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<!DOCTYPE html>
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en"><head>
<meta charset="utf-8">
<meta name="generator" content="quarto-1.3.294">
<meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes">
<title>Structural Dynamics - 2 Free Vibrations of Single Degree of Freedom Systems</title>
<style>
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ul.task-list{list-style: none;}
ul.task-list li input[type="checkbox"] {
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margin: 0 0.8em 0.2em -1em; /* quarto-specific, see https://github.com/quarto-dev/quarto-cli/issues/4556 */
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</head><body class="nav-sidebar floating">$$
% 05/16 change, define new env to convert multlined to multline* in HTML
\newenvironment{multlined}{\begin{multline*}}{\end{multline*}}
% define the command for indenting the first line of paragraphs after the first
\newcommand{\newpara}{ }
%\DeclareMathOperator{\expo}{e}
\def\expo{\mathrm{e}}
\def\expon#1{\expo^{#1}}
% \renewcommand\pi{\uppi} % date: 03/29 change: comment out this line since \uppi cannot be understood by qmd
\def\abs#1{\left| #1 \right|}
\def\matabs#1{\Bigl| #1 \Bigr|} % determinant
\def\RB#1{\mathcal{#1}}
% \def\rem#1{{\bfseries{#1}}} % regular emphasis
% \def\mem#1{{\bfseries{#1}}} % much emphasis
% date: 03/30 change \bfseries to \textbf
\def\rem#1{{\textit{#1}}} % regular emphasis
\def\mem#1{{\textit{#1}}} % much emphasis
% date: 03/30 change \emph to ** in qmd
% \def\lem#1{\emph{#1}} % less emphasis
\def\unit#1{\mathrm{\, #1}}
\def\punit#1{\mathrm{#1}}
\DeclareMathOperator{\dif}{d}
\def\diff{\dif \!}
\def\dt{\diff t} % dt
%%%%%%%%%%% VECTORS
\def\vctr#1{\underline{\mathrm{#1}}} % vectors with roman letters
\def\svctr#1{\underline{#1}} % vectors with symbols
\def\pvctr#1{\, \underline{\mathrm{#1}}} % vectors with roman letters and prior spacing
\def\vmag#1{\left| #1 \right|} % magnitude of a vector
\def\subvec#1#2{\vctr{#1}_{#2}} % vector (roman letter) with subscript
%%%%%%%%%%%
%%%%%%%%%%% MATRICES
\def\mtrx#1{[\mathrm{#1}]} % matrices with roman letters
\def\smtrx#1{[\mathnormal{#1}]} % matrices with symbols
\def\mmat{\mtrx{M}} % mass matrix
\def\cmat{\mtrx{C}} % damping matrix
\def\kmat{\mtrx{K}} % stiffness matrix
\def\nvec#1{\underline{{#1}}} % column matrix (vector) for symbols
\def\snvec#1{\underline{{#1}}} % column matrix (vector) for symbols
% \def\nvec#1{\underset{\sim}{\mathrm{#1}}} % column matrix (vector) for symbols
% \def\snvec#1{\underset{\sim}{\mathnormal{#1}}} % column matrix (vector) for symbols
\def\colmat#1{\left\{ \begin{array}{c} #1 \end{array} \right\}} % column matrix array
% \def\nvec#1{\undertilde{\mathrm{#1}}} % column matrix (vector) for symbols
% \def\snvec#1{\undertilde{#1}} % column matrix (vector) for symbols
% 03/31 \undertilde not found
\def\onecol{\nvec{1}}
\def\idmat{\mtrx{I}}
\def\zerocol{\nvec{0}}
\def\zeromat{\mtrx{0}}
\def\barmmat{\mtrx{{M'}}} % mass matrix in barred coordinates
\def\barcmat{\mtrx{{C'}}} % damping matrix in barred coordinates
\def\barkmat{\mtrx{{K'}}} % stiffness matrix in barred coordinates
%%%%%%%%%%%
\def\sint{\int \!}
\def\lsint#1#2{\int_{#1}^{#2} \hspace{-1ex}}
\def\divp#1#2{\frac{\diff #1}{\diff #2}}
\def\divt#1{\frac{\diff \, #1}{\diff t}}
\def\ddivt#1{\frac{\diff^2 #1}{\diff t^2}}
\def\pardiv#1#2{\frac{\partial #1}{\partial #2}}
\def\pdivt#1{\dot{#1}}
\def\pddivt#1{\ddot{#1}}
%\def{\ssum}{\mathsmaller{\sum}}
\DeclareMathOperator{\ssum}{\scaleobj{0.8}{\sum}}
\def\lowsum#1{\displaystyle{\ssum\limits_{#1}^{}}}
\def\allsum#1#2{\ssum\limits_{#1}^{#2}}
\def\volume{V}
\def\area{A}
\def\dV{\diff \volume} % differential volume
\def\dA{\diff \area} % differential area
\def\dm{\diff m} % differential mass
\def\totm{{m}} % total mass of a system of particles
\def\mden{\varrho} % mass density
\def\mlen{\widehat{m}} % mass per unit length
\def\gravity{\mathrm{g}}
\def\com{\mathrm{cm}} % center of mass
\def\cok{\mathrm{ck}} % center of stiffness
%%%%%%%%%%% GENERALIZED COORDINATE
\def\gc{q} % generalized coordinate
\def\dgc{\dot{q}} % dot-time derivative of gen. coord.
\def\ddgc{\ddot{q}} % second dot-time derivative of gen. coord.
\def\gct{\gc (t)}
\def\dgct{\dgc (t)}
\def\ddgct{\ddgc (t)}
\def\gcic{\gc_o} % \initial condition for gen. coord.
\def\dgcic{\dgc_o} % \initial condition for gen. vel.
%%%%%%%%%%%
%%%%%%%%%%% GENERALIZED SDOF SYSTEM
\def\gengc{q^{*}} % generalized coordinate in generalized SDOF sys
\def\dgengc{\dot{q}^{*}} % dot-time derivative of gen. coord. in generalized SDOF sys
\def\ddgengc{\ddot{q}^{*}} % second dot-time derivative of gen. coord. in generalized SDOF sys
\def\gengct{\gengc(t)}
\def\dgengct{\dgengc(t)}
\def\ddgengct{\ddgengc(t)}
\def\gengcic{\gc_o} % initial condition for gen. coord. in generalized SDOF sys
\def\gendgcic{\dgc_o} % initial condition for gen. vel. in generalized SDOF sys
\def\genm{m^{\ast}} % generalized mass
\def\genc{c^{\ast}} % generalized damping
\def\genk{k^{\ast}} % generalized stiffness
\def\genf{f^{\ast}} % generalized force
\def\gendamp{\damp^{\ast}} % generalized damping ratio
\def\shpf{\psi} % continuous shape function in generalized SDOF approach
% \def\vshpf{\snvec{\uppsi}} % shape function column matrix in generalized SDOF approach
%% \uppsi cannot be understood by qmd for html, replace \uppsi with \Psi
\def\vshpf{\snvec{\psi}} % shape function column matrix in generalized SDOF approach
\def\pvshpf#1{\psi_{#1}} % component of shape function column matrix in generalized SDOF approach
\def\genfreq{\freq^{\ast}} % undamped natural frequency in generalized SDOF sys.
%%%%%%%%%%%
% \def\u{u}
% \def\uvec{\nvec{\u}}
% \def\uvect{\nvec{\u}(t)}
% \def\du{\dot{\u}}
% \def\ddu{\ddot{\u}}
% \def\duvec{\dot{\nvec{\u}}}
% \def\duvect{\dot{\nvec{\u}}(t)}
% \def\dduvec{\ddot{\nvec{\u}}}
% \def\dduvect{\ddot{\nvec{{\u}}}(t)}
% \def\dmprat{\zeta}
% \def\ut{{\u}(t)}
% \def\dut{\dot{\u}(t)}
% \def\ddut{\ddot{\u}(t)}
% \def\gdis{\u_{g}}
% \def\gvel{\du_{g}}
% \def\gacc{\ddu_{g}}
% \def\gdist{\u_{g}(t)}
% \def\gvelt{\du_{g}(t)}
% \def\gacct{\ddu_{g}(t)}
% \def\maxgdis{D_{g}}
% \def\maxgvel{{V}_{g}}
% \def\maxgacc{{A}_{g}}
%\def\maxgdis{\u_{g,\mathrm{max}}}
%\def\maxgvel{\du_{g,\mathrm{max}}}
%\def\maxgacc{\ddu_{g,\mathrm{max}}}
%%%%%%%%%%% GROUND MOTION
\def\gdis{g} % ground displacement
\def\gvel{\dot{\gdis}} % ground velocity
\def\gacc{\ddot{\gdis}} % ground acceleration
\def\gdist{\gdis(t)} % ground displacement with time
\def\gvelt{\gvel(t)} % ground velocity with time
\def\gacct{\gacc(t)} % ground acceleration with time
\def\maxgdis{\mathsf{D}_{\! \gdis}} % absolute maximum ground displacement
\def\maxgvel{\mathsf{V}_{\!\gdis}} % absolute maximum ground velocity
\def\maxgacc{\mathsf{A}_{\!\gdis}} % absolute maximum ground acceleration
\def\adis{\alpha} % absolute displacement (earthquake analysis)
\def\avel{\dot{\adis}} % absolute velocity
\def\aacc{\ddot{\adis}} % absolute acceleration
\def\adisvec{\snvec{{\adis}}} % absolute acceleration
\def\aaccvec{\snvec{\ddot{\adis}}} % absolute acceleration
\def\gdisvec{\nvec{\gdis}} % multiple support motion displacement column
\def\gaccvec{\nvec{\gacc}} % multiple support motion acceleration column
\def\ininfmat{\mtrx{b}} % input influence matrix
\def\ininfvec{\nvec{b}} % input influence vector
%%%%%%%%%%%
%%%%%%%%%%% EARTHQUAKE RESPONSE
\def\baseshear{V_{b}} % base shear
\def\overturn{M_{b}} % overturning moment
\def\estat{f_{es}} % equivalent static force
\def\estati#1{f_{es,#1}} % equivalent static force
\def\estatvec{\nvec{f_{es}}} % equivalent static force
\def\modbaseshear#1{V_{b}^{(#1)}} % modal base shear
\def\modoverturn#1{M_{b}^{(#1)}} % modal overturning moment
\def\modestatvec#1{\nvec{f_{es}^{(#1)}}} % modal equivalent static force
\def\ccor#1{\varrho_{#1}}
\def\excifact{L} % earthquake excitation factor
\def\modpart{\Gamma} % modal participation factor
%%%%%%%%%%% SPECTRA
\def\dspec{\mathsf{D}} % disp spectral resp
\def\vspec{\mathsf{V}} % vel spectral resp
\def\aspec{\mathsf{A}} % acc spectral resp
\def\maxstifforce{\extforce_{el}} % max spring force (SDOF)
\def\maxlindisp{\gc_{el}} % maximum linear disp
\def\maxtotdisp{\gc_{max}} % maximum total disp
\def\yielddisp{\gc_{yl}} % yield displacement
\def\redfac{R} % Yield strength reduction factor
\def\duct{\mu} % ductility factor
\def\ydspec{\dspec_{yl}}
\def\yvspec{\vspec_{yl}}
\def\yaspec{\aspec_{yl}}
%%%%%%%%%%%
%%%%%%%%%%% FORCE
\def\extforce{f} % external force
\def\extforcet{f (t)} % external force with time (t)
\def\extf{\nvec{\extforce}} % external force vector
\def\extft{\extf (t)} % external force vector with time (t)
\def\fvec{\vctr{f}} % resultant force vector
\def\compf{f} % scalar components of the resultant force
\def\sforce{F} % scalar single external force, amplitude of external force
\def\force{\vctr{\mathrm{\sforce}}} % single external force vector
\def\stifforce{\extforce_{S}}
\def\intf{\nvec{\extforce_{I}}} % inertia force vector
\def\stff{\nvec{\extforce_{S}}} % stiffness force vector
\def\dmpf{\nvec{\extforce_{D}}} % damping force vector
\def\ldvec{\nvec{\extforce}} % load (force) vector in MDOF systems
\def\ldtvec{\ldvec (t)} % load (force) vector with time in MDOF systems
\def\barldvec{\nvec{\extforce'}} % load (force) vector in MDOF systems barred coord
\def\resistforce{\extforce_{R}} % resisting force (stiffness or stiffness+damping)
%%%%%%%%%%%
%%%%%%%%%%% GENERALIZED COORDINATE VECTORS
\def\gcvec{\nvec{\gc}} % generalized coordinate vector
\def\dgcvec{\dot{\gcvec}} % generalized velocity vector
\def\ddgcvec{\ddot{\gcvec}} % generalized acceleration vector
\def\gctvec{\gcvec (t)} % generalized coordinate vector with time
\def\dgctvec{\dgcvec (t)} % generalized velocity vector with time
\def\ddgctvec{\ddgcvec (t)} % generalized acceleration vector with time
\def\adisvec{\snvec{\alpha}} % absolute displacement column (earthquake analysis)
\def\avelvec{\dot{\adisvec}} % absolute velocity column
\def\aaccvec{\ddot{\adisvec}} % absolute acceleration column
\def\gcvecic{\nvec{\gc_o}} % initial displacement column
\def\dgcvecic{\nvec{\dot{\gc}_o}} % initial displacement column
\def\bargcvec{\nvec{{\gc}}'} % barred generalized coordinates
\def\bardgcvec{\nvec{{\dot{\gc}}}'} % barred generalized velocities
\def\barddgcvec{\nvec{{\ddot{\gc}}}'} % barred generalized accelerations
%%%%%%%%%%%
%%%%%%%%%%% MODAL COORDINATES
\def\modcor{z} % modal coordinates
\def\modcort{\modcor(t)} % modal coordinate with time
\def\dmodcor{\dot{\modcor}} % modal velocity
\def\ddmodcor{\ddot{\modcor}} % modal acceleration
\def\modcorvec{\nvec{\modcor}} % modal coordinate vector
\def\modcortvec{\nvec{\modcor}(t)} % modal coordinate vector with time
\def\dmodcorvec{\dot{\modcorvec}} % modal velocity vector
\def\ddmodcorvec{\ddot{\modcorvec}} % modal acceleration vector
\def\modm{\widehat{M}} % modal mass
\def\modc{\widehat{C}} % modal damping matrix coefficient
\def\modk{\widehat{K}} % modal stiffness
\def\modmmat{\mtrx{\widehat{M}}} % modal mass matrix
\def\modcmat{\mtrx{\widehat{C}}} % modal damping matrix
\def\modcmatm{\mtrx{\widehat{C}^{\sharp}}} % modal damping matrix modified
\def\modkmat{\mtrx{\widehat{K}}} % modal stiffness matrix
\def\modcoric#1{{\modcor_{#1 o}}} % initial ith modal coordinate
\def\dmodcoric#1{{\dmodcor_{#1 o}}} % initial ith modal velocity
\def\modcorvecic{\nvec{\modcor_o}} % initial modal coordinate vector
\def\dmodcorvecic{\nvec{\dot{\modcor}_{o}}} % initial modal velocity vector
\def\modcoramp{Z} % amplitude of free vibration in modal coordinates
\def\modextforce{\widehat{\extforce}} % load (force) component in modal coords
\def\modldvec{\nvec{\modextforce}} % load (force) vector in modal coords
\def\modgcvec#1{\nvec{\gc}^{(#1)}}
%%%%%%%%%%%
% absolute disp variable ua MDOF (currently same as gen coord.)
% \def\ua{q}
% \def\dua{\dot{\ua}}
% \def\ddua{\ddot{\ua}}
% \def\dmprat{\zeta}
% \def\uat{\ua(t)}
% \def\duat{\dot{\ua}(t)}
% \def\dduat{\ddot{\ua}(t)}
% \def\uavec#1{\nvec{\ua}_{#1}}
% \def\xuavec{\nvec{\ua}} % modified on 06/20 because $\uavec{}_{N\times1}$ does not work in html
% \def\duavec#1{\dot{\nvec{\ua}_{#1}}}
% %\def\dduavec#1{\nvec{\ddua}_{#1}}
% \def\xduavec{\dot{\nvec{\ua}}}
% \def\xdduavec{\ddot{\nvec{\ua}}}
% \def\uavect{\xuavec (t)}
% \def\duavect{\xduavec (t)}
% \def\dduavect{\xdduavec (t)}
%%%%% EIGENVALUES; EIGENVECTORS; RAYLEIGH-RITZ
\def\eigvecs{\phi} % mode shape symbol
\def\eigvec{\snvec{\eigvecs}} % mode shape vector
\def\eigveci#1{\snvec{\eigvecs_{#1}}} % mode shape vector for the ith mode
\def\meigveci#1{\snvec{\overline{\eigvecs_{#1}}}} % mass normalized mode shape
\def\teigveci#1{\snvec{\eigvecs_{#1}}^T} % mode shape vector transposed for the ith mode
\def\rayvecs{\psi} % rayleigh vector symbol
\def\rayvec{\snvec{\rayvecs}} % rayleigh vector
\def\trayvec{\snvec{\rayvecs}^T} % rayleigh vector
\def\ritzvec#1{\snvec{\rayvecs_{#1}}}
\def\tritzvec#1{\snvec{\rayvecs_{#1}}^T}
\def\ritzcandvec#1{\nvec{u_{#1}}}
\def\tritzcandvec#1{\snvec{u_{#1}}^T}
\def\ritzmat{\mtrx{U}}
\def\rayfreq{\freq^{\ast}} % Rayleigh's quotient
\def\ritzfreq#1{\freq^{\ast}_{#1}} % Ritz frequencies
\def\ritzmmat{\mtrx{M^{\ast}}}
\def\ritzkmat{\mtrx{K^{\ast}}}
\def\deigvec{\snvec{\widehat{\eigvecs}}}
\def\deigveci{\snvec{\widehat{\eigvecs_{i}}}}
\def\eigval{\lambda}
% \def\modmat{\smtrx{\Phiup}}
\def\modmat{\mtrx{\Phi}} % date: 05/01 change
\def\spectmat{\mtrx{\omega^2}} % date: 05/01 change
\def\dmodmat{\mtrx{\widehat{\Phi}}}
%%%%%%%%%%
%%%%%%%%%%% TIME STEPPING & NONLINEARITY
% \def\dsct#1#2{#1_{#2}}
% \def\edsct#1#2#3{#1^{(#3)}_{#2}}
\def\dsct#1#2{#1_{[#2]}}
\def\edsct#1#2#3{#1^{(#3)}_{[#2]}}
\def\yieldforce{\extforce_{yl}}
% \def\residual{\Delta \stifforce}
\def\residual{\Delta \resistforce}
%%%%%%%%%%%%
\def\puvec#1{\, \vctr{u}_{#1}}
% Cartesian rectangular unit vectors
\def\xuvec{\, \vctr{i}}
\def\yuvec{\, \vctr{j}}
\def\zuvec{\, \vctr{k}}
\def\pos{r} % position variable
\def\vel{v} % velocity
\def\acc{a} % acceleration
\def\posrelcom{R} % position relative to the center of mass
\def\pvec{\vctr{\pos}} % position vector
\def\vvec{\vctr{\vel}} % velocity vector
\def\avec{\vctr{\acc}} % acceleration vector
\def\zerovec{\vctr{0}} % zero vector
%\def\pdpvec{\vctr{\pdivt{\pos}}} % time (dot) derivative of the position vector
%\def\pdvvec{\vctr{\pdivt{\vel}}} % time (dot) derivative of the velocity vector
\def\pdpvec{\pdivt{\pvec}} % time (dot) derivative of the position vector
\def\pdvvec{\pdivt{\vvec}} % time (dot) derivative of the velocity vector
\def\prelcom{\vctr{\posrelcom}} % position vector relative to the center of mass
\def\vrelcom{\pdivt{\prelcom}} % velocity vector relative to the center of mass
\def\arelcom{\pddivt{\prelcom}} % velocity vector relative to the center of mass
\def\dpvec{\diff \pvec} % differential of position vector
\def\dvvec{\diff \vvec} % differential of velocity vector
\def\angvel{\omega} % scalar angular velocity
\def\dtangvel{\pdivt{\angvel}} % dot-time derivative of scalar angular velocity
%\def\vecangvel{\svctr{\angvel}} % angular velocity vector
\def\angvelvec{\svctr{\omega}} % angular velocity vector
\def\dtangvelvec{\pdivt{\svctr{\angvel}}} % dot-time derivative of angular velocity vector
\def\angacc{\alpha} % scalar angular acceleration
\def\angaccvec{\vctr{\angacc}} % angular acceleration vector
%\def\om{\Omega}
%\def\omvec{\vctr{\om}}
\def\vprod{\times} % vector product
\def\sprod{\cdot} % scalar product
\def\spvprod{\!\! \times} % vector product with spacing adjusted
\def\spsprod{\!\! \cdot} % scalar product with spacing adjusted
\def\linmom{\vctr{\mathrm{L}}} % linear momentum vector
\def\angmom#1{\vctr{\mathrm{H}}_{#1}} % angular momentum vector with respect to point #1
\def\dlinmom{\pdivt{\vctr{\mathrm{L}}}} % dot-time derivative of the linear momentum vector
\def\dangmom#1{\pdivt{\vctr{\mathrm{H}}}_{#1}} % dot-time derivative of the angular momentum vector with respect to point #1
\def\smoment{{M}}
\def\moment#1{\vctr{\mathrm{\smoment}}_{#1}}
\def\pmoment#1#2{\vctr{\mathrm{\smoment}}_{#1}^{(#2)}}
%%%%%%%%%%%%%%% FREQUENCIES etc.
