6. Description of document versions

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 \(k\), mass \(m\), and viscous damping \(c\), the equation of motion is of the form:

for which the traditional notations are:

• The natural pulsation of the undamped system: \({\mathrm{\omega }}_{0}=\sqrt{\frac{k}{m}}\);

• critical damping: \({c}_{\text{critique}}=2m{\mathrm{\omega }}_{0}\);

• reduced damping, expressed as a percentage of critical depreciation): \(\mathrm{\xi }=\frac{c}{{c}_{\text{critique}}}=\frac{c}{2m{\mathrm{\omega }}_{0}}\);

• the natural pulsation of the damped system: \({\mathrm{\omega }}_{0}\text{'}={\mathrm{\omega }}_{0}\sqrt{(1-{\mathrm{\xi }}^{2})}\);

• static deflection for a force \({F}_{0}\): \({\mathrm{\delta }}_{\text{st}}=\frac{{F}_{0}}{k}\);

• the reduced frequency: \(\mathrm{\eta }=\frac{\mathrm{\omega }}{{\mathrm{\omega }}_{0}}\);

• the reduced system equation: \(\ddot{X}+2\mathrm{\xi }{\mathrm{\omega }}_{0}\dot{X}+{\mathrm{\omega }}_{0}^{2}X=0\).

The global response to a harmonic excitation of the form \(F(t)={F}_{0}\mathrm{cos}(\omega t)\) is the sum of:

• of a free response \({X}_{1}(t)\) damped oscillatory general solution where \({X}_{\mathrm{l0}}\) and \({\varphi }_{0}\) are determined by the initial conditions:

\({X}_{1}(t)={X}_{0}{e}^{-\xi {\omega }_{0}t}\mathrm{cos}(\omega {\text{'}}_{0}t+{\varphi }_{0})\)

• of a forced response \({X}_{f}(t)\) permanent special solution \({X}_{f}(t)={X}_{\mathrm{f0}}\mathrm{cos}(\omega t-\varphi )\)

\({X}_{\mathrm{f0}}=\frac{{F}_{0}}{\sqrt{{(k-m{\omega }^{2})}^{2}+{(c\omega )}^{2}}}\varphi =\mathrm{arctg}(\frac{c\omega }{k-m{\omega }^{2}})\) eq A1.1-1

Which is written in reduced form:

\(\frac{{X}_{\mathrm{f0}}}{{\delta }_{\mathrm{st}}}=\frac{k{X}_{\mathrm{f0}}}{{F}_{0}}=\frac{1}{\sqrt{{(1-{\eta }^{2})}^{2}+{(2\xi \eta )}^{2}}}\varphi =\mathrm{arctg}(\frac{2\xi \eta }{1-{\eta }^{2}})\) eq A1.1-2

Figure A1.1-a: Response of an oscillator in imposed force (module and phase)

The response to a harmonic excitation of the form \(F(t)={F}_{0}{e}^{j\omega t}\) is written with a forced response (special permanent solution \({X}_{f}(t)={X}_{\mathrm{f0}}{e}^{(j\omega t-\varphi )}\))

\({X}_{\mathrm{f0}}=\frac{{F}_{0}}{\sqrt{{(k-m{\omega }^{2})}^{2}+{(c\omega )}^{2}}}\varphi =\text{arctg}(\frac{c\omega }{k-m{\omega }^{2}})\) eq A1.1-3

Which is written in reduced form:

\(\frac{k{X}_{f0}}{{F}_{0}}=\frac{1}{1-{\eta }^{2}+j2\xi \eta }\equiv H(j\omega )\varphi =\text{arctg}(\frac{2\xi \eta }{1-{\eta }^{2}})\) eq A1.1-4

where \(H(j\omega )\) is the complex harmonic response of a simple oscillator:

\(H(j\omega )=\frac{1}{\sqrt{{(1-{\eta }^{2})}^{2}+{(2\xi \eta )}^{2}}}\)

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 \(k\), mass \(m\), and viscous damping \(c\), the absolute equation of motion is of the form:

The forced response to an imposed harmonic movement of the form \(s(t)={s}_{0}\mathrm{cos}(\omega t)\) is of the form \({X}_{m}(t)={X}_{\mathrm{m0}}\mathrm{cos}(\omega t-{\varphi }_{1}-{\varphi }_{2})\) sum of two response terms, particular permanent solutions:

• term induced by excitement on the go \({X}_{d0}\mathrm{cos}(\omega t-{\varphi }_{d})\)

\({X}_{\mathrm{d0}}=\frac{k{s}_{0}}{\sqrt{{(k-m{\omega }^{2})}^{2}+{(c\omega )}^{2}}}{\varphi }_{d}=\text{arctg}(\frac{c\omega }{k-m{\omega }^{2}})\)

• term induced by excitement in speed \({X}_{v0}\mathrm{cos}(\omega t-{\varphi }_{v})\)

\({X}_{\mathrm{v0}}=\frac{\omega c{s}_{0}}{\sqrt{{(k-m{\omega }^{2})}^{2}+{(c\omega )}^{2}}}{\varphi }_{v}=\text{arctg}(\frac{c\omega }{k-m{\omega }^{2}})\)

which leads to a total forced response:

\({X}_{m}(t)={X}_{m}\mathrm{cos}(\omega t-{\varphi }_{1}-{\varphi }_{2})\equiv {s}_{0}\sqrt{\frac{{k}^{2}+{(c\omega )}^{2}}{[{(k-m{\omega }^{2})}^{2}+{(c\omega )}^{2}]}}\mathrm{cos}(\omega t-{\varphi }_{1}-{\varphi }_{2})\)

Hence the reduced form of the absolute amplitude:

