AA242B: MECHANICAL VIBRATIONS
AA242B: MECHANICAL VIBRATIONS1 / 41AA242B: MECHANICAL VIBRATIONSDirect Time-Integration MethodsThese slides are based on the recommended textbook: M. Géradin and D. Rixen, “MechanicalVibrations: Theory and Applications to Structural Dynamics,” Second Edition, Wiley, John &Sons, Incorporated, ISBN-13:97804719754651 / 41
AA242B: MECHANICAL VIBRATIONS2 / 41Outline1 Stability and Accuracy of Time-Integration Operators2 Newmark’s Family of Methods3 Explicit Time Integration Using the Central Difference Algorithm2 / 41
AA242B: MECHANICAL VIBRATIONS3 / 41Stability and Accuracy of Time-Integration OperatorsMultistep Time-Integration MethodsLagrange’s equations of dynamic equilibrium (p(t) 0)Mq̈ Cq̇ Kq 0q(0) q0q̇(0) q̇0(1)First-order form q̈ M 0q̇00 M 0Kq0q̇M C{z} {z } {z} {z } {z } ABu̇ u̇ Au AAwhereu0A A 1B AADirect time-integration3 / 41
AA242B: MECHANICAL VIBRATIONS4 / 41Stability and Accuracy of Time-Integration OperatorsMultistep Time-Integration MethodsGeneral multistep time-integration method for first-order systems ofthe form u̇ AummXXun 1 αj un 1 j hβj u̇n 1 jj 1j 0where h tn 1 tn is the computational time-step, un u(tn ), and q̇n 1un 1 qn 1is the state-vector calculated at tn 1 from the m preceding statevectors and their derivatives as well as the derivative of thestate-vector at tn 1β0 6 0 leads to an implicit scheme — that is, a scheme where theevaluation of un 1 requires the solution of a system of equationsβ0 0 corresponds to an explicit scheme — that is, a scheme wherethe evaluation of un 1 does not require the solution of any system ofequations and instead can be deduced directly from the results at theprevious time-steps4 / 41
AA242B: MECHANICAL VIBRATIONS5 / 41Stability and Accuracy of Time-Integration OperatorsMultistep Time-Integration MethodsGeneral multistep integration method for first-order systems(continue)mmXXαj un 1 j hβj u̇n 1 jun 1 j 1j 0trapezoidal rule (implicit)un 1 un hhh(u̇n u̇n 1 ) ( A I)un 1 un u̇n222backward Euler formula (implicit)un 1 un hu̇n 1 (hA I)un 1 unforward Euler formula (explicit)un 1 un hu̇n un 1 (I hA)un5 / 41
AA242B: MECHANICAL VIBRATIONS6 / 41Stability and Accuracy of Time-Integration OperatorsNumerical Example: the One-Degree-of-Freedom OscillatorConsider an undamped one-degree-of-freedom oscillatorq̈ ω02 q 0with ω0 π rad/s and the initial displacementq(0) 1, q̇(0) 0exact solutionq(t) cos ω0 tassociated first-order systemu̇ Auwhere A ω02001 u [q̇, q]T , and initial condition u(0) 01 6 / 41
AA242B: MECHANICAL VIBRATIONS7 / 41Stability and Accuracy of Time-Integration OperatorsNumerical Example: the One-Degree-of-Freedom OscillatorNumerical solutionTT 3s, h 323Exact solutionTrapezoidal ruleEuler backwardEuler forward21q0 1 2 3 400.511.5t22.537 / 41
