IA: 1P1: Mechanical Vibrations Flashcards
What is the general expression for an “nth” order differential equation
What are the 3 mechanical components for translational motion?
- Mass
- Damper (Dashpot)
- Spring
In terms of forces, what role does a mass play in a mechanical system?
For translational motion
include relevant equations
Mass provides a resistive force proportional to acceleration:
In terms of forces, what role does a damper (dashpot) play in a mechanical system?
For translational motion
include relevant equations
A damper/dashpot provides a resistive force proportional to velocity:
In terms of forces, what role does a spring play in a mechanical system?
For translational motion
include relevant equations
A spring provides a resistive force proportional to displacement:
In terms of energy, what role does a mass play in a mechanical system?
For translational motion
include relevant equations
A mass stores kinetic energy when in motion, it does not dissipate any energy. Equation for kinetic energy stored:
In terms of energy, what role does a damper/dashpot play in a mechanical system?
For translational motion
include relevant equations
A damper/dashpot dissipates energy, it does not store any energy. Equation for the power dissipated:
In terms of energy, what role does a spring play in a mechanical system?
For translational motion
include relevant equations
A spring stores potential energy when there is a displacement, it does not dissipate any energy. The equation for the potential energy stored:
Draw a diagram for a mass in translational motion:
Draw a diagram for a damper/dashpot in translational motion:
Draw a diagram for a spring in translational motion:
What are the 3 mechanical components for rotational motion?
- Intertia
- Torsional Damper
- Torsional Spring
In terms of forces, what role does interia play in a mechanical system?
For rotational motion
Include relevant equations
Provides a torque proportional to angular acceleration:
In terms of forces, what role does a torsional damper play in a mechanical system?
For rotational motion
Include relevant equations
Provides a torque proportional to angular velocity:
In terms of forces, what role does a torsional spring play in a mechanical system?
For rotational motion
Include relevant equations
Provides a torque proportional to angular displacement:
In terms of energy, what role does interia play in a mechanical system?
For rotational motion
Include relevant equations
It stores rotational kinetic energy, it does not dissipate any energy. Equation for rotational kinetic energy:
In terms of energy, what role does a torsional damper play in a mechanical system?
For rotational motion
Include relevant equations
It dissipates energy, it does not store any energy. Equation for the power dissipated:
In terms of energy, what role does a torsional spring play in a mechanical system?
For rotational motion
Include relevant equations
It stores rotational potential energy, it does not dissipate any energy. Equation for rotational potential energy:
How can components of an electrical system be analagous to a mechanical system?
- Inductance ≡ Mass (L ≡ m)
- Resistance ≡ Damper (R ≡ λ)
- Capacitance ≡ Spring (1/C ≡ k)
When will a mass-spring-damper system produce a first order differential equation and when will it produce a second order differential equation?
- 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).
What are “compatible” motions?
Compatible motions are when both move by the same amount, i.e. a spring and damper in parallel.
What must you consider when a spring and damper are in series?
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.
What is the standard form of first order differential equation?
For a first order differential equation in the form below, what is T?
T is the time constant, it has the units of time.
When determining the governing equation of an electrical system, what is the concept of equilibrium in mechanical circuits analagous to?
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
What is the standard form for the solution to the first order differential equation in the form:
- 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.
What is the signifigance of plotting the complementary function (yᴄғ = Ae⁻ᵗ/ᵀ) to the below equation?
- Initial slope = -A/T
- Initial slope intercepts the asymptode at t = T
- t = T when y = Ae⁻¹
How can you determine the particular integral for the below equation for the step response (x = H(t))?
yₚᵢ = x₀
How can you determine the particular integral for the below equation for an impulse response (x = δ(t))?
yₚᵢ = 0
How do you determine the initial conditions for the below equation for an impulse response (x = δ(t))?
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.
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?
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
How can you determine the particular integral for the below equation for a ramp response (x = αt)?
This is just a standard method for finding a particular integral! see example below:
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?
The derivative of the ramp function is the step function. Therefore you can simply integrate the step response to determine the ramp response!
What are the methods for determining the particular integral for the harmonic response (x = Xcos(ωt)) to the equation?
- Solution by cos(ωt) method
- Solution by eᶦʷᵗ method
- eᶦʷᵗ method + phasor diagrams
The particular integral will be in the form:
How can you determine the particular integral for the harmonic response (x = Xcos(ωt)) to the equation using the cos(ωt) method?
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:
How can you determine the particular integral for the harmonic response (x = Xcos(ωt)) to the equation using the eᶦʷᵗ method?
φₓ assumes that the input is phase shifted. Therefore this may be zero if there is no phase shift.
How can you determine the particular integral for the harmonic response (x = Xcos(ωt)) to the equation using the eᶦʷᵗ method and phasor diagrams?
This is the best method
What is the identity for “Acos(ωt) + Bsin(ωt)”?
NOT IN THE DATABOOK
Why is the harmonic response to a differential equation so useful?
Any function can be expressed as a series of sines and cosines (Fourier series), therefore a harmonic response can be used for any input
For a block of mass m, create an expression for the power within the system:
What is the standard form of the differential equation?
What does each constant in this equation represent? (ωₙ, ζ, x)
ωₙ = Natural frequency [rad s⁻¹]
ζ = Damping ratio [dimensionless]
x = “Displacement” input [m]
What is the equation for the natural frequency, ωₙ?
What is the equation for the damping ratio, ζ?
What is the equation for the “displacement” input?
What is the complementary function for the below differential equation?
How does the value of ζ effect the complementary function and hence the response of the system?
- ζ > 1: “Over damped”
- ζ = 1: “Critically damped”
- ζ < 1: “Under damped”
What is the complementary function like for an “over damped” scenario (ζ > 1)?
What is the complementary function like for a “critically damped” scenario (ζ = 1)?
What is the complementary function like for an “under damped” scenario (ζ < 1)?
Third form is preferred (single phase shifted cosine)
What is the “damped natural frequency”, ωd?
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.
What happens when ζ = 0?
There is no damping, therefore there is just simple harmonic motion at the natural frequency ωₙ
How do you determine the particular integral for the step response (x = x₀H(t))?
yₚᵢ = x₀
Given that the system is under damped (ζ < 1), how do you determine the initial conditions for the step response?
Given that the system is under damped (ζ < 1), how do you determine the initial conditions for the impulse response (x = βδ(t)) ?
Determine the initial conditions for the unit step input, and then just differentiate the unit step response to find the unit impulse response
Remember to multiply unit impulse response by β
How can you measure the decay rate of a vibrating system?
Using the Logarithmic Decrement:
Derive the expressions for the logarithmic decrement at any 2 peaks
What are examples of vibrating systems where the “spring” is less obvious?
- A simple pendulum
- A trifilar pendulum
