Mechanical vibrations Definition and 35 Threads

The "Vibrations and Waves" problem-solving course on MIT OCW has a section on driven harmonic oscillators which can be seen here. I would like to do the first of the two problems. Unfortunately, there are two issues 1) The latex is not rendering on that website (relatively minor issue, I think...

If ##\omega_0## is given (by the nature of the physical system under consideration, for example the spring constant and mass of a simple pendulum) then ##A## can be thought of as a function of the driving angular frequency ##\omega_d##. We can differentiate (1) and find the ##\omega_d## that...

I have a question about solving this system. I (naively, I think) initially did the following Trial solution: ##\vec{x}=e^{\lambda t}\vec{a}##. Sub this into the system (2) $$\lambda^2 e^{\lambda t}\vec{a}=M^{-1}Ke^{\lambda t}\vec{a}\tag{3}$$ $$(M^{-1}K-\lambda^2I)\vec{a}=0\tag{4}$$ I then...

Consider a system of ##n## coupled oscillators. ##n## particles are constrained to move in the ##x## direction. ##x_j## measures the displacement of the ##j##th particle from equilibrium. From Newton's 2nd law $$m_j\ddot{x_j}=F_j\tag{1}$$ "Because the system is linear, we expect that we can...

[I do not know if this is the right subforum] The answer to the question to the title is: for very long time. However the tuning fork clearly has to stop at some point because some of the energy will turn into heat. However I want to quantify for how long. More specifically I am interested on...

Is there a math that describes these shapes at least one frequency?

Hi, I am trying to get the transfer function from a wall between rooms. From one side I have the force of a hammer as an input ,and in the other side of the wall (next room) I have an accelerometer. Is it possible to get the TF without know the damping, stiffness and mass of the wall partition...

Summary: How to express ωₙ in terms of only mass (m) and stiffness (k)? I tried doing it with F=kx but it is out of my ability to simplify it to only m and k. Here is my approach:

Good Morning These documents suggest a good text on Vibrations. https://www.academia.edu/34400682/Mechanical_Vibrations_4600_431_Example_Problems?email_work_card=reading-history If you read the PDF, you see MANY example problems. However, each and every one is an abstract schematic. I keep...

Hi All, Anyone willing to help out in explaining what eigenfreuqncy for this oscilatory system, would be? Also if anybody knows the equation to calulate this stuff please, if you're willing to share I'd be greatful! Thanks, regards.

Hi, So the question is to: derive the equations of motion for the following in terms of x1 and x2? The bar is assumed to be light and rigid. (NB. I know I posted another vibrations problem earlier in which I tried to use an energy approach to get to the equations of motion. However, we haven't...

So I was trying to learn how to build a mathematical model of an oscillating system. The system and FBD is shown below. I just got confused why I got a different value of k from both x force and torsional equilibrium equation? Am I missing something?

I saw this general formula: ##w_{0} = \sqrt{\frac{k}{m}}## In my case both masses after collision create connected system, so ##w_{0} = \sqrt{\frac{k}{m+M}}## Plugging it into ##\omega = \sqrt{\omega_{0}^2 - \beta^2}## gives : ##\omega = \sqrt{\frac{k}{m+M} - \beta^2} = \sqrt{80 - 21^2} <...

Can someone double check my calculations? I will skip ##\theta## shift angle in all calculations for simplicity. ##x(t) = Acos(\omega t) \quad \land \quad \omega = \frac{2\pi}{T} \quad \land \quad A = \frac{1}{50}m = h## 1.1: ##x(\frac{1}{4}T)## = ? ##x(\frac{1}{4}T) = Acos(\frac{2\pi}{T}...

So I am almost sure I know how to solve this, just curious about the maximum velocity. Anyway, if you could double check my calculations, here it is. ##T = \frac{t}{n} = \frac{10s}{15} = \frac{2}{3}s## ##\omega = \frac{2\pi}{T} = 2\pi \frac{3}{2} = 3\pi## a). position at ##t = 0.8s##...

I'm now taking classes on mechanical vibrations and fluid dynamics, and those are two fields that are very interesting to me. I've always liked the subject of aerodynamics, but now I'm really liking to study mechanical vibrations, very interesting field of study. I'm looking for some specialty...

How can we determine the system properties (natural frequencies & damping ratio ) practically using forced vibration? One way is to measure the phase angle, using the bode plot, the exciting frequency corresponding to 90 (deg) phase shift should be the natural frequency, are there any other...

