What is Damped harmonic oscillator: Definition and 51 Discussions
In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force F proportional to the displacement x:
F
→
=
−
k
x
→
,
{\displaystyle {\vec {F}}=-k{\vec {x}},}
where k is a positive constant.
If F is the only force acting on the system, the system is called a simple harmonic oscillator, and it undergoes simple harmonic motion: sinusoidal oscillations about the equilibrium point, with a constant amplitude and a constant frequency (which does not depend on the amplitude).
If a frictional force (damping) proportional to the velocity is also present, the harmonic oscillator is described as a damped oscillator. Depending on the friction coefficient, the system can:
Oscillate with a frequency lower than in the undamped case, and an amplitude decreasing with time (underdamped oscillator).
Decay to the equilibrium position, without oscillations (overdamped oscillator).The boundary solution between an underdamped oscillator and an overdamped oscillator occurs at a particular value of the friction coefficient and is called critically damped.
If an external time-dependent force is present, the harmonic oscillator is described as a driven oscillator.
Mechanical examples include pendulums (with small angles of displacement), masses connected to springs, and acoustical systems. Other analogous systems include electrical harmonic oscillators such as RLC circuits. The harmonic oscillator model is very important in physics, because any mass subject to a force in stable equilibrium acts as a harmonic oscillator for small vibrations. Harmonic oscillators occur widely in nature and are exploited in many manmade devices, such as clocks and radio circuits. They are the source of virtually all sinusoidal vibrations and waves.
I have been able to cut it down quite a bit and when I worked out the uncertainties and made the conclusion my teacher was unsure about my uncertainties.
Homework Statement
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Let us assume that neutral atoms or molecules can be modeled as harmonic oscillators in some cases. Then, the equation of the displacement between nucleus and electron cloud can be written as
$$\mu\left(\frac{d^x}{dt^2}+\gamma\frac{dx}{dt}+\omega_0^2x\right)=qE.$$
where...
Hello,
I have a question regarding Damped Harmonic Motion and I was wondering if anyone out there could help me out? Under normal conditions, gravity will not have an affect on a damped spring oscillator that goes up and down. Gravity will just change the offset, and the normal force equation...
Homework Statement
Homework Equations
Complex number solutions
z= z0eαt
Energy equations and Q (Quality Factor)
The Attempt at a Solution
For this question, I followed my book's "general solution" for dampened harmonic motions, where z= z0eαt, and then you can solve for α and eventually...
Homework Statement
An oscillator when undamped has a time period T0, while its time period when damped. Suppose after n oscillations the amplitude of the damped oscillator drops to 1/e of its original value (value at t = 0).
(a) Assuming that n is a large number, show that...
If you consider b^2/m > 4*k, you can get the solution by using classic method (b = damping constant, m = mass and k = spring constant) otherwise you have to use complex numbers. How have the references books proved the solution for this differential equation?
Homework Statement
The acceleration amplitude of a damped harmonic oscillator is given by
$$A_{acc}(\omega) = \frac{QF_o}{m} \frac{\omega}{\omega _o} \sqrt{\it{R}(\omega)}$$
Show that as ##\lim_{\omega\to\infty}, A_{acc}(\omega) = \frac{F_o}{m}##
Homework Equations
$$\it{R}(\omega) =...
Homework Statement
A damped harmonic oscillator consists of a block (m = 2.72 kg), a spring (k = 10.3 N/m), and a damping force (F = -bv). Initially, it oscillates with an amplitude of 28.5 cm; because of the damping, the amplitude falls to 0.721 of the initial value at the completion of 7...
Homework Statement
A damped harmonic oscillator is driven by an external force of the form $$F_{ext}=F_0sin(\omega t)$$
Show that the steady state solution is given by $$x(t)=A(\omega)sin(\omega t-\phi)$$
where $$ A(\omega)=\frac{F_0/m}{[(\omega_0^2-\omega^2)^2+4\gamma^2\omega^2]^{1/2}} $$
and...
Homework Statement
On June 10, 2000, the Millennium Bridge, a new footbridge over the River Thames in London, England, was opened to the public. However, after only two days, it had to be closed to traffic for safety reasons. On the opening day, in fact, so many people were crossing it at the...
I uploaded a picture of what I am stuck on. I understand the equation of motion 3.4.5a for a damped oscillator but I don't understand how to use binomial theorem to get the expanded equation 3.4.5b. I am no where near clever enough to figure this one out. I know how to use binomial theorem to...
Homework Statement
consider any damped harmonic oscillator equation
m(d2t/dt2 +bdy/dt +ky=0
a. show that a constant multiple of any solution is another solution
b. illustrate this fact using the equation
(d2t/dt2 +3dy/dt +2y=0
c. how many solutions to the equation do you get uf you use this...
