CodeCogs

Determines the free vibration of a single-degree-of-freedom system with viscous damping

Interface

#include <codecogs/engineering/structures/free_viscous.h>

using namespace Engineering::Structures;

inline double free_viscous(double t, double omega, double eta, double init_displacement = 0, double init_velocity = 0)
inline double free_viscous(double t, double m, double k, double c, double init_displacement, double init_velocity)

Overview

Consider the diagram below in which you have an object of mass m sliding with no friction due to the action of the spring in the left side of the diagram having spring constant k.

This module computes the displacement u of the object at any given moment of time, relative to the static equilibrium point. The constant c is called the coefficient of viscous damping and the values of $\omega$ and $\eta$ are calculated using the following rules

$\omega^2 = \frac{k}{m}, \qquad \eta = \frac{c}{c_{cr}},$

(5)

where $c_{cr}$ is the critical damping coefficient and may be calculated using the formula:

$c_{cr} = 2 m \omega.$

(6)

Since the governing equation of this type of motion is given by

$\ddot{u} + 2 \eta \omega \dot{u} + \omega u = 0,$

(7)

this module basically evaluates the solution to the above second-order linear differential equation with constant coefficients.

We distinguish four types of solutions, based on the value of the viscous damping factor $\eta$ . Also consider the initial conditions $u_0 := u(0)$ and $\dot{u}_0 := \dot{u}(0)$ .

the undamped case ( $\eta = 0$ )

$u(t) = u_0 \cos \omega t + \frac{\dot{u}_0}{\omega} \sin \omega t$

(1)

the underdamped case ( $\eta < 1$ )

$u(t) = \mathrm{e}^{-\eta \omega t} \left( u_0 \cos \omega_1 t + \frac{\dot{u}_0 + \eta \omega u_0}{\omega_1} \sin \omega_1 t\right), \qquad \omega_1 = \omega \sqrt{1 - \eta^2}$

(2)

the critically damped case ( $\eta = 1$ )

$u(t) = \mathrm{e}^{-\eta \omega t} \left[u_0 + \left(\dot{u}_0 + \eta \omega u_0\right) t\right]$

(3)

the overdamped case ( $\eta > 1$ )

$u(t) = \mathrm{e}^{-\eta \omega t} \left( u_0 \cosh \omega_2 t + \frac{\dot{u}_0 + \eta \omega u_0}{\omega_2} \sinh \omega_2 t\right), \qquad \omega_2 = \omega \sqrt{\eta^2 - 1}$

(4)

References

"Structural Dynamics - An Introduction to Computer Methods", Roy R. Craig, Jr.

Note:

This module can also be used to study vertical free vibration, relative to the appropriate point of static equilibrium determined by the cancellation of the weight of the object and the other forces in the system.

FUNCTION

free_viscous

The example below calculates the displacement of a single-degree-of-freedom system having an undamped natural frequency $\omega = 5$ rad/s and a damping factor $\eta = 20\%$ . Also the initial displacement $u_0$ is considered null, while the initial velocity is $\dot{u}_0 = 20$ m/s. The solution is evaluated over a period of 10 seconds with steps of half a second.

Example 1

#include <codecogs/engineering/structures/free_viscous.h>
#include <stdio.h>

int main() {
  double omega = 5, eta = 0.2,
  init_displacement = 0, init_velocity = 20;

  for (double t = 0; t < 10.5; t += 0.5)
    printf("t = %lf\tu(t) = %lf\n", t, 
    Engineering::Structures::free_viscous(t, omega, eta, 
    init_displacement, init_velocity));

  return 0;
}

Output

t = 0.000000    u(t) = 0.000000
t = 0.500000    u(t) = 1.580175
t = 1.000000    u(t) = -1.475793
t = 1.500000    u(t) = 0.796992
t = 2.000000    u(t) = -0.201432
t = 2.500000    u(t) = -0.105072
t = 3.000000    u(t) = 0.172233
t = 3.500000    u(t) = -0.122202
t = 4.000000    u(t) = 0.050769
t = 4.500000    u(t) = -0.002460
t = 5.000000    u(t) = -0.016380
t = 5.500000    u(t) = 0.016203
t = 6.000000    u(t) = -0.009107
t = 6.500000    u(t) = 0.002544
t = 7.000000    u(t) = 0.000974
t = 7.500000    u(t) = -0.001846
t = 8.000000    u(t) = 0.001365
t = 8.500000    u(t) = -0.000596
t = 9.000000    u(t) = 0.000055
t = 9.500000    u(t) = 0.000168
t = 10.000000   u(t) = -0.000177

Parameters

the moment of time at which to evaluate the displacement of the object (seconds)

omega

undamped circular natural frequency (radians per second)

eta

viscous damping factor (dimensionless)

init_displacement

Default value = 0

init_velocity

Default value = 0

Returns

the displacement of the object from the static equilibrium point at time t (meters)

Interactive Calculator

omega

eta

init_displacement

init_velocity

Result –

FUNCTION

free_viscous

The example below calculates the displacement of a single-degree-of-freedom system having an object of mass $m = 0.008$ kg, a spring constant $k = 0.2$ N/m and a coefficient of viscous damping $c = 0.016$ Ns/m. Also the initial displacement $u_0$ is considered null, while the initial velocity is $\dot{u}_0 = 20$ m/s. The solution is evaluated over a period of 10 seconds with steps of half a second.

Example 1

#include <codecogs/engineering/structures/free_viscous.h>
#include <stdio.h>

int main() {
  double m = 0.008, k = 0.2, c = 0.016,
  init_displacement = 0, init_velocity = 20;

  for (double t = 0; t < 10.5; t += 0.5)
    printf("t = %lf\tu(t) = %lf\n", t, 
    Engineering::Structures::free_viscous(t, m, k, c, 
    init_displacement, init_velocity));

  return 0;
}

Output

t = 0.000000    u(t) = 0.000000
t = 0.500000    u(t) = 1.580175
t = 1.000000    u(t) = -1.475793
t = 1.500000    u(t) = 0.796992
t = 2.000000    u(t) = -0.201432
t = 2.500000    u(t) = -0.105072
t = 3.000000    u(t) = 0.172233
t = 3.500000    u(t) = -0.122202
t = 4.000000    u(t) = 0.050769
t = 4.500000    u(t) = -0.002460
t = 5.000000    u(t) = -0.016380
t = 5.500000    u(t) = 0.016203
t = 6.000000    u(t) = -0.009107
t = 6.500000    u(t) = 0.002544
t = 7.000000    u(t) = 0.000974
t = 7.500000    u(t) = -0.001846
t = 8.000000    u(t) = 0.001365
t = 8.500000    u(t) = -0.000596
t = 9.000000    u(t) = 0.000055
t = 9.500000    u(t) = 0.000168
t = 10.000000   u(t) = -0.000177

Parameters

the moment of time at which to evaluate the displacement of the object (seconds)

the mass of the object (kilograms)

the spring constant (Newtons per meter)

the coefficient of viscous damping (Newtons per meter per second)

init_displacement

the initial displacement of the object from the static equilibrium point (meters)

init_velocity

the initial velocity of the object (meters per second)

Returns

the displacement of the object from the static equilibrium point at time t (meters)

Interactive Calculator

init_displacement

init_velocity

Result –