Domain

Explanation

Vibration?

• The physical oscillations (even or uneven) with respect to time, space or any desired variable of a well-defined structure due to either internal disturbances or external excitations or a combination of both
• The system is literally swinging to and fro with different amplitudes (max offset from rest position), frequencies (cycles in unit time) and inducing various engineering situations within the structural components
• A famous example is the student's vibrating ruler

What causes these vibrations?

• Literally anything that directly or indirectly contact the structure
• To excite a structure externally, the requirements include:

1. Direct contact: impacts (hits, knocks, quakes, explosions or crashes) & traction / point
2. Indirect contact: force fields including gravitational, electromagnetic or nuclear origins
3. Excitation energy/structure mass ratio: the higher the more & intense the vibrations
4. Location(s) of the excitations: excitations at the dynamically weak spots of a structure induce heavier vibrations
5. Intensity of the excitations: certain intensity excites the structure much more than others given the same energy input => resonance (max disturbed response) occurs when excited intensity equals the structural dynamic frequency
6. Phase difference between the excitation frequency and vibration frequency: the greater the phase difference, the more the increase in induced vibrations => max when the "excitation wave" is out of phase with the "vibration wave" (wave interaction)

• To excite a structure internally, the requirements are the same as above - the Principle of Transmissibility (sources different, effects equivalent)

What are the types of excitations?

• In the following forms:

1. Static: invariant with time
2. Periodic-harmonic: sin, cos, sinh, cosh loads
3. Periodic-general: time domain, frequency domain
4. Stochastic: fits known stochastic distribution(s)
5. Random: stochastic & stationary Gaussian - mean, definite
6. Arbitrary: earthquakes & contingencies - Duhamel integral with Dirac delta
7. White noise: complete spectrum of all available disturbances - ideal, filtered/unfiltered, non-white, differential distribution

Modeling

• The process of modeling is an oxymoron (irony) in itself
• On one hand, models are fundamental for engineers to understand a phenomenal reality and its attributes essential for engineering
• On the other, the mere representation of reality with models is inherently flawed - assumptions are made, reduction into a finite set of performance variables & relationship between these variables (laws, theories, theorems, principles, techniques)
• It is interesting to note that researchers are cracking their brains to form & improve models when their eyes should have been directed to the phenomenal reality itself - however, the structure is often yet to be built but there are always similar ones in existence whether artificial or natural
• The modeling process:

1. Phenomenal reality
2. Model representation: assumptions
3. Mathematical model: continuous/distributed (exact towards model) & discrete (approximate towards model)
4. Solution techniques: mixture of mathematics & engineering methodologies with issues of consistency, convergence, stability, etc.
5. Results: interpretations & appropriate applications

Model representation

• To formulate & understand the identified parameters that are significant to the phenomenon in question

1. Overall structure, members, connections & loads
2. Since continuum, use free-body diagram (FBD) of finite dimension with infinitesimal loads, forces and dimensions wrt. space & time
3. Boundary-initial-value conditions
4. Assumptions of relationships in terms of equilibrium, compatibility and trial solutions
5. Governing equations
6. Solution principles & techniques
7. Approximations with truncations, rounding-off and boundedness concerns

Vibration aspects?

• For a distinct member:

1. Free vibration: ODE problem - modal frequencies (w_i) & mode shapes (y _i)
2. Axial
3. Flexural
4. Shear
5. Torsional
6. Orthogonality of mode shapes: [f _r ^T][M][f _s] = 0, [f _r ^T][K][f _s] = 0, r¹ s

Approximate method of determining natural frequencies?

• Natural frequencies are the intrinsic structural frequencies where resonance occurs when equal to the excitation / disturbance frequencies

1. Rayleigh method: COE - KE + SE = constant, assuming v(x,t)=f (x).Y(t)
2. Rayleigh principle: with reasonable mode shape f (x) assumed, satisfying the slope & deflection BC, good approximation of natural frequencies (slightly higher - conservative side) w² = K^*/M^*
3. Rayleigh-Ritz method: reduces into finite number of dof's, v(x)=_{NS j=1}A_j.y _j , with y _j satisfying Rayleigh's principle
4. Principle of Virtual Work: W_E = W_I with Rayleigh-Ritz method

• Note the need for appropriate trial functions to be assumed (unnecessary in FEM, replaced by the polynomial shape interpolation function)

Uncoupling equations of motion

• Process of mode superposition method:

1. Assume mode shape: y _j wrt. BC (slope & deflection), v(x,t)=_{NS j=1y j}.Y(t)
2. Stiffness: k_ij = ò EI.y _i. y _j.dx + k.q _i. q _j = (distributed) + (lumped)
3. Mass: m_ij = ò r A.y _i. y _j.dx + M.y _i. y _j = (distributed) + (lumped)
4. Rayleigh damping: [C]=a .[M]+b .[K], thus orthogonality condition applies where V _r = a /2w_r + b .w_r/2 with lightly damped: [C]=2.V _r.w_r.[M]
5. Form governing equation of motion
6. Det( [K]-w²[M] ) = 0 for modal frequencies wi & mode shapes y _i
7. Form the uncoupled N modal equations of motion
8. Solve accordingly

• Note: earthquake analysis (higher energy input, more modes) & wind analysis (lower energy input, larger surface area, fewer modes needed)

Limitations

• Modal superposition method only applies to linear systems due to the linear combination of trial solutions (mode shapes )
• Considerations of only the first N modes - truncation error
• Mode shapes form basis (linearly indept.) of displacement vector field, but incomplete basis
• Orthogonality of mode shapes applies to [M] & [K] only with assumption of energy-dissipating damping with Rayleigh damping [C]=a .[M]+b .[K]

Strengths

• Computationally efficient
• Flexible & allows generalization of user-defined considered modes
• Enables Duhamel integral (for arbitrary excitations) and modal participation factor (G _i=ò m(x).f _i(x)dx/ò m(x).f _i²(x)dx) to be used for analysis