Laplace

Question 1

What is the first shiftng theorem?

Answer 1

L{e^-aty(t)}=y(s+a)

Question 2

How would you use partial fractions with a denomenator of s(s²+4)?

Answer 2

(A/s)+((BS+c)/(s²+4))

Question 3

With SHM, what is one of the definitions of ω?

Answer 3

ω=(k/m)^0.5

Question 4

What is the heaviside function?

Answer 4

sometimes denoted by u

Question 5

What is the 2nd shifting theorem?

Answer 5

L{H(t-a) y(t-a)}=e^-asy(s)

Question 6

what is the delta function?

Answer 6

Known as an impulse function

Question 7

What is the convolution theorem?

Answer 7

Question 8

What is the transfer function?

Answer 8

G(s)=Y(s)/U(s)

Where G is the transfer function, Y the output and U the input.

Need to find Laplace transform Y(s) with an unknown input u(t) L{u(t)}=U(s)

Question 9

What is the fourier series equation?

Answer 9

Question 10

What are the equations for a_n and b_n ?

Answer 10

Question 11

How do you find L for a fourier series?

Answer 11

period/2 =L

Question 12

How is an odd and even function definded?

Answer 12

Even f(x)=f(-x)

Odd f(x)=-f(-x)

Average of an odd function=0

Question 13

If a function is of the form f(x)=f(x+π), what does that mean for the harmonics?

Answer 13

Only even harmonics

f(x)=-f(x+π) has odd harmonics

Question 14

What are the equations for the sum of a a fourier series?

Answer 14

Question 15

How do you find the fourier transform of a function?

Answer 15

Question 16

What is the quotient rule?

Answer 16

Question 17

Integration by parts

Answer 17

Question 18

How would you start this in partial fractions?

Answer 18

Question 19

How would you find the laplace transform of t²cos(t)

Answer 19

Apply the laplace transform to cos(t) and then differentiate twice. Look at the formula at the bottom of the sheet.

Question 20

What does a₀/2 denonte?

Answer 20

The average of a function

Question 21

If you did a fourier transform on an even fuction, how would this show in the result?

Answer 21

It would only contain cos functions

Question 22

What is the fourier transform?

Answer 22

Question 23

What is the average of an odd function?

Answer 23

0

Question 24

what is the inverse fourier transform?

Answer 24

G(ω) is the fourier transform

f(t) is the inverse

Question 25

If a function has a fourier transform with sin and cos terms, is it odd, even or neither?

Answer 25

neither

An even one will only have cos terms

An odd one will only have sin terms

Question 26

How do you find the fundamental frequeny of a fourier series?

Answer 26

Find the greatest common divisor of the frequencies in the sin and cos terms.

Question 27

If you where to graph the frequency spectrum, what would be on the axis?

Answer 27

X axis- frequency

Y axis- intensity

Question 28

What happens when you multiply a function and multiply by a delta function?

Answer 28

It samples it

Question 29

Formulas to find the centre of mass of a shape?

Answer 29

Question 30

Formulas to find the moment of area of a shape?

Answer 30

Question 31

Having found an fourier transform, how do you find the amplitude spectrum?

Answer 31

square and square root the function and plot.