7/21/2019 10 Spring With Mass

http://slidepdf.com/reader/full/10-spring-with-mass 1/12

1

VERTICAL SPRING WITH MASSCONSIDER

Mass at end of mass-less spring

Spring with homogenous mass distribution

Spring with mass and point mass

At end of springAt arbitrary location

Spring with several point masses.

No damping

Transient response (breakage)

Harmonic loading

7/21/2019 10 Spring With Mass

http://slidepdf.com/reader/full/10-spring-with-mass 2/12

2

Point mass at end. Static deflection

7/21/2019 10 Spring With Mass

http://slidepdf.com/reader/full/10-spring-with-mass 3/12

3

mL << M

 Natural freq.

Transient resp.

Time dep. load 

7/21/2019 10 Spring With Mass

http://slidepdf.com/reader/full/10-spring-with-mass 4/12

4

Homogenous mass distribution (wire only)

• Equilibrium of element

• Separation of variables

• Eigenvalues (fixed at x=0, no forceat x=L)

7/21/2019 10 Spring With Mass

http://slidepdf.com/reader/full/10-spring-with-mass 5/12

5

Example. Transient response.(wire only)

Eigenmodes and modal participation

• L= 1000m

• E=2.1E11 N/m2

• R=0.03m

• Rho = 0.6*7800kg/m3

• U(L,0) = 0.44m (F(L,0)=264kN)

• Ut(L,0) = 0

7/21/2019 10 Spring With Mass

http://slidepdf.com/reader/full/10-spring-with-mass 6/12

6

Transient response cont’d(wire only)

Response at x=L

7/21/2019 10 Spring With Mass

http://slidepdf.com/reader/full/10-spring-with-mass 7/12

7

Transient response cont’d. (wire only)

Shock behaviour 

7/21/2019 10 Spring With Mass

http://slidepdf.com/reader/full/10-spring-with-mass 8/12

8

Transient response cont’d (wire only)

7/21/2019 10 Spring With Mass

http://slidepdf.com/reader/full/10-spring-with-mass 9/12

9

Transient response. Point mass at x=L.

(M=2mL)

7/21/2019 10 Spring With Mass

http://slidepdf.com/reader/full/10-spring-with-mass 10/12

10

Transient response cont’d Point mass at x=L.

(M=2mL) Response at x=L

7/21/2019 10 Spring With Mass

http://slidepdf.com/reader/full/10-spring-with-mass 11/12

11

Transient response cont’d Point mass at x=L.

(M=2mL)

7/21/2019 10 Spring With Mass

http://slidepdf.com/reader/full/10-spring-with-mass 12/12

12

M=0

M=0.1mL

M=2mL