\def\freq{\omega} % undamped natural frequency
\def\cfreq{f} % undamped cyclic frequency
\def\period{T} % period
\def\dfreq{\overline{\freq}} % damped frequency
\def\dperiod{\overline{\period}} % damped period
\def\phs{\theta} % phase angle
\def\damp{\zeta} % damping ratio
\def\extfreq{\Omega} % frequency of external harmonic excitation
\def\extphs{\varphi} % phase of external harmonic excitation
\def\ratfreq{\rho}
\def\ratdur{\beta}
%%%%%%%%%%%%% INERTIA
%\DeclareMathOperator{\inrt}{\mathds{I}}
\def\inrt{{I}} % symbol for inertia
\def\inertia#1{\mkern1mu \inrt_{#1}} % inertia with subscript
\def\dtinertia#1{\mkern1mu \pdivt{\inrt}_{#1}} % dot-time derivative of inertia with subscript
\def\ke{\mathscr{T}} % kinetic energy
\def\pe{\mathscr{V}} % potential energy
\def\te{\mathscr{E}} % total energy
\def\work{\mathscr{W}} % work
\def\virt{\delta} % virtual (variation, work, etc.)
% \def\virt{\updelta} % virtual (variation, work, etc.) date: 03/29 change: comment out this line
\def\vgc{\virt q} % (virtual) variation in gen. coord.
\def\vgengc{\virt \gengc} % (virtual) variation in gen. coord. in generalized SDOF sys
\def\vwork{\virt \mathscr{W}} % virtual work
\def\lagrangian{\mathscr{L}}
\def\vforce{\mathscr{F}} % Lagrangian force; coefficient of the virtual disp. in virtual work.
\def\wnc{{\mathscr{W}}^{nc}} % work of non-conservative forces
\def\vwnc{{\mathscr{W}}^{nc}} % work of non-conservative forces
\def\pwork#1{\mathscr{W}_{#1}} % work on some particle called #1
\def\pwnc#1{{\mathscr{W}}^{nc}_{#1}} % work of non-conservative forces on some particle called #1
\def\dbar{{\mathchar'26\mkern-12mu \mathrm{d}}} % not total derivative
\def\grad{\vctr{\nabla}} % gradient operator
\def\imag{\mathrm{j}}
\def\curv{\kappa} % curvature (as in moment-curvature)
\def\ratio#1#2{\displaystyle{\frac{#1}{#2}}}
\def\dfrm{\Delta} % deformation
\def\maxdfrm{\Delta_{\mathrm{max}}} % deformation
% \def\impresp{\mathscr{h}} %impulse response function
% \def\frf{\mathscr{H}} %frequency response function
\def\impresp{\mathsf{h}} %impulse response function
\def\frf{\mathsf{H}} %frequency response function
\def\tshift{t_{\star}}
\def\cosp#1{\cos \left( #1 \right)}
\def\sinp#1{\sin \left( #1 \right)}
\def\dynamp{\mathbb{D}}
\def\dynampr{\dynamp (\ratfreq,\damp)}
\def\tstep{t_{\Delta}}
\def\ststep{{t^2_{\Delta}}}
%%% State Space Models
% \def\dscs#1{\nvec{{x}_{#1}}}
%\def\dscs#1{\nvec{x}_{#1}}
%\def\dscf#1{\nvec{{f}}_{#1}}
\def\dscs#1{\nvec{x}_{[#1]}}
\def\dscf#1{\nvec{{f}}_{[#1]}}
\def\dscA{\mtrx{A}}
\def\dscB{\mtrx{B}}
\def\eigval{\lambda}
\def\eigvecmat{\mtrx{V}}
% \def\eigvalmat{\mtrx{\uplambda}}
\def\eigvalmat{\mtrx{\lambda}} %05/01 change
$$
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<ul>
<li><a href="#sec-Motivation" id="toc-sec-Motivation" class="nav-link active" data-scroll-target="#sec-Motivation"><span class="header-section-number">2.1</span> Motivation and Preliminary Discussions</a></li>
<li><a href="#sec-ThePrototype" id="toc-sec-ThePrototype" class="nav-link" data-scroll-target="#sec-ThePrototype"><span class="header-section-number">2.2</span> The Prototype</a></li>
<li><a href="#sec-Undamped" id="toc-sec-Undamped" class="nav-link" data-scroll-target="#sec-Undamped"><span class="header-section-number">2.3</span> Undamped Free Vibrations</a></li>
<li><a href="#sec-Damping" id="toc-sec-Damping" class="nav-link" data-scroll-target="#sec-Damping"><span class="header-section-number">2.4</span> Damping</a>
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<li><a href="#sec-ViscousDampingModel" id="toc-sec-ViscousDampingModel" class="nav-link" data-scroll-target="#sec-ViscousDampingModel"><span class="header-section-number">2.4.1</span> Viscous Damping Model</a></li>
<li><a href="#sec-Estimating" id="toc-sec-Estimating" class="nav-link" data-scroll-target="#sec-Estimating"><span class="header-section-number">2.4.2</span> Estimating Viscous Damping from Free Vibration Data</a></li>
<li><a href="#sec-EnergyDissipated" id="toc-sec-EnergyDissipated" class="nav-link" data-scroll-target="#sec-EnergyDissipated"><span class="header-section-number">2.4.3</span> Energy Dissipated by Viscous Damping</a></li>
<li><a href="#sec-Nonviscous" id="toc-sec-Nonviscous" class="nav-link" data-scroll-target="#sec-Nonviscous"><span class="header-section-number">2.4.4</span> Non-viscous Damping Models</a></li>
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<li><a href="#sec-GeneralizedSDOF" id="toc-sec-GeneralizedSDOF" class="nav-link" data-scroll-target="#sec-GeneralizedSDOF"><span class="header-section-number">2.5</span> Generalized SDOF Systems</a>
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<li><a href="#continuous-systems" id="toc-continuous-systems" class="nav-link" data-scroll-target="#continuous-systems"><span class="header-section-number">2.5.1</span> Continuous Systems</a></li>
<li><a href="#discrete-systems" id="toc-discrete-systems" class="nav-link" data-scroll-target="#discrete-systems"><span class="header-section-number">2.5.2</span> Discrete Systems</a></li>
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<li><a href="#epilogue" id="toc-epilogue" class="nav-link" data-scroll-target="#epilogue"><span class="header-section-number">2.5.4</span> Epilogue</a></li>
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<!--
import matplotlib.animation as animation
import matplotlib.pyplot as plt
import numpy as np
from matplotlib import gridspec
plt.rcParams["animation.html"] = "jshtml"
plt.rcParams['figure.dpi'] = 150
plt.ioff()
fig = plt.figure(figsize=(8, 4))
gs = gridspec.GridSpec(1, 2, width_ratios=[1 ,4])
X = np.linspace(0, 2*np.pi, 100)
Y = np.sin(X)
Xc = np.cos(X)
Yc = np.sin(X)
ax0 = plt.subplot(gs[0], aspect=1.0)
ax0.set_xlim([-1.1, 1.1])
ax0.set_ylim([-1.1, 1.1])
ax0.set_ylabel('$y$')
ax0.set_xlabel('$x$')
ax0.hlines(0, -1.1, 1.1, linewidth=0.5, color='k')
ax0.vlines(0, -1.1, 1.1, linewidth=0.5, color='k')
cirgraph, = ax0.plot([], [])
cirdot, = ax0.plot([], [], 'o', color='red')
ax1 = plt.subplot(gs[1])
ax1.set_xlim([0, 2*np.pi])
ax1.set_ylim([-1.1, 1.1])
ax1.set_ylabel('$y$=sin($t$)', ha='left', y=1, rotation=0, labelpad=0)
ax1.set_xlabel('$t$')
ax1.hlines(0, 0, 2*np.pi, linewidth=0.5, color='k')
sinegraph, = ax1.plot([], [])
dot, = ax1.plot([], [], 'o', color='red')
def sine(i):
sinegraph.set_data(X[:i],Y[:i])
dot.set_data(X[i],Y[i])
cirgraph.set_data(Xc[:i],Yc[:i])
cirdot.set_data(Xc[i],Yc[i])
return sinegraph, dot, cirgraph, cirdot,
animation.FuncAnimation(fig, sine, frames=len(X), interval=50, blit=True)
``` -->
<section id="sec-Motivation" class="level2" data-number="2.1">
<h2 data-number="2.1" data-anchor-id="sec-Motivation"><span class="header-section-number">2.1</span> Motivation and Preliminary Discussions</h2>
<p>The fundamental step in the analysis of any system is idealization and conceptual modeling. In engineering analysis, the idealized model should be simple enough so as to allow extraction of predictive results, but it should still have sufficient complexity to reflect all the salient features of the particular problem studied. The claim to validity may only be tested against experiments which are neither feasible nor even possible for many structural systems. Some simple models have however provided acceptable predictive accuracy in numerous previous trials so that structural engineers take many such models for granted while thinking about real, complex structures.</p>
<p>The static analysis of structures makes ample use of simple behavioural models; trusses, beams, plates or shells, elastic or inelastic, with or without time dependent properties have been used in the design and analysis of many structures still standing. The degree of freedom of a structural element in static analysis is associated with deformation patterns that are included in the model. A truss element is assumed, based on connection details and loading patterns, to deform only axially so that the final form of the bar is completely determined by translations of its two end points. A beam element, on the other hand, has rotational deformations so that its deformed shape is determined by end translations, end rotations, and the deformational patterns imposed by the loads applied inbetween its two ends.</p>
<p>Since the concept of degree of freedom in dynamics is intimately connected to mass distribution and forces associated with accelerations, it may be the case that static and dynamic degrees of freedom of a system do not map one-to-one. In dynamics, inertia-related degrees of freedom govern and the static degrees of freedom will often need to be condensed since, in developing simple models of deformable structures, we will often assume that the mass is concentrated at certain locations.<a href="#fn1" class="footnote-ref" id="fnref1" role="doc-noteref"><sup>1</sup></a> To discuss how such a simplification may proceed, let us consider the portal frame shown in <a href="#fig-portalframe">Figure <span>2.1</span></a>.</p>
<div id="fig-portalframe" class="quarto-figure quarto-figure-center">
<figure class="figure">
<p><img src="images/portalframe.png" class="quarto-discovered-preview-image img-fluid figure-img" style="width:95.0%"></p>
<p></p><figcaption class="figure-caption">Figure 2.1: (a) Degrees of freedom in static analysis, (b) masses lumped at the nodes, (c) dynamic degree of freedom, (d) free body diagram of the mass.</figcaption><p></p>
</figure>
</div>
<p>Under the action of a general static loading, one needs to keep track of the three degrees of freedom shown in <a href="#fig-portalframe">Figure <span>2.1</span></a>(a) to be able to specify the deformations that take place: The lateral translation <span class="math inline">\(u_1\)</span> of the joints, and the joint rotations <span class="math inline">\(u_2\)</span> and <span class="math inline">\(u_3\)</span>.<a href="#fn2" class="footnote-ref" id="fnref2" role="doc-noteref"><sup>2</sup></a> The dominant issue in modeling a system for dynamic analysis, however, is the distribution of the mass or, relatedly, the inertial effects produced by the distribution of the mass and their accelerations in the system (think of d’Alembert forces). It is common to consider that in such structural frames most of the mass will be concentrated in the level of the floor plate and the beams attached to it; furthermore, the accelerations at this level will also be relatively more significant than say the accelerations near the supports. Should one wish to incorporate the contribution of the columns to the mass, it may be argued that some , but not all, of the column mass will have an significant effect on the dynamics. With this line of thought, a model in which masses are lumped to the two joints may be developed. An important question that must be raised is whether or not rotational accelerations will have a significant affect. This is a question without an obvious a priori answer but past experiences indicate that under the action of common load effects, rotational accelerations at the joints and the inertial effects thereby generated remain inconsequential compared to those related to lateral translation; in a sense these rotations are predominantly secondary effects caused by lateral translation and compatibility. If rotational accelerations are thus neglected, angular momentum equations for the joints will be in essence static equilibrium equations, and no mass need to be assigned to the rotational degrees of freedom. This model, commonly employed in finite element analysis of frame structures, is referred to as the <em>lumped mass model</em>.</p>