I know the equation for damped oscillation where the damping force depends on velocity. In that case the damped oscillation has a fixed angular frequency and thus time period! I am wondering if there are any types of damped oscillation where the time period is not constant i.e. the motion is not...
Relaxation time is defined as the time taken for mechanical energy to decay to 1/e of its original value.
Why do we take a specific ratio of 1/e? What is its significance?
Homework Statement
An oscillator with mass 0.5 kg, stiffness 100 N/m, and mechanical resistance 1.4 kg/s is driven by a sinusoidal force of amplitude 2 N. Plot the speed amplitude and the phase angle between the displacement and speed as a function of the driving frequency and find the...
Homework Statement
The terminal speed of a freely falling object is v_t (assume a linear form of air resistance). When the object is suspended by a spring, the spring stretches by an amount a. Find the formula of the frequency of oscillation in terms of g, v_t, and a. Homework Equations
the...
Hi,
in this article:
http://dx.doi.org/10.1016/S0021-9991(03)00308-5
damped molecular dynamics is used as a minimization scheme.
In formula No. 9 the author gives an estimator for the optimal damping frequency:
Can someone explain how to find this estimate?
best,
derivator
Homework Statement
For a lightly damped harmonic oscillator and driving frequencies close to the natural frequency \omega \approx \omega_{0}, show that the power absorbed is approximately proportional to
\frac{\gamma^{2}/4}{\left(\omega_{0}-\omega\right)^{2}+\gamma^{2}/4}
where \gamma is...
The model of damped harmonic oscillator is given by the composite system with the hamiltonians ##H_S\equiv\hbar \omega_0 a^\dagger a##, ##H_R\equiv\sum_j\hbar\omega_jr_j^\dagger r_j##, and ##H_{SR}\equiv\sum_j\hbar(\kappa_j^*ar_j^\dagger+\kappa_ja^\dagger...
Homework Statement
I have a ball of 20 kg describing a damped harmonic movement, ie,
m*∂^2(x)+R*∂x+K*x=0,
with m=mass, R=resistance, K=spring constant.
The initial position is x(0)=1, the initial velocity is v(0)=0.
Knowing that v(1)=0.5, v(2)=0.3, I have to calculate K and R...
This is a problem I've been trying to solve for quite some time now. Any help would be appreciated.
Homework Statement
When a person with the mass of 105kg sits in a car, the body of the car descends by 2,5cm in total. In the car there are four shock absorbers filled with oil and a spring...
Homework Statement
The displacement amplitude of a lightly damped oscillator with m=0.250kg and k=6400N/m is observed to decrease by 15% in exactly five minutes
a) Calculate the fraction (in%0 of the initial mechanical energy of the oscillator that has been converted to other forms of energy...
Homework Statement
I was wondering if there was a general method for finding a function that fits a set of data for a damped harmonic oscillator
I'm currently writing up a presentation on the experiment for the gravitational constant and the way i did the experiment was to use a torsion...
Homework Statement
A damped harmonic oscillator is displaced a distance xo from equilibrium and released with zero initial velocity. Find the motion in the underdamped, critically damped, and overdamped case.
Homework Equations
d2x/dt2 + 2K dx/dt + ω2x = 0
Underdamped: x =...
Homework Statement
A spring is elastically stretched 10 cm if a force of 3 Newtons is imposed. A 2 kg mass is hung from the spring and is also attached to a viscous damper that exerts a restraining force of 3 Newtons when the velocity of the mass is 5 m/sec. An external force time function...
Homework Statement
The equation for a damped oscillator is d2x/dt2+2βdx/dt +ω02 x = 0. Let ω0=1.0 s−1 and β = 0.54 s−1. The initial values are x(0) = x0 and v(0)=0.
Determine x(t)/x0 at t = 2π/ω0.
Homework Equations
the solution to equation is given by...
Homework Statement
The logarithmic decrement δ of a lightly damped oscillator is defined to be the natural logarithm of the ratio of successive maximum displacements (in the same direction) of a free damped oscillator. That is, δ = ln(An/An+1) where An is the maximum displacement of the n-th...
Homework Statement
A damped harmonic oscillator is being forced. I have to say whether it is direct forcing or forcing by displacement. I have the equation of motion which is expressed in terms of the particle's height above the equilibrium point and an expression for the force being...
Hi there,
I just started an intermediate classical mechanics course at university and was smacked upside the head with this question that I don't know how to even start.
Homework Statement
We are to find the response function of a damped harmonic oscillator given a Forcing function. The...
Homework Statement
a block of mass m=.5kg is sliding on a horizontal table with coefficients of static and kinetic friction of .8 and .5 respectively. It is attached to a wall with a spring of unstretched length l=.13m and force constant 200 n/m. The block is released from rest at t=0 when...
Homework Statement
There is a block attached to the wall via a spring. The only damping force is friction, where there is kinetic and static.
Homework Equations
m(d^2x/dt^2)=-kx-?