AA242B: MECHANICAL VIBRATIONS8 / 41Stability and Accuracy of Time-Integration OperatorsStability Behavior of Numerical SolutionsAnalysis of the characteristic equation of a time-integration methodconsider the first-order system u̇ Aufor this problem, the general multistep method can be written asun 1 mXαj un 1 j hj 1mXβj u̇n 1 j j 0mX[αj I hβj A] un 1 j 0,α0 1j 0let {µr }rr n 1 be the eigenvalues of A and X be the matrix of associated eigenvectors X 1 AX diag(µ1 , · · · , µr , · · · , µn )the characteristic equation associated withmP[αj I hβj A] un 1 j 0isj 0obtained by searching for a solution of the formun 1 m Xa(decomposition on an eigen basis)u(n 1 m) 1 λun 1 m λXa λun · · · λk un 1 k · · · λm Xa(solution form).un 1where λ C is called the solution amplification factor8 / 41
AA242B: MECHANICAL VIBRATIONS9 / 41Stability and Accuracy of Time-Integration OperatorsStability Behavior of Numerical SolutionsAnalysis of the characteristic equation of a time-integration method(continue)hencemX[αj I hβj A] λm j Xa 0j 0since X 1 AX diag(µ1 , · · · , µr , · · · , µn ), premultiplying the aboveresult by X 1 leads to" m#Xm j[αj I hβj diag(µ1 , · · · , µr , · · · , µn )] λa 0j 0 mX[αj hβj µr ] λm j 0, r 1, 2, · · · , nj 0hence, the numerical response un 1 λm Xa remains bounded if eachsolution of the above characteristic equation of degree m satisfies λk 1, k 1, · · · , m9 / 41
AA242B: MECHANICAL VIBRATIONS10 / 41Stability and Accuracy of Time-Integration OperatorsStability Behavior of Numerical SolutionsAnalysis of the characteristic equation of a time-integration method(continue)the stability limit is a circle of unit radiusin the complex plane of µr h, the stability limit is therefore given bywriting λ e iθ , 0 θ 2πmX µr h j 0mXαj e i(m j)θβj e i(m j)θj 0one-step schemes (m 1)µr h α0 e iθ α1 e iθ α1 β0 e iθ β1β0 e iθ β110 / 41
AA242B: MECHANICAL VIBRATIONS11 / 41Stability and Accuracy of Time-Integration OperatorsStability Behavior of Numerical SolutionsAnalysis of the characteristic equation of a time-integration method(continue)one-step schemes (m 1) (continue)µr h α0 e iθ α1 e iθ α1 iθβ0 e β1β0 e iθ β1forward Euler: α1 1, β0 0, β1 1 µr h e iθ 1the solution is unstable in the entire plane except inside the circle ofunit radius and center 1backward Euler: α1 1, β0 1, β1 0 µr h 1 e iθthe solution is stable in the entire plane except inside the circle ofunit radius and center 1112i sin θtrapezoidal rule: α1 1, β0 , β1 µr h 221 cos θthe solution is stable in the entire left-hand plane11 / 41
AA242B: MECHANICAL VIBRATIONS12 / 41Stability and Accuracy of Time-Integration OperatorsStability Behavior of Numerical SolutionsAnalysis of the characteristic equation of a time-integration method(continue)application to the single degree-of-freedom oscillator 0 ω02q̈ ω02 q 0,A 10the eigenvalues are µr iω0the roots µr h are located in the unstable region of the forward Eulerscheme amplification of the numerical solutionthe roots µr h are located in the stable region of the backward Eulerscheme decay of the numerical solutionthe roots µr h are located on the stable boundary of the trapezoidalrule scheme the amplitude of the oscillations is preserved12 / 41