<p>Since the lateral translation is assumed to be common to both joints, we could consider a single mass <span class="math inline">\(m=m_1 + m_2\)</span> which would be dynamically equivalent, and the motion of this mass may be completely defined by the single generalized coordinate <span class="math inline">\(\gc \equiv u_1\)</span> as in <a href="#fig-portalframe">Figure <span>2.1</span></a>(c). What is left to determine for the formulation of the equation governing the motion of this mass is the lateral stiffness of the frame, i.e. the resistance with which it tries to pull the mass back to the initial equilibrium position.<a href="#fn3" class="footnote-ref" id="fnref3" role="doc-noteref"><sup>3</sup></a> This cumulative resistance will stem from the deformations of columns and beams, and from static analysis we know that under the action of the lateral load <span class="math inline">\(\extforce\)</span>, static equilibrium equations for the joints may be expressed as</p>
<p><span id="eq-statcond1"><span class="math display">\[
\begin{bmatrix} k_{11} & k_{12} & k_{13} \\ k_{21} & k_{22} & k_{23} \\k_{31} & k_{32} & k_{33} \end{bmatrix} \begin{Bmatrix} u_1 \\ u_2 \\ u_3 \end{Bmatrix} = \begin{Bmatrix} \extforce \\ 0 \\ 0 \end{Bmatrix}
\qquad(2.1)\]</span></span> where <span class="math inline">\(k_{ij}\)</span> are stiffness coefficients. Since rotational accelerations and the associated d’Alembert moments are to be neglected, the right hand sides of the second and third equations that represent moment equilibrium of the joints will remain zero in dynamic equilibrium analysis. Consequently, <span class="math inline">\(\{u_2,u_3\}\)</span> are not independent: noting that <span class="math inline">\(\gc \equiv u_1\)</span>, the relation between <span class="math inline">\(\gc\)</span> and <span class="math inline">\(\{u_2,u_3\}\)</span> at any instant will be given by <span id="eq-statcond2"><span class="math display">\[
- \begin{bmatrix} k_{22} & k_{23} \\k_{32} & k_{33} \end{bmatrix}^{-1}\begin{bmatrix} k_{21} \\k_{31} \end{bmatrix} \gc = \begin{Bmatrix} u_2 \\ u_3 \end{Bmatrix}
\qquad(2.2)\]</span></span> so that eliminating the rotational degrees of freedom from <a href="#eq-statcond1">Equation <span>2.1</span></a> with the help of <a href="#eq-statcond2">Equation <span>2.2</span></a> leads to a single expression relating the lateral displacement directly to the lateral force, <span id="eq-statcondfin"><span class="math display">\[
\left( k_{11}- \begin{bmatrix} k_{12} & k_{13} \end{bmatrix} \begin{bmatrix} k_{22} & k_{23} \\k_{32} & k_{33} \end{bmatrix}^{-1}\begin{bmatrix} k_{21} \\k_{31} \end{bmatrix} \right) \gc = \extforce \rightarrow k \gc = \extforce
\qquad(2.3)\]</span></span> where the coefficient <span class="math inline">\(k\)</span> is called the <em>lateral stiffness of the frame</em>. This procedure of eliminating the dynamically indirect degrees of freedom from the model is referred to as <em>static condensation</em> and it is widely used to simplify the structural models through elimination of those degrees of freedom onto which no mass or moment of inertia is assigned. It is important to note that even though inertial properties are alluded to, especially with respect to their existence or nonexistence, determination of the stiffness properties of a system is essentially a matter of static analysis.</p>
<p>Now that the conceptual model is finalized, we may use any of the frameworks previously reviewed to derive the equation that will govern the motion of the mass, i.e. the time variation of <span class="math inline">\(\gc (t)\)</span>. For example, making use of d’Alembert’s principle and the free body diagram shown in <a href="#fig-portalframe">Figure <span>2.1</span></a>(d), the sum of forces in the horizontal direction would lead to <span class="math display">\[
m \ddgc (t) + k \gc (t) = \extforce (t)
\]</span> which, as we will so often see, is the canonical equation governing the dynamics of undamped linear single degree of freedom systems under the action of external load effects.</p>
</section>
<section id="footnotes" class="footnotes footnotes-end-of-section" role="doc-endnotes">
<hr>
<ol>
<li id="fn1"><p>There are of course models in which mass is distributed as well as flexibility. Such models will be discussed in latter sections and chapters.<a href="#fnref1" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
<li id="fn2"><p>Even this model has an abundance of implicit assumptions. As an example, the axial deformations are considered to be at least an order of magnitude smaller than flexural deformations, so that: i. the translations of the two joints along the longitudinal axes of the columns are neglected, and ii. the joints are assumed to translate laterally by the same amount since the beam’s axial deformations are neglected.<a href="#fnref2" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
<li id="fn3"><p>There is another important component called <em>damping</em> which is used to model energy dissipation during motion; we leave aside for the moment the introduction of this component.<a href="#fnref3" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
</ol>
</section>
<section id="sec-ThePrototype" class="level2" data-number="2.2">
<h2 data-number="2.2" data-anchor-id="sec-ThePrototype"><span class="header-section-number">2.2</span> The Prototype</h2>
<p>As the equation of motion has the same form for all single degree of freedom (SDOF) systems, a single simple model suffices to conceptually represent all undamped SDOF systems. This model, sketched in <a href="#fig-sdofmk">Figure <span>2.2</span></a>, comprises a mass, representing the inertial features that contribute to the equation of motion through accelerations, and a spring, representing flexiblility features that contribute to the equation of motion through deformations. The mass is assumed to be travelling on a frictionless surface (the wheels are there to hint, admittedly erroneously, at motion unhindered by interaction with the ground) and there are no other dissipative sources either. Physical properties of the system are assumed to be invariant in time. Each component of the model may indeed correspond to a single entity of some system but most likely each will represent the contribution of multiple sources. It may therefore be more appropriate to think of the mass and stiffness coefficients as <em>equivalent mass</em> and <em>equivalent stiffness</em> of the system, but we refrain from emphasizing this distinction through symbols or scripts to in an effort to keep notational complexity at a minimum.</p>
<div id="fig-sdofmk" class="quarto-figure quarto-figure-center">
<figure class="figure">
<p><img src="images/sdofmk.png" class="img-fluid figure-img" style="width:50.0%"></p>
<p></p><figcaption class="figure-caption">Figure 2.2: Prototype for undamped single degree of freedom systems.</figcaption><p></p>
</figure>
</div>
<p>To provide a few examples on how such a model could be so vastly representative, let us first derive the equation of motion for the system of <a href="#fig-sdofmk">Figure <span>2.2</span></a>. The free body diagram of the system including the d’Alembert force is provided in <a href="#fig-sdofmkfbd">Figure <span>2.3</span></a>. Sum of forces in the horizontal direction leads to the following equation of motion governing the time variation of the single coordinate <span class="math inline">\(\gc\)</span>: <span id="eq-eomsdof"><span class="math display">\[
\extforce (t) - m \ddgc (t) - k \gc (t) = 0 \quad \rightarrow \quad m \ddgc (t) + k \gc (t) = \extforce (t)
\qquad(2.4)\]</span></span></p>
<div id="fig-sdofmkfbd" class="quarto-figure quarto-figure-center">
<figure class="figure">
<p><img src="images/sdofmkfbd.png" class="img-fluid figure-img" style="width:65.0%"></p>
<p></p><figcaption class="figure-caption">Figure 2.3: Free body diagram of the prototype single degree of freedom system including the d’Alembert force.</figcaption><p></p>
</figure>
</div>
<p>Alternatively we could use the Lagrangian approach which would proceed as follows: The kinetic and potential energies of the system are given by <span class="math display">\[
\ke = \frac{1}{2}m \dgc^2, \quad \pe = \frac{1}{2} k \gc^2
\]</span> while the virtual work done by the external forces through a virtual displacement is given by <span class="math display">\[
\vwork_{e} = \extforce \vgc = \vforce \vgc
\]</span> so that the generalized force is simply <span class="math inline">\(\vforce = \extforce\)</span>. Therefore Lagrange’s equations lead to <span class="math display">\[
\frac{\diff}{\diff t} \left(\frac{\partial \ke}{\partial \dgc}\right) - \frac{\partial \ke}{\partial \gc} + \frac{\partial \pe}{\partial \dgc}= \vforce \quad \rightarrow \quad m \ddgc (t) + k \gc (t) = \extforce (t)
\]</span> which, of course, is identical to the equation of motion in <a href="#eq-eomsdof">Equation <span>2.4</span></a>.</p>
<p>Let us now consider the systems shown in <a href="#fig-sdofmodels">Figure <span>2.4</span></a>. The simply supported beam in <a href="#fig-sdofmodels">Figure <span>2.4</span></a>(a), assumed to behave linear elastically, is supposed to be very light compared to the heavy mass <span class="math inline">\(m\)</span> so that it is to be modeled as massless. The system has a single degree of freedom since the mass may only translate vertically (assuming the deflections are small so that lateral motion due to beam deformations is negligible). The critical issue in modeling this system is the determination of its stiffness, which is due to the deformation resistance of the beam. By definition, a stiffness coefficient is numerically equal to the external force that should be imposed along the degree of freedom to induce a unit displacement.<a href="#fn4" class="footnote-ref" id="fnref4" role="doc-noteref"><sup>4</sup></a> If the beam is undergoing uniaxial bending with bending rigidity <span class="math inline">\(EI\)</span> (i.e. the product of modulus of elasticity <span class="math inline">\(E\)</span> and the second moment of the cross sectional area about the bending axis <span class="math inline">\(I\)</span>) and length <span class="math inline">\(\ell\)</span>, from structural analysis we know that the relationship between a force <span class="math inline">\(\extforce\)</span> applied at the midspan of the beam and the deflection <span class="math inline">\(\Delta\)</span> under the force is given by</p>
<p><span class="math display">\[
\extforce = \frac{48 EI}{\ell^3} \Delta
