StructuralDynamics/multidoffreevib.html at main · keten-group/StructuralDynamics · GitHub

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% \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

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% \def\rem#1{{\bfseries{#1}}} % regular emphasis
% \def\mem#1{{\bfseries{#1}}} % much emphasis
% date: 03/30 change \bfseries to \textbf
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\DeclareMathOperator{\dif}{d}
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\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 \!}
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\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 &amp; 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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<section id="harmonic-motion-and-the-eigenvalue-problem" class="level2" data-number="7.1">
<h2 data-number="7.1" data-anchor-id="harmonic-motion-and-the-eigenvalue-problem"><span class="header-section-number">7.1</span> Harmonic Motion and the Eigenvalue Problem</h2>
<p>Solving the coupled equations of motion of an MDOF system is not a trivial issue. If one wants to find how each generalized coordinate evolves in time, one has to find <span class="math inline">\(n\)</span> many time functions in an <span class="math inline">\(n\)</span>-DOF system that will somehow satisfy the coupled equations at all times. Most often this task will have to be dealt with numerically, if it can be dealt with at all. Numerical results, however, tend to be not so conducive in helping us develop physical insights, which we do need in order to be able to judge the validity of any analysis.</p>
<p>So we may ask: is there any insight or experience that we have gained from analyses of SDOF systems that may help us get started with MDOF systems. One predominant characteristic of an SDOF system was its unique period (or frequency) with which it oscillated if it were perturbed form its original equilibrium configuration. Would an MDOF system do the same, and, what does this exactly mean? By the <em>period of the MDOF system</em> we shall understand the period, if it exists, with which the whole system repeats its motion in that during free vibrations all generalized coordinates execute harmonic motions with the same period. Since there are multiple degrees of freedom, the motions along each may have different <em>amplitudes</em>, but all motion should bear the same period. Consider the three masses hung along the same rope as shown in <a href="#fig-dof3pendulum">Figure&nbsp;<span>7.1</span></a>. We may intuitively feel that if we were to pull this pendulum to some initial angle and let go, the masses would move in harmony, swinging back and forth while keeping their initial relative appearance so that if we were to take consecutive snapshots, the pendulum would look pretty much as it did at the first instant albeit at smaller angles, somewhat like the sketch in <a href="#fig-dof3pendulum">Figure&nbsp;<span>7.1</span></a> (a). We may also feel that after such an initial perturbance the snapshots depicted in <a href="#fig-dof3pendulum">Figure&nbsp;<span>7.1</span></a> (b) are probably unlikely, that the masses are unlikely to do as they please and swing any way they want at any speed they want.</p>
<div id="fig-dof3pendulum" class="quarto-figure quarto-figure-center">
<figure class="figure">
<p><img src="images/dof3pendulum.png" class="quarto-discovered-preview-image img-fluid figure-img" style="width:70.0%"></p>
<p></p><figcaption class="figure-caption">Figure&nbsp;7.1: Pendulum with three masses.</figcaption><p></p>
</figure>
</div>
<p>We can’t quit at intuition of course and so the question is whether or not such harmonic motion is mathematically possible. For the whole system to move with the same frequency but different amplitudes along the generalized coordinates, the equations of motion should admit a solution that may be expressed as <span id="eq-mdofharm"><span class="math display">\[
\gctvec = \colmat{\gc_1 (t) \\ \gc_2 (t) \\ \vdots \\ \gc_n (t)} = \colmat{\eigvecs_{1} \\ \eigvecs_{2} \\ \vdots \\ \eigvecs_{n}} \cosp{\freq t - \phs} = \eigvec \cosp{\freq t - \phs}
\qquad(7.1)\]</span></span> where <span class="math inline">\(\eigvecs_{j}\)</span> is the time independent amplitude at the <em>j</em>th generalized coordinate, <span class="math inline">\(\freq\)</span> is the frequency with which the whole system moves, and <span class="math inline">\(\phs\)</span> is some possible lag. Note that the way the system looks as it moves is governed by <span class="math inline">\(\eigvec\)</span> or, more precisely, <em>the ratios of <span class="math inline">\(\eigvecs_{j}\)</span> to each other</em>. As we assume these amplitudes to be time invariant, the <em>relative shape of deformation is constant</em> by assumption, with instantaneous positions communally regulated by the time dependent cosine component.</p>
<p>Admissibility of such a motion depends on whether the equation of motion for free vibrations, given by <span id="eq-eomudfreevib"><span class="math display">\[
\mmat \ddgctvec + \kmat \gctvec = \zerocol
\qquad(7.2)\]</span></span> may be satisfied by this solution. The damping matrix is conspicuously missing of course, but we will come back to that in a little while. To see whether or not the solution works we plug the proposed expression <a href="#eq-mdofharm">Equation&nbsp;<span>7.1</span></a> into <a href="#eq-eomudfreevib">Equation&nbsp;<span>7.2</span></a> to get <span id="eq-soludfreevib1"><span class="math display">\[
-\freq^2 \mmat \eigvec \cosp{\freq t - \phs} + \kmat \eigvec \cosp{\freq t - \phs} = \left(\kmat - \freq^2 \mmat \right) \eigvec \cosp{\freq t - \phs} = \zerocol
\qquad(7.3)\]</span></span> If such a solution is to exist, <a href="#eq-soludfreevib1">Equation&nbsp;<span>7.3</span></a> should be satisfied at all times. Obviously the cosine term can take on nonzero values, and so there are only two possibilities. It could be that <span class="math inline">\(\eigvec \equiv \zerocol\)</span>, which means that the system is not moving at all! Equilibrium is naturally satisfied in that case (remember that initially the system is at an equilibrium state) but this is not the option we are seeking; this is called the <em>trivial solution</em>. The nontrivial solution is the possibility of <span class="math display">\[
\left(\kmat - \freq^2 \mmat \right) \eigvec = \zerocol
\]</span> which means that, for a nonzero <span class="math inline">\(\eigvec\)</span>, the matrix <span class="math inline">\(\left(\kmat - \freq^2 \mmat \right)\)</span> is (to borrow from linear algebra) rank deficient, i.e.&nbsp;its determinant is zero and its inverse does not exist; for if it did, then the only possibility would be <span class="math inline">\(\eigvec = \left(\kmat - \freq^2 \mmat \right)^{-1} \zerocol = \zerocol\)</span>.</p>
<p>What does this mean? Well, we were seeking for the possibility of harmonic motion and it turns out that: yes, it is possible, but only for those values of <span class="math inline">\(\freq\)</span> which make <span id="eq-detfreevib"><span class="math display">\[
\left| \kmat - \freq^2 \mmat \right| = 0
\qquad(7.4)\]</span></span> where <span class="math inline">\(\left| \mtrx{A} \right|\)</span> denotes the determinant of matrix <span class="math inline">\(\mtrx{A}\)</span>. For an <span class="math inline">\(n\)</span>-DOF system, <span class="math inline">\(\mmat\)</span> and <span class="math inline">\(\kmat\)</span> are <span class="math inline">\(n \times n\)</span> matrices, and the characteristic equation of <a href="#eq-detfreevib">Equation&nbsp;<span>7.4</span></a> is of order <span class="math inline">\(n\)</span> in <span class="math inline">\(\freq^2\)</span>. The characteristic equation, therefore, has <span class="math inline">\(n\)</span> roots, meaning there are <span class="math inline">\(n\)</span> values of <span class="math inline">\(\freq\)</span> which may make the determinant zero. From a more physical perspective: an MDOF system has multiple frequencies, as many as the number of degrees of freedom, with which it may execute harmonic motion.</p>