Homework Statement Homework EquationsThe Attempt at a Solution I found this solution for the nature frequency but here it does not include the Ball weight and centrifugal force in the moment balance equation about the pivot (O), it is wrong answer...is not it? I tried to solve the problem...

Hello, My name is Emre and I am a MSc student in mechanical engineering.I am looking for PhD in US.My research interests are mechanical vibrations,rail vehicles,finite element method and control theory.In fact I have 2.68/4 GPA in Undergrad and 3.79/4 GPA in Master.So if anyone has any advice...

Homework Statement Determine the equations of motions in terms of x and gamma. Assume small angles and that the wheel rolls without slip. The mass of the thin homogeneous large disk of radius 2R is 2m. The mass of the thin homogeneous inner disk of radius R is m. The rod of length 2R is...

Homework Statement Determine the equations of motions in terms of x and gamma. Assume small angles and that the wheel rolls without slip. The mass of the thin homogeneous large disk of radius 2R is 2m. The mass of the thin homogeneous inner disk of radius R is m. The rod of length 2R is...

Dear friends, I have designed a robotic arm driven by servo motor via a planetary gear box.I have used taper roller bearing & deep groove bearing to hold the driven shaft.The distance between two bearings is 10 mm .When I turn ON the motor, the motion of arm is jerky.Also, I've used lamina...

Homework Statement I am currently taking mechanical vibration course and am given coursework. A wooden block (20X200mm) floats half submerged in water. Determine the frequency of small oscillations of the block rolling from side to side. In this motion, the centre of mass remains in the plane...

hi! i'm beginner in the subject of vibrations. i'm trying to understand the waterfall plots. i have attached a waterfall plot. can someone please help me to understand it? waterfall plots are used when the operating speeds are varying (run up or coast down). in this plot, as the...

Here is my problem verbatim. A mass weighing 100g stretches a spring 5cm. If the mass is set in motion from its equilibrium position with a downward velocity of 10cm/s, and if there is no damping, determine the position \(u\) of the mass at any time \(t\). When does the mass first return to its...

The title may be incorrect, I named this after the section of my book in which this is located. My problem is as follows. Determine \(\omega_0\), R, and \(\delta\) so as to write the given expression in the form \(u=R\cos(\omega_0 t-\delta)\) \(\color{blue}{u=4\cos(3t)-2\sin(3t)},~\text{My...

I was wondering what the common methods for solving such a system are: 2 m \ddot{x} - m l \ddot{θ} θ + k x = 0 m l^{2} \ddot{θ} - m l \ddot{x} θ + m g l θ = 0

Homework Statement FYI - I don't have any knowledge on the mechanical aspects of electric motors. you're given a 400 hp electric motor that needs to run at varying rpms. The armature is 1200 lbs and is 800mm long with a diameter of 500 mm. the air gap that exists is 2.5 mm. At 800 rpm the air...

Homework Statement what distance x from the axis of variation O should be 1 kg slider, the system oscillations period is 0.9 s? T=0.9 s; the spring stiffness k=75 N/m; slider mass m=1 kg; beam mass m=3 kg; Homework Equations I\varphi(over letter "..")+k\varphi=0 \varphi(over...

Homework Statement Find the general solution for the differential equation Homework Equations y'' + 16y = tan(4t) The Attempt at a Solution I get C1cos(4t) + C2sin(4t) = 0 for my homogeneous equation. I did the usual method of undetermined coefficients (I think I took all of the...

Homework Statement A spring and dashpot system is to be designed for a 32lb weight so that the overall system is critically damped. (a) How must the damping constant γ and spring constant k be related? (b) Assume the system is to be designed so that the mass, when given an initial velocity of...

Homework Statement A mass m=4 is attached to both a spring, with spring constant k=37, and a dash-pot with damping constant c=4. The ball is started in motion with initial position x0=1 and initial velocity v0=8 . Determine the position function x(t). Homework Equations The...

Homework Statement Assume that the differential equation of a simple pendulum of length L is L\Theta'' + g\Theta=0 where g=GM/R^2 is the gravitational acceleration at the location of the pendulum. Two pendulums are of lengths L1 and L2 and when located at the respective distances R1 and...

Homework Statement A mass m is attached to a spring(massless) that is located inside a massless box. The box is falling under gravity. When the box starts to fall the spring is in it's equilibrium position and the box sticks to the ground when it hits it. -The box is a distance H from the...

Hi, I don't understand how to go about solving a problem like this; A weight stretches a spring 3 inches. It is set in motion at a point 4 inches below it's equilibrium point with zero velocity. -Find the maximum amplitude -When does it reach (the first time) it's highest point -Find...