\]</span> so that if <span class="math inline">\(\Delta=1\)</span>, then the force that has to be applied is numerically equal to <span class="math inline">\({48 EI}/{\ell^3}\)</span>. While this interpretation is more direct for experimental determination of stiffness, in the analytical derivation the stiffness is simply identified as the coefficient that multiplies the displacement in the linear force-displacement relationship, i.e. <span class="math inline">\({48 EI}/{\ell^3}\)</span>. Whichever way it is interpreted the bottom line is that, as far the vertical motion of the mass is concerned, the massless beam is equivalent to a linear spring with equivalent spring coefficient <span class="math display">\[
k = \frac{48 EI}{\ell^3}
\]</span> and the equation of motion governing the time variation of <span class="math inline">\(\gc\)</span> may be shown to be <span class="math display">\[
m \ddgc (t) + \frac{48 EI}{\ell^3} \gc (t) = 0
\]</span> since there are no external forces acting on the mass. This statement may immediately trigger the question ‘what happened to gravity?’ In structural dynamics, it is generally assumed that the deformations due to gravity loads and the deformations due to dynamic loads may be analyzed separately and the results so obtained may be superposed afterwards. This of course is strictly possible only if the deformations resulting from either is relatively small so that the geometry is not significantly altered and the response remains linear. There are important exceptions to this approach, most notably for systems in which second order effects and/or nonlinearity is pronounced.</p>
<div id="fig-sdofmodels" class="quarto-figure quarto-figure-center">
<figure class="figure">
<p><img src="images/sdofmodels.png" class="img-fluid figure-img" style="width:90.0%"></p>
<p></p><figcaption class="figure-caption">Figure 2.4: (a) A beam carrying a heavy mass, (b) a heavy disc attached to a shaft, (c) truss and springs supporting a heavy mass.</figcaption><p></p>
</figure>
</div>
<p>The shaft in <a href="#fig-sdofmodels">Figure <span>2.4</span></a>(b) is again assumed to be massless. It is supporting a disc with moment of inertia <span class="math inline">\(\inertia{\ell}\)</span> about the longitudinal axis of the shaft. When torsional vibrations are to be studied, the system has a single degree of freedom since the disc will only be rotating about the longitudinal axis. Consider now an instant at which the disc has rotated by an amount <span class="math inline">\(\gc\)</span>. If the shaft of length <span class="math inline">\(\ell\)</span> is assumed to behave linear elastically with torsional rigidity <span class="math inline">\(GJ\)</span> (i.e. the product of the shear modulus <span class="math inline">\(G\)</span> and the polar moment of the cross sectional area <span class="math inline">\(J\)</span>), then the resisting moment acted upon the disc by the shaft at that instant may be shown to be in opposite direction to <span class="math inline">\(\gc\)</span> and of magnitude <span class="math display">\[
\smoment = \frac{GJ}{\ell} \gc
\]</span> Sketching the free body diagram of the disc and applying the principle of angular momentum will then yield the equation of motion <span class="math display">\[
\inertia{\ell} \ddgc (t) + \frac{GJ}{\ell} \gc (t) = 0
\]</span> which once again has a form identical to that of <a href="#eq-eomsdof">Equation <span>2.4</span></a>: If one could build a physical model that looks like the system shown in <a href="#fig-sdofmk">Figure <span>2.2</span></a>, adjust its mass <span class="math inline">\(m\)</span> to be numerically equal to <span class="math inline">\(\inertia{\ell}\)</span>, and adjust its stiffness <span class="math inline">\(k\)</span> to be numerically equal to <span class="math inline">\(GJ/\ell\)</span>, then the response of that built model would perfectly match the response of the disc-shaft system under numerically similar load effects. Such similarities are generally referred to as <em>mechanical analogies</em> and with this terminology, the system of <a href="#fig-sdofmk">Figure <span>2.2</span></a> may be referred to as the mechanical analogue for all undamped linear single degree of freedom systems.</p>
<div id="fig-equivsprings" class="quarto-figure quarto-figure-center">
<figure class="figure">
<p><img src="images/equivsprings.png" class="img-fluid figure-img" style="width:100.0%"></p>
<p></p><figcaption class="figure-caption">Figure 2.5: Equivalent springs with parallel and series connections.</figcaption><p></p>
</figure>
</div>
<p>Slightly more complicated is the system shown <a href="#fig-sdofmodels">Figure <span>2.4</span></a>(c), where a three member truss, supported by two identical springs, is carrying a heavy mass attached to its top node. To simplify the problem, it is assumed that the truss members have negligible mass, and that the supports are restrained to move vertically. Due to symmetry, the mass will translate in the vertical direction alone, assuming no external forces are present to explicitly force any other motion. The system therefore again has only one degree of freedom. If we were to consider an instant at which the mass has moved down an amount of <span class="math inline">\(\gc\)</span>, it is clear that some of that displacement will be due to the deformation of the truss bars and the rest will be due to the deformation of the springs. A detailed analysis may be undertaken to analyze the bar forces and associated displacements to find the equivalent stiffness of the system but the following analogy may help while thinking about this and similar problems. The combined system of <a href="#fig-equivsprings">Figure <span>2.5</span></a>(a), with the massless truss and springs, is mechanically analogous to the springs shown in <a href="#fig-equivsprings">Figure <span>2.5</span></a>(b) where the truss has been replaced by a spring with stiffness <span class="math inline">\(k_t\)</span>. This spring is representative of the force-displacement relationship of the truss alone, with displacement here referring to the relative vertical translation of node <span class="math inline">\(c\)</span> with respect to nodes <span class="math inline">\(a\)</span> and <span class="math inline">\(b\)</span> which, due to symmetry, will behave identically. It may be shown through structural analysis that the relative vertical displacement <span class="math inline">\(\Delta\)</span> of node <span class="math inline">\(c\)</span> under the action of a vertical force <span class="math inline">\(\extforce\)</span> applied at <span class="math inline">\(c\)</span> is given by <span class="math display">\[
\Delta = \frac{2\ell}{3EA}\extforce
\]</span> so that the equivalent spring coefficient <span class="math inline">\(k_t\)</span> may be defined as <span class="math display">\[
k_t = \frac{3EA}{2\ell}
\]</span> As for the two springs supporting the truss, they may be said to be connected in <em>parallel</em>; their displacement is the same but each will carry a portion of the total force in proportion to its stiffness. Consider any two linear springs, with coefficients <span class="math inline">\(k_1\)</span> and <span class="math inline">\(k_2\)</span>, connected in parallel. Let the total force they transmit be <span class="math inline">\(\extforce\)</span>, and assume they both displace by the same amount <span class="math inline">\(\Delta\)</span>. The relation between force and displacement for such a system would be <span class="math display">\[
\extforce = \underbrace{k_1 \Delta}_{{\text{force in $1$}}} + \underbrace{k_2 \Delta}_{{\text{force in $2$}}} = (k_1 + k_2) \Delta
\]</span> so that the response of such a system would be equivalent to a system with a single spring of stiffness <span class="math inline">\(k_1 + k_2\)</span>.<a href="#fn5" class="footnote-ref" id="fnref5" role="doc-noteref"><sup>5</sup></a> Therefore the system of <a href="#fig-equivsprings">Figure <span>2.5</span></a>(b) may be replaced by the system of <a href="#fig-equivsprings">Figure <span>2.5</span></a>(c).</p>
<p>Now we have two springs in <em>series</em>, with stiffness coefficients <span class="math inline">\(k_t\)</span> and <span class="math inline">\(2k_s\)</span>. For any two linear springs, say with stiffness coefficients <span class="math inline">\(k_1\)</span> and <span class="math inline">\(k_2\)</span>, that are connected in series, the force transmitted through each will be the same, but each will contribute to the displacement in proportion to their stiffness. The relationship between the force <span class="math inline">\(\sforce\)</span> through such a system and the total displacement <span class="math inline">\(\Delta\)</span> is given by <span class="math display">\[
\Delta = \underbrace{\frac{\extforce}{k_1}}_{{\text{deformation in $1$}}} + \underbrace{\frac{\extforce}{k_2}}_{{\text{deformation in $2$}}} = \left(\frac{1}{k_1} + \frac{1}{k_2}\right) \extforce
\]</span> so that such a series system would be equivalent to a single spring with spring coefficient<a href="#fn6" class="footnote-ref" id="fnref6" role="doc-noteref"><sup>6</sup></a> <span class="math display">\[
\frac{1}{\frac{1}{k_1} + \frac{1}{k_2}}
\]</span> Applying this result to our system leads finally to the single equivalent spring shown in <a href="#fig-equivsprings">Figure <span>2.5</span></a>(d), with equivalent spring coefficient <span class="math inline">\(k\)</span> given by <span class="math display">\[
k = \frac{1}{\frac{1}{k_t} + \frac{1}{2k_s}} = \frac{6EAk_s}{4\ell k_s + 3 EA}
\]</span> and the equation of motion for the system of <a href="#fig-sdofmodels">Figure <span>2.4</span></a>(c) would finally be derived as <span class="math display">\[
m \ddgc (t) + k \gc (t) = m \ddgc (t) + \frac{6EAk_s}{4\ell k_s + 3 EA} \gc (t) = 0
\]</span></p>
<p>We should recall that, although various details on a number of issues have been mentioned, the main point of this exercise was to motivate the use of the prototype single degree of freedom model. Whether or not a system may be effectively represented by a single degree of freedom model, however, is a question that does not have a straightforward answer, as hopefully illustrated by even the relatively simple systems discussed above. There is no shortcut or a single foolproof recipe that we know of which would be applicable in all situations; as such, this book will mainly focus on the methods of analysis that will be applicable to different models, assuming that those models are representative of whatever real life system they are aspiring to replicate the behavior of. The success or failure of the modeling approach will eventually boil down to how accurately the model will be able to predict the real response. The only way we know of to increase one’s model developing abilities is simply by <em>doing</em>: working on many textbook exercises and more realistic problems to get acquainted with a diverse set of cases, identifying the strengths and caveats of modeling assumptions and approximations, so as to be able to make rational choices based on reliable experience and information. We therefore highly recommend the reader to attempt all the solved and unsolved exercises that will be provided in the text in an effort to start on this arduous journey which will in all likelihood continue, as ours still do, indefinitely.</p>
