Description of document versions ==================================== .. csv-table:: "**Version** **Aster**", "**Author (s)** **Organization (s)**", "**Description of changes**" "5", "*J.R. LEVESQUE, L. VIVAN, L., D., D. SELIGMANN, D., EDF -* R&D/ AMA ", "Initial text" "04/06/11", "*J.R. LEVESQUE, L., L. VIVAN,* *Y. PONS* *EDF -* R&D/ AMA ", "" "9.4", "S. AUDEBERT EDF -R&D/ AMA ", "" "10", "S. AUDEBERT EDF -R&D/ AMA ", "Taking into account sheet REX 12005: modification of intra-group accumulation §4.6.6" "11", "S. AUDEBERT EDF -R&D/ AMA ", "Taking into account sheet REX 17054: introduction of the modal recombination method according to GUPTA" **Transient response of a damped simple oscillator** **A1.1** **Forced vibration of a system with a degree of translational freedom For a simple oscillator with stiffness :math:`k`, mass :math:`m`, and viscous damping :math:`c`, the equation of motion is of the form: .. image:: images/1000144C000069D50000281AD27EE751DA7EF8F6.svg :width: 285 :height: 108 .. _RefImage_1000144C000069D50000281AD27EE751DA7EF8F6.svg: for which the traditional notations are: * The natural pulsation of the undamped system: :math:`{\mathrm{\omega }}_{0}=\sqrt{\frac{k}{m}}`; * critical damping: :math:`{c}_{\text{critique}}=2m{\mathrm{\omega }}_{0}`; * reduced damping, expressed as a percentage of critical depreciation): :math:`\mathrm{\xi }=\frac{c}{{c}_{\text{critique}}}=\frac{c}{2m{\mathrm{\omega }}_{0}}`; * the natural pulsation of the damped system: :math:`{\mathrm{\omega }}_{0}\text{'}={\mathrm{\omega }}_{0}\sqrt{(1-{\mathrm{\xi }}^{2})}`; * static deflection for a force :math:`{F}_{0}`: :math:`{\mathrm{\delta }}_{\text{st}}=\frac{{F}_{0}}{k}`; * the reduced frequency: :math:`\mathrm{\eta }=\frac{\mathrm{\omega }}{{\mathrm{\omega }}_{0}}`; * the reduced system equation: :math:`\ddot{X}+2\mathrm{\xi }{\mathrm{\omega }}_{0}\dot{X}+{\mathrm{\omega }}_{0}^{2}X=0`. The **global** response to a harmonic excitation of the form :math:`F(t)={F}_{0}\mathrm{cos}(\omega t)` is the sum of: * of a free response :math:`{X}_{1}(t)` damped oscillatory general solution where :math:`{X}_{\mathrm{l0}}` and :math:`{\varphi }_{0}` are determined by the initial conditions: :math:`{X}_{1}(t)={X}_{0}{e}^{-\xi {\omega }_{0}t}\mathrm{cos}(\omega {\text{'}}_{0}t+{\varphi }_{0})` * of a forced response :math:`{X}_{f}(t)` permanent special solution :math:`{X}_{f}(t)={X}_{\mathrm{f0}}\mathrm{cos}(\omega t-\varphi )` .. _RefEquation A1.1-1: :math:`{X}_{\mathrm{f0}}=\frac{{F}_{0}}{\sqrt{{(k-m{\omega }^{2})}^{2}+{(c\omega )}^{2}}}\varphi =\mathrm{arctg}(\frac{c\omega }{k-m{\omega }^{2}})` eq A1.1-1 Which is written in reduced form: .. _RefEquation A1.1-2: :math:`\frac{{X}_{\mathrm{f0}}}{{\delta }_{\mathrm{st}}}=\frac{k{X}_{\mathrm{f0}}}{{F}_{0}}=\frac{1}{\sqrt{{(1-{\eta }^{2})}^{2}+{(2\xi \eta )}^{2}}}\varphi =\mathrm{arctg}(\frac{2\xi \eta }{1-{\eta }^{2}})` eq A1.1-2 .. image:: images/10004CB000001E7D00002ACA6DC2B1961A3655A9.svg :width: 285 :height: 108 .. _RefImage_10004CB000001E7D00002ACA6DC2B1961A3655A9.svg: **Figure A1.1-a: Response of an oscillator in imposed force (module and phase)** The response to a harmonic excitation of the form :math:`F(t)={F}_{0}{e}^{j\omega t}` is written with a forced response (special permanent solution :math:`{X}_{f}(t)={X}_{\mathrm{f0}}{e}^{(j\omega t-\varphi )}`) .. _RefEquation A1.1-3: :math:`{X}_{\mathrm{f0}}=\frac{{F}_{0}}{\sqrt{{(k-m{\omega }^{2})}^{2}+{(c\omega )}^{2}}}\varphi =\text{arctg}(\frac{c\omega }{k-m{\omega }^{2}})` eq A1.1-3 Which is written in reduced form: .. _RefEquation A1.1-4: :math:`\frac{k{X}_{f0}}{{F}_{0}}=\frac{1}{1-{\eta }^{2}+j2\xi \eta }\equiv H(j\omega )\varphi =\text{arctg}(\frac{2\xi \eta }{1-{\eta }^{2}})` eq A1.1-4 where :math:`H(j\omega )` is the complex harmonic response of a simple oscillator: :math:`H(j\omega )=\frac{1}{\sqrt{{(1-{\eta }^{2})}^{2}+{(2\xi \eta )}^{2}}}` **Imposed movement of a system to a degree of freedom in translation** **A2.1** **Absolute movement of a system with one degree of liberty** For a simple oscillator with stiffness :math:`k`, mass :math:`m`, and viscous damping :math:`c`, the **absolute** equation of motion is of the form: .. image:: images/10001BFA000069D5000019A23A832206CCDED6B1.svg :width: 285 :height: 108 .. _RefImage_10001BFA000069D5000019A23A832206CCDED6B1.svg: The **forced** response to an imposed harmonic movement of the form :math:`s(t)={s}_{0}\mathrm{cos}(\omega t)` is of the form :math:`{X}_{m}(t)={X}_{\mathrm{m0}}\mathrm{cos}(\omega t-{\varphi }_{1}-{\varphi }_{2})` sum of two response terms, particular permanent solutions: * term induced by excitement on the go :math:`{X}_{d0}\mathrm{cos}(\omega t-{\varphi }_{d})` :math:`{X}_{\mathrm{d0}}=\frac{k{s}_{0}}{\sqrt{{(k-m{\omega }^{2})}^{2}+{(c\omega )}^{2}}}{\varphi }_{d}=\text{arctg}(\frac{c\omega }{k-m{\omega }^{2}})` * term induced by excitement in speed :math:`{X}_{v0}\mathrm{cos}(\omega t-{\varphi }_{v})` :math:`{X}_{\mathrm{v0}}=\frac{\omega c{s}_{0}}{\sqrt{{(k-m{\omega }^{2})}^{2}+{(c\omega )}^{2}}}{\varphi }_{v}=\text{arctg}(\frac{c\omega }{k-m{\omega }^{2}})` which leads to a total forced response: :math:`{X}_{m}(t)={X}_{m}\mathrm{cos}(\omega t-{\varphi }_{1}-{\varphi }_{2})\equiv {s}_{0}\sqrt{\frac{{k}^{2}+{(c\omega )}^{2}}{[{(k-m{\omega }^{2})}^{2}+{(c\omega )}^{2}]}}\mathrm{cos}(\omega t-{\varphi }_{1}-{\varphi }_{2})` Hence the reduced form of the absolute amplitude: :math:`\frac{{X}_{m}}{{s}_{0}}=\sqrt{\frac{1+{(2\xi \eta )}^{2}}{[{(1-\text{}{\mathrm{êta}}^{2})}^{2}+{(2\xi \eta )}^{2}]}}{\varphi }_{1}=\text{arctg}(\frac{2\xi \eta }{1-{\eta }^{2}}){\varphi }_{2}=\text{arctg}(\frac{1}{2\xi \eta })` If the movement imposed at the base is expressed in complex form :math:`s(t)=\Re ({s}_{0}{e}^{j\omega t})`, the relative amplitude or transmittability can be written from the complex harmonic response of a simple oscillator :math:`H(j\omega )` .. _RefEquation A2.1-1: :math:`\frac{{X}_{m}}{{s}_{0}}=\sqrt{1+{(2\xi \eta )}^{2}}\mid H(j\omega )\mid` eq A2.1-1 **A2.2** **Relative movement of a system to a degree of liberty** The problem of responding to an imposed movement can be treated in relative displacement of the mass with respect to the base by asking :math:`x=X-s` The **relative** equation of motion for an imposed harmonic motion of the form :math:`s(t)={s}_{0}\mathrm{cos}(\omega t)` is then of the form :math:`m\ddot{x}+c\dot{x}+kx=-m\ddot{s}` or in reduced form: .. _RefEquation A2.2-1: :math:`\ddot{x}+2\xi {\omega }_{0}\dot{x}+{\omega }_{0}^{2}x=-\ddot{s}={\omega }^{2}{s}_{0}\mathrm{cos}(\omega t)` eq A2.2-1 The relative **forced** response is then, for a permanent solution :math:`{x}_{\mathrm{m0}}\mathrm{cos}(\omega t-\varphi )`, .. _RefEquation A2.2-2: :math:`{x}_{\mathrm{m0}}=\frac{m{\omega }^{2}{s}_{0}}{\sqrt{{(k-m{\omega }^{2})}^{2}+{(c\omega )}^{2}}}\varphi =\text{arctg}(\frac{c\omega }{k-m{\omega }^{2}})` eq A2.2-2 Which is written in reduced form: .. _RefEquation A2.2-3: :math:`\frac{{x}_{\mathrm{m0}}}{{s}_{0}}=\frac{{\eta }^{2}}{\sqrt{{(1-{\eta }^{2})}^{2}+{(2\xi \eta )}^{2}}}` eq A2.2-3 .. image:: images/1000331C0000242C0000196D20E41C7F8F5E0644.svg :width: 285 :height: 108 .. _RefImage_1000331C0000242C0000196D20E41C7F8F5E0644.svg: **Figure A2.2-a: Response of an oscillator in imposed motion (relative displacement module)** **Non-periodic imposed movement of a system to a degree of liberty** The problem dealt with earlier was limited to a periodic imposed movement. For a non-periodic excitation, with an amplitude that varies over time, and is exerted over a finite period of time, the response to a series of pulses is considered. **A3.1** **Impulse response** The simplest form is the unit impulse force, which when applied to a mass at rest before the impulse is applied (:math:`x=\dot{x}=0` for :math:`t<0` or :math:`t={0}^{\text{-}}`) can be written as: :math:`\tilde{f}\mathrm{=}\underset{\Delta t\to 0}{\mathrm{lim}}{\mathrm{\int }}_{t}^{t+\Delta t}F\mathit{dt}\mathrm{=}F\mathrm{\cdot }\mathit{dt}\mathrm{=}1\mathrm{=}m\dot{X}(t\mathrm{=}0)\mathrm{-}m\dot{X}(t\mathrm{=}{0}^{\text{-}})\mathrm{=}m\dot{{X}_{0}}` The initial conditions are then noted :math:`X(t=0)={X}_{0}` and :math:`\dot{X}(t=0)=\dot{{X}_{0}}=\frac{1}{m}` The general equation for the free vibration response of a system at one degree of freedom: :math:`{X}_{l}(t)={e}^{-\xi {\omega }_{0}t}({X}_{0}\mathrm{cos}\omega {\text{'}}_{0}t+\frac{\dot{{X}_{0}}+\xi {\omega }_{0}{x}_{0}}{\omega {\text{'}}_{0}}\mathrm{sin}\omega {\text{'}}_{0}t)` then becomes the :math:`g(t)` impulse response of a system with one degree of freedom .. _RefEquation A3.1-1: :math:`{X}_{l}(t)=g(t)=\frac{{e}^{-\xi {\omega }_{0}t}}{m\omega {\text{'}}_{0}}\mathrm{sin}\omega {\text{'}}_{0}t` eq A3.1-1 For a non-unit impulse :math:`\tilde{F}=F\cdot \Delta t`, the initial speed is :math:`\dot{{X}_{0}}=\frac{F}{m}` and the response becomes: .. _RefEquation A3.1-2: :math:`{X}_{l}(t)=\frac{\tilde{F}{e}^{-\xi {\omega }_{0}t}}{m\omega {\text{'}}_{0}}\mathrm{sin}\omega {\text{'}}_{0}t=\tilde{F}g(t)` eq A3.1-2 If the impulse force is applied at any time :math:`\tau`, the answer is: :math:`{X}_{l}(t)=\tilde{F}g(t-\tau )` **A3.2** **Any forced vibration response** The excitation force :math:`F(t)` can be broken down into a series of pulses of variable amplitude :math:`F(\tau )` applied at the instant :math:`\tau` for a time :math:`\tau`. If :math:`\Delta \tau \to 0`, the answer at a time :math:`t` is obtained by: :math:`X(t)={\int }_{0}^{t}F(\tau )g(t-\tau )d\tau` and by replacing with the expression for the impulse response [:ref:`éq A.3-2 <éq A.3-2>`], we obtain the convolution equation for a system at rest at time 0 of the form: .. _RefEquation A3.2-1: :math:`X(t)\mathrm{=}\frac{1}{m\omega {\text{'}}_{0}}{\mathrm{\int }}_{0}^{t}F(\tau ){e}^{\mathrm{-}\xi {\omega }_{0}(t\mathrm{-}\tau )}\mathrm{sin}\omega {\text{'}}_{0}(t\mathrm{-}\tau )d\tau` eq A3.2-1 known as the integral of DUHAMEL. **A3.3** **Any imposed motion response** For a relative motion analysis represented by [:ref:`éq A2.2-1 <éq A2.2-1>`]: :math:`\ddot{x}+2\xi {\omega }_{0}\dot{x}+{\omega }_{0}^{2}x=-\ddot{s}={\omega }^{2}{s}_{0}\mathrm{cos}(\omega t)` The integral of DUHAMEL becomes: .. _RefEquation A3.3-1: :math:`x(t)=\frac{1}{\omega {\text{'}}_{0}}{\int }_{0}^{t}\ddot{s}(\tau ){e}^{-\xi {\omega }_{0}(t-\tau )}\mathrm{sin}\omega {\text{'}}_{0}(t-\tau )d\tau` eq A3.3-1