\(\frac{{X}_{m}}{{s}_{0}}=\sqrt{\frac{1+{(2\xi \eta )}^{2}}{[{(1-\text{}{\mathrm{êta}}^{2})}^{2}+{(2\xi \eta )}^{2}]}}{\varphi }_{1}=\text{arctg}(\frac{2\xi \eta }{1-{\eta }^{2}}){\varphi }_{2}=\text{arctg}(\frac{1}{2\xi \eta })\)

If the movement imposed at the base is expressed in complex form \(s(t)=\Re ({s}_{0}{e}^{j\omega t})\), the relative amplitude or transmittability can be written from the complex harmonic response of a simple oscillator \(H(j\omega )\)

\(\frac{{X}_{m}}{{s}_{0}}=\sqrt{1+{(2\xi \eta )}^{2}}\mid H(j\omega )\mid\) 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 \(x=X-s\)

The relative equation of motion for an imposed harmonic motion of the form \(s(t)={s}_{0}\mathrm{cos}(\omega t)\) is then of the form \(m\ddot{x}+c\dot{x}+kx=-m\ddot{s}\) or in reduced form:

\(\ddot{x}+2\xi {\omega }_{0}\dot{x}+{\omega }_{0}^{2}x=-\ddot{s}={\omega }^{2}{s}_{0}\mathrm{cos}(\omega t)\) eq A2.2-1

The relative forced response is then, for a permanent solution \({x}_{\mathrm{m0}}\mathrm{cos}(\omega t-\varphi )\),

\({x}_{\mathrm{m0}}=\frac{m{\omega }^{2}{s}_{0}}{\sqrt{{(k-m{\omega }^{2})}^{2}+{(c\omega )}^{2}}}\varphi =\text{arctg}(\frac{c\omega }{k-m{\omega }^{2}})\) eq A2.2-2

Which is written in reduced form:

\(\frac{{x}_{\mathrm{m0}}}{{s}_{0}}=\frac{{\eta }^{2}}{\sqrt{{(1-{\eta }^{2})}^{2}+{(2\xi \eta )}^{2}}}\) eq A2.2-3

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 (\(x=\dot{x}=0\) for \(t<0\) or \(t={0}^{\text{-}}\)) can be written as:

\(\tilde{f}\mathrm{=}\underset{\Delta t\to 0}{\mathrm{lim}}{\mathrm{\int }}_{t}^{t+\Delta t}F\mathit{dt}\mathrm{=}F\mathrm{\cdot }\mathit{dt}\mathrm{=}1\mathrm{=}m\dot{X}(t\mathrm{=}0)\mathrm{-}m\dot{X}(t\mathrm{=}{0}^{\text{-}})\mathrm{=}m\dot{{X}_{0}}\)

The initial conditions are then noted \(X(t=0)={X}_{0}\) and \(\dot{X}(t=0)=\dot{{X}_{0}}=\frac{1}{m}\)

The general equation for the free vibration response of a system at one degree of freedom:

\({X}_{l}(t)={e}^{-\xi {\omega }_{0}t}({X}_{0}\mathrm{cos}\omega {\text{'}}_{0}t+\frac{\dot{{X}_{0}}+\xi {\omega }_{0}{x}_{0}}{\omega {\text{'}}_{0}}\mathrm{sin}\omega {\text{'}}_{0}t)\)

then becomes the \(g(t)\) impulse response of a system with one degree of freedom

\({X}_{l}(t)=g(t)=\frac{{e}^{-\xi {\omega }_{0}t}}{m\omega {\text{'}}_{0}}\mathrm{sin}\omega {\text{'}}_{0}t\) eq A3.1-1

For a non-unit impulse \(\tilde{F}=F\cdot \Delta t\), the initial speed is \(\dot{{X}_{0}}=\frac{F}{m}\) and the response becomes:

\({X}_{l}(t)=\frac{\tilde{F}{e}^{-\xi {\omega }_{0}t}}{m\omega {\text{'}}_{0}}\mathrm{sin}\omega {\text{'}}_{0}t=\tilde{F}g(t)\) eq A3.1-2

If the impulse force is applied at any time \(\tau\), the answer is:

\({X}_{l}(t)=\tilde{F}g(t-\tau )\)

A3.2 Any forced vibration response

The excitation force \(F(t)\) can be broken down into a series of pulses of variable amplitude \(F(\tau )\) applied at the instant \(\tau\) for a time \(\tau\). If \(\Delta \tau \to 0\), the answer at a time \(t\) is obtained by:

\(X(t)={\int }_{0}^{t}F(\tau )g(t-\tau )d\tau\)

and by replacing with the expression for the impulse response [éq A.3-2], we obtain the convolution equation for a system at rest at time 0 of the form:

\(X(t)\mathrm{=}\frac{1}{m\omega {\text{'}}_{0}}{\mathrm{\int }}_{0}^{t}F(\tau ){e}^{\mathrm{-}\xi {\omega }_{0}(t\mathrm{-}\tau )}\mathrm{sin}\omega {\text{'}}_{0}(t\mathrm{-}\tau )d\tau\) eq A3.2-1

known as the integral of DUHAMEL.

A3.3 Any imposed motion response

For a relative motion analysis represented by [éq A2.2-1]:

\(\ddot{x}+2\xi {\omega }_{0}\dot{x}+{\omega }_{0}^{2}x=-\ddot{s}={\omega }^{2}{s}_{0}\mathrm{cos}(\omega t)\)

The integral of DUHAMEL becomes:

\(x(t)=\frac{1}{\omega {\text{'}}_{0}}{\int }_{0}^{t}\ddot{s}(\tau ){e}^{-\xi {\omega }_{0}(t-\tau )}\mathrm{sin}\omega {\text{'}}_{0}(t-\tau )d\tau\) eq A3.3-1