</section>
<section id="footnotes" class="footnotes footnotes-end-of-section" role="doc-endnotes">
<hr>
<ol start="4">
<li id="fn4"><p>When more than one degree of freedom are present, one should consider forces at all degrees of freedom so that while a unit displacement occurs in one, there are no displacements at any other degrees of freedom. More on this in latter chapters.<a href="#fnref4" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
<li id="fn5"><p>Which may easily be generalized to any number of springs connected in parallel.<a href="#fnref5" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
<li id="fn6"><p>Which again generalizes easily to multiple springs connected in series.<a href="#fnref6" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
</ol>
</section>
<section id="sec-Undamped" class="level2" data-number="2.3">
<h2 data-number="2.3" data-anchor-id="sec-Undamped"><span class="header-section-number">2.3</span> Undamped Free Vibrations</h2>
<p>If a system in natural (static) equilibrium is set to motion by imposing some initial displacement and/or velocity but then left to move around the initial equilibrium configuration on its own, without imposing any external load effects, then the system is said to be executing <em>free vibrations</em>.</p>
<p>In order to introduce some basic concepts and definitions, let us first analyze a system in which there is no energy dissipation. Such a system would be governed by <a href="#eq-eomsdof">Equation <span>2.4</span></a> but with no external force so that <span class="math display">\[
m \ddgc (t) + k \gc (t) = 0
\]</span> but this equation alone is not sufficient to describe the problem completely. To be able to determine the time variation of <span class="math inline">\(\gc (t)\)</span>, one needs to know the <em>state of the system</em> at some time so as to be able to track the system thereafter. The state of a single degree of freedom system at some instant <span class="math inline">\(t=t_o\)</span> refers collectively to its displacement and velocity, i.e. <span class="math inline">\(\{\gc(t_o),\dgc(t_o)\}\)</span>, since these two variables are sufficient to determine all that happens to the system along with the governing equation of motion. This ‘instant’ <span class="math inline">\(t_o\)</span> is generally taken to be the start of the time interval considered corresponding to <span class="math inline">\(t=0\)</span>, and the values of displacement and velocity at this initial instant are called <em>initial conditions</em> for the system. These initial conditions are so often utilized that we shall denote them by special symbols: the initial displacement will be denoted by <span class="math inline">\(\gcic\)</span> and the initial velocity by <span class="math inline">\(\dgcic\)</span>. A more proper statement of the problem may now be expressed as: <span id="eq-eomsdofic"><span class="math display">\[
m \ddgc (t) + k \gc (t) = 0 \, ; \quad \left\{\gc(0)= \gcic,\dgc(0) = \dgcic\right\}
\qquad(2.5)\]</span></span></p>
<p><a href="#eq-eomsdofic">Equation <span>2.5</span></a> is a second order linear differential equation with constant coefficients. Its solution may be expressed in the form <span id="eq-eomsdofsol1"><span class="math display">\[
\gc (t) = A \expon{st}
\qquad(2.6)\]</span></span> where <span class="math inline">\(A\)</span> and <span class="math inline">\(s\)</span> are in general complex valued and as yet undetermined. Substituting this proposed solution to the equation of motion in <a href="#eq-eomsdofic">Equation <span>2.5</span></a> yields <span class="math display">\[
(m s^2 + k) A \expon{st} = 0
\]</span> If this equality is to be satisfied for all time <span class="math inline">\(t\)</span>, then there are two possibilities that should be considered. It could be that <span class="math inline">\(A = 0\)</span>, in which case equilibrium is automatically satisfied since the system is not moving; this is called the <em>trivial solution</em>. The second possibility, which is the non-trivial solution, requires <span class="math inline">\(s\)</span> to be such that <span class="math display">\[
ms^2 + k = 0
\]</span> which yields two solutions for <span class="math inline">\(s\)</span>: <span class="math display">\[
s_1 = + \sqrt{-\frac{k}{m}} = + \imag \freq, \quad s_2 = - \sqrt{-\frac{k}{m}} = - \imag \freq
\]</span> where <span class="math inline">\(\imag\)</span>, defined through <span class="math inline">\(\imag^2 \equiv -1\)</span>, is the unit imaginary number, and we have introduced <span class="math inline">\(\freq \equiv + \sqrt{k/m}\)</span>. This variable <span class="math inline">\(\freq\)</span> is of fundamental importance; it is called the <em>(undamped) natural frequency</em> of the system for reasons that will soon be clear. Since both of the solutions for <span class="math inline">\(s\)</span> will satisfy the equation of motion, the general solution will be the superposition of the two, so that <span id="eq-noshown"><span class="math display">\[
\gc (t) = A_1 \expon{s_1 t} + A_2 \expon{s_2 t} = A_1 \expon{\imag \freq t} + A_2 \expon{-\imag \freq t}
\qquad(2.7)\]</span></span> and finally, since <span class="math inline">\(\gc (t)\)</span> is real valued, it must be that <span class="math inline">\(A_1\)</span> and <span class="math inline">\(A_2\)</span> are complex conjugates, so that with <span class="math inline">\(A = A_1 = a+\imag b = A_2^* = (a-\imag b)^*\)</span>, we have the following general solution: <span id="eq-eomsdofsol2"><span class="math display">\[
\gc (t) = A \expon{\imag \freq t} + A^* \expon{-\imag \freq t}
\qquad(2.8)\]</span></span> <a href="#eq-eomsdofsol2">Equation <span>2.8</span></a> is referred to as the general solution since the response of all SDOF systems to initial conditions will be of this form. The two coefficients <span class="math inline">\(a\)</span> and <span class="math inline">\(b\)</span> in <span class="math inline">\(A = a + \imag b\)</span> will have to be determined based on the specific initial conditions of the particular problem being studied. This form of the solution, however, is not very conducive to direct physical interpretation, and most often expressing the solution in terms of well known trigonometric functions is preferred. Euler’s formula says that <span id="eq-eulersform"><span class="math display">\[
\expon{\pm \imag \theta} = \cos \theta \pm \imag \sin \theta
\qquad(2.9)\]</span></span> and when this expansion is used in <a href="#eq-eomsdofsol2">Equation <span>2.8</span></a> with <span class="math inline">\(\theta = \freq t\)</span>, we arrive after some algebra to the following form for the general solution: <span id="eq-eomsdofsol3"><span class="math display">\[
\gc (t) = C_1 \cos \freq t + C_2 \sin \freq t
\qquad(2.10)\]</span></span> When expressed in this form, the nature of the time variation of <span class="math inline">\(\gc\)</span> is much more obvious. The displacements that will occur in a SDOF system after it is set to motion will be of sinusoidal nature, with frequency <span class="math inline">\(\freq\)</span> and an amplitude that depends on the particular initial conditions. In fact, for <span class="math inline">\(\gc(0) = \gcic\)</span> and <span class="math inline">\(\dgc (0) = \dgcic\)</span>, the coefficients <span class="math inline">\(C_1\)</span> and <span class="math inline">\(C_2\)</span> will be evaluated as <span class="math display">\[\begin{align*}
\gc (0) = \gcic & = C_1 \cos 0 + C_2 \sin 0 \quad \rightarrow \quad C_1 = \gcic \\
\dgc (0) = \dgcic & = - \freq C_1 \sin 0 + \freq C_2 \cos 0 \quad \rightarrow \quad C_1 = \frac{\dgcic}{\freq}
\end{align*}\]</span> so that the general solution now takes the form <span id="eq-eomsdofsol4"><span class="math display">\[
\gc (t) = \gcic \cos \freq t + \frac{\dgcic}{\freq} \sin \freq t
\qquad(2.11)\]</span></span> That this solution corresponds in fact to a single sinusoidal wave may be shown mathematically through the use of the following expansion formula: <span id="eq-cosexp"><span class="math display">\[
\cos (\theta \pm \varphi) = \cos \theta \cos \varphi \mp \sin \theta \sin \varphi
\qquad(2.12)\]</span></span> A function of the form <span class="math inline">\(Q \cos (\freq t - \phs)\)</span> is therefore equivalent to <span class="math display">\[
Q \cos (\freq t - \phs) = (Q \cos \phs) \cos \freq t + (Q \sin \phs) \sin \freq t = C_1 \cos \freq t + C_2 \sin \freq t
\]</span> so that with <span id="eq-eomsdofsol5a"><span class="math display">\[
C_1 = Q \cos \phs , \qquad C_2 = Q \sin \phs
\qquad(2.13)\]</span></span> the general solution may be expressed as <span id="eq-eomsdofsol5b"><span class="math display">\[
\gc (t) = Q \cos (\freq t - \phs)
\qquad(2.14)\]</span></span></p>
<p>This form is arguably the one that is most easily visualized. The response to initial conditions is in the form of a single cosine wave with <em>amplitude</em> <span class="math inline">\(Q\)</span> and <em>phase angle</em><a href="#fn7" class="footnote-ref" id="fnref7" role="doc-noteref"><sup>7</sup></a> <span class="math inline">\(\phs\)</span>. The amplitude and phase are given, through <a href="#eq-eomsdofsol5a">Equation <span>2.13</span></a> and <a href="#eq-eomsdofsol4">Equation <span>2.11</span></a>, by <span id="eq-eomsdofsol6a"><span class="math display">\[
Q = \sqrt{C_1^2 + C_2^2} = \sqrt{\gcic^2 + \left(\frac{\dgcic}{\freq}\right)^2}
\qquad(2.15)\]</span></span> <span id="eq-eomsdofsol6"><span class="math display">\[
\phs = \arctan \frac{\sin \phs}{\cos \phs} = \arctan \frac{C_2/Q}{C_1 / Q} = \arctan \frac{\dgcic / (\freq Q)}{\gcic / Q}
\qquad(2.16)\]</span></span> The response of an undamped linear single degree of freedom system to some initial conditions <span class="math inline">\(\gcic > 0\)</span> and <span class="math inline">\(\dgcic > 0\)</span> is shown in <a href="#fig-dof1undampedic">Figure <span>2.6</span></a>. This sinusoidal response starts from the initial displacement <span class="math inline">\(\gcic\)</span> with slope <span class="math inline">\(\dgcic\)</span>. The first peak is reached at time <span class="math inline">\(t=\hat{t} = \phs / \freq\)</span>. The peak amplitude <span class="math inline">\(Q\)</span> is the same at all peaks, in other words the vibrations do not decay since there is no energy dissipation in the system and once it gets going with a certain total mechanical energy, given by <span class="math display">\[
\te_o = \ke_o + \pe_o = \frac{1}{2} m \dgcic^2 + \frac{1}{2} k \gcic^2
\]</span> its mechanical energy remains at this constant level. The maximum displacement and velocity (in an absolute sense) that occur during the motion, inferred from <a href="#eq-eomsdofsol5b">Equation <span>2.14</span></a> and its time derivative, are given by <span class="math display">\[
