Linear Vibrations: Linear Vibrations Home Page

Example 1 - Derive the differential equation of motion for the system shown in the following figure.
$$\begin{align} & {{m}_{1}}{{{\ddot{x}}}_{1}}=-k{{x}_{1}}-5k\left( {{x}_{1}}-{{x}_{3}} \right)-k\left( {{x}_{1}}-{{x}_{2}} \right)+{{F}_{1}}\left( t \right) \\ & {{m}_{2}}{{{\ddot{x}}}_{2}}=-k\left( {{x}_{2}}-{{x}_{1}} \right)-k\left( {{x}_{...

In this method the approximate formula used for obtaining x_i+1 from x_i is made to coincide with Taylor’s series expansion of x and x_i+1 up to terms of order (∆t)^n. The Taylor’s series expansion of x(t) at t+∆t is given by:
$$x\left( t+\Delta t \right)=x\left( t \right)+\dot{x}\left( t \right)\Delta t+\ddot{x}\left( t \right)\frac{{{\left( \Delta t \right)}^{2}}}{2!}+\left( t \right)\frac{{{\left( \Delta t \right)}^{3}}}{3!}+\left( t \right)\frac{{{\left( \Delta t \right...

Let’s say that the vibrations of the system are described by the following governing equation:
$$m\frac{{{d}^{2}}x}{d{{t}^{2}}}+c\frac{dx}{dt}+kx=F\left( t \right)$$
Let the duration over which the solution of the previous equation is required be divided into n equal parts of interval h = ∆t each. To obtain a satisfactory solution, we must select a time step ∆t that is smaller than critical time step (∆t_cri)^2....

Example 1 - Find the natural frequencies of the system shown in the following figure with:
$${{m}_{1}}=m,{{m}_{2}}=2m,{{k}_{1}}=k,{{k}_{2}}=2k.$$
Determine the response of the system when k = 1000 N/m, m = 20 kg  and when initial values of displacements of the masses m_1 and m_2 are 1 and -1, respectively....