Harmonic Motion Flashcards
Describe harmonic motion.
Harmonic motion involves oscillatory movement where the restoring force is directly proportional to the displacement from the equilibrium position.
What are the different classes of damping in harmonic motion?
The classes of damping in harmonic motion are underdamped, overdamped, and critically damped.
Define simple harmonic motion (SHM).
Simple harmonic motion is a type of harmonic motion where the restoring force is directly proportional to the displacement and acts towards the equilibrium position.
How is forced harmonic motion different from free harmonic motion?
Forced harmonic motion is when an external force is applied to the system, causing it to deviate from its natural oscillation, unlike free harmonic motion which occurs without external influence.
What is resonance in the context of harmonic motion?
Resonance in harmonic motion refers to the phenomenon where the amplitude of oscillations becomes significantly large when the frequency of the driving force matches the natural frequency of the system.
How is harmonic motion illustrated best by an example?
Harmonic motion is best illustrated by the example of a sinusoidal oscillatory motion equation, where s represents displacement, t represents time, a is the amplitude, ω is the circular frequency, and φ is the phase angle.
Describe the motion shown schematically with equation provided.
The motion schematically is characterized by a sinusoidal function with an initial phase angle of 0.
Describe amplitude in the context of oscillation.
Amplitude is the maximum displacement of a body from its equilibrium position during oscillation.
Define period in the context of oscillation.
Period is the time taken for a complete cycle of oscillation to occur.
How is frequency calculated in the context of oscillation?
Frequency is calculated as the number of cycles completed in one second, where frequency (f) equals the reciprocal of the period (T) or the angular frequency (ω) divided by 2π.
Describe the concept of phase in the context of motion.
Phase in motion refers to the difference in timing or position between two oscillatory motions, often measured in terms of phase angles.
Define phase angles in the context of motion.
Phase angles, denoted as φ and α, represent the angular positions within an oscillatory motion, determining the relationship and alignment between different motions.
How are two motions described as being in phase or out of phase?
Two motions are considered in phase when their phase angles φ and α are equal, while they are out of phase when the phase difference (φ - α) is not zero.
How do you find the time ‘t’ between two peaks of phi and alpha?
t_0=(φ−α)/ω
Describe free-undamped harmonic motion.
It is a type of motion where an object is displaced from its equilibrium position and released with an initial velocity, experiencing oscillations without any damping forces.
Define equilibrium position in the context of harmonic motion.
Equilibrium position is the point where the displacement of the object is zero (x=0) and there is no net force acting on the object.
How is the free-body diagram for a position vector r=x i represented in the given context?
It is represented by the sum of the spring force (Fs = kx) and the weight force (mg) acting on the object in the direction of motion.
How are the force vertically in a free-undamped system?
balanced
Define Newton’s 2nd law of motion.
It states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass.
How is the differential equation for free-undamped harmonic motion characterized in mechanics?
It is described as a homogeneous, linear second-order ODE with constant coefficients, represented by the equation ¨x + ω^2x = 0, where ω is the natural circular frequency.
Describe Euler’s formula.
Euler’s formula states that e^(iθ) = cos(θ) + i sin(θ), where θ is a real number.
Describe the process of obtaining coefficients B1 and B2 in the real solution of the differential equation.
The coefficients B1 and B2 are obtained by applying the initial conditions to the general solution, which is in the form of B1 cos(ωt) + B2 sin(ωt).
Define the initial conditions given the content.
The initial conditions are x = x0 and ẋ = ẋ0 at time t = 0
Define amplitude and phase angle in the context of the solution.
Amplitude (C) is √(x0^2 + (ẋ0/ω)^2) and phase angle (φ) is arctan(ωx0/ẋ0).
What is involved in a free-damped harmonic motion system?
Viscous dashpot
What is the force of the viscous dashpot denoted by?
rẋ
What is the equation of motion obtained for free-damped?
−kxi − rx ̇i = mx ̈i
How can the damping ratio be obtained?
r/2mω
What are the features of an over damped system?
- 2 real, distinct and negative eigenvalues λ1 and λ2
- No oscillations
- ξ > 1
What are the features of critically damped system?
- 2 repeated, real and negative eigenvalues λ1 = λ2 = −ω
- ξ = 1
- No oscillations
What are the features of underdamped system?
2 complex conjugate eigenvalues
- ξ < 1
- λ =−ξω ± iω_d
How can you calculate the damped natural circular frequency?
ω_d =ω√1−ξ^2
How do oscillations occur in underdamped systems?
T_d=2pi/ω_d
How do you work out a numerical value for damping ratio?
By measuring successive peak values of deflection of the decaying oscillation
What is the ratio of two successive peak displacements?
e^((2πξ)/√( 1−ξ^2))
What os the logarithmic decrement denoted by?
δ
How can you fine the logarithmic decrement?
Taking nature log of ratio of two succesive peaks
What are examples of forced motion?
- Structures shaken by wind or earthquakes.
- Rotating machinery vibrating floors.
- Heavy traffic passing on raised surfaces such as roads and bridges.
What forces are involved in force-damped motion in horizontal direction?
- F_s = kx
- F_d = rẋ
- F_0sinΩt
What is the RHS of the ODE called?
Forcing term
What can the ODE of the force-damped system be described as?
Non-homogenous
What are the terms called in the damped case when there is a non-homogenous equation>
- transient component
2 and 3: stead state
What is the maxim dynamic force on the mass m?
F_max=kx_max
How can the DMF be obtained?
X/d=1/(1-β^2)
What is called when DMF tend to infinity?
Resonance
What happens when β = 1?
DMF function having a smaller peak value as the damping ratio ξ is increased
When does max DMF occur?
β ≤ 1
How can frequency of structure change?
By altering the mass and stiffness during design;
What are Tuned-Mass Damping systems?
- Can be designed at concept stage
- Can also be a retrofitting measure
