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free viscous

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

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Contents

Interface
Free Viscous
Free Viscous
Page Comments

Private project under development, to help contact the author:

Interface

C++

Excel

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.

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1/free_viscous.png cannot be found in /users/1/free_viscous.png. Please contact the submission author.

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 $\inline \omega$ and $\inline \eta$ are calculated using the following rules

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

(2)

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

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

(3)

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

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

(4)

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

Available to Associate member only.

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.

Authors

Lucian Bentea (July 2007)

Free Viscous

doublefree_viscous(	double	`t`
	double	`omega`
	double	`eta`
	double	`init_displacement` `= 0`
	double	`init_velocity` `= 0`	)[inline]

The example below calculates the displacement of a single-degree-of-freedom system having an undamped natural frequency $\inline \omega = 5$ rad/s and a damping factor $\inline \eta = 20\%$ . Also the initial displacement $\inline u_0$ is considered null, while the initial velocity is $\inline \dot{u}_0 = 20$ 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

t	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)

Source Code

This module is private, for owner's use only.

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