Mechanical Engineering Formula Sheet

 

BEng (Hons) Mechanical Engineering Network formula sheet

Sine and Cosine Rules B

a  SinA   b SinB

c

 2ab cosC    c a b   2

2

2

A

Moments

 M     Fd 

Mo = r x F 

Varignon’s theorem

 M O



 Rd 

   

 pP   qQ

Mo = r x P + r x Q  

Equilibrium

  F  x

 0  

  F  y  0     M o  0  

(1)  (2)  (3) 

Stress – Strain Relationships

E  

F   A

   

V 

 

  

 

G

  x      z  

 A

   

 

 

   

 

E   2G   1    

 y      z  

C b

2 2 2  2ab cosD   c a b  

  

a

Lo

 

D

 

Area Moments and Centroids





i y     xdA  

i  x     ydA ydA  

 x    

i y   A

y   

 

i x   A

 

Area Moments of Inertia (general)

I x    y  dA  

Iy    x    dA  

2

2





Area Moments of Inertia (standard cases) 3

bd 

I x  

  Rectangle dimensions b x  d.  d. Note: the x-axis passes though the centroid of the area and

12

is parallel to side b 4

 d 

I x  

  Solid circular shaft shaft having a diameter, d. The x-axis is any diameter

64



    d o  d i  4

I x  

4

  

64

Hollow circular shaft having an outer diameter, do and inner diameter di. The x-axis is any diameter

Bending of Beams

M



I

 



y 

E  R

 

Torsion of Circular Sections

T 



 J 

  



G 

r 

T   

 

 L

 P 

 

 

Solid circular section having an outer radius, r. The z-axis is the longitudinal (polar) axis)

 Jz 

  r 4

 

2

Hollow circular section having an outer radius, r o and inner radius ri. The z-axis is the longitudinal (polar) axis

 Jz 



   r o  r i  4

2

4

 

 

Constant acceleration equations  

v  u  at  v  u  2as   2

2

 s  ut  

1 2

2

at 

Friction Friction force (f)=µN (f)=µN  

(where N = normal force, µ = coefficient of friction)

Motion in a circle V = rω 

at = rα  ar = rω θ=

 s r 

2

  2

Centripetal force = m.r. m.r.ω ω 

(where m = mass, r = radius, ω = angular velocity)  velocity) 

θ = ωt  ωt 

    i t  

1 2

 t 2

   f     i   t 

 

   f  2   i2  2 

Energy Equations Potential Energy:

   mgh    P . E   P . E     

Kinetic Energy:

     K . E 

1 2

2 kx  

1 2 1

2 mv  

 K . E       I     2

2

(for height h above a datum)

(for spring stiffness k)

(for translational motion)

(for rotational motion)

 

 

Equations of motion

  kx  M    x  n Displacement Velocity Acceleration

Amplitude

 x  x



0

k  

m

   A cos( n t    )   n  A sin( n t    ) 

   x

 n



A =

2

 x0

 A cos( n t    )

2

  x        n  

2

0

   x       x     n  

  1    tan 

 

0

0

 n    2 f  n  

Torsional Systems

 n   

    k   I        0  

k   I 

 

Effective spring stiffness Parallel connections:

Series connections:

k eff     k 1   k 2   1

k eff  



1

k 1



1

k 2

 

or

k eff   

k 1k 2 k 1  k 2