AA242B: MECHANICAL VIBRATIONS13 / 41Newmark’s Family of MethodsThe Newmark MethodTaylor’s expansion of a function f0f (tn h) f (tn ) hf (tn ) h2 00hs (s)1f (tn ) · · · f (tn ) 2s!s!Ztn hf(s 1)s(τ )(tn h τ ) dτtnApplication to the velocities and displacementsZf q̇, s 0 q̇n 1 q̇n tn 1q̈(τ )dτZ tn 1qn h q̇n q̈(τ )(tn 1 τ )dτtnf q, s 1 qn 1 (2)tnGiven ) of q̈(τ ) in the time-interval [tn , tn 1 ] andany approximation q̈(τanypairofquadratureZ tn 1Z tn 1rules for approximating the resulting integrals )dτ and )(tn 1 τ )dτq̈(τq̈(τtntnorpair of directof the time-integralsZ tanyZ approximationstn 1n 1q̈(τ )dτ andq̈(τ )(tn 1 τ )dτtntn(9) leads to a numerical time-integration scheme for solving (1)13 / 41
AA242B: MECHANICAL VIBRATIONS14 / 41Newmark’s Family of MethodsThe Newmark MethodTaylor expansions of q̈n and q̈n 1 around τ [tn , tn 1 ]q̈n q̈n 1 (tn τ )2 ···(3)2(tn 1 τ )2q̈(τ ) q(3) (τ )(tn 1 τ ) q(4) (τ ) · · · (4)2q̈(τ ) q(3) (τ )(tn τ ) q(4) (τ )Combine (1 γ) (3) γ (4) and extract q̈(τ ) q̈(τ ) (1 γ)q̈n γq̈n 1 q(3) (τ )(τ hγ tn ) O(h2 q(4) )Combine (1 2β) (3) 2β (4) and extract q̈(τ ) q̈(τ ) (1 2β)q̈n 2βq̈n 1 q(3) (τ )(τ 2hβ tn ) O(h2 q(4) )14 / 41
AA242B: MECHANICAL VIBRATIONS15 / 41Newmark’s Family of MethodsThe Newmark Methodtn 1ZstSubstitute the 1 expression of q̈(τ ) inq̈(τ )dτtnZ tn 1tn 1Zq̈(τ )dτ tn (1 γ)q̈n γq̈n 1 q(τ )(τ hγ tn ) O(h q ) dτ(3)3 (4)(3)2 (4)tnZ (1 γ)h q̈n γh q̈n 1 tn 1q(τ )(τ hγ tn )dτ O(h q)tnApply the mean value theoremZ tn 1"q̈(τ )dτ (1 γ)h q̈n γh q̈n 1 q(3)(τ̃ )tn (1 γ)h q̈n γh q̈n 1 (Substitute the 2nd expression of q̈(τ ) in(τ hγ tn )22#tn 13 (4) O(h q)tn12 (3)3 (4) γ)h q (τ̃ ) O(h q )2Ztn 1q̈(τ )(tn 1 τ )dτtnZ tn 1tnq̈(τ )(tn 1 τ )dτ (11223 (3)4 (4) β)h q̈n βh q̈n 1 ( β)h q (τ̃ ) O(h q )2615 / 41
AA242B: MECHANICAL VIBRATIONS16 / 41Newmark’s Family of MethodsThe Newmark MethodIn summary γ and β, the following holds truetn 1Zq̈(τ )dτ (1 γ)h q̈n γh q̈n 1 rntnZtn 1 q̈(τ )(tn 1 τ )dτ tn1 β2 202h q̈n βh q̈n 1 rnwhere rn 1 γ2 2 (3)h q3 (4)(τ̃ ) O(h q);0 rn 1 β6 3 (3)h q4 (4)(τ̃ ) O(h q)and tn τ̃ tn 1neglecting each of rn and rn0 on the ground that they are higher-orderfunctions of the time-step h leads to the following family oftime-integration schemes (Newmark’s family) for solving (1)q̇n 1 qn 1 q̇n (1 γ)h q̈n γh q̈n 1 1qn h q̇n h2 β q̈n h2 βq̈n 12(5)(6)where γ and β are quadrature parameters16 / 41
AA242B: MECHANICAL VIBRATIONS17 / 41Newmark’s Family of MethodsThe Newmark MethodParticular values of the parameters γ and β11and β corresponds to linearly interpolating q̈(τ ) in26[tn , tn 1 ] q̈n 1 q̈n q̈ln (τ ) q̈n (τ tn )h11γ and β corresponds to averaging q̈(τ ) in [tn , tn 1 ]24γ av (τ ) q̈n 1 q̈nq̈217 / 41
AA242B: MECHANICAL VIBRATIONS18 / 41Newmark’s Family of MethodsThe Newmark MethodApplication to the direct time-integration of Mq̈ Cq̇ Kq p(t)write the equilibrium equation at tn 1 and substitute the expressions(5) and (6) into it [M γhC βh2 K]q̈n 1 pn 1 C[q̇n (1 γ)hq̈n ] 1 K qn hq̇n β h2 q̈n2if the time-step h is uniform, M γhC βh2 K can be factored oncesolve the above system of equations for q̈n 1substitute the result into the expressions (5) and (6) to obtain q̇n 1and qn 118 / 41
