Dynamic Analysis of Structures Questions and Answers

Answer: A

Correct answer: \( m\ddot{u} + c\dot{u} + ku = p(t) \)

The equation states dynamic equilibrium among four forces: the inertia force \( m\ddot{u} \), the damping force \( c\dot{u} \), the restoring force \( ku \), and the applied load.

Each term pairs a property with the derivative it acts on - mass with acceleration, damping with velocity, stiffness with displacement - and that pairing is what the distractors disturb. Interchanging the derivatives, or moving a term to the other side, gives an equation with no physical meaning.

Answer: B

Correct answer: a fictitious inertia force opposing the acceleration

D'Alembert's principle adds an inertia force of magnitude \( m\ddot{u} \) acting opposite to the acceleration, after which the system may be treated as though it were in static equilibrium.

The advantage is method rather than physics: the whole apparatus of statics - free bodies, equilibrium equations, virtual work - becomes available for a dynamic problem. The force is fictitious in the sense that no external agent applies it; it is the mass's resistance to acceleration written as though it were a load.

Answer: C

Correct answer: \( -m\ddot{u}_g \)

Writing the equation in coordinates relative to the moving base gives

\[ m\ddot{u} + c\dot{u} + ku = -m\ddot{u}_g \]

so the ground acceleration appears as an equivalent force proportional to the mass. That is why base excitation and applied loading can be handled by the same machinery, and why seismic demand scales with mass - a heavier building attracts a larger effective force from the same ground motion. The negative sign follows from the change of reference frame.

Answer: D

From \( \omega_n = \sqrt{k/m} \) the period follows as \( T = 2\pi/\omega_n \). Inverting the ratio inside the root is the most frequent slip, and it makes stiff systems appear slow. Doubling the mass therefore lengthens the period by about forty-one per cent.

Answer: A

Equivalently \( c_c = 2m\omega_n \), and the damping ratio is \( \xi = c/c_c \). Expressing damping as this ratio makes the figure comparable across structures of any size. Both forms follow from setting the discriminant of the characteristic equation to zero.

Answer: B

An overdamped system returns to rest without oscillating and more slowly than a critically damped one. Real structures are strongly underdamped, at a few per cent of critical. Isolation bearings and supplemental dampers are the components that approach such values.

Answer: C

Correct answer: \( \omega_d = \omega_n\sqrt{1-\xi^2} \)

Damping lowers the frequency of free oscillation, and the reduction goes as the square root of \( 1-\xi^2 \).

The practical consequence is that the correction is negligible at civil engineering damping levels: at \( \xi = 0.05 \) the factor is 0.99875, so the damped and undamped periods differ by about an eighth of one per cent. The distinction matters mathematically and almost never numerically, which is why codes work with the undamped period throughout.

Answer: D

Correct answer: 400 kN·s/m

Critical damping is

\[ c_{cr} = 2\sqrt{km} = 2\sqrt{4 \times 10^{6} \times 10^{4}} \]

\[ = 2\sqrt{4 \times 10^{10}} = 2 \times 2 \times 10^{5} = 4 \times 10^{5}\ \mathrm{N\,s/m} = 400\ \mathrm{kN\,s/m} \]

Keeping the units consistent is the whole difficulty: stiffness must be in newtons per metre before the root is taken. Critical damping is the value at which free motion returns to rest without oscillating, and it is the reference against which real damping is expressed.

Answer: A

Correct answer: 5 per cent

The damping ratio is the actual damping over the critical value:

\[ \xi = \frac{c}{c_{cr}} = \frac{20}{400} = 0.05 = 5\% \]

That is a typical figure for a reinforced concrete building, and it is the value codes assume when constructing a design response spectrum. Steel structures are usually taken lower, at 2 to 3 per cent, which raises the spectral demand - the two questions together show how the ratio is arrived at from measurable quantities.

Answer: B

Correct answer: two successive peak amplitudes

The logarithmic decrement is

\[ \delta = \ln\frac{u_i}{u_{i+1}} \]

taken from a free vibration record. The ratio is constant from cycle to cycle for viscous damping, which is what makes a single measurement meaningful and lets several cycles be averaged for accuracy.

It is the simplest experimental route to the damping ratio: displace the structure, release it, and measure the decay. No forcing equipment is needed, unlike the half-power bandwidth method.