[{"data":1,"prerenderedAt":32},["ShallowReactive",2],{"topic-en-theoretical-mechanics\u002Fdynamics\u002Fanalytical-mechanics-and-oscillations\u002Fsmall-oscillations-one-degree-of-freedom-system":3},{"topic":4,"trail":13,"children":27,"tasks":28,"alternates":29},{"id":5,"name":6,"locale":7,"path":8,"seo_title":6,"seo_description":9,"seo_text":10,"content_html":11,"content_chunks":12},269,"Small Oscillations of a One-Degree-of-Freedom System","en","theoretical-mechanics\u002Fdynamics\u002Fanalytical-mechanics-and-oscillations\u002Fsmall-oscillations-one-degree-of-freedom-system","Free and forced small oscillations of a one-degree-of-freedom mechanical system, natural frequency, and resonance.","This topic covers linearized free and forced oscillations of a one-degree-of-freedom system, natural frequency, damping, and resonance.","\u003Cp>\u003Cstrong>Small oscillations\u003C\u002Fstrong> occur near a stable equilibrium when deviations are small enough for the equations of motion to be linearized. A one-degree-of-freedom system is described by one generalized coordinate.\u003C\u002Fp>\u003Ch2>Undamped free oscillations\u003C\u002Fh2>\u003Cp>The standard linear equation is:\u003C\u002Fp>\u003Cp>$$m\\ddot x+kx=0.$$\u003C\u002Fp>\u003Cp>The natural circular frequency is:\u003C\u002Fp>\u003Cp>$$\\omega_n=\\sqrt{\\frac{k}{m}},$$\u003C\u002Fp>\u003Cp>and the period is $T=2\\pi\u002F\\omega_n$.\u003C\u002Fp>\u003Ch2>Free-oscillation response\u003C\u002Fh2>\u003Cp>The solution may be written $x=C_1\\cos\\omega_nt+C_2\\sin\\omega_nt$ or $x=A\\cos(\\omega_nt+\\varphi)$. Amplitude and initial phase are determined from the initial conditions.\u003C\u002Fp>\u003Ch2>Viscous damping\u003C\u002Fh2>\u003Cp>With linear resistance $c\\dot x$:\u003C\u002Fp>\u003Cp>$$m\\ddot x+c\\dot x+kx=0.$$\u003C\u002Fp>\u003Cp>The type of motion depends on damping relative to its critical value. With light damping, the system undergoes decaying oscillations.\u003C\u002Fp>\u003Ch2>Forced oscillations\u003C\u002Fh2>\u003Cp>For harmonic excitation $F_0\\cos\\Omega t$:\u003C\u002Fp>\u003Cp>$$m\\ddot x+c\\dot x+kx=F_0\\cos\\Omega t.$$\u003C\u002Fp>\u003Cp>The steady-state response has excitation frequency $\\Omega$, while its amplitude depends on frequency ratio and damping.\u003C\u002Fp>\u003Ch2>Resonance\u003C\u002Fh2>\u003Cp>In an ideal undamped system, harmonic excitation at $\\Omega=\\omega_n$ produces resonant growth of amplitude. With damping, the amplitude remains finite and the frequency-response maximum shifts depending on damping.\u003C\u002Fp>\u003Ch2>Linearization near equilibrium\u003C\u002Fh2>\u003Cp>For a general system, the coordinate is measured from stable equilibrium and only first-order terms in the small deviation are retained. The result has the form $m_{eq}\\ddot q+c_{eq}\\dot q+k_{eq}q=Q(t)$.\u003C\u002Fp>\u003Ch2>Example\u003C\u002Fh2>\u003Cp>For $m=2$ kg and $k=50$ N\u002Fm without damping, $\\omega_n=\\sqrt{50\u002F2}=5$ rad\u002Fs and $T=2\\pi\u002F5\\approx1.26$ s.\u003C\u002Fp>\u003Ch2>Common mistakes\u003C\u002Fh2>\u003Cul>\u003Cli>confusing circular frequency in rad\u002Fs with ordinary frequency $f=\\omega\u002F(2\\pi)$ in hertz;\u003C\u002Fli>\u003Cli>using a linear small-oscillation model for large deviations without checking validity;\u003C\u002Fli>\u003Cli>ignoring damping when estimating resonant amplitude of a real system;\u003C\u002Fli>\u003Cli>using a physical mass directly instead of equivalent inertia for a compound mechanism without derivation.\u003C\u002Fli>\u003C\u002Ful>",[],[14,18,22,26],{"id":15,"name":16,"path":17},81,"Theoretical Mechanics","theoretical-mechanics",{"id":19,"name":20,"path":21},138,"Dynamics","theoretical-mechanics\u002Fdynamics",{"id":23,"name":24,"path":25},296,"Analytical Mechanics and Oscillations","theoretical-mechanics\u002Fdynamics\u002Fanalytical-mechanics-and-oscillations",{"id":5,"name":6,"path":8},[],[],{"en":30,"uk":31},"https:\u002F\u002Fmechclassroom.com\u002Fen\u002Ftopics\u002Ftheoretical-mechanics\u002Fdynamics\u002Fanalytical-mechanics-and-oscillations\u002Fsmall-oscillations-one-degree-of-freedom-system","https:\u002F\u002Fmechclassroom.com\u002Ftopics\u002Fteoretychna-mekhanika\u002Fdynamika\u002Fanalitychna-mekhanika-ta-kolyvannia\u002Fmali-kolyvannia-systemy-z-odnym-stupenem-vilnosti",1787712537979]