AA242B: MECHANICAL VIBRATIONS19 / 41Newmark’s Family of MethodsConsistency of a Time-Integration MethodA time-integration scheme is said to be consistent ifun 1 un u̇(tn )h 0hlimThe Newmark time-integration method is consistent (1 γ)q̈ n γq̈n 1 un 1 unq̈n 1lim lim q̇nq̇n β h q̈n βh q̈n 1h 0h 0h2Consistency is one necessary condition for convergence19 / 41
AA242B: MECHANICAL VIBRATIONS20 / 41Newmark’s Family of MethodsStability of a Time-Integration MethodA time-integration scheme is said to be stable if there exists anintegration time-step h0 0 so that for any h [0, h0 ], a finitevariation of the state vector at time tn induces only a non-increasingvariation of the state-vector un j calculated at a subsequent timetn jStability is the other necessary condition for convergence20 / 41
AA242B: MECHANICAL VIBRATIONS21 / 41Newmark’s Family of MethodsStability of a Time-Integration MethodPremultiplying Eq. (5) and Eq. (6) by M and taking into accountthe equations of equilibrium (1) at tn and tn 1 leads after somealgebraic manipulations toMq̇n 1Mqn 1Mq̇n h(1 γ)[ Cq̇n Kqn pn ]γh[ Cq̇n 1 Kqn 1 pn 1 ]1 Mqn hMq̇n ( β)h2 [ Cq̇n Kqn pn ]2 βh2 [ Cq̇n 1 Kqn 1 pn 1 ](7) 21 / 41
AA242B: MECHANICAL VIBRATIONS22 / 41Newmark’s Family of MethodsStability of a Time-Integration MethodEquations (7) can be re-written in matrix form asun 1 A(h)un gn 1 (h)where A is the amplification matrix associated with the integrationoperator 1A(h) H 11 (h)H0 (h), gn 1 H1 (h)bn 1 (h) bn 1 n γhpn 1 (1 γ)hpM γhCγhK , H1 1 2222 β h pn βh pn 1βh CM βh K2 (1 γ)hC(1 γ)hK M 11H0 22 β h C hM M β h K2222 / 41
AA242B: MECHANICAL VIBRATIONS23 / 41Newmark’s Family of MethodsStability of a Time-Integration MethodEffect of an initial disturbanceδu0 u00 u0 δun 1 A(h)δun A2 (h)δun 1 · · · A(h)n 1 δu0consider the eigenpairs of A(h)(λr , xr )thenδun 1 An 1 (h)2NXs 1as xs 2NXas λn 1xsss 1where N is the dimension of the semi-discrete second-orderdynamical system δun 1 will be amplified by the time-integration operator only ifthe modulus of an eigenvalue of A(h) is greater than unity δun 1 will not be amplified by the time-integration operator if allmoduli of all eigenvalues of A(h) are less than unity23 / 41
AA242B: MECHANICAL VIBRATIONS24 / 41Newmark’s Family of MethodsStability of a Time-Integration MethodUndamped casedecouple the equations of equilibrium by writing them (for the purpose of analysis) inthe modal basisNX2q Qy yi qai ÿi ωi yi pi (t)i 1apply the Newmark scheme to the i-th modal equation recalled aboveamplification matrix ωi2 h2ωi2 h2 ωi2 h2 1 γ22 1 γ 1 βω2 h221 βω hiiA(h) ωi2 h2h1 122 22 21 βω hito obtain the 1 βω hi2characteristic equation is λ λ 2 (γ 2η 212 )η 1 (γ 21 )η 2 0 whereωi2 h21 βωi2 h2characteristic equation has:a pair of complex conjugate roots λ1 and λ2 if γ 12 2 4β 24 2 γ 12 2 η 2 4, i 1, · · · , N (case 1)ω hi two identical real roots if γ 12 2 η 2 4 (case 2) two distinct real roots if γ 12 2 η 2 4 (case 3)24 / 41
AA242B: MECHANICAL VIBRATIONS25 / 41Newmark’s Family of MethodsStability of a Time-Integration MethodUndamped case (continue)it can be shown that case 1 is the limiting case, in which caseλ1,2 ρe iφwheresρ φ 11 γ η22 q η 1 14 (γ 12 )2 η 2 arctan 1 21 (γ 12 )η 2then, the Newmark scheme is stable ifρ 1 γ 12and 214γ 4β 2 2 , i 1, · · · , N2ωi h limitation on the maximum time-step25 / 41