<p>In linear algebra, this problem is usually cast as the <em>generalized eigenvalue problem</em>: find all <span class="math inline">\(\freq^2\)</span> and <span class="math inline">\(\eigvec\)</span> for which <span id="eq-eigprob"><span class="math display">\[
\freq^2 \mmat \eigvec = \kmat \eigvec
\qquad(7.5)\]</span></span> is satisfied. When matrix dimensions are very small, it may be possible to solve this problem by hand. After a while it becomes tedious and outright impossible to proceed by hand and so the eigenvalue problems are almost always solved numerically.<a href="#fn1" class="footnote-ref" id="fnref1" role="doc-noteref"><sup>1</sup></a> In the structural dynamics jargon, the eigenvalues of <a href="#eq-eigprob">Equation&nbsp;<span>7.5</span></a>, or rather their positive square roots the <span class="math inline">\(\freq\)</span>’s, are called the <em>natural frequencies</em> or <em>modal frequencies</em>, and its eigenvectors the <span class="math inline">\(\eigvec\)</span>’s are called the <em>mode shapes</em> of the system. This terminology is well established and we will adhere to it.</p>
<p>To give a flavor of what this all means, let us work on the two-story shear building shown in <a href="#fig-dof2shearbldg">Figure&nbsp;<span>7.2</span></a>. It is relatively easy to show that the mass and stiffness matrices for this simple model are given by <span class="math display">\[
\mmat = \begin{bmatrix} m_1 &amp; 0 \\ 0 &amp; m_2 \end{bmatrix}, \quad \kmat = \begin{bmatrix} k_1 + k_2 &amp; -k_2 \\ -k_2 &amp; k_2 \end{bmatrix}
\]</span></p>
<div id="fig-dof2shearbldg" class="quarto-figure quarto-figure-center">
<figure class="figure">
<p><img src="images/dof2shearbldg.png" class="img-fluid figure-img" style="width:35.0%"></p>
<p></p><figcaption class="figure-caption">Figure&nbsp;7.2: Two story shear building.</figcaption><p></p>
</figure>
</div>
<p>Let us simplify the algebra by considering a particular case with <span class="math inline">\(m_1 = m_2 = m\)</span>, <span class="math inline">\(k_1 = 2k\)</span> and <span class="math inline">\(k_2 = k\)</span>. The eigenvalue problem becomes <span id="eq-dof2eig"><span class="math display">\[
\left(\begin{bmatrix} 3k &amp; -k \\ -k &amp; k \end{bmatrix} - \freq^2 \begin{bmatrix} m &amp; 0 \\ 0 &amp; m \end{bmatrix}\right) \eigvec = \zerocol
\qquad(7.6)\]</span></span> with the characteristic equation given by <span id="eq-dof2det"><span class="math display">\[
\left| \kmat - \freq^2 \mmat \right| =\left| \begin{bmatrix} 3k-\freq^2 m &amp; -k \\ -k &amp; k - \freq^2 m \end{bmatrix} \right| = m^2 (\freq^2)^2 - 4 {k}{m} \freq^2 + 2 k^2 = 0
\qquad(7.7)\]</span></span> which yields the following two roots via the well-known quadratic formula: <span class="math display">\[
\freq_1^2 = (2 - \sqrt{2}) \ratio{k}{m}, \quad \freq_2^2 = (2 + \sqrt{2}) \ratio{k}{m}
\]</span> By convention, when numbering the multiple frequencies of a system, they are numbered in increasing order so that <span class="math display">\[
\freq_1 \leq \freq_2 \leq \ldots \leq \freq_{n-1} \leq \freq_n
\]</span> This convention is almost always adhered to, with the lowest frequency <span class="math inline">\(\freq_1\)</span> generally referred to as the <em>fundamental frequency</em> of the system; the corresponding period <span class="math inline">\(\period_1 = 2 \pi/ \freq_1\)</span>, which happens to be the longest period of the system, is analogously called the <em>fundamental period</em>. Experience has shown that in most cases the largest amplitudes of motion occur in relation to this fundamental period, and therefore its proper estimation is of prime importance.</p>
<p>So what kind of a deformed shape does the system take on while moving with any of these frequencies; in other words, what are the corresponding <span class="math inline">\(\eigvec\)</span>’s? We have to go back to <a href="#eq-dof2det">Equation&nbsp;<span>7.7</span></a> and try to figure out the <span class="math inline">\(\eigveci{i}\)</span> that accompanies the particular frequency <span class="math inline">\(\freq_i\)</span>. For our system we have two frequencies and two such mode shapes. We will denote the related mode shapes by <span class="math inline">\(\eigveci{1}\)</span> and <span class="math inline">\(\eigveci{2}\)</span>, and their components as <span class="math display">\[
\eigveci{1} = \colmat{\eigvecs_{11} \\ \eigvecs_{21}}, \quad \eigveci{2} = \colmat{\eigvecs_{12} \\ \eigvecs_{22}}
\]</span> where we take <span class="math inline">\(\eigvecs_{ji}\)</span> to mean the <em>j</em>th component of the <em>i</em>th mode shape. In short, <span class="math inline">\(\eigvecs_{ji}\)</span> is the relative amplitude at generalized coordinate <em>j</em> while the system moves in mode <em>i</em>. For <span class="math inline">\(\freq_1^2\)</span>, the eigenvalue equation is given by <span class="math display">\[
\left(\begin{bmatrix} 3k &amp; -k \\ -k &amp; k \end{bmatrix} - \freq_1^2 \begin{bmatrix} m &amp; 0 \\ 0 &amp; m \end{bmatrix}\right) \eigveci{1} = \begin{bmatrix}3k - (2-\sqrt{2})k &amp; -k \\ -k &amp; k - (2-\sqrt{2})k \end{bmatrix} \colmat{\eigvecs_{11} \\ \eigvecs_{21}} = \zerocol
\]</span> which seemingly yields two equations as <span class="math display">\[
(1+\sqrt{2}) \eigvecs_{11} - \eigvecs_{21} = 0, \quad \eigvecs_{11} - (1 -\sqrt{2}) \eigvecs_{21} = 0
\]</span> but these two equations are in fact one and the same: <span class="math display">\[
\eigvecs_{21} = 2.414 \eigvecs_{11}
\]</span> How do we determine two unknowns with one equation? We can’t. Mathematically this result is to be expected since, for the eigenvalue problem to work out, <span class="math inline">\(\eigvec\)</span> must lie in the null space of the rank deficient coefficient matrix <span class="math inline">\((\kmat-\freq^2 \mmat)\)</span>; consequently, if some <span class="math inline">\(\eigvec\)</span> satisfies <span class="math inline">\((\kmat-\freq^2 \mmat)\eigvec=\zerocol\)</span>, then so does <span class="math inline">\(a\eigvec\)</span> where <span class="math inline">\(a\)</span> is an arbitrary multiplier. Does this make sense physically? Think about it this way: What we have determined is the <em>relative shape</em> the system maintains as it moves with a certain frequency, but the absolute amplitude of motion should depend on the initial conditions (e.g.&nbsp;how large is the initial perturbation?). We have not considered such information yet and it will have to wait until we discuss the initiation of motion. Until then, we have to be satisfied with only the relative ratios of what will occur along the generalized coordinates. We may therefore choose to represent the relative amplitudes according to any particular scaling at this point. We may, for example, assign a magnitude of one to a particular generalized coordinate; say we assign <span class="math inline">\(\eigvecs_{11}=1\)</span>, in which case we have <span class="math display">\[
\eigveci{1}=\colmat{1 \\ 2.414}
\]</span> Alternatively we may adjust the values of the coefficients so that the mode shape has unit magnitude, i.e.&nbsp;<span class="math inline">\({\eigveci{1}^T}\eigveci{1}=1\)</span>, in which case we would have <span class="math display">\[
\eigveci{1}=\colmat{0.383 \\ 0.924}
\]</span> Both of these choices are fine in that the actual response that we will observe as the system oscillates with the fundamental frequency will be given by <span class="math display">\[
\gctvec = A_1 \eigveci{1} \cosp{\freq_1 t - \phs_1}
\]</span> where, depending on the particular scaling we choose, the coefficient <span class="math inline">\(A_1\)</span> may take on different values, but at the end all choices will yield the same <span class="math inline">\(\gctvec\)</span>.</p>
<div id="fig-dof2shearbldgms" class="quarto-figure quarto-figure-center">
