Dynamic Analysis of Structures Questions and Answers

Answer: A

Correct answer: 0.32 s

Under a load equal to the weight, \( k = mg/\delta \), so the period reduces to a function of the static deflection alone:

\[ T = 2\pi\sqrt{\frac{\delta}{g}} = 2\pi\sqrt{\frac{0.025}{9.81}} \]

\[ = 2\pi \times 0.0505 = 0.317\ \mathrm{s} \]

Mass and stiffness both cancel, which is what makes the relation so useful: a deflection measurement gives the period directly, and it is the basis of Rayleigh's method applied to a self-weight deflected shape.

Answer: B

Correct answer: an independent displacement needed to define the position of the system

A degree of freedom is one of the independent coordinates required to specify the configuration completely - and the number of them fixes the size of the mass and stiffness matrices, and hence the number of modes.

A shear building idealised with one horizontal displacement per floor has as many degrees of freedom as storeys. A mode is a result of the analysis rather than an input to it, and a restrained joint is precisely a point where a degree of freedom has been removed.

Answer: C

Correct answer: after an initial disturbance with no further force applied

Free vibration is the motion that follows an initial displacement or velocity once the disturbance is removed, so \( p(t) = 0 \) and the structure oscillates at its own natural frequency.

Damping causes the amplitude to decay, which is what a logarithmic decrement measurement exploits, so constant amplitude describes only the idealised undamped case. Self weight is a static load producing no oscillation, and a continuously applied harmonic force is forced vibration by definition.

Answer: D

Correct answer: while an external time-varying force or support motion acts on it

Forced vibration is motion under a continuing excitation, whether an applied load or a moving support - and base motion counts, since in relative coordinates it appears as the equivalent force \( -m\ddot{u}_g \).

The response consists of a transient at the natural frequency, which damping removes, and a steady state at the forcing frequency, which persists. Motion after the force is removed is free vibration, and neither damping nor the number of degrees of freedom enters the definition.

Answer: A

Correct answer: the force required to produce unit displacement

Stiffness is force per unit displacement, in newtons per metre, and it is the property multiplying \( u \) in the equation of motion.

Its reciprocal is flexibility, the displacement produced by unit force, which is what the second option describes. Mass per unit length is a quite different property, and energy dissipated per cycle is a measure of damping. Stiffness and mass together fix the natural frequency as \( \sqrt{k/m} \).

Answer: B

Correct answer: radians per second

Circular frequency \( \omega \) is measured in radians per second, and it relates to the cyclic frequency \( f \) in hertz by

\[ \omega = 2\pi f \]

with the period \( T = 1/f = 2\pi/\omega \) in seconds. Keeping the two apart is essential in numerical work: the factor of \( 2\pi \) between them is the commonest source of error in dynamics calculations, and the units are what identify which quantity a formula expects.

Answer: C

Correct answer: 0.25 s

Period is the reciprocal of frequency:

\[ T = \frac{1}{f} = \frac{1}{4} = 0.25\ \mathrm{s} \]

Note that \( f \) is in hertz, cycles per second, not the circular frequency \( \omega \) in radians per second - the two differ by \( 2\pi \) and confusing them is the commonest slip in dynamics. A 4 Hz structure is fairly stiff, corresponding roughly to a low-rise building of three or four storeys.

Answer: D

Taking the reciprocal gives \( 1/0.8 = 1.25 \) Hz. Confidence in moving between period and frequency is worth having, since codes and instruments use both. The circular frequency of the same system is \( 2\pi/0.8 = 7.85 \) rad/s.

Answer: A

Correct answer: opposite to the direction of its acceleration

The inertia force is \( -m\ddot{u} \), so it opposes the acceleration - the mass's resistance to being accelerated, written as though it were a force.

That sign is what makes D'Alembert's principle work: adding the inertia force to the real forces produces a set that sums to zero, so the dynamic problem can be handled with the methods of statics. Its direction has nothing to do with gravity, which enters separately as a static load.

Answer: B

Micro-cracking, friction at joints and interaction with non-structural elements all remove energy, and the viscous model is a convenient idealisation of the total. It is not a property that can be computed from the geometry alone.