AA242B: MECHANICAL VIBRATIONS26 / 41Newmark’s Family of MethodsStability of a Time-Integration MethodUndamped case (continue)the algorithm is conditionally stable ifγ 12it is unconditionally stable if furthermore β 14 21γ — that2is,γ 12andβ 14 21γ 211the choice γ and β leads to an unconditionally stable24time-integration operator of maximum accuracy26 / 41
AA242B: MECHANICAL VIBRATIONS27 / 41Newmark’s Family of MethodsStability of a Time-Integration MethodUndamped case (continue)Stability of the Newmark scheme27 / 41
AA242B: MECHANICAL VIBRATIONS28 / 41Newmark’s Family of MethodsStability of a Time-Integration MethodDamped case (C 6 0)consider the case of modal dampingthen, the uncoupled equations of motion areÿi 2ξi ωi ẏi ωi2 yi pi (t)where ξi is the modal damping coefficient11consider the case γ , β 24an analysis similar to that performed in the undamped case revealsthat in this case, the Newmark scheme remains stable as long asξi 1in general, damping has a stabilizing effect for moderate values of ξi28 / 41
AA242B: MECHANICAL VIBRATIONS29 / 41Newmark’s Family of MethodsAmplitude and Periodicity ErrorsFree-vibration of an undamped linear oscillator 2ÿ ω y 0andy (0) y0 , ẏ (0) 0A 01 ω020 the above problem has an exact solution y (t) y0 cos ωt which canbe written in complex discrete form as yn 1 e iωh yn the exactamplification factor is ρex 1 and the exact phase is φex ωhthe numerical solution satisfies ẏn 1un 1 A(h)unyn 1let λ1,2 (β, γ) be the eigenvalues of A(h, β, γ) 2when γ 21 4β ω24h2 , λ1 and λ2 are complex-conjugateiλ1,2 (β, γ) ρ(β, γ)e iφ(β,γ)wheresρ 1 1γ 2 η2 , q 11 2 2 η 1 4 (γ 2 ) η φ arctan ,1 12 (γ 12 )η 22η ω 2 h21 βω 2 h229 / 41
AA242B: MECHANICAL VIBRATIONS30 / 41Newmark’s Family of MethodsAmplitude and Periodicity ErrorsFree-vibration of an undamped linear oscillator (continue)amplitude errorρ ρex1 ρ 1 2 1γ ω 2 h2 O(h4 )2relative periodicity error φ1 T 1 Tφ1φ 1φex1φex ωh1 1 φ2 β 112 ω 2 h2 O(h3 )30 / 41
AA242B: MECHANICAL VIBRATIONS31 / 41Newmark’s Family of MethodsAmplitude and Periodicity ErrorsAlgorithmγβStabilitylimitωhPurely explicitCentral difference0120002Fox & Goodwin12112Linear acceleration12Average constantacceleration12Table:Amplitudeerrorρ 1ω 2 h24Periodicityerror TT0—2 2 ω24h2.450O(h3 )163.460ω 2 h22414 0ω 2 h212Time-integration schemes of the Newmark familyThe purely explicit scheme (γ 0, β 0) is uselessThe Fox & Godwin scheme has asymptotically the smallest phaseerror but is only conditionally stable11The average constant acceleration scheme (γ , β ) is the24unconditionally stable scheme with asymptotically the highestaccuracy31 / 41
AA242B: MECHANICAL VIBRATIONS32 / 41Newmark’s Family of MethodsTotal Energy ConservationConservation of total energydynamic system with scleronomic constraintsnsXd(T V) mD Qs q̇sdts 111T q̇T Mq̇ and V qT Kq22the dissipation function D is a quadratic function of the velocities(m 2)1D q̇T Cq̇2external force component of the power balancensXQs q̇s q̇T ps 1integration over a time-step [tn , tn 1 ]Z tn 1t[T V]tn 1 ( q̇T Cq̇ q̇T p)dtntn32 / 41