<figure class="figure">
<p><img src="images/dof2shearbldgms.png" class="img-fluid figure-img" style="width:70.0%"></p>
<p></p><figcaption class="figure-caption">Figure&nbsp;7.3: Mode shapes of the two story shear building.</figcaption><p></p>
</figure>
</div>
<p>There is also a second frequency with which harmonic motion is possible. The mode shape corresponding to <span class="math inline">\(\freq_2\)</span> is prescribed by <span class="math display">\[
\left(\begin{bmatrix} 3k &amp; -k \\ -k &amp; k \end{bmatrix} - \freq_2^2 \begin{bmatrix} m &amp; 0 \\ 0 &amp; m \end{bmatrix}\right) \eigveci{2} = \begin{bmatrix}3k - (2+\sqrt{2})k &amp; -k \\ -k &amp; k - (2+\sqrt{2})k \end{bmatrix} \colmat{\eigvecs_{12} \\ \eigvecs_{22}} = \zerocol
\]</span> which yields the following relationship: <span class="math display">\[
\eigvecs_{22} = -0.414 \eigvecs_{12}
\]</span> If we again assign a unit amplitude to the first generalized coordinate, we have <span class="math display">\[
\eigveci{2}=\colmat{1 \\ -0.414}
\]</span> Finally as there are two possibilities of harmonic motion, the general solution is the superposition the two, giving as the final answer the following expression for our example of the two story shear building: <span id="eq-dof2modsol"><span class="math display">\[
\gctvec = A_1 \eigveci{1} \cosp{\freq_1 t - \phs_1} + A_2 \eigveci{2} \cosp{\freq_2 t - \phs_2}
\qquad(7.8)\]</span></span> Before we move on, it is worthwhile to visualize the mode shapes for such a structure, schematically sketched in <a href="#fig-dof2shearbldgms">Figure&nbsp;<span>7.3</span></a>. In the fundamental mode, the relative movements of the masses are such that the building sways to one side as a whole. The second mode, on the other hand, comprises movement of masses in opposite directions, with one sign change or one cross-over. We’ll see in numerous cases that such modal motions are typical of many similar systems, with number of crossovers increasing with increasing mode numbers and the fundamental mode comprising a unidirectional movement.</p>
</section>
<section id="footnotes" class="footnotes footnotes-end-of-section" role="doc-endnotes">
<hr>
<ol>
<li id="fn1"><p>The more commonly encountered version is the <em>eigenvalue problem</em> expressed as <span class="math inline">\(\lambda \nvec{x} = \mtrx{A} \nvec{x}\)</span>. Eigenvalue problems are so common and important in mathematical physics that there are extremely efficient and well-established algorithms available in computing platforms. We therefore will not delve into how this problem is solved numerically.<a href="#fnref1" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
</ol>
</section>
<section id="orthogonality-of-modes" class="level2" data-number="7.2">
<h2 data-number="7.2" data-anchor-id="orthogonality-of-modes"><span class="header-section-number">7.2</span> Orthogonality of Modes</h2>
<p>A property known as orthogonality of the mode shapes leads to a very well-known and extremely useful analyses tool for MDOF systems. To demonstrate this property, let us assume we have an <em>n</em>-DOF system with symmetric <span class="math inline">\(\mmat\)</span> and <span class="math inline">\(\kmat\)</span>, with frequencies <span class="math inline">\(\freq_i\)</span> and mode shapes <span class="math inline">\(\eigveci{i}\)</span> for <span class="math inline">\(i=1,2,\ldots,n\)</span>. For ease of proof let us further consider the case with no repeated roots so that <span class="math inline">\(\freq_i \neq \freq_j\)</span> for <span class="math inline">\(i \neq j\)</span>; in structural dynamics this is almost always the case except when there is perfect symmetry so that the structure has identical periods in two directions but even in such cases the following conclusions will hold.</p>
<p>For such a system, the eigenvalue problem for two different modes, modes <span class="math inline">\(i\)</span> and <span class="math inline">\(j\)</span>, will be given by the following equations: <span id="eq-eigmodei"><span class="math display">\[
\kmat\eigveci{i} = \freq_i^2 \mmat \eigveci{i}
\qquad(7.9)\]</span></span> <span id="eq-eigmodej"><span class="math display">\[\kmat\eigveci{j} = \freq_j^2 \mmat \eigveci{j}
\qquad(7.10)\]</span></span> If we pre-multiply <a href="#eq-eigmodei">Equation&nbsp;<span>7.9</span></a> by <span class="math inline">\(\eigveci{j}^T\)</span> and <a href="#eq-eigmodej">Equation&nbsp;<span>7.10</span></a> by <span class="math inline">\(\teigveci{i}\)</span>, we obtain: <span id="eq-eigmodeij"><span class="math display">\[
\eigveci{j}^T \kmat\eigveci{i} = \eigveci{j}^T \freq_i^2 \mmat \eigveci{i}
\qquad(7.11)\]</span></span> <span id="eq-eigmodeji"><span class="math display">\[
\teigveci{i} \kmat\eigveci{j} = \freq_j^2 \teigveci{i}  \mmat \eigveci{j}
\qquad(7.12)\]</span></span> Since the mass and stiffness matrices are symmetric, the transpose of <a href="#eq-eigmodeji">Equation&nbsp;<span>7.12</span></a> yields <span id="eq-eigmodejiT"><span class="math display">\[
(\teigveci{i} \kmat\eigveci{j})^T = \eigveci{j}^T \kmat^T\eigveci{i} = \eigveci{j}^T \kmat \eigveci{i} = \freq_j^2 \eigveci{j}^T  \mmat \eigveci{i}
\qquad(7.13)\]</span></span> so that the difference of <a href="#eq-eigmodeij">Equation&nbsp;<span>7.11</span></a> and <a href="#eq-eigmodejiT">Equation&nbsp;<span>7.13</span></a> leads to: <span id="eq-eigmodeijdif"><span class="math display">\[
0 = (\freq_i^2 - \freq_j^2) \eigveci{j}^T  \mmat \eigveci{i}
\qquad(7.14)\]</span></span> The equality in <a href="#eq-eigmodeijdif">Equation&nbsp;<span>7.14</span></a> demands that either <span class="math inline">\((\freq_i^2 - \freq_j^2)=0\)</span> or <span class="math inline">\(\eigveci{j}^T \mmat \eigveci{i}=0\)</span>; but if there are no repeated roots, <span class="math inline">\(\freq_i^2 - \freq_j^2 \neq 0\)</span> and so it must be true that <span id="eq-massort"><span class="math display">\[
\eigveci{j}^T  \mmat \eigveci{i} = 0
\qquad(7.15)\]</span></span> whenever <span class="math inline">\(i \neq j\)</span>. This condition also implies by the eigenvalue equation that <span id="eq-stiffort"><span class="math display">\[
\eigveci{j}^T  \kmat \eigveci{i} = 0 \quad \text{for } i\neq j.
\qquad(7.16)\]</span></span> When <span class="math inline">\(i=j\)</span>, on the other hand, these products may yield nonzero values. Let us denote those values by <span id="eq-modalmk"><span class="math display">\[
\teigveci{i}  \mmat \eigveci{i} = \modm_i, \quad \teigveci{i}  \kmat \eigveci{i} = \modk_i
\qquad(7.17)\]</span></span> where <span class="math inline">\(\modm_i\)</span> is called the modal mass and <span class="math inline">\(\modk_i\)</span> the modal stiffness for the <em>i</em>-th mode. The orthogonality conditions are generally expressed more concisely via the Kronecker-<span class="math inline">\(\delta\)</span> <span class="math inline">\(\delta_{ij}\)</span> defined as <span class="math display">\[
\delta_{ij} = \begin{cases}1 &amp; i=j \\ 0 &amp; i \neq j \end{cases}
\]</span> so that modal orthogonality may be simply written as follows: <span id="eq-ortfinal"><span class="math display">\[
\eigveci{j}^T  \mmat \eigveci{i} = \modm_i \delta_{ij}, \quad \eigveci{j}^T  \kmat \eigveci{i} = \modk_i \delta_{ij}
\qquad(7.18)\]</span></span> It is important to note that these relations imply a formulation to calculate the frequencies. Considering the eigenvalue equation for the <em>i</em>-th mode and premultiplying it with <span class="math inline">\(\teigveci{i}\)</span> we have <span id="eq-ortfreq"><span class="math display">\[
\teigveci{i} \kmat\eigveci{i} = \freq_i^2 \teigveci{i} \mmat \eigveci{i} \rightarrow \freq_i^2 = \ratio{\teigveci{i} \kmat\eigveci{i}}{\teigveci{i} \mmat\eigveci{i}} = \ratio{\modk_i}{\modm_i}
\qquad(7.19)\]</span></span> The particular values of the modal masses and stiffnesses will naturally depend on the particular scaling chosen for the mode shapes. Consider for example the two story shear building of <a href="#fig-dof2shearbldg">Figure&nbsp;<span>7.2</span></a> discussed previously, with the mass and stiffness matrices given by <span class="math display">\[