\gc_{\max} = Q, \quad \dgc_{\max} = \freq Q
\]</span></p>
<div id="fig-dof1undampedic" class="quarto-figure quarto-figure-center">
<figure class="figure">
<p><img src="images/dof1undampedic.png" class="img-fluid figure-img" style="width:100.0%"></p>
<p></p><figcaption class="figure-caption">Figure 2.6: Initial condition response of an undamped single degree of freedom system.</figcaption><p></p>
</figure>
</div>
<p>The maximum kinetic energy occurs whenever the displacement is zero and at a value of <span class="math display">\[
\ke_{\max} = \frac{1}{2} m (\dgc_{\max})^2 = \frac{1}{2} m (\freq Q)^2 = \frac{1}{2} m \left(\freq^2 \gcic^2 + \dgcic^2\right)
\]</span> whereas the maximum potential energy occurs whenever the velocity is zero and at a value of <span class="math display">\[
\pe_{\max} = \frac{1}{2} k (\gc_{\max})^2 = \frac{1}{2} k Q^2 = \frac{1}{2} k \left(\gcic^2 + \frac{\dgcic^2}{\freq^2}\right)
\]</span> The maximum force that occurs in the spring, possibly important for design considerations, is given by <span class="math display">\[
\text{maximum spring force} = k \gc_{\max} = k Q = k \sqrt{\gcic^2 + \left(\frac{\dgcic}{\freq}\right)^2}
\]</span></p>
<p>The response is said to be <em>periodic with period <span class="math inline">\(\period\)</span></em> so that the motion of the mass repeats itself in patterns of duration <span class="math inline">\(\period\)</span>; periodicity is expressed mathematically by the condition <span class="math display">\[
\gc(t) = \gc(t+\period) \quad \forall t
\]</span> where <span class="math inline">\(\period\)</span> is said to be the period of the system, given by <span class="math display">\[
\period = \frac{2\pi}{\freq}
\]</span> as may be easily inferred from <a href="#eq-eomsdofsol5b">Equation <span>2.14</span></a>. The period, which may be defined as the time it takes for motion to complete one full cycle, is a fundamental quantity in structural dynamics and it is more often referred to than the frequency due to its more direct physical implication. It is often stated in units of seconds [<span class="math inline">\(\punit{sec}\)</span> or <span class="math inline">\(\punit{s}\)</span>] since the periods of most systems encountered in structural dynamics are of the order of seconds. What is called the <em>cyclic frequency</em> is defined as the number of cycles completed in a unit of time, i.e. <span class="math display">\[
\text{cyclic frequency} = \frac{1}{\period} = \frac{\freq}{2 \pi}
\]</span> If time is measured in seconds, than the unit of cyclic frequency is named <span class="math inline">\(\textrm{Hertz}\)</span>, denoted by <span class="math inline">\(\punit{Hz}\)</span>, so that <span class="math inline">\(1 \unit{Hz} = \punit{no. of cycles} / 1 \unit{sec}\)</span>. To distinguish between the two frequencies, <span class="math inline">\(\freq\)</span> (with units of <span class="math inline">\(\punit{rad/sec}\)</span>) is often referred to as <em>angular frequency</em> since it corresponds to the angle the cosine function in the response goes through in a unit of time. Here we will simply refer to <span class="math inline">\(\freq\)</span> as <em>frequency</em> to economize on adjectives, and use the term cyclic frequency whenever we need to explicitly refer to it.</p>
</section>
<section id="footnotes" class="footnotes footnotes-end-of-section" role="doc-endnotes">
<hr>
<ol start="7">
<li id="fn7"><p>A phase angle is also referred to as a lead or a lag. When written in this form, <span class="math inline">\(\phs\)</span> may be referred to as a lag in the sense that the time to first peak is determined by the value of <span class="math inline">\(\phs\)</span> and for any <span class="math inline">\(\phs > 0\)</span>, the first peak occurs sometime after <span class="math inline">\(t=0\)</span>, as opposed to an unshifted cosine wave (<span class="math inline">\(\phs = 0\)</span>) for which the first peak occurs exactly at <span class="math inline">\(t=0\)</span>.<a href="#fnref7" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
</ol>
</section>
<section id="sec-Damping" class="level2" data-number="2.4">
<h2 data-number="2.4" data-anchor-id="sec-Damping"><span class="header-section-number">2.4</span> Damping</h2>
<p>Although indispensable for introducing basic concepts, one of the major drawbacks of the undamped model is that it does not reflect energy dissipation, which is a phenomenon observed in all real life systems. Energy dissipation may occur do to various mechanisms. Friction between the system and its surroundings, radiation, internal friction and heat generation, cracking and plastic deformations, to list some prominent causes, may all contribute at varying degrees to dissipation. The simple fact is that explicit identification and modeling of all such causes is not feasible if at all possible, unless some of them are dominating and the aim is to specifically address and study them.</p>
<section id="sec-ViscousDampingModel" class="level3" data-number="2.4.1">
<h3 data-number="2.4.1" data-anchor-id="sec-ViscousDampingModel"><span class="header-section-number">2.4.1</span> Viscous Damping Model</h3>
<p>A simple model for dissipation has proved both mathematically convenient and practically applicable, so as to provide acceptable representations of experimentally observed responses. The model is based phenomenologically on the response of a linear viscous dashpot in which the applied force is proportional to the velocity of deformation. Amalgamation of energy dissipation mechanisms to a single <em>equivalent linear viscous dashpot</em> provides a model that is capable of producing a certain type of amplitude decay often observed in free vibrations of real structures. This model is therefore generally accepted as a reasonable prototype to model energy dissipation in small amplitude vibrations, so much so that it is explicitly referred to in many structural codes and guidelines. With due caution we will simply refer to this model as <em>viscous damping</em> and represent it graphically with a dashpot as shown in <a href="#fig-sdofmckfbd">Figure <span>2.7</span></a>(a).</p>
<div id="fig-sdofmckfbd" class="quarto-figure quarto-figure-center">
<figure class="figure">
<p><img src="images/sdofmckfbd.png" class="img-fluid figure-img" style="width:100.0%"></p>
<p></p><figcaption class="figure-caption">Figure 2.7: (a) Viscously damped single degree of freedom system and (b) its free body diagram including the d’Alembert force.</figcaption><p></p>
</figure>
</div>
<p>The force across the damper is given by <span class="math inline">\(c \dgc\)</span> where <span class="math inline">\(\dgc\)</span> is the rate of deformation and <span class="math inline">\(c\)</span> is called the (linear) viscous dashpot coefficient: it is this coefficient, or some other one derived form it, that must be adjusted properly if the model is to represent accurately the amplitude decrement of the real system. The force acted upon the mass by the damper is therefore equal in magnitude to <span class="math inline">\(c \dgc\)</span> and opposite in direction to the direction of the velocity. The free body diagram of the mass, including the damping force and the d’Alembert force, is shown in <a href="#fig-sdofmckfbd">Figure <span>2.7</span></a>(b). The sum of the horizontal forces leads to <span id="eq-eomdsdof"><span class="math display">\[
m \ddgc (t) + c \dgc (t) + k \gc (t) = \extforce (t)
\qquad(2.17)\]</span></span> which is the equation of motion that represents the dynamics of all viscously damped single degree of freedom systems. The unforced vibrations will hence be governed by <span id="eq-eomdsdofic"><span class="math display">\[
m \ddgc (t) + c \dgc (t) + k \gc (t) = 0 \, ; \quad \left\{\gc(0) = \gcic, \; \dgc(0) = \dgcic\right\}
\qquad(2.18)\]</span></span> which is once again a second order linear differential equation with constant coefficients.</p>
<p>Since <a href="#eq-eomdsdofic">Equation <span>2.18</span></a> is of the same form as <a href="#eq-eomsdofic">Equation <span>2.5</span></a>, we may expect <a href="#eq-eomsdofsol1">Equation <span>2.6</span></a> to work in this case as well. Using the candidate solution in <a href="#eq-eomdsdofic">Equation <span>2.18</span></a> leads to <span class="math display">\[
(ms^2 + cs + k)A\expon{st} = 0
\]</span> and the non-trivial solution requires <span class="math display">\[
ms^2 + cs + k = 0
\]</span> There are two solutions for <span class="math inline">\(s\)</span> that satisfy this quadratic equation, given by <span class="math display">\[
s_1 = -\frac{c}{2m} + \frac{1}{2m}\sqrt{c^2 - 4mk}, \quad s_2 = -\frac{c}{2m} - \frac{1}{2m}\sqrt{c^2 - 4mk}
\]</span> It is possible to rewrite these expressions in somewhat simpler, familiar terms. Remembering that <span class="math inline">\(\freq^2 = k/m\)</span>, and defining <span id="eq-defdamprat1"><span class="math display">\[
2 \damp \freq \equiv \frac{c}{m}
\qquad(2.19)\]</span></span> where <span class="math inline">\(\damp\)</span> is a new but dependent variable derived from the damping properties, the roots may also be expressed as <span id="eq-dampedroots"><span class="math display">\[
s_1 = -\damp \freq + \freq \sqrt{\damp^2 - 1}, \quad s_2 = -\damp \freq - \freq \sqrt{\damp^2 - 1}
\qquad(2.20)\]</span></span> It is somewhat easier to observe that the nature of the solution depends very much on the value of the coefficient <span class="math inline">\(\damp\)</span>. Although only one of the cases is mostly relevant, for completeness we note the following three possibilities:</p>
<ol type="i">
<li><em>Overdamped systems</em>: Consider that <span class="math inline">\(\damp > 1\)</span>. In this case both roots are real valued and the exponentials lead to hyperbolic sines and cosines such that no oscillation is possible. In fact, one has <span class="math display">\[\begin{align*}
\gc (t) & = A_1\expon{\left(-\damp \freq + \freq \sqrt{\damp^2 - 1}\right)t} + A_2\expon{\left(-\damp \freq - \freq \sqrt{\damp^2 - 1}\right)t} \\
& = \expon{-\damp \freq t}\left[A_1\expon{\freq t \sqrt{\damp^2 - 1}} + A_2\expon{- \freq t \sqrt{\damp^2 - 1}}\right] \\
& = \expon{-\damp \freq t}\left[(A_1+A_2)\cosh \left(\freq t \sqrt{\damp^2 - 1}\right) + (A_1 - A_2)\sinh \left(\freq t \sqrt{\damp^2 - 1}\right) \right] \\
\gc (t) & = \expon{-\damp \freq t}\left[C_1 \cosh \left(\freq t \sqrt{\damp^2 - 1}\right) + C_2 \sinh \left(\freq t \sqrt{\damp^2 - 1}\right) \right]
\end{align*}\]</span> To simplify the presentation, let us define <span class="math display">\[
\vartheta \equiv \freq \sqrt{\damp^2 -1}
\]</span> so that <span id="eq-overdampedic"><span class="math display">\[
\gc (t) = \expon{-\damp \freq t}\left(C_1 \cosh \vartheta t + C_2 \sinh \vartheta t \right) %= \expon{-\damp \freq t} Q \cosh \left(\mu t - \phs \right)
\qquad(2.21)\]</span></span> To satisfy initial conditions one must have <span class="math display">\[\begin{align*}