AA242B: MECHANICAL VIBRATIONS33 / 41Newmark’s Family of MethodsTotal Energy ConservationConservation of total energy (continue)note that because M and K are symmetric (MT M and KT K)1T(q̇n 1 q̇n ) M(q̇n 1 q̇n )21T (qn 1 qn ) K(qn 1 qn )2when time-integration is performed using the Newmark algorithm with 11γ , β , the above variation becomes see (5) and (6)24t[T V]tn 1 [Tn 1 Tn ] [Vn 1 Vn ]nt [T V]tn 1n 1hTT(qn 1 qn ) (pn pn 1 ) (q̇n 1 q̇n ) C(q̇n 1 q̇n )24when applied to a conservative system (C 0 and p 0), preserves the total energyRtt tnn 1 ( q̇T Cq̇ q̇T p)dt and thereforefor non-conservative systems, [T V]tn 1nboth terms in the right-hand side of the above formula result from numericalquadrature relationships that are consistent with the time-integration operator Z t Z tn 1 Tn 1 Tpn pn 11Tq̇ dtq̇ pdt (qn 1 qn ) (pn pn 1 )22tntn Z Z ttn 1n 1 Tq̇n q̇n 11q̇n q̇n 1TTq̇ Cq̇dt q̇ dt C (qn 1 qn ) C222tntn hT(q̇n 1 q̇n ) C(q̇n 1 q̇n )433 / 41
AA242B: MECHANICAL VIBRATIONS34 / 41Explicit Time Integration Using the Central Difference AlgorithmAlgorithm in Terms of VelocitiesCentral Difference (CD) scheme Newmark’s with γ 12 , β 0q̇n 1 qn 1 q̈n q̈n 1)22hqn hn 1 q̇n n 1 q̈n2(8)q̇n hn 1 (where hn 1 tn 1 tnEquivalent three-step formstart withqn qn 1 hn q̇n 1 hnhn2q̈n 1 qn 1 hn q̇n 1 q̈n 1(9)22 {z}q̇n 12divide by hn and subtract the result from qn 1 divided by hn 1account for the relationship (8) q̈n hn (qn 1 qn ) hn 1 (qn qn 1 )hn 1 hn hn 1(10)2where hn 1 2hn hn 1234 / 41
AA242B: MECHANICAL VIBRATIONS35 / 41Explicit Time Integration Using the Central Difference AlgorithmAlgorithm in Terms of VelocitiesCase of a constant time-step hq̈n qn 1 2qn qn 1h2Efficient implementationuse a lumped mass matrix Mh1q̈02 hn q̇n 1 see (9)initialize: q̈0 M 1 (p0 Kq0 ) and q̇ 1 q̇0 2increment the displacement: qn qn 12compute the acceleration: q̈n M 1 (pn Kqn ) (enforce equilibrium at tn )increment the velocity at half time-step (formula results from (10))q̇n 1 q̇n 1 hn 1 q̈n q̈n 22q̇n 1 q̇n 1222hn 12Stability condition: for γ 1/2 and β 0, γ 1 22 4β 42 h2ωcr ωcr h 2 whereωcr is the highest frequency contained in the model – this condition is also known as theCourant condition2hcr is referred to here as the maximum Courant stability time-stepωcr35 / 41
AA242B: MECHANICAL VIBRATIONS36 / 41Explicit Time Integration Using the Central Difference AlgorithmApplication Example: the Clamped-Free Bar Excited by an End LoadClamped bar subjected to a step load at its free endModel made of N 20 finite elements with equal length l 121321931718LN201920lumped mass matrixEigenfrequencies of the continuous systemr r πEA2r 1 π EA2r 1 πωcontr (2r 1) 2 mL2N2 ml 2N236 / 41
AA242B: MECHANICAL VIBRATIONS37 / 41Explicit Time Integration Using the Central Difference AlgorithmApplication Example: the Clamped-Free Bar Excited by an End LoadFinite element stiffness and mass matrices mlM 2 22 1 EA K l 20 2.021 12 1 12.00. 1 12 1 1 1(11)Analytical frequencies of the discrete systemrωr 2EAsinml 2 2r 12N π2 2r 12N π2 ,r 1, 2, · · · N 2 sin ωcr ωcr (r N, N ) 2Critical time-step for the CD algorithmωcr hcr 2 hcr 137 / 41