\mmat = \begin{bmatrix} m &amp; 0 \\ 0 &amp; m \end{bmatrix}, \quad \kmat = \begin{bmatrix} 3k &amp; -k \\ -k &amp; k \end{bmatrix}
\]</span> One option that was considered was to scale the mode shapes so that the amplitude corresponding to the first generalized coordinate was of unit magnitude, leading to <span class="math display">\[
\eigveci{1}=\colmat{1 \\ 2.414}, \quad \eigveci{2}=\colmat{1 \\ -0.414}
\]</span> With these mode shapes, <a href="#eq-ortfinal">Equation&nbsp;<span>7.18</span></a> yields <span class="math display">\[
\modm_1 = \eigveci{1}^T \mmat \eigveci{1} = 6.83 m, \quad \modm_2 = \eigveci{2}^T \mmat \eigveci{2} = 1.17 m, \quad \eigveci{1}^T \mmat \eigveci{2} = 0
\]</span> and <span class="math display">\[
\modk_1 = \eigveci{1}^T \kmat \eigveci{1} = 4 k, \quad \modk_2 = \eigveci{2}^T \kmat \eigveci{2} = 4k, \quad \eigveci{1}^T \kmat \eigveci{2} = 0
\]</span> One way to check the algebra is to make sure that the modal masses and stiffness lead to the calculated frequencies: <span class="math display">\[
\ratio{\modk_1}{\modm_1} = 0.586 \frac{k}{m} = \freq_1^2, \quad \ratio{\modk_2}{\modm_2} = 3.41 \frac{k}{m} = \freq_2^2
\]</span> An alternative scaling for the first mode shape was also considered subject to <span class="math inline">\(\eigveci{1}^T\eigveci{1} = 1\)</span> in which case the mode shapes were given by <span class="math display">\[
\eigveci{1}=\colmat{0.383 \\ 0.924}, \quad \eigveci{2}=\colmat{1 \\ -0.414}
\]</span> so that with these mode shapes, the orthogonality conditions would in this case yield <span class="math display">\[
\modm_1 = \eigveci{1}^T \mmat \eigveci{1} = m, \quad \modm_2 = \eigveci{2}^T \mmat \eigveci{2} = 1.17 m, \quad \eigveci{1}^T \mmat \eigveci{2} = 0
\]</span> and <span class="math display">\[
\modk_1 = \eigveci{1}^T \kmat \eigveci{1} = 0.586 k, \quad \modk_2 = \eigveci{2}^T \kmat \eigveci{2} = 4k, \quad \eigveci{1}^T \kmat \eigveci{2} = 0
\]</span> and once again we have <span class="math display">\[
\ratio{\modk_1}{\modm_1} = 0.586 \frac{k}{m} = \freq_1^2, \quad \ratio{\modk_2}{\modm_2} = 3.41 \frac{k}{m} = \freq_2^2
\]</span> as we should. If the mode shapes are scaled so that the modal masses are equal to <span class="math inline">\(1\)</span>, then the mode shapes are said to be <em>mass normalized</em>. Recall that scaling of a mode shape is essentially multiplying it with a non-zero scalar. If <span class="math inline">\(\eigveci{i}\)</span> is the arbitrarily scaled mode shape for the <span class="math inline">\(i\)</span>-th mode, the mass normalized mode shape <span class="math inline">\(\meigveci{i}\)</span> for that mode is related to <span class="math inline">\(\eigveci{i}\)</span> through <span class="math inline">\(\meigveci{i}=s_i \eigveci{i}\)</span> where <span class="math inline">\(s_i\)</span> is some scalar. To find out what it should be, we can use the condition that mass normalized mode shapes should lead to unit modal masses, so that <span id="eq-massnormcoef"><span class="math display">\[
\meigveci{i}^T \mmat \meigveci{i} = 1 = s_i^2 \teigveci{i} \mmat  \eigveci{i} = s_i^2 \modm_i \quad \rightarrow \quad s_i = \ratio{1}{\sqrt{\modm_i}}
\qquad(7.20)\]</span></span> Our aim is to introduce the terminology of mass normalized mode shapes and to show how they can be constructed. From now on we will refrain from introducing a special symbol for mass normalized mode shapes in an effort to curb symbolic overcrowding and simply state, if necessary, whether a particular mode shape is mass normalized.<br>
Finally, we should note that the eigenvalue problem may be collectively stated for all the modes with a relatively simple matrix notation. Collecting the eigenvalue equation for each mode in consecutive columns we get <span class="math display">\[
\left[ \kmat \eigveci{1} \; \kmat \eigveci{2} \; \cdots \; \kmat \eigveci{n} \right] = \left[ \mmat \eigveci{1} \freq_1^2 \; \kmat \eigveci{2} \freq_2^2 \; \cdots \; \kmat \eigveci{n}\freq_n^2 \right]
\]</span> which may also be written as <span class="math display">\[
\kmat \left[ \eigveci{1} \; \eigveci{2} \; \cdots \; \eigveci{n}\right] = \mmat \left[ \eigveci{1} \; \eigveci{2} \; \cdots \; \eigveci{n}\right] \begin{bmatrix} \freq_1^2 &amp; 0 &amp; 0 &amp; \cdots &amp; 0 \\ 0 &amp; \freq_2^2 &amp; 0 &amp; \cdots &amp; 0 \\ \vdots &amp; \vdots &amp; \ddots &amp; \cdots &amp; \vdots \\ 0 &amp; 0 &amp; \cdots &amp; \freq_{n-1}^2 &amp; 0\\ 0 &amp; 0 &amp; 0 &amp; \cdots &amp; \freq_n^2\end{bmatrix}
\]</span> Defining the <em>mode shape matrix</em> (or the <em>modal matrix</em> as it is referred to in linear algebra) <span class="math inline">\(\modmat\)</span> as <span class="math display">\[
\modmat = \left[ \eigveci{1} \; \eigveci{2} \; \cdots \; \eigveci{n}\right]
\]</span> and the diagonal <em>spectral matrix</em> <span class="math inline">\(\spectmat\)</span> as <span class="math display">\[
\spectmat = \begin{bmatrix} \freq_1^2 &amp; 0 &amp; 0 &amp; \cdots &amp; 0 \\ 0 &amp; \freq_2^2 &amp; 0 &amp; \cdots &amp; 0 \\ \vdots &amp; \vdots &amp; \ddots &amp; \cdots &amp; \vdots \\ 0 &amp; 0 &amp; \cdots &amp; \freq_{n-1}^2 &amp; 0\\ 0 &amp; 0 &amp; \cdots &amp; 0 &amp; \freq_n^2\end{bmatrix}
\]</span> the eigenvalue problem may be collectively expressed as <span id="eq-spectral"><span class="math display">\[
\kmat \modmat = \mmat \modmat \spectmat
\qquad(7.21)\]</span></span> Furthermore, the orthogonality conditions may now be expressed as <span class="math display">\[
\modmat^T \mmat \modmat = \begin{bmatrix} \modm_1 &amp; 0 &amp; 0 &amp; \cdots &amp; 0 \\ 0 &amp; \modm_2 &amp; 0 &amp; \cdots &amp; 0 \\ \vdots &amp; \vdots &amp; \ddots &amp; \cdots &amp; \vdots \\ 0 &amp; 0 &amp; \cdots &amp; \modm_{n-1} &amp; 0\\ 0 &amp; 0 &amp; 0 &amp; \cdots &amp; \modm_n\end{bmatrix} = \modmmat, \quad \modmat^T \kmat \modmat = \begin{bmatrix} \modk_1 &amp; 0 &amp; 0 &amp; \cdots &amp; 0 \\ 0 &amp; \modk_2 &amp; 0 &amp; \cdots &amp; 0 \\ \vdots &amp; \vdots &amp; \ddots &amp; \cdots &amp; \vdots \\ 0 &amp; 0 &amp; \cdots &amp; \modk_{n-1} &amp; 0\\ 0 &amp; 0 &amp; 0 &amp; \cdots &amp; \modk_n\end{bmatrix} = \modkmat
\]</span> where the diagonal matrices <span class="math inline">\(\modmmat\)</span> and <span class="math inline">\(\modkmat\)</span> are called the <em>modal mass matrix</em> and the <em>modal stiffness matrix</em>, respectively.</p>
</section>
<section id="coordinate-transformations-and-the-eigenvalue-problem" class="level2" data-number="7.3">
<h2 data-number="7.3" data-anchor-id="coordinate-transformations-and-the-eigenvalue-problem"><span class="header-section-number">7.3</span> Coordinate Transformations and the Eigenvalue Problem</h2>
<p>Do coordinate transformations affect frequencies and mode shapes? The answer is probably relatively easy to reason: frequencies characterize how fast the harmonic motions of the whole system take place and therefore they should not depend on which coordinates are used to describe the motion; mode shapes, on the other hand, depict what the system looks like so that different coordinates may require different coefficients to specify the shape of the system. How shall we approach this issue mathematically? Consider two sets of generalized coordinates, <span class="math inline">\(\gcvec\)</span> and <span class="math inline">\(\gcvec'\)</span>, both of which may be used to define the dynamics of a system. Assume these two coordinate sets are related through <span class="math display">\[
\gcvec = \mtrx{T} \gcvec'
\]</span> with the eigenvalue problem in <span class="math inline">\(\gcvec\)</span> coordinates given by: <span class="math display">\[
(\kmat -\freq^2 \mmat) \eigvec = \zerocol
\]</span> Remembering that <span class="math inline">\(\matabs{\mtrx{A}\mtrx{B}\mtrx{C}}=\matabs{\mtrx{A}}\matabs{\mtrx{B}}\matabs{\mtrx{C}}\)</span>, we note that the characteristic equation <span class="math display">\[
\matabs{\kmat -\freq^2 \mmat} = 0
