Harmonic Motion

Question 1

Describe harmonic motion.

Answer 1

Harmonic motion involves oscillatory movement where the restoring force is directly proportional to the displacement from the equilibrium position.

Question 2

What are the different classes of damping in harmonic motion?

Answer 2

The classes of damping in harmonic motion are underdamped, overdamped, and critically damped.

Question 3

Define simple harmonic motion (SHM).

Answer 3

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.

Question 4

How is forced harmonic motion different from free harmonic motion?

Answer 4

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.

Question 5

What is resonance in the context of harmonic motion?

Answer 5

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.

Question 6

How is harmonic motion illustrated best by an example?

Answer 6

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.

Question 7

Describe the motion shown schematically with equation provided.

Answer 7

The motion schematically is characterized by a sinusoidal function with an initial phase angle of 0.

Question 8

Describe amplitude in the context of oscillation.

Answer 8

Amplitude is the maximum displacement of a body from its equilibrium position during oscillation.

Question 9

Define period in the context of oscillation.

Answer 9

Period is the time taken for a complete cycle of oscillation to occur.

Question 10

How is frequency calculated in the context of oscillation?

Answer 10

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.

Question 11

Describe the concept of phase in the context of motion.

Answer 11

Phase in motion refers to the difference in timing or position between two oscillatory motions, often measured in terms of phase angles.

Question 12

Define phase angles in the context of motion.

Answer 12

Phase angles, denoted as φ and α, represent the angular positions within an oscillatory motion, determining the relationship and alignment between different motions.

Question 13

How are two motions described as being in phase or out of phase?

Answer 13

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.

Question 14

How do you find the time ‘t’ between two peaks of phi and alpha?

Answer 14

t_0=(φ−α)/ω

Question 15

Describe free-undamped harmonic motion.

Answer 15

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.

Question 16

Define equilibrium position in the context of harmonic motion.

Answer 16

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.

Question 17

How is the free-body diagram for a position vector r=x i represented in the given context?

Answer 17

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.

Question 18

How are the force vertically in a free-undamped system?

Answer 18

balanced

Question 19

Define Newton’s 2nd law of motion.

Answer 19

It states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass.

Question 20

How is the differential equation for free-undamped harmonic motion characterized in mechanics?

Answer 20

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.

Question 21

Describe Euler’s formula.

Answer 21

Euler’s formula states that e^(iθ) = cos(θ) + i sin(θ), where θ is a real number.

Question 22

Describe the process of obtaining coefficients B1 and B2 in the real solution of the differential equation.

Answer 22

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).

Question 23

Define the initial conditions given the content.

Answer 23

The initial conditions are x = x0 and ẋ = ẋ0 at time t = 0

Question 24

Define amplitude and phase angle in the context of the solution.

Answer 24

Amplitude (C) is √(x0^2 + (ẋ0/ω)^2) and phase angle (φ) is arctan(ωx0/ẋ0).

Question 25

What is involved in a free-damped harmonic motion system?

Answer 25

Viscous dashpot

Question 26

What is the force of the viscous dashpot denoted by?

Answer 26

rẋ

Question 27

What is the equation of motion obtained for free-damped?

Answer 27

−kxi − rx ̇i = mx ̈i

Question 28

How can the damping ratio be obtained?

Answer 28

r/2mω

Question 29

What are the features of an over damped system?

Answer 29

• 2 real, distinct and negative eigenvalues λ1 and λ2
• No oscillations
• ξ > 1

Question 30

What are the features of critically damped system?

Answer 30

• 2 repeated, real and negative eigenvalues λ1 = λ2 = −ω
• ξ = 1
• No oscillations

Question 31

What are the features of underdamped system?

Answer 31

2 complex conjugate eigenvalues
- ξ < 1
- λ =−ξω ± iω_d

Question 32

How can you calculate the damped natural circular frequency?

Answer 32

ω_d =ω√1−ξ^2

Question 33

How do oscillations occur in underdamped systems?

Answer 33

T_d=2pi/ω_d

Question 34

How do you work out a numerical value for damping ratio?

Answer 34

By measuring successive peak values of deflection of the decaying oscillation

Question 35

What is the ratio of two successive peak displacements?

Answer 35

e^((2πξ)/√( 1−ξ^2))

Question 36

What os the logarithmic decrement denoted by?

Answer 36

δ

Question 37

How can you fine the logarithmic decrement?

Answer 37

Taking nature log of ratio of two succesive peaks

Question 38

What are examples of forced motion?

Answer 38

• Structures shaken by wind or earthquakes.
• Rotating machinery vibrating floors.
• Heavy traffic passing on raised surfaces such as roads and bridges.

Question 39

What forces are involved in force-damped motion in horizontal direction?

Answer 39

• F_s = kx
• F_d = rẋ
• F_0sinΩt

Question 40

What is the RHS of the ODE called?

Answer 40

Forcing term

Question 41

What can the ODE of the force-damped system be described as?

Answer 41

Non-homogenous

Question 42

What are the terms called in the damped case when there is a non-homogenous equation>

Answer 42

1. transient component
   2 and 3: stead state

Question 43

What is the maxim dynamic force on the mass m?

Answer 43

F_max=kx_max

Question 44

How can the DMF be obtained?

Answer 44

X/d=1/(1-β^2)

Question 45

What is called when DMF tend to infinity?

Answer 45

Resonance

Question 46

What happens when β = 1?

Answer 46

DMF function having a smaller peak value as the damping ratio ξ is increased

Question 47

When does max DMF occur?

Answer 47

β ≤ 1

Question 48

How can frequency of structure change?

Answer 48

By altering the mass and stiffness during design;

Question 49

What are Tuned-Mass Damping systems?

Answer 49

• Can be designed at concept stage
• Can also be a retrofitting measure