generate.2order {doremi}

Generation of the second order differential equation solution with deSolve

Description

generate.2order returns a data frame containing the time (supplied as input) and a simulated signal generated as a solution to a second order differential equation with constant coefficients that are provided as inputs:

\frac{d^2y}{dt} + 2\xi\omega_{n}\frac{dy}{dt} + \omega_{n}^2 y = \omega_{n}^2 k*u(t)

Where: y(t) is the signal, \frac{dy}{dt} its derivative and \frac{d^2y}{dt} its second derivative

• \omega_{n} = \frac{2\pi}{T} -where T is the period of the oscillation- is the system's natural frequency, the frequency with which the system would vibrate if there were no damping. The term \omega_{n}^2 represents thus the ratio between the attraction to the equilibrium and the inertia. If we considered the example of a mass attached to a spring, this term would represent the ratio of the spring constant and the object's mass.

• \xi is the damping ratio. It represents the friction that damps the oscillation of the system (slows the rate of change of the variable). The term 2\xi\omega_n thus represents the respective contribution of the inertia, the friction and the attraction to the equilibrium. The value of \xi determines the shape of the system time response, which can be: \xi<0 Unstable, oscillations of increasing magnitude \xi=0 Undamped, oscillating 0<\xi<1 Underdamped or simply "damped": the oscillations are damped by an exponential of damping rate \xi\omega_{n} \xi=1 Critically damped \xi>1 Over-damped, no oscillations in the return to equilibrium

• k is the gain

• u(t) is an external excitation perturbing the dynamics

The excitation is also provided as input and it can be null (then the solution will be a damped linear oscillator when the initial condition is different from 0)

Usage

generate.2order(
  time = 0:100,
  excitation = NULL,
  y0 = 0,
  v0 = 0,
  t0 = NULL,
  xi = 0.1,
  period = 10,
  k = 1,
  yeq = 0
)

Arguments

time

is a vector containing the time values corresponding to the excitation signal.

excitation

Is a vector containing the values of the excitation signal.

y0

is the initial condition for the variable y(t=t0), (0, by default), it is a scalar.

v0

is the initial condition for the derivative dy(t=t0), (0, by default), it is a scalar.

t0

is the time corresponding to the initial condition y0 and v0. Default is the minimum value of the time vector.

xi

is the damping factor. A negative value will produce divergence from equilibrium.

period

is the period T of the oscillation, T = \frac{2*\pi}{\omega_{n}} as mentioned

k

Default is 1. It represents the proportionality between the stationary increase of signal and the excitation increase that caused it. Only relevant if the excitation is non null.

yeq

is the signal equilibrium value, i.e. the stationary value reached when the excitation term is 0.

Value

Returns a data.table containing four elements:

• t is a vector containing the corresponding time values

• y is a vector containing the values calculated with deSolve so that y is a solution to a second order differential equation with constant coefficients (provided as input) evaluated at the time points given by t

• dy is a vector containing the values of the derivative calculated at the same time points

• exc is the excitation vector

Examples

generate.2order(time=0:249,excitation=c(rep(0,10),rep(1,240)),period=10)
generate.2order(y0=10)