\]</span> will have the same roots as <span class="math display">\[
\matabs{\mtrx{T}^{-T}\mtrx{T}^{T}(\kmat -\freq^2 \mmat)\mtrx{T}\mtrx{T}^{-1}}=0
\]</span> since <span class="math inline">\(\mtrx{T}\mtrx{T}^{-1}=\mtrx{T}^{-T}\mtrx{T}^{T}=\idmat\)</span> where <span class="math inline">\(\idmat\)</span> is the identity matrix (the determinant of which is equal to 1) and <span class="math inline">\(\mtrx{T}^{-T}\)</span> is the inverse of <span class="math inline">\(\mtrx{T}^{T}\)</span>. Proceeding with this line of inquiry, we have <span class="math display">\[
\mtrx{T}^{-T} (\mtrx{T}^{T}\kmat\mtrx{T} -\freq^2 \mtrx{T}^{T}\mmat\mtrx{T}) \mtrx{T}^{-1}\eigvec = \mtrx{T}^{-T} (\barkmat -\freq^2 \barmmat) \mtrx{T}^{-1}\eigvec = \zerocol
\]</span> which implies <span id="eq-coordtranseig1"><span class="math display">\[
\mtrx{T}^{-T} (\barkmat -\freq^2 \barmmat) \mtrx{T}^{-1}\eigvec = \zerocol \quad \rightarrow \quad (\barkmat -\freq^2 \barmmat) \mtrx{T}^{-1}\eigvec = \zerocol
\qquad(7.22)\]</span></span> since <span class="math inline">\(\mtrx{T}^{-T} \neq \zeromat\)</span>. Noting that the eigenvalue problem in the transformed coordinates would be given by, <span id="eq-coordtranseig2"><span class="math display">\[
(\barkmat -(\freq')^2 \barmmat) \eigvec' = \zerocol
\qquad(7.23)\]</span></span> we conclude, by comparing <a href="#eq-coordtranseig1">Equation&nbsp;<span>7.22</span></a> and <a href="#eq-coordtranseig2">Equation&nbsp;<span>7.23</span></a>, that <span class="math display">\[
(\freq')^2 = \freq^2
\]</span> and hence the eigenvalues are unique; i.e., the values of the frequencies of a system are independent of the generalized coordinates employed in writing the equations of motion. It also follows by comparison that <span class="math display">\[
\eigvec' = \mtrx{T}^{-1}\eigvec
\]</span> so that in general the mode shapes in the transformed coordinates will be different than those in the original coordinate as we had foreseen.</p>
</section>
<section id="modal-analysis-of-free-vibrations" class="level2" data-number="7.4">
<h2 data-number="7.4" data-anchor-id="modal-analysis-of-free-vibrations"><span class="header-section-number">7.4</span> Modal Analysis of Free Vibrations</h2>
<p>As in the SDOF case, free vibrations in the context of MDOF systems refers to motion that takes place due to some initial perturbation from the system’s original equilibrium configuration, with no other external disturbance present. Let’s first consider an undamped system subjected to a set of initial displacements and velocities at time <span class="math inline">\(t=0\)</span>: <span class="math display">\[
\mmat \ddgctvec + \kmat \gctvec = \zerocol, \quad \{\gcvec (0) = \gcvecic, \dgcvec (0) = \dgcvecic\}
\]</span> where <span class="math display">\[
\gcvecic = \colmat{\gc_{1o} \\ \gc_{2o} \\ \vdots \\ \gc_{no}}, \quad \dgcvecic = \colmat{\dgc_{1o} \\ \dgc_{2o} \\ \vdots \\ \dgc_{no}}
\]</span> are the generalized coordinate and generalized velocity vectors (column matrices), respectively, at time <span class="math inline">\(t=0\)</span>; we will refer to these two collectively as the initial conditions for the MDOF system. Since this matrix equation is a coupled set of equations, coming up with solutions simultaneously for all generalized coordinates directly is not an easy feat and in most cases simply not feasible. One way to approach this problem is to inquire if there is any way we can decouple the <span class="math inline">\(n\)</span> equations of motion describing our system. Let’s imagine an alternative set of coordinates, <span class="math display">\[
\modcorvec (t) = \begin{bmatrix} \modcor_1(t) \\ \modcor_2(t) \\ \vdots \\ \modcor_N(t) \end{bmatrix}
\]</span> such that if the equations were to be written in these coordinates, the mass and stiffness matrices would all become perfectly diagonal. In that case, the first equation would only involve terms with <span class="math inline">\(\modcor_1\)</span> and <span class="math inline">\(\ddmodcor_1\)</span>, the second equation with <span class="math inline">\(\modcor_2\)</span> and <span class="math inline">\(\ddmodcor_2\)</span>, and so forth. If this were possible, one would essentially have <span class="math inline">\(n\)</span>-many uncoupled single degree of freedom systems, each of which would execute simple harmonic motions due to their initial conditions. We have implicitly solved this problem when we derived the eigenvalue problem. Recall that we had asked if it were possible for an MDOF system to execute simple harmonic motion, and the answer we got was that the system had certain frequencies and mode shapes with which it indeed could execute such motion. Furthermore we also showed that these mode shapes were orthogonal in that <span class="math display">\[
\modmat^T \mmat \modmat = \modmmat, \quad \modmat^T \kmat \modmat = \modkmat
\]</span> where <span class="math inline">\(\modmmat\)</span> and <span class="math inline">\(\modkmat\)</span> were diagonal matrices! If we recall the coordinate transformation rules given by <a href="multidof.html#eq-coordtrans4">Equation&nbsp;<span>6.20</span></a>, we may guess where this discussion is going. Let us define a set of coordinates <span class="math inline">\(\modcorvec (t)\)</span> such that <span class="math display">\[
\gctvec = \modmat \modcorvec (t)
\]</span> The equations of motion when transformed to these coordinates would yield <span class="math display">\[
\modmat^T \mmat \modmat \ddmodcorvec (t) + \modmat^T \kmat \modmat \modcorvec (t) = \modmmat \ddmodcorvec (t) + \modkmat \modcorvec (t) = \zerocol, \quad \{\modcorvec (0) = \modcorvecic, \dmodcorvec (0) = \dmodcorvecic\}
\]</span> where <span class="math display">\[
\modcorvecic = \colmat{\modcor_{1o} \\ \modcor_{2o} \\ \vdots \\ \modcor_{no}}, \quad \dmodcorvecic = \colmat{\dmodcor_{1o} \\ \dmodcor_{2o} \\ \vdots \\ \dmodcor_{no}},
\]</span> are the initial conditions in the <span class="math inline">\(\modcorvec\)</span> coordinates. These will have to be calculated via the initial conditions in the <span class="math inline">\(\gcvec\)</span> coordinates: since the two sets of coordinates are related through the same transformation at all times, we have <span class="math display">\[
\gcvecic = \modmat \modcorvecic \; \rightarrow \; \modcorvecic = \modmat^{-1} \gcvecic
\]</span> and <span class="math display">\[
\dgcvecic = \modmat \dmodcorvecic \; \rightarrow \; \dmodcorvecic = \modmat^{-1} \dgcvecic
\]</span> Calculating the initial conditions via matrix inversion is costly, especially if the dimensions are large; furthermore, it may be that we are interested in only a handful of coordinates <span class="math inline">\(\modcor_i\)</span> (for reasons that will be soon discussed), for which evaluating the whole set by inverting the mode shape matrix is too inefficient. Instead we may proceed as follows: the orthogonality of the mode shapes lead to <span class="math display">\[\begin{align*}
\teigveci{i} \mmat \gcvecic &amp; = \teigveci{i} \mmat \modmat \modcorvecic \\
&amp; = \colmat{\teigveci{i} \mmat \eigveci{1}^T \;\; \teigveci{i} \mmat \eigveci{2} \;\; \cdots \;\; \teigveci{i} \mmat \teigveci{i} \;\; \cdots \;\; \teigveci{i} \mmat \eigveci{n}} \modcorvecic \\
&amp; = \colmat{0 \;\; 0 \;\; \cdots \;\; \modm_{i} \;\; \cdots \;\; 0} \modcorvecic\\
\teigveci{i} \mmat \gcvecic &amp; = \modm_i \modcor_{io}
\end{align*}\]</span> so that <span id="eq-modic1"><span class="math display">\[
\modcor_{io} = \ratio{\teigveci{i} \mmat \gcvecic}{\modm_i}
\qquad(7.24)\]</span></span> and similarly, <span id="eq-modic2"><span class="math display">\[
\dmodcor_{io} = \ratio{\teigveci{i} \mmat \dgcvecic}{\modm_i}
\qquad(7.25)\]</span></span> Both these equations may be collectively written for all modal coordinates via matrix notation as <span id="eq-modicmat"><span class="math display">\[
\modcorvecic = \modmmat^{-1} \modmat^{T} \mmat \gcvecic, \quad \dmodcorvecic = \modmmat^{-1} \modmat^{T} \mmat \dgcvecic
