Modify according using P.O.S. (linearly elastic) for varying geometry, material properties & assumed modeshapes
• Rayleigh-Ritz Method:

1. Assume scalar combination of mode shape functions:
2. Rayleigh’s quotient:
3. Minimization:
4. Formulae for K_ij & M_ij
5. 

: w_i, i=1-p

• Flexure beams: ,
• Axial rod: ,
• Torsional systems: ,