Solutions Manual Dynamics Of Structures 3rd Edition Ray W
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Problem Statement
A water tower is idealized as a SDOF system with mass ( m = 5000\ \text{kg} ), lateral stiffness ( k = 2\times 10^5\ \text{N/m} ), and negligible damping.
(a) Determine the natural period (T_n) and circular natural frequency (\omega_n).
(b) If an initial displacement (u(0) = 0.05\ \text{m}) and initial velocity (\dot u(0) = 0.2\ \text{m/s}) are imposed, write the free vibration response (u(t)).
(c) A harmonic force (F(t) = F_0 \sin(\omega t)) with (F_0 = 1000\ \text{N}) and (\omega = 0.8,\omega_n) is then applied starting at (t=0) with zero initial conditions. Find the steady‑state amplitude and the total response.
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General solution: ( u(t) = A \cos(\omega_n t) + B \sin(\omega_n t) )
Apply (u(0)=0.05 \Rightarrow A = 0.05)
(\dot u(t) = -A\omega_n \sin(\omega_n t) + B\omega_n \cos(\omega_n t))
(\dot u(0)=0.2 = B\omega_n \Rightarrow B = 0.2 / 6.3249 = 0.03162\ \text{m})
Thus
[
u(t) = 0.05\cos(6.3249 t) + 0.03162\sin(6.3249 t)\ \text{m}
]
Amplitude ( = \sqrt{0.05^2 + 0.03162^2} = 0.05916\ \text{m} ).
(Phase angle (\phi = \tan^{-1}(B/A) = 32.3^\circ).)
The Challenge: Non-linear damping response. The Manual’s Value: Shows the iterative linear acceleration method step-by-step, which most textbooks gloss over. If you are studying for the PE (Professional
Author: [Your Name]
Date: April 25, 2026
Subject: Structural Dynamics Instruction