\gc(0) & = \gcic = C_1 \quad \rightarrow \quad C_1 = \gcic \\
\dgc(0) & = \dgcic = - \damp \freq C_1 + \vartheta C_2 \quad \rightarrow \quad C_2 = \frac{\dgcic + \damp \freq \gcic}{\vartheta}
\end{align*}\]</span> and hence the general solution for initial condition response of overdamped SDOF systems may be expressed as <span class="math display">\[
\gc (t) = \expon{-\damp \freq t}\left[\gcic \cosh \vartheta t + \frac{\dgcic + \damp \freq \gcic}{\vartheta} \sinh \vartheta t \right]
\]</span></li>
</ol>
<div id="fig-overdampedic" class="quarto-figure quarto-figure-center">
<figure class="figure">
<p><img src="images/overdampedic.png" class="img-fluid figure-img" style="width:85.0%"></p>
<p></p><figcaption class="figure-caption">Figure 2.8: Response of overdamped systems subjected to initial conditions <span class="math inline">\(\{\gcic \neq 0, \dgcic=0\}\)</span>.</figcaption><p></p>
</figure>
</div>
<p>How the response varies for some values of <span class="math inline">\(\damp\)</span> may be seen in <a href="#fig-overdampedic">Figure <span>2.8</span></a>. Each curve in this figure shows the response of an overdamped SDOF system with a specific value of <span class="math inline">\(\damp\)</span> due to nonzero initial displacement <span class="math inline">\(\gcic \neq 0\)</span> and zero initial velocity <span class="math inline">\(\dgcic = 0\)</span>. The plots are non-dimensionalized, with the horizontal axis corresponding to time scaled with <span class="math inline">\(\period\)</span> (i.e. the period the system would have if it were undamped). Evidently all responses simply decay towards the static equilibrium position. What is slightly counter intuitive is that the higher the damping coefficient in an overdamped system, the longer it takes for the system to go back to the original configuration. Higher damping limits the velocities to smaller values so that recovery takes longer.</p>
<ol start="2" type="i">
<li><p><em>Critically damped systems</em>: The nature of the solution will eventually change as damping gets smaller and the transition occurs when <span class="math inline">\(c^2 - 4mk = 0\)</span> or, equivalently, when <span class="math inline">\(\damp = 1\)</span>. In this case the roots in <a href="#eq-dampedroots">Equation <span>2.20</span></a> yield <span class="math display">\[
s_1 = s_2 = - \freq
\]</span> and with repeated roots the solution takes the form <span class="math display">\[
\gc (t) = A_1 \expon{- \freq t} + A_2 t \expon{- \freq t}
\]</span> For initial displacement <span class="math inline">\(\gcic\)</span> and initial velocity <span class="math inline">\(\dgcic\)</span>, the coefficients are evaluated as <span class="math display">\[\begin{align*}
\gc(0) & = \gcic = A_1 \quad \rightarrow \quad A_1 = \gcic \\
\dgc(0) & = \dgcic = - \freq A_1 + A_2 \quad \rightarrow \quad A_2 = \dgcic + \freq \gcic
\end{align*}\]</span> so that the general solution for critically damped single degree of freedom systems may be expressed as <span class="math display">\[
\gc (t) = \gcic \expon{- \freq t} + (\dgcic + \freq \gcic) t \expon{- \freq t}
\]</span> The response is once again a simple decay with no oscillation around the static equilibrium configuration but the recovery will be faster than the curves shown in <a href="#fig-dof1undampedic">Figure <span>2.6</span></a>. This particular case allows us to define a <em>critical damping</em> for SDOF systems, given by <span class="math display">\[
c_{cr} = 2\sqrt{km}
\]</span> since for <span class="math inline">\(c > c_{cr}\)</span> the system will be overdamped, and the for <span class="math inline">\(c < c_{cr}\)</span> it will be underdamped. This critical value depends on the mass and stiffness properties of the system and thus it may be estimated for design purposes. Employing the critical damping concept, the coefficient <span class="math inline">\(\damp\)</span> initially defined via <a href="#eq-defdamprat1">Equation <span>2.19</span></a> may also be defined as the ratio of the available damping to the critical damping, i.e. <span class="math display">\[
\damp = \frac{c}{c_{cr}}
\]</span> and so <span class="math inline">\(\damp\)</span> is referred to as the <em>critical damping ratio</em> or simply as the <em>damping ratio</em>, which is the name we will use to address it.</p></li>
<li><p><em>Underdamped systems</em>: When <span class="math inline">\(\damp < 1\)</span>, the square roots in <a href="#eq-dampedroots">Equation <span>2.20</span></a> lead to complex numbers, and harmonic motion becomes possible. Let us express the two roots as <span class="math display">\[\begin{align*}
s_1 & = -\damp \freq + \freq \sqrt{-(1-\damp^2)} = -\damp \freq + \imag \dfreq , \\
s_2 & = -\damp \freq - \freq \sqrt{-(1-\damp^2)} = -\damp \freq - \imag \dfreq = s_1^*
\end{align*}\]</span> where <span id="eq-dfreq"><span class="math display">\[
\dfreq \equiv \freq\sqrt{1-\damp^2}
\qquad(2.22)\]</span></span> is called the <em>damped frequency</em> of the system. Remembering the discussion about complex coefficients in undamped vibrations, the general solution may be written as <span class="math display">\[
\gc (t) = \expon{-\damp \freq t}\left(A \expon{\imag \dfreq t} + A^* \expon{-\imag \dfreq t} \right)
\]</span> which, with the help of <a href="#eq-eulersform">Equation <span>2.9</span></a>, may be recast in the following form: <span class="math display">\[
\gc (t) = \expon{-\damp \freq t}\left(C_1 \cos \dfreq t + C_2 \sin \dfreq t \right)
\]</span> Applying the initial conditions leads to <span class="math display">\[\begin{align*}
\gc(0) & = \gcic = C_1 \quad \rightarrow \quad C_1 = \gcic \\
\dgc(0) & = \dgcic = - \damp \freq C_1 + \dfreq C_2 \quad \rightarrow \quad C_2 = \frac{\dgcic + \damp \freq \gcic}{\dfreq}
\end{align*}\]</span> so that <span id="eq-underdampedic1"><span class="math display">\[
\gc (t) = \expon{-\damp \freq t}\left(\gcic \cos \dfreq t + \frac{\dgcic + \damp \freq \gcic}{\dfreq} \sin \dfreq t \right)
\qquad(2.23)\]</span></span> Alternatively, via <a href="#eq-cosexp">Equation <span>2.12</span></a> and defining <span class="math display">\[
Q \cos \phs = C_1, \quad Q \sin \phs = C_2
\]</span> the solution may be expressed as <span id="eq-underdampedic2"><span class="math display">\[
\gc (t) = Q \expon{-\damp \freq t} \cos (\dfreq t - \phs)
\qquad(2.24)\]</span></span> where <span id="eq-underdampedic3a"><span class="math display">\[
Q = \sqrt{C_1^2 + C_2^2} = \frac{\sqrt{\gcic^2 + 2 \damp \gcic \left(\ratio{\dgcic}{\freq}\right) + \left(\ratio{\dgcic}{\freq}\right)^2}}{\sqrt{1-\damp^2}}
\qquad(2.25)\]</span></span> <span id="eq-underdampedic3b"><span class="math display">\[
\phs = \arctan \frac{\sin \phs}{\cos \phs} = \arctan \frac{C_2/Q}{C_1 / Q} = \arctan \frac{(\dgcic+\damp\freq\gcic) / (\dfreq Q)}{\gcic / Q}
\qquad(2.26)\]</span></span> Evidently the oscillations in underdamped systems will take place with an exponentially decaying amplitude defined by <span class="math inline">\(Q\expon{-\damp \freq t}\)</span>. The response is not strictly harmonic but we still talk of a period and frequency of the system: the oscillatory part, <span class="math inline">\(\cos (\dfreq t - \phs)\)</span>, is a harmonic wave with frequency equal to the damped frequency of the system, and the <em>damped period</em> of the system is defined as <span class="math display">\[
\dperiod = \frac{2\pi}{\dfreq}
\]</span> Since the damping ratio <span class="math inline">\(\damp < 1\)</span>, <span class="math display">\[
\dperiod = \frac{2\pi}{\freq \sqrt{1-\damp^2}}= \frac{\period}{\sqrt{1-\damp^2}} > \period
\]</span> so that the damped period is slightly longer than the undamped period. The difference, however, is generally small for common structures, in which damping ratios are generally on the order of <span class="math inline">\(1\%\)</span> to <span class="math inline">\(15\%\)</span>. Even for a relatively high damping ratio of <span class="math inline">\(10\%\)</span>, the difference between the two periods is about <span class="math inline">\(0.5 \%\)</span>; in most applications, this difference may be neglected and the undamped period may be referred to.</p></li>
</ol>
<div id="fig-underdampedic" class="quarto-figure quarto-figure-center">
<figure class="figure">
<p><img src="images/underdampedic.png" class="img-fluid figure-img" style="width:90.0%"></p>
<p></p><figcaption class="figure-caption">Figure 2.9: Response of underdamped systems subjected to initial conditions: (a) Typical response of an underdamped system to initial conditions <span class="math inline">\(\{\gcic > 0, \dgcic>0\}\)</span>, (b) effects of damping ratio on the response to initial conditions <span class="math inline">\(\{\gcic \neq 0, \dgcic=0\}\)</span>.</figcaption><p></p>
</figure>
</div>
<p>The responses of underdamped systems are visualized in <a href="#fig-underdampedic">Figure <span>2.9</span></a>. The typical response of an underdamped system, subject to initial conditions <span class="math inline">\(\gcic > 0\)</span> and <span class="math inline">\(\dgcic > 0\)</span> is shown in <a href="#fig-underdampedic">Figure <span>2.9</span></a>(a). The response at any instance is the product of the harmonic wave <span class="math inline">\(\cos (\dfreq t - \phs)\)</span> with the envelope <span class="math inline">\(Q\expon{-\damp\freq t}\)</span>, resulting in an oscillation about the natural equilibrium configuration with an exponentially decaying amplitude. The damped period is easily identified as the duration between any two consecutive down-crossings (or up-crossings), or any two consecutive peaks (or troughs). How fast the response decays in a single cycle depends on the value of <span class="math inline">\(\zeta\)</span>; this dependence is clearly seen in <a href="#fig-underdampedic">Figure <span>2.9</span></a>(b), which shows the responses of underdamped systems with three different values of <span class="math inline">\(\zeta\)</span>, all plotted in non-dimensionalized form to allow a direct comparison.</p>
</section>
<section id="sec-Estimating" class="level3" data-number="2.4.2">
<h3 data-number="2.4.2" data-anchor-id="sec-Estimating"><span class="header-section-number">2.4.2</span> Estimating Viscous Damping from Free Vibration Data</h3>
<p>Increasing damping ratio in underdamped systems leads obviously to faster decay in the response, which may be quantified by the decrease in amplitudes of peaks. This relation is used directly in a well known method used to estimate the damping ratio of SDOF systems. Consider a local peak, say the <em>i</em>th local peak, that occurs at some time <span class="math inline">\(t=t_i\)</span>. The amplitude <span class="math inline">\(Q_i\)</span> of this peak will be given by <span class="math display">\[
Q_i = \gc (t_i) = Q \expon{-\damp \freq t_i} \cos (\dfreq t_i - \phs)
\]</span> as shown in <a href="#fig-logdec">Figure <span>2.10</span></a>.</p>
<div id="fig-logdec" class="quarto-figure quarto-figure-center">
<figure class="figure">
<p><img src="images/logdec.png" class="img-fluid figure-img" style="width:100.0%"></p>