The Attempt at a Solution
I can solve this, except usually the damping force is given as...
I am supposed to find the number of mircostates for the following Hamiltonian
\
\begin{equation}
\Sigma {(q_n+mwp_n)^2}<2mE
\end{equation}
So I am attempting to take the integral as follows
\
\int e^{(q_n+mwp_n)^2} d^{3n}q d^{3n} p
[tex\]
I found a solution that tells me
\...
Hi everyone,
I'm dealing with system identification for the first time in my life and am in desperate need of help :) The system is spring-mounted and I'm analyzing the vertical and torsional displacements. However, it seems like the vertical and torsional oscillations are coupled (shouldn't...
Homework Statement
"Show that the ratio of two successive maxima in the displacement of a damped harmonic oscillator is constant."Homework Equations
x = a e^(-\upsilont/2) cos (\omegat - \vartheta)The Attempt at a Solution
So I want to find when this beast has its maximum values, so I take the...
damped harmonic oscillator, urgent help needed!
Homework Statement
for distinct roots (k1, k2) of the equation k^2 + 2Bk + w^2 show that x(t) = Ae^(k1t) + Be^(k2t) is a solution of the following differential equation: (d^2)x/dt^2 + 2B(dx/dt) + (w^2)x = 0
Homework Equations
The...
I have trouble understanding how damping affects the period (of a torsion pendulum). I know that damping affects the amplitude of the oscillator, however how would damping change the period then?
I have a feeling this has to do with angular frequency, w, given by:
w = sqrt( (k/m) -...
Homework Statement
A damped harmonic oscillator originally at rest and in its equilibrium position is subject to a periodic driving force over one period by F(t)=-\tau^2+4t^2 for -\tau/2<t<\tau/2 where \tau =n\pi/\omega
a.) Obtain the Fourier expansion of the function in the integral...
1. The equation of motion is Ma(t) +rv(t) + Kx(t)=0
a) Look for a solution of this equation with x(t) proportional exp(-Ct) and find two possible values of C.
Homework Equations
3. No clue... Please help if you can!
Homework Statement
Show that the fractional energy lost per period is
\frac{\Delta E}{E} = \frac{2\pi b}{m\omega_0} = \frac{2\pi}{Q}
where \omega_0 = \srqt{k/m} and Q = m\omega_0 / b
Homework Equations
E = 1/2 k A^2 e^{-(b/m)t} = E_0 e^{-(b/m)t}
The Attempt at a Solution
\Delta E = 1/2 k A^2...
A mass m moves along the x-axis subject to an attractive force given by \frac {17} {2} \beta^2 m x and a retarding force given by 3 \beta m \dot{x}, where x is its distance from the origin and \beta is a constant. A driving force given by m A \cos{\omega t} where A is a constant, is applied to...
Driven Damped Harmonic Oscillator, f != ma??
Let's say I've got a driven damped harmonic oscillator described by the following equation:
A \ddot{x} + B \dot{x} + C x = D f(t)
given that f = ma why can't I write
A \ddot{x} + B \dot{x} + C x = D ma
substitute \ddot{x} = a to get
A \ddot{x}...
I'm trying to find the work done by a harmonic oscillator when it moves from x_{0} = 0 m to x_{max} = 1 m.
The oscillator has initial velocity v_{0}, a maximum height of x_{max} = 1 m, initial height of x_{0} = 0 m, a spring constant of k, a mass of m = 1 kg, and a damping factor of b.
It can...
Please I don't understand this problem at all:
Consider a driven damped harmonic oscillator.Calculate the power dissipated by the damping force?
calculate the average power loss, using the fact that the average of (sin(wt+phi) )^2 over a cycle is one half?
Please can I have some help for...
Driven, damped harmonic oscillator -- need help with particular solution
Consider a damped oscillator with Beta = w/4 driven by
F=A1cos(wt)+A2cos(3wt). Find x(t).
I know that x(t) is the solution to the system with the above drive force.
I know that if an external driving force applied...
Question:
(a) Show that the total mechanical energy of a lightly damped harmonic oscillator is
E = E_0 e^{-bt/m}
where E_0 is the total mechanical energy at t = 0.
(b) Show that the fractional energy lost per period is
\frac{\Delta E}{E} = \frac{2 \pi b}{m \omega_0} = \frac{2...
Question: A damped harmonic oscillator loses 5.0 percent of its mechanical energy per cycle. (a) By what percentage does its frequency differ from the natural frequency \omega_0 = \sqrt{k/m}? (b) After how may periods will the amplitude have decreased to 1/e of its original value?
So, for...
Hi,
I'm having a lot of trouble with a damped harmonic oscillator problem:
A damped harmonic oscillator consists of a block (m=2.00kg), a spring (k=10 N/m), and a damping force (F=-bv). Initially it oscillates with an amplitude of 25.0cm. Because of the damping force, the amplitude falls...