In case of unsteady motion of bodies underwater or unsteady flow around objects, we must consider an additional effect (force) acting on the structure when formulating the system equation.
A typical mass-spring-dashpot system can be described by the following equation:
mx˙˙ + bx˙ + kx = f (t )
where m is the system mass, b is the linear damping coefficient, k is the spring coefficient, f(t) is a driving force acting on the mass, and x is the displacement of the mass. The natural frequency ω of the system is simply:
w=SQRT(k/m): 1.1
Given an object of mass m attached to a spring, you can determine the spring constant by
setting the mass in motion and observing the frequency of oscillation, then using equation
(1.1) to solve for k. Alternately we can also determine the spring coefficient k by simply
applying a force f and measuring the displacement x:
f = kx
The apparent mass of an object in air differs from the apparent mass of an object in water.
Statically, the buoyancy force acting on the body makes it appear less massive. Using a
load (force) cell to measure the weight Mg of an object will reveal this disparity quite
clearly. It is important to take this into account when formulating the natural frequency
of a spring-mass system in water. In this lab we will compare the natural frequency of
such a spring-mass system in air and in water. In addition to the buoyancy effect, an
added mass term must be considered.
In a physical sense, this added mass is the weight added to a system due to the fact that an
accelerating or decelerating body must move some
volume of surrounding fluid with it as it moves. The added mass force opposes the
motion and can be factored into the system equation as follows:
mx˙˙ + bx˙ + kx = f (t )-ma x˙˙
where ma is the added mass. Reordering the terms the system equation becomes:
(m+ma )x˙˙ + bx˙ + kx = f (t )
From here we can treat this again as a simple spring-mass-dashpot system with a new
mass a
m'= m+ ma
such that the natural frequency of the system is now
w=SQRT(k/m+ma):
A typical mass-spring-dashpot system can be described by the following equation:
mx˙˙ + bx˙ + kx = f (t )
where m is the system mass, b is the linear damping coefficient, k is the spring coefficient, f(t) is a driving force acting on the mass, and x is the displacement of the mass. The natural frequency ω of the system is simply:
w=SQRT(k/m): 1.1
Given an object of mass m attached to a spring, you can determine the spring constant by
setting the mass in motion and observing the frequency of oscillation, then using equation
(1.1) to solve for k. Alternately we can also determine the spring coefficient k by simply
applying a force f and measuring the displacement x:
f = kx
The apparent mass of an object in air differs from the apparent mass of an object in water.
Statically, the buoyancy force acting on the body makes it appear less massive. Using a
load (force) cell to measure the weight Mg of an object will reveal this disparity quite
clearly. It is important to take this into account when formulating the natural frequency
of a spring-mass system in water. In this lab we will compare the natural frequency of
such a spring-mass system in air and in water. In addition to the buoyancy effect, an
added mass term must be considered.
In a physical sense, this added mass is the weight added to a system due to the fact that an
accelerating or decelerating body must move some
volume of surrounding fluid with it as it moves. The added mass force opposes the
motion and can be factored into the system equation as follows:
mx˙˙ + bx˙ + kx = f (t )-ma x˙˙
where ma is the added mass. Reordering the terms the system equation becomes:
(m+ma )x˙˙ + bx˙ + kx = f (t )
From here we can treat this again as a simple spring-mass-dashpot system with a new
mass a
m'= m+ ma
such that the natural frequency of the system is now
w=SQRT(k/m+ma):
No comments:
Post a Comment