AA242B: MECHANICAL VIBRATIONS38 / 41Explicit Time Integration Using the Central Difference AlgorithmApplication Example: the Clamped-Free Bar
AA242B: MECHANICAL VIBRATIONS 1/41 AA242B: MECHANICAL VIBRATIONS Direct Time-Integration Methods These slides are based on the recommended textbook: M. G eradin and D. Rixen, \Mechanical Vibrations: Theory and Applications to Structural Dynamics," Second Edition, Wiley, John & Sons, Incorporated, ISBN-13:9780471975465 1/41
AA242B: MECHANICAL VIBRATIONS 1/50 AA242B: MECHANICAL VIBRATIONS Undamped Vibrations of n-DOF Systems These slides are based on the recommended textbook: M. G eradin and D. Rixen, \Mechanical Vibrations: Theory and Applications to Structural Dynamics," Second Edition, Wiley, John & Sons, Incorporated, ISBN-13:9780471975465 1/50
Mechanical vibrations. (Allyn and Bacon series in Mechanical engineering and applied mechanics) Includes index. 1. Vibrations. I. Morse, Ivan E., joint author. Hinkle, Theodore, joint author. Title. 1978 620.3 77-20933 ISBN ISBN (International)
Vibration is a continuous cyclic motion of a structure or a component. Generally, engineers try to avoid vibrations, because vibrations have a number of unpleasant effects: . It is thought that the vibrations were a form of self-excited vibration known as flutter, or ‘galloping’’ A similar form of vibration is known to occur .
Introduction to Vibrations Free Response Part 1: Spring-mass systems Vibration is a sub-discipline of dynamics that deals with repetitive motions. Some familiar examples are the vibrations of automobiles, guitar strings, cell phones and pendulums. Vibrations can be unwanted or wanted. For example,
Mechanical Vibrations Lecture #1 Derivation of equations of motion (Newton-Euler Laws) Derivation of Equation of Motion Define the vibrations of interest -Degrees of freedom (translational, rotat
The approach has been applied to evaluate the mechanical vibrations of the motors in associate with the geometry of the rotor slots. Also, the effects of nonsinusoidal supply voltages on the current harmonics have . comparison studies of the rotor slot shapes affecting the motor lateral vibrations in associate with three IEEE standard slots .
Experimental evidence has shown that a human body can be injured by vibrations. It was reported that about 12 million workers in the USA were affected by vibrations . [2] According to that, researchers have been working hard to reduce this dangerous phe-nomenon, and therefore they have written a lot of studies on how to avoid the effects of
API Recommended Practice 500, Classification of Locations for Electrical Installations at Petroleum Facilities Classified as Class 1, Division 1 and Division 2 API Recommended Practice 505, Classification of Locations for Electrical Installations at Petroleum Facilities Classified as Class 1, Zone 0, Zone 1 and Zone 2 ASSE Z359.1, The Fall Protection Code ASTM F2413, Standard Specification for .