\qquad(7.26)\]</span></span> This looks like a lot of work compared to a one-step matrix inversion but we must remember that with increasing dimensions, matrix multiplication becomes much less costly compared to matrix inversions (since the modal mass matrix is diagonal, its inversion is trivial). Keeping in mind that the mass matrix is frequently diagonal, the approach summarized above, which also allows for picking specific subsets, turns out to be quite an efficient way to calculate the initial conditions in the transformed coordinates.</p>
<p>In any case, this is the brief summary of what happened so far: we aim to solve for the free vibration response of an MDOF system, governed by <span id="eq-eomudfree"><span class="math display">\[
\mmat \ddgctvec + \kmat \gctvec = \zerocol \, ; \quad \{\gcvec (0) = \gcvecic, \dgcvec (0) = \dgcvecic\}
\qquad(7.27)\]</span></span> and we attempt the solution via the transformation <span id="eq-modtrans"><span class="math display">\[
\gcvec (t) = \modmat \modcorvec (t)
\qquad(7.28)\]</span></span> with the governing equations in the new coordinates given by <span id="eq-modeomud"><span class="math display">\[
\modmmat \ddmodcorvec (t) + \modkmat \modcorvec (t) = \zerocol \, ; \quad \{\modcorvec (0) = \modcorvecic, \dmodcorvec (0) = \dmodcorvecic\}
\qquad(7.29)\]</span></span> It may be better to introduce some terminology to reduce the burden of keeping track of which coordinates are which. When we initially write the equations of motion, the generalized coordinates that we employ almost always have a physical correspondence; for example, a generalized coordinate may be the displacement of a particular point or a rotation of a rigid body about a particular axis. Therefore we will call the initial set of generalized coordinates <span class="math inline">\(\gcvec\)</span> that we use to derive the governing equations of motion the <em>physical coordinates</em>. In contrast, the transformation of <a href="#eq-modtrans">Equation&nbsp;<span>7.28</span></a> relates these physical coordinates to a set of coordinates that have no direct physical correspondence; the <span class="math inline">\(\modcorvec\)</span> coordinates are essentially a set of abstract variables which help us uncouple the initial set of equations. We will refer to this set of coordinates which are related to the physical coordinates via the mode shape matrix the <em>modal coordinates</em>.</p>
<p>We already discussed that the mass and stiffness matrices in the modal coordinates are diagonal by virtue of mode shape orthogonality conditions. The set of equations of <a href="#eq-modeomud">Equation&nbsp;<span>7.29</span></a> are therefore <span class="math inline">\(n\)</span> uncoupled equations, each of the form: <span class="math display">\[
\modm_i \ddmodcor_i (t) + \modk_i \modcor_i (t) = 0 \, ; \quad \{\modcor_i (0) = \modcor_{io}, \dmodcor_i (0) = \dmodcor_{io} \}
\]</span> So by transforming to modal coordinates, we have converted our <span class="math inline">\(n\)</span>-DOF system to <span class="math inline">\(n\)</span> uncoupled SDOF systems! The good news is that we have solved the SDOF problem before: the free vibration response of an undamped SDOF system is given by <a href="singledof.html#eq-eomsdofsol4">Equation&nbsp;<span>2.11</span></a> or equivalently by <a href="singledof.html#eq-eomsdofsol5b">Equation&nbsp;<span>2.14</span></a>, which may be written for the <span class="math inline">\(i\)</span>-th modal coordinate as <span id="eq-modsolfreevib"><span class="math display">\[
\modcor_i (t) = \modcor_{io} \cosp{\freq_i t} + \frac{\dmodcor_{io}}{\freq_i} \sinp{\freq_i t} = \modcoramp_i \cosp{\freq_i t - \phs_i}
\qquad(7.30)\]</span></span> where <span id="eq-modsolfreevibampphs"><span class="math display">\[
\freq_i = \sqrt{\ratio{\modk_i}{\modm_i}}, \quad \modcoramp_i = \sqrt{\modcor_{io}^2 + \left(\frac{\dmodcor_{io}}{\freq_i}\right)^2}, \quad \tan \phs_i = \frac{\left(\dmodcor_{io}/{\freq_i}\right)/\modcoramp_i}{\modcor_{io}/\modcoramp_i}
\qquad(7.31)\]</span></span> We are not done yet. What we want is the response of the system in physical coordinates, i.e.&nbsp;the solution to the original set of equations of <a href="#eq-eomudfree">Equation&nbsp;<span>7.27</span></a>. Hence we must transform back to the physical coordinates via <span id="eq-modtophys"><span class="math display">\[
\begin{array}{rcl}
\gcvec (t) &amp; \!\!\! = &amp; \!\!\! \modmat \modcorvec (t) = \eigveci{1} \modcor_1 (t) + \eigveci{2} \modcor_2 (t) + \cdots + \eigveci{n} \modcor_n (t) \\
&amp; \!\!\! = &amp; \!\!\! \modcoramp_1 \eigveci{1} \cosp{\freq_1 t - \phs_1}+ \modcoramp_2 \eigveci{2} \cosp{\freq_2 t - \phs_2} + \cdots + \modcoramp_n \eigveci{n} \cosp{\freq_n t - \phs_n}
\end{array}
\qquad(7.32)\]</span></span> and we should note that the solution in <a href="#eq-modtophys">Equation&nbsp;<span>7.32</span></a> is essentially the same (except for the number of modes involved) as the one we obtained at first when we were seeking the possibility of harmonic motion, i.e. <a href="#eq-dof2modsol">Equation&nbsp;<span>7.8</span></a>.</p>
<p>The expansion <span class="math display">\[
\gcvec (t) = \modmat \modcorvec (t) =  \sum_{i=1}^{n} \eigveci{i} \modcor_{i} (t)
\]</span> is generally referred to as the <em>modal expansion</em>. This equation provides a particular interpretation of the response as being the superposition of some variables which are in some sense independent, this said independence being related to the orthogonality of the mode shapes. Consider, for example, what would happen if the system were to be set in motion via an initial displacement pattern that looked like one of the mode shapes, say the <span class="math inline">\(j\)</span>th mode shape; i.e.&nbsp;consider the following initial conditions: <span class="math display">\[
\gcvecic = \epsilon \eigveci{j}, \; \dgcvecic = \zerocol
\]</span> where <span class="math inline">\(\epsilon\)</span> is a non-zero real number. The initial conditions in modal coordinates are then given by <span class="math display">\[
\modcor_{io} = \ratio{\teigveci{i} \mmat \gcvecic}{\modm_i} = \ratio{\teigveci{i} \mmat \epsilon \eigveci{j}}{\modm_i} = \epsilon \delta_{ij}
\]</span> since <span class="math inline">\(\teigveci{i} \mmat \eigveci{j} = \modm_{i} \delta_{ij}\)</span> due to the orthogonality conditions. The initial velocities in modal coordinates, i.e.&nbsp;<span class="math inline">\(\modcorvecic\)</span>, are zero since all initial velocities in physical coordinates are zero. Therefore, when the initial displacements are a scaled version of the <span class="math inline">\(j\)</span>th mode shape, the only non-zero response comes from the <span class="math inline">\(j\)</span>th modal coordinate, and it is given by <span class="math display">\[
\modcor_{j} (t) = \epsilon \cosp{\freq_j t}
\]</span> with all other <span class="math inline">\(\modcor_{i} (t) \equiv 0\)</span> for <span class="math inline">\(i \neq j\)</span>. The response of the system is then given by <span class="math display">\[
\gcvec (t) = \sum_{i=1}^{n} \eigveci{i} \modcor_{i} (t) = \epsilon \eigveci{j} \cosp{\freq_j t}
\]</span> This result is important and it is worthwhile to state it again in plain language: if an MDOF system is set in motion by an initial perturbation that coincides with one of the mode shapes of the system, then the system will oscillate with only the frequency of that particular mode to which that mode shape belongs. No other mode will contribute to the response. It is yet speculative but somewhat foreseeable that if an initial displacement pattern does not fully comply with a particular mode shape but mostly resembles it, than the biggest contribution to the response may be expected to come from that particular mode, with smaller but possibly non-zero contributions from the others. This pattern we shall demonstrate through some examples.</p>
<section id="sec-mdofxo" class="level4 unnumbered">
<h4 class="unnumbered" data-anchor-id="sec-mdofxo">EXAMPLE 7 .1</h4>
