Domain

Explanation

Shock?!

• In dynamics (study of oscillations), shock is similar to that of biological systems – a short, sudden, heavy impact

Vibration?

• Vibrations are oscillations that move about certain positions (stable equilibrium state)
• They can be periodic (steadily oscillating), non-periodic (unsteadily oscillating) or arbitrary (randomly oscillating)

Control?

• No one would like to be shocked or disturbed (like mosquito bites, noise, pain or discomfort)
• Neither do our machines, tools, buildings & other structures
• Structures do have some inherent resistance & capabilities to withstand shocks & vibrations below certain range (design loads)
• However, in order to protect, preserve & enhance our structures, any structural empowerment must be included
• These controls come in the form of reinforcements, springs, dashpots, dampers, isolators, actuators, etc.

SDOF

• Natural frequency:
• Effective mass, m_eff: T=1/2m_effv²
• Rayleigh’s method: T_max=U_max
• By Newton’s 2^nd Law, motion eq is
• Damping,
• Damping ratio,
• Underdamped: , susceptible to oscillations
• Critically damped: , at most one oscillation before rest
• Overdamped: , no oscillation, decay exponentially
• Damped frequency:
• Response: sinusoidal response
• Amplitude: determined by initial conditions &
• Decay of peak values: determined by (of substrate)
• Oscillation frequency & period: by (of system)
• Measurement of damping: log decrement of n peaks
• Coulomb damping: natural damping through friction (opposes motion, negative to direction of motion x)

Forced harmonic vibration

• Harmonic: a form of periodic oscillation that is sinusoidal in nature, whose phase remains constant and displacement & acceleration have the same forms
• or
• 2 forms of external excitations:

1. Force applied directly to mass:
2. Indirect force transmitted through mass support:

, where x is relative motion & z the ground motion

• Vibration isolation: aims to minimize transmissibility TR
• Vibration measurement: aims for TR=1, with low

Transient vibration

• Under free vibration, systems with damping would eventually reach zero steady state equilibrium & come to rest
• For these systems, transient vibration response is crucial
• Using impulse function: time integral of the force
• Define unit impulse, h(t)
• Response
• Forcing functions:
• Step: , ramp: , exponential decay: , a>0, velocity pulse:

, triangular pulse: ramp1+ ramp2+free, sine pulse

Shock isolation

• System idealization: isolators’ locations (With symmetry & C.M.=C.G. for uncoupled motion) or (With non-symmetry & C.M./=C.G. for coupled motion)
• (Un)coupled motion: for translational & rotational modes
• Classification of shock isolation problems:

1. Class I: foundation motion mitigation, e.g. shock strut (aircraft landing gear), objective: limit shock-induced stresses of the equipment
2. Class II: equipment force mitigation, e.g. recoil cylinders (military), objective: limit forces transmitted to the support

• Solutions of both classes are equivalent – solve one, solves the other
• Velocity Step method for isolation design:

1. Discretisation of forcing acceleration spectrum into discrete pulses
2. Each acceleration pulse is approximated by 1) rectangular pulse, 2) sine pulse or 3) versed sine pulse
3. Equation of motion (Rayleigh’s Principle) with , where is restoring force by isolator spring, d the isolator deflection & u_m dot the maximum velocity step
4. Linear spring isolator: where xdd is the isolated transmitted acceleration &
5. Hardening spring isolator: hardening stiffness with a limit to the maximum isolator deflection
6. Softening spring isolator: softening stiffness, maximise energy absorption
7. Use analytical expressions & charts inter-changeably
8. Velocity change where , tau the acceleration pulse duration and the approximated rectangular pulse duration
9. Negligible effect of velocity step approximation ~5% error
10. *For effective shock isolation:

• Note that for stiff, rigid system with high w_n, pulse duration must be very small, else use other controls (avoid isolation: since it would amplify acceleration: )
• For higher-order system with dof>1, the further the masses away from the supports & isolators, the higher the transmissibility ratio => loose components apt to fail

Stability

• From model derive governing equation:
• Assimilate F(t) into
• Stability criterion: criteria
• Unstable: violate any of the above
• Results in instability, increasing responses, flutter instability, self-excited oscillations