1.1
If the weight w is displaced bỵ amount, ỵ, the beam and the springs will exert a total force
on the mass of
3𝐸𝐼
𝑃=# + 2𝑘, 𝑢
𝐿(
The beam and springs act in parallel. The equivalent stiffness is:
𝑃 3𝐸𝐼
𝑘 = =# + 2𝑘,
.
𝑢 𝐿(
Natural frequencỵ:
𝑘 