<div id="fig-xdof2mkfree" class="quarto-figure quarto-figure-center">
<figure class="figure">
<p><img src="images/xdof2mkfree.png" class="img-fluid figure-img" style="width:60.0%"></p>
<p></p><figcaption class="figure-caption">Figure&nbsp;7.4: Two DOF system subject to free-vibration.</figcaption><p></p>
</figure>
</div>
<p>Consider a 2-DOF system with masses and spring constants defined as shown in <a href="#fig-xdof2mkfree">Figure&nbsp;<span>7.4</span></a>. We would like to calculate its free vibration response to initial conditions <span class="math display">\[
\gcvec (0) = \colmat{Q/2\\ Q} = Q \colmat{1/2\\ 1}, \; \dgcvec(0) = \colmat{0 \\ 0}
\]</span> We first have to derive the governing equations of motion. Say we use the Lagrangian approach to construct the mass and stiffness matrices. The kinetic and potential energies are given by <span class="math display">\[
\ke = \frac{1}{2} 2m \dgc_1^2 + \frac{1}{2} m \dgc_2^2, \quad \pe = \frac{1}{2} 2k \gc_1^2 + \frac{1}{2} k (\gc_2 - \gc_1)^2
\]</span> so that, by <a href="multidof.html#eq-lagrangianmij">Equation&nbsp;<span>6.8</span></a> and <a href="multidof.html#eq-lagrangiankij">Equation&nbsp;<span>6.10</span></a>, <span class="math display">\[
m_{11} = \ratio{\partial^2 \ke}{\partial \dgc_1 \partial \dgc_1} = 2m, \; m_{22} = \ratio{\partial^2 \ke}{\partial \dgc_2 \partial \dgc_2} = m, \, m_{12} = \ratio{\partial^2 \ke}{\partial \dgc_1 \partial \dgc_2} = 0 = m_{21}
\]</span> and <span class="math display">\[
k_{11} = \ratio{\partial^2 \pe}{\partial \gc_1 \partial \gc_1} = 3k, \; k_{22} = \ratio{\partial^2 \pe}{\partial \gc_2 \partial \gc_2} = k, \, k_{12} = \ratio{\partial^2 \pe}{\partial \gc_1 \partial \gc_2} = -k = k_{21}
\]</span> The equations governing the free vibration are therefore given by <span class="math display">\[
\begin{bmatrix} 2m &amp; 0 \\ 0 &amp; m \end{bmatrix} \colmat{\ddgc_1 \\ \ddgc_2} + \begin{bmatrix} 3k &amp; -k \\ -k &amp; k \end{bmatrix} \colmat{\gc_1 \\ \gc_2} = \colmat{0 \\ 0}
\]</span> Next, we find the eigenvalues by solving the characteristic equation that is obtained from <span class="math inline">\(\left|\kmat -\freq^2 \mmat \right|=0\)</span>. The characteristic equation is given by <span class="math display">\[
\left|\begin{bmatrix} 3k &amp; -k \\ -k &amp; k \end{bmatrix} -\freq^2 \begin{bmatrix} 2m &amp; 0 \\ 0 &amp; m \end{bmatrix}\right| = (3k-\freq^2 2m)(k-\freq^2 m)-k^2 = 2m(\freq^2)^2 - 5 k m \freq^2 + 2 k^2= 0
\]</span> or equivalently by <span class="math display">\[
(\freq^2)^2 - \frac{5 k}{2 m} \freq^2 + \left(\frac{k}{m}\right)^2 = 0
\]</span> and therefore the squares of the frequencies are calculated as <span class="math display">\[
\freq_1^2 = \ratio{1}{2}\ratio{k}{m}, \quad \freq_2^2 = 2 \ratio{k}{m}
\]</span> The mode shapes are computed by plugging back each eigenvalue into the equation one at a time and solving for the corresponding eigenvector. For the first one we have <span class="math display">\[
\left[\begin{bmatrix}3k &amp; -k \\ -k &amp; k \end{bmatrix} -\ratio{1}{2}\ratio{k}{m} \begin{bmatrix} 2m &amp; 0 \\ 0 &amp; m \end{bmatrix}\right]\colmat{\eigvecs_{11} \\ \eigvecs_{21}} = \colmat{0 \\ 0}
\]</span> leading to: <span class="math display">\[
\eigvecs_{21} = 2 \eigvecs_{11}
\]</span> If per choice we set <span class="math inline">\(\eigvecs_{11} = 1\)</span>, we get <span class="math display">\[
\eigveci{1} = \colmat{1 \\ 2}
\]</span> Similarly for the second mode shape we have <span class="math display">\[
\left[\begin{bmatrix}3k &amp; -k \\ -k &amp; k \end{bmatrix} -2\ratio{k}{m} \begin{bmatrix} 2m &amp; 0 \\ 0 &amp; m \end{bmatrix}\right]\colmat{\eigvecs_{12} \\ \eigvecs_{22}} = \colmat{0 \\ 0}
\]</span> so that <span class="math display">\[
\eigvecs_{22} = - \eigvecs_{12}
\]</span> and with <span class="math inline">\(\eigvecs_{12} = 1\)</span> we get <span class="math display">\[
\eigveci{2} = \colmat{1 \\ -1}
\]</span> The modal masses with these mode shapes are given by <span class="math display">\[
\modmmat = \modmat^T \mmat \modmat = \begin{bmatrix} 1 &amp; 2 \\ 1 &amp; -1 \end{bmatrix} \begin{bmatrix} 2m &amp; 0 \\ 0 &amp; m \end{bmatrix} \begin{bmatrix} 1 &amp; 1 \\ 2 &amp; -1 \end{bmatrix} = \begin{bmatrix} 6m &amp; 0 \\ 0 &amp; 3m \end{bmatrix}
\]</span> and the modal stiffnesses are given by <span class="math display">\[
\modkmat = \modmat^T \kmat \modmat = \begin{bmatrix} 1 &amp; 2 \\ 1 &amp; -1 \end{bmatrix} \begin{bmatrix} 3k &amp; -k \\ -k &amp; k \end{bmatrix} \begin{bmatrix} 1 &amp; 1 \\ 2 &amp; -1 \end{bmatrix} = \begin{bmatrix} 3k &amp; 0 \\ 0 &amp; 6k
\end{bmatrix}
\]</span> whence we can check <span class="math display">\[
\freq_1^2 = \ratio{\modk_1}{\modm_1}=\ratio{1}{2}\ratio{k}{m}, \quad \freq_2^2 = \ratio{\modk_2}{\modm_2}=2\ratio{k}{m}
\]</span> If instead we would like to work with mass normalized mode shapes, those we can obtain by scaling the mode shapes above according to <a href="#eq-massnormcoef">Equation&nbsp;<span>7.20</span></a> so that <span class="math display">\[
\meigveci{1}=\ratio{1}{\sqrt{6m}}\colmat{1 \\ 2}, \quad \meigveci{2}=\ratio{1}{\sqrt{3m}}\colmat{1 \\ -1}
\]</span> and the modal mass and stiffness matrices in this case would be given by <span class="math display">\[
\modmmat = \begin{bmatrix} 1 &amp; 0 \\ 0 &amp; 1 \end{bmatrix}, \quad \modkmat = \begin{bmatrix} \frac{1}{2}\frac{k}{m} &amp; 0 \\ 0 &amp; 2\frac{k}{m} \end{bmatrix}
\]</span> Since there is no particular advantage to use mass normalized mode shapes for this system, we will proceed with our original set of mode shapes.</p>
<p>To solve for the response in modal coordinates, we must find the initial conditions in those coordinates. By <a href="#eq-modic1">Equation&nbsp;<span>7.24</span></a> and <a href="#eq-modic2">Equation&nbsp;<span>7.25</span></a> we have, <span class="math display">\[
\modcor_{1o} = \frac{1}{\modm_1}\eigveci{1}^T \mmat \gcvecic = \frac{1}{6m} \left\{1 \;\;\; 2\right\} \begin{bmatrix} 2m &amp; 0 \\ 0 &amp; m \end{bmatrix} \colmat{Q/2 \\ Q} = \ratio{Q}{2}
\]</span> and <span class="math display">\[
\modcor_{2o} = 0, \quad \dmodcor_{1o} = \dmodcor_{2o} = 0
\]</span> which the reader may easily verify. This result should, of course, be expected as per our discussion preceding this example: the initial condition shape coincides with the first mode in that <span class="math inline">\(\gcvecic = (Q/2) \eigveci{1}\)</span>, and therefore it sets in motion only the corresponding first mode and does not induce any motion in the other mode. The amplitudes and the phase angles in modal coordinates are calculated via <a href="#eq-modsolfreevibampphs">Equation&nbsp;<span>7.31</span></a> to obtain <span class="math display">\[
\modcoramp_1 = \frac{Q}{2}, \quad \phs_1 = \arctan \ratio{0}{1} = 0, \quad \modcoramp_2 = 0
\]</span> so that <span class="math display">\[
\modcor_1 (t) = \ratio{Q}{2} \cosp{\freq_1 t}, \quad \modcor_2 (t) \equiv 0
\]</span> The response in physical coordinates is then given by <span class="math display">\[
\gcvec (t) = \colmat{\gc_1 (t) \\ \gc_2 (t)} = \eigveci{1} \modcor_1 (t) + \eigveci{2} \modcor_2 (t) = Q \colmat{1/2 \\ 1} \cosp{\freq_1 t} = \colmat{\frac{Q}{2} \cosp{\freq_1 t} \\ Q \cosp{\freq_1 t}}
\]</span> The response of the two masses are shown <a href="#fig-xdof2freevibic1">Figure&nbsp;<span>7.5</span></a>, plotted against dimensionless time <span class="math inline">\(\tau = t/\period_1 = (\freq_1 t)/(2 \pi)\)</span>. Due to the given initial conditions both masses move in phase, but the displacement of the first mass is half of that of the second mass.</p>
<div id="fig-xdof2freevibic1" class="quarto-figure quarto-figure-center">
<figure class="figure">
<p><img src="images/xdof2freevibic1.png" class="img-fluid figure-img" style="width:100.0%"></p>
<p></p><figcaption class="figure-caption">Figure&nbsp;7.5: Free vibration response for the system shown in <a href="#fig-xdof2mkfree">Figure&nbsp;<span>7.4</span></a>, subject to initial conditions <span class="math inline">\(\gcvecic = \left\{Q/2 \;\;\; Q\right\}^T, \;\; \dgcvecic = \zerocol\)</span>. The plots show time histories with respect to normalized time <span class="math inline">\(\tau = t/\period_1\)</span>.</figcaption><p></p>
