Energy of motion in general
The energy of motion is usually called “kinetic energy” and is the amount of energy that an object has due to its motion.
Energy is a scalar, so it is independent of the direction of the motion for the object.
The size of the energy of motion is given by Ek=(mv2)/2, where Ek is the energy of motion, m [kg] is the mass of the object and v [m/s] is the velocity. The unit of energy in SI is Joule [J] and imperial units calories [kcal].
Energy of motion for an oscillating system
For an oscillating system given by a mass suspended on a spring the energy of motion is shifted back and forth between kinetic energy, Ek=(mv2)/2, and gravitational potential energy, Epg=mgh + spring potential energy, Epk=(kx2)/2, throughout the sinusoidal motion.
Since the energy is conserved we have: Epg+Epk+Ek=mgh+(kx2)/2+(mv2)/2=constant, where g 9.81[m/s2] is the gravitational acceleration constant and x [m] is the amplitude of the motion. In such a system no energy is transferred to the environment.
For an oscillating system given by a mass suspended on a spring and damper some of the energy is continually lost to dissipation, the conversion of kinetic energy to heat. The energy dissipated through one cycle of vibration is given by: Ed=πcωh2)/2, where c [m/s/N] is the damping coefficient and ω [rad/s] is the vibration frequency.
Energy of motion for fluid flow
For incompressible fluid flow the energy is conserved in a flowline and can be described by:
Ek+Ep+Ei=(mv2)/2+mgh+pV=constant, where Ei is the internal energy of the fluid, p [Pa] is the fluid pressure and V [m3] is the volume of the fluid. This equation is an alternative form of Bernoulli’s equation.
A more used form of the equation is found when dividing it by the volume V, giving:
(ρv2)/2+ρgh+p=constant, where ρ [kg/m3] is the density of the fluid. This last equation is a great form of Bernoulli’s equation given that many flows can be described as incompressible. The energies are now described in units of energy density (per unit volume). With the equation you can track how the energy is shifting to diffferent forms in a pipeline.
The first expression (ρv2)/2, is describing the energy of motion/kinetic energy density of the fluid and is highly correlated to vibration in pipelines. Different mechanisms can occur due to high energy of motion including flow induced turbulence (FIT), flow induced pulsation (FIP/FLIP) and rapid changes of density for multiphase flow (slug flow).
High pressure drops across a valve or from intrusive elements can create large acoustic sources, high frequency excitation, and cause cavitation and flashing for liquids.
Momentum Technologies pipeline vibration products and services
Understanding the mechanisms causing vibration, including the energy of motion, is the core in a root cause analysis of vibration. Momentum Technologies has great experience in understanding and solving vibration problems and provide services for vibration measurement and structural analysis of pipelines:
Simple harmonic motion in general
Simple harmonic motion is motion where the only force acting on a body is a restoring force proportional to the displacement of the body is present. The restoring force is acting in the opposite direction of the motion.
What does this mean?
A force that is proportional to the displacement of a body is in mathematical terms: F=kx, where F is the acting force, x is the displacement and k is a factor that says how large the force is. If the restoring force is a spring it is called the spring constant which is a simplification of a mechanical spring where the spring force is linearly increasing with the displacement of the spring. So by doubling the displacement of the spring you double the force.
When the restoring force is acting in the opposite direction of the motion it simply means that you have a force that is given by F=-kx. This describes the force that a spring has when you push or pull it out of its equilibrium position.
But we are not talking about statics, but dynamics when we are using the word “motion”, and you know from Newton’s second law that the sum of forces acting on a body is equal to mass times the acceleration: ΣF=ma. So simple harmonic motion describes the motion where the only force acting on the body is F=-kx.
Put into Newton’s second law: -kx=ma.
Vibration and harmonic motion
Observing such a system experiencing simple harmonic motion you would see it moving back and forth in a sinusoidal way, x=A*sin(ω*t), where x is the displacement from the equilibrium position, A is the amplitude of the motion, ω is the angular frequency and t is the time.
If you differentiate the displacement twice you get the acceleration:
v=Aω*cos(ω*t) => a=-Aω2*sin(ω*t).
If you put the expression for the acceleration into the previous expression you get:
-kx=ma => -k*(A*sin(ω*t))=m*(-Aω2*sin(ω*t)).
Simplifying you get: ω2=k/m.
This last expression describes the natural way of such a simple system to move, simple harmonic motion, and ω is then called the natural frequency or eigenfrequency of the system.
Below you can see a simple mass attached to a spring modelled in OpenModelica, where the mass has simple harmonic motion:
Simple harmonic motion and Momentum Technologies
Understanding natural frequencies is the core in measurement and analysis of vibration. Momentum Technologies has great experience in understanding and solving vibration problems and provide services for vibration measurement and structural analysis of pipelines: