IA: 1P1: Mechanical Vibrations

Question 1

What is the general expression for an “nth” order differential equation

Answer 1

Question 2

What are the 3 mechanical components for translational motion?

Answer 2

• Mass
• Damper (Dashpot)
• Spring

Question 3

In terms of forces, what role does a mass play in a mechanical system?

For translational motion

include relevant equations

Answer 3

Mass provides a resistive force proportional to acceleration:

Question 4

In terms of forces, what role does a damper (dashpot) play in a mechanical system?

For translational motion

include relevant equations

Answer 4

A damper/dashpot provides a resistive force proportional to velocity:

Question 5

In terms of forces, what role does a spring play in a mechanical system?

For translational motion

include relevant equations

Answer 5

A spring provides a resistive force proportional to displacement:

Question 6

In terms of energy, what role does a mass play in a mechanical system?

For translational motion

include relevant equations

Answer 6

A mass stores kinetic energy when in motion, it does not dissipate any energy. Equation for kinetic energy stored:

Question 7

In terms of energy, what role does a damper/dashpot play in a mechanical system?

For translational motion

include relevant equations

Answer 7

A damper/dashpot dissipates energy, it does not store any energy. Equation for the power dissipated:

Question 8

In terms of energy, what role does a spring play in a mechanical system?

For translational motion

include relevant equations

Answer 8

A spring stores potential energy when there is a displacement, it does not dissipate any energy. The equation for the potential energy stored:

Question 9

Draw a diagram for a mass in translational motion:

Answer 9

Question 10

Draw a diagram for a damper/dashpot in translational motion:

Answer 10

Question 11

Draw a diagram for a spring in translational motion:

Answer 11

Question 12

What are the 3 mechanical components for rotational motion?

Answer 12

• Intertia
• Torsional Damper
• Torsional Spring

Question 13

In terms of forces, what role does interia play in a mechanical system?

For rotational motion

Include relevant equations

Answer 13

Provides a torque proportional to angular acceleration:

Question 14

In terms of forces, what role does a torsional damper play in a mechanical system?

For rotational motion

Include relevant equations

Answer 14

Provides a torque proportional to angular velocity:

Question 15

In terms of forces, what role does a torsional spring play in a mechanical system?

For rotational motion

Include relevant equations

Answer 15

Provides a torque proportional to angular displacement:

Question 16

In terms of energy, what role does interia play in a mechanical system?

For rotational motion

Include relevant equations

Answer 16

It stores rotational kinetic energy, it does not dissipate any energy. Equation for rotational kinetic energy:

Question 17

In terms of energy, what role does a torsional damper play in a mechanical system?

For rotational motion

Include relevant equations

Answer 17

It dissipates energy, it does not store any energy. Equation for the power dissipated:

Question 18

In terms of energy, what role does a torsional spring play in a mechanical system?

For rotational motion

Include relevant equations

Answer 18

It stores rotational potential energy, it does not dissipate any energy. Equation for rotational potential energy:

Question 19

How can components of an electrical system be analagous to a mechanical system?

Answer 19

• Inductance ≡ Mass (L ≡ m)
• Resistance ≡ Damper (R ≡ λ)
• Capacitance ≡ Spring (1/C ≡ k)

Question 20

When will a mass-spring-damper system produce a first order differential equation and when will it produce a second order differential equation?

Answer 20

• A first order differential equation is produced when the mass is negligibly small and so the acceleration term can be ignored
• A second order differential equation is produced when the mass is of signifigance and so the acceleration term must be included
• A first order equation can also be produced when the effect of the spring is negligible, you are then left with an acceleration and a velocity term. Since acceleration is the first derivative of velocity this second order differential equation (in terms of displacement) can become a first order differential equation (in terms of velocity).

Question 21

What are “compatible” motions?

Answer 21

Compatible motions are when both move by the same amount, i.e. a spring and damper in parallel.

Question 22

What must you consider when a spring and damper are in series?

Answer 22

As they are not in parallel, they are not compatible and so do not move by the same amount. Therefore you must consider equilibrium at 2 positions.

Question 23

What is the standard form of first order differential equation?

Answer 23

Question 24

For a first order differential equation in the form below, what is T?

Answer 24

T is the time constant, it has the units of time.

Question 25

When determining the governing equation of an electrical system, what is the concept of equilibrium in mechanical circuits analagous to?

Answer 25

The condition that the voltages around the circuit must sum to zero: * Voltage is analagous to force * Charge is analagous to displacement * Current is analagous to velocity

Question 26

What is the standard form for the solution to the first order differential equation in the form:

Answer 26

* A is determined from initial conditions * In practice we often do not need to evaluate the integral for the particular integral as there are alternative (easier) means to find it.

Question 27

What is the signifigance of plotting the complementary function (yᴄғ = Ae⁻ᵗ/ᵀ) to the below equation?

Answer 27

* Initial slope = -A/T * Initial slope intercepts the **asymptode** at t = T * t = T when y = Ae⁻¹

Question 28

How can you determine the particular integral for the below equation for the step response (x = H(t))?

Answer 28

yₚᵢ = x₀

Question 29

How can you determine the particular integral for the below equation for an impulse response (x = δ(t))?

Answer 29

yₚᵢ = 0

Question 30

How do you determine the initial conditions for the below equation for an impulse response (x = δ(t))?

Answer 30

Integrating over the equation of motion over a small time interval in the region of t = 0 (0- and 0+), allows you to find the initial conditions.

Question 31

What is an alternative method to finding the impulse response to the below first order differential equation rather than finding the particular integral and integrating to determine the initial conditions?

Answer 31

Since the impulse function is the derivative of the step function, you could determine the step response and then differentiate it to determine the impulse response. Impulse response = derivative of step response

Question 32

How can you determine the particular integral for the below equation for a ramp response (x = αt)?

Answer 32

**This is just a standard method for finding a particular integral!** see example below:

Question 33

What is an alternative method for finding the ramp response to the below equation (x = αt) rather than using the standard method to find the particular integral?

Answer 33

The derivative of the ramp function is the step function. Therefore you can simply integrate the step response to determine the ramp response!

Question 34

What are the methods for determining the particular integral for the harmonic response (x = Xcos(ωt)) to the equation?

Answer 34

* Solution by cos(ωt) method * Solution by eᶦʷᵗ method * eᶦʷᵗ method + phasor diagrams The particular integral will be in the form:

Question 35

How can you determine the particular integral for the harmonic response (x = Xcos(ωt)) to the equation **using the cos(ωt) method**?

Answer 35

This is just the standard method for finding the particular integral, an identity is then used to make it be in the form of a single phase shifted cosine:

Question 36

How can you determine the particular integral for the harmonic response (x = Xcos(ωt)) to the equation **using the eᶦʷᵗ method**?

Answer 36

φₓ assumes that the input is phase shifted. Therefore this may be zero if there is no phase shift.

Question 37

How can you determine the particular integral for the harmonic response (x = Xcos(ωt)) to the equation using the eᶦʷᵗ method **and phasor diagrams**? ## Footnote **This is the best method!!!**

Answer 37

Question 38

What is the identity for "**Acos(ωt) + Bsin(ωt)**"? ## Footnote NOT IN THE DATABOOK

Answer 38

Question 39

Why is the harmonic response to a differential equation so useful?

Answer 39

Any function can be expressed as a series of sines and cosines (Fourier series), therefore a harmonic response can be used for any input

Question 40

For a block of mass m, create an expression for the power within the system:

Answer 40

Question 41

What is the standard form of the differential equation?

Answer 41

Question 42

What does each constant in this equation represent? (ωₙ, ζ, x)

Answer 42

ωₙ = Natural frequency [rad s⁻¹] ζ = Damping ratio [dimensionless] x = "Displacement" input [m]

Question 43

What is the equation for the natural frequency, ωₙ?

Answer 43

Question 44

What is the equation for the damping ratio, ζ?

Answer 44

Question 45

What is the equation for the "displacement" input?

Answer 45

Question 46

What is the complementary function for the below differential equation?

Answer 46

Question 47

How does the value of ζ effect the complementary function and hence the response of the system?

Answer 47

* ζ > 1: "Over damped" * ζ = 1: "Critically damped" * ζ < 1: "Under damped"

Question 48

What is the complementary function like for an "over damped" scenario (ζ > 1)?

Answer 48

Question 49

What is the complementary function like for a "critically damped" scenario (ζ = 1)?

Answer 49

Question 50

What is the complementary function like for an "under damped" scenario (ζ < 1)?

Answer 50

Third form is preferred (single phase shifted cosine)

Question 51

What is the "damped natural frequency", ωd?

Answer 51

The frequency at which a damped system oscillates when it is displaced and then released. It is always lower than the undamped natural frequency due to the presence of damping. It is only relevant for under damping (ζ < 1) as it is the only **damped** response which shows oscillatory motion.

Question 52

What happens when ζ = 0?

Answer 52

There is no damping, therefore there is just simple harmonic motion at the natural frequency ωₙ

Question 53

How do you determine the particular integral for the step response (x = x₀H(t))?

Answer 53

yₚᵢ = x₀

Question 54

Given that the system is under damped (ζ < 1), how do you determine the initial conditions for the step response?

Answer 54

Question 55

Given that the system is under damped (ζ < 1), how do you determine the initial conditions for the impulse response (x = βδ(t)) ?

Answer 55

Determine the initial conditions for the unit step input, and then just differentiate the unit step response to find the unit impulse response ## Footnote Remember to multiply unit impulse response by β

Question 56

How can you measure the decay rate of a vibrating system?

Answer 56

Using the **Logarithmic Decrement**:

Question 57

Derive the expressions for the logarithmic decrement at any 2 peaks

Answer 57

Question 58

What are examples of vibrating systems where the "spring" is less obvious?

Answer 58

* A simple pendulum * A trifilar pendulum

Question 59

What does the base motion, z(t), represent?

Answer 59

It represents real-world practical situations such as an earthquake input to a building, or the effect of ground roughness on a vehicle

Question 60

What is the governing differential equation for this system? ## Footnote Express it in standard form, i.e. natural frequency ωₙ, damping ratio ζ, and displacement input x

Answer 60

In many cases z = 0 and so the right hand side is simply x

Question 61

How can you find the harmonic response function for the following system?

Answer 61

Using the eᶦʷᵗ and phasor diagram method

Question 62

What is the response ratio (and amplitude) of the harmonic resposne to this system?

Answer 62

Question 63

What is the phase between the input and output of the harmonic response to this system?

Question 64

How can you determine the phase and amplitude of the response of a second order system without using fairly complicated equations (such as those below)?

Answer 64

Use the data book graphs for the appropriate response. The databook also contains graphs for: * The amplitude of the step response of a linear second-order system initially at rest * The amplitude of the impulse response of a linear second-order system initially at rest * The amplitude and phase for 3 separate cases of the harmonic response of a linear second-order system

Question 65

What is the significance of the case of ζ = 1 for the amplitude of the harmonic response?

Answer 65

ζ = 1 is the condition for critical damping, it exhibits no resonance peak

Question 66

How does the resonance peak shift with an increase in damping?

Answer 66

It shifts towards lower frequencies

Question 67

How can you determine the expression for the **damped resonant frequency**?

Answer 67

Question 68

What is the expression for the **damped resonant frequency**?

Answer 68

Question 69

What is the **damped resonant frequency**?

Answer 69

It is the frequency at which the response amplitude peaks

Question 70

Whats the difference between the **damped resonant frequency** and the **damped natural frequency**?

Answer 70

* Damped Natural Frequency (ωd): The frequency at which a system oscillates on its own after being disturbed, without continuous external forcing. * Damped Resonant Frequency (ωr): The frequency at which a system shows the greatest response amplitude when it is continuously driven by an external sinusoidal force

Question 71

How can you determine the expression for the **maximum damped resonant amplitude**?

Answer 71

Question 72

What is the expression for the **maximum damped resonant amplitude**?

Answer 72

Question 73

What is the Q factor?

Answer 73

The maximum damped resonant amplitude when ζ is very small (ζ << 1)