</figure>
</div>
<p>What if the system was subjected to some other initial conditions? Consider first what would happen if the initial displacement pattern didn’t exactly match one of the modes but was given by <span class="math display">\[
\gcvecic = \colmat{2Q/3 \\ Q}, \quad \dgcvecic = \zerocol
\]</span> Note that the initial displacement pattern still resembles the first mode more than the second one. In this case the initial conditions in modal coordinates will be given by <span class="math display">\[\begin{align*}
\modcor_{1o} &amp; = \frac{1}{6m} \left\{1 \;\;\; 2\right\} \begin{bmatrix} 2m &amp; 0 \\ 0 &amp; m \end{bmatrix} \colmat{2Q/3 \\ Q} = \ratio{5}{9}Q \\
\modcor_{2o} &amp; = \frac{1}{3m} \left\{1 \;\;\; -1\right\} \begin{bmatrix} 2m &amp; 0 \\ 0 &amp; m \end{bmatrix} \colmat{2Q/3 \\ Q} = \ratio{1}{9}Q
\end{align*}\]</span> and <span class="math inline">\(\dmodcor_{1o} = \dmodcor_{2o} = 0\)</span>. The modal amplitudes and phase angles will be <span class="math display">\[
\modcoramp_1 = \frac{5}{9}Q, \quad \phs_1 = \arctan \ratio{0}{1} = 0, \quad \modcoramp_2 = \frac{1}{9}Q, \quad \phs_2 =\arctan \ratio{0}{1} = 0
\]</span> with the modal responses given by <span class="math display">\[
\modcor_1 (t) = \ratio{5}{9}Q \cosp{\freq_1 t}, \quad \modcor_2 (t) =\ratio{1}{9}Q \cosp{\freq_2 t}
\]</span> leading to the following displacement time histories for the two masses: <span class="math display">\[
\gcvec (t) = \colmat{\gc_1 (t) \\ \gc_2 (t)} = \eigveci{1} \modcor_1 (t) + \eigveci{2} \modcor_2 (t) = \colmat{\frac{5}{9}Q \cosp{\freq_1 t} + \frac{1}{9}Q \cosp{\freq_2 t} \\ \frac{10}{9}Q \cosp{\freq_1 t} - \frac{1}{9}Q \cosp{\freq_2 t}}
\]</span> In terms of normalized time <span class="math inline">\(\tau = t/\period_1 = (\freq_1 t)/(2 \pi)\)</span>, the responses are given by <span class="math display">\[
\colmat{\gc_1 (\tau) \\ \gc_2 (\tau)} = \colmat{\frac{5}{9}Q \cosp{2 \pi \tau} + \frac{1}{9}Q \cosp{4 \pi \tau} \\ \frac{10}{9}Q \cosp{2 \pi \tau} - \frac{1}{9}Q \cosp{4 \pi \tau}}
\]</span> wherein we have incorporated the information that for our system, <span class="math inline">\(\freq_2 = 2 \freq_1\)</span>. These displacement time histories plotted against normalized time <span class="math inline">\(\tau\)</span> are shown in <a href="#fig-xdof2freevibic2">Figure&nbsp;<span>7.6</span></a>. It is noteworthy that even though there is no exact match between the initial displacement pattern and the first mode, they are relatively similar and therefore most of the physical response is contributed by the first mode.</p>
<div id="fig-xdof2freevibic2" class="quarto-figure quarto-figure-center">
<figure class="figure">
<p><img src="images/xdof2freevibic2.png" class="img-fluid figure-img" style="width:100.0%"></p>
<p></p><figcaption class="figure-caption">Figure&nbsp;7.6: Free vibration response for the system shown in <a href="#fig-xdof2mkfree">Figure&nbsp;<span>7.4</span></a>, subject to initial conditions <span class="math inline">\(\gcvecic = \left\{2Q/3 \;\;\; Q\right\}^T, \dgcvecic = \zerocol\)</span>. The plots show time histories with respect to normalized time <span class="math inline">\(\tau = t/\period_1\)</span>.</figcaption><p></p>
</figure>
</div>
<p>Finally, consider what would happen if the system were now given a set of initial velocities, so that the initial conditions are given by <span class="math display">\[
\gcvecic = \zerocol, \; \dgcvecic = \colmat{v \\ -v} = v \colmat {1 \\ -1}
\]</span> In this case the initial conditions in modal coordinates are calculated to be <span class="math inline">\(\modcor_{1o} = \modcor_{2o}=0\)</span>, <span class="math display">\[\begin{align*}
\dmodcor_{1o} &amp; = \frac{1}{6m}\left\{1 \;\;\; 2\right\} \begin{bmatrix} 2m &amp; 0 \\ 0 &amp; m \end{bmatrix} \colmat{1 \\ -1} v = 0\\
\dmodcor_{2o} &amp; = \frac{1}{3m} \left\{1 \;\;\; -1\right\} \begin{bmatrix} 2m &amp; 0 \\ 0 &amp; m \end{bmatrix} \colmat{1 \\ -1}v = v
\end{align*}\]</span> so that modal amplitudes and phase angles obtained via <a href="#eq-modsolfreevibampphs">Equation&nbsp;<span>7.31</span></a> are <span class="math display">\[
\modcoramp_1 = 0, \quad \modcoramp_2 = \frac{v}{\freq_2}, \quad \phs_2 = \arctan \ratio{1}{0} = \frac{\pi}{2}
\]</span> leading to <span class="math display">\[
\modcor_1 (t) \equiv 0, \quad \modcor_2 (t) = \frac{v}{\freq_2} \cosp{\freq_2 t - \frac{\pi}{2}} = \frac{v}{\freq_2} \sinp{\freq_2 t}
\]</span> As the initial velocity distribution coincides with the second mode shape (i.e.&nbsp;since <span class="math inline">\(\modcorvecic = \zerocol\)</span> and <span class="math inline">\(\dmodcorvecic = v \eigveci{2}\)</span>), only the second mode gets excited and the first mode does not get excited at all. Accordingly, the displacements of the two masses will be given by <span class="math display">\[
\colmat{\gc_1 (t) \\ \gc_2 (t)} = \underbrace{\eigveci{1} \modcor_1 (t)}_{\equiv \zerocol} + \eigveci{2} \modcor_2 (t) = \colmat{\frac{v}{\freq_2} \sinp{\freq_2 t}\\ - \frac{v}{\freq_2} \sinp{\freq_2 t}}
\]</span> the plots of which are shown in <a href="#fig-xdof2freevibic3">Figure&nbsp;<span>7.7</span></a>. In this case the two masses start to move in opposite directions with the initial velocities imposed on them and oscillate with frequency <span class="math inline">\(\freq_2\)</span>, with no contribution form the first mode.</p>
<div id="fig-xdof2freevibic3" class="quarto-figure quarto-figure-center">
<figure class="figure">
<p><img src="images/xdof2freevibic3.png" class="img-fluid figure-img" style="width:100.0%"></p>
<p></p><figcaption class="figure-caption">Figure&nbsp;7.7: Free vibration response for the system shown in <a href="#fig-xdof2mkfree">Figure&nbsp;<span>7.4</span></a>, subject to initial conditions <span class="math inline">\(\gcvecic = \zerocol, \dgcvecic = \left\{v \;\;\; -v\right\}^T\)</span>. The plots show time histories with respect to normalized time <span class="math inline">\(\tau = t/\period_1\)</span>, with <span class="math inline">\(\period_2 = \period_1 / 2\)</span>.</figcaption><p></p>
</figure>
</div>
</section>
</section>
<section id="approximations-for-frequencies" class="level2" data-number="7.5">
<h2 data-number="7.5" data-anchor-id="approximations-for-frequencies"><span class="header-section-number">7.5</span> Approximations for Frequencies</h2>
<section id="rayleighs-quotient" class="level3" data-number="7.5.1">
<h3 data-number="7.5.1" data-anchor-id="rayleighs-quotient"><span class="header-section-number">7.5.1</span> Rayleigh’s Quotient</h3>
<p>Consider a particular mode of an MDOF system, with mode shape <span class="math inline">\(\eigveci{i}\)</span> and frequency <span class="math inline">\(\freq_i\)</span>. We have already seen that the following equation holds exactly true for undamped systems: <span id="eq-rayleighex"><span class="math display">\[
\freq_i^2 = \ratio{\modk}{\modm} = \ratio{\teigveci{i} \kmat \eigveci{i}}{\teigveci{i} \mmat \eigveci{i}}
\qquad(7.33)\]</span></span> Note that this ratio is also a statement about the equality of maximum kinetic and potential energies that the system attains as it oscillates harmonically in the <span class="math inline">\(i\)</span>-th mode. When the system oscillates as such, the generalized displacements and velocities are given by <span class="math display">\[
\gcvec (t) = A_i \eigveci{i} \cosp{\freq_i t - \phs_i}, \quad \dgcvec (t) = -\freq_i A_i \eigveci{i} \sinp{\freq_i t - \phs_i}
\]</span> so that the kinetic energy and potential energy at any instant are calculated as <span class="math display">\[\begin{align*}
\ke  &amp; = \frac{1}{2} \dgcvec^T \mmat \dgcvec = \frac{1}{2} A_i^2 \freq_i^2 \teigveci{i} \mmat \eigveci{i}  \sin^2\left(\freq_i t - \phs_i\right)\\
\pe  &amp; = \frac{1}{2} \gcvec^T \mmat \gcvec = \frac{1}{2} A_i^2 \teigveci{i} \kmat \eigveci{i}  \cos^2\left(\freq_i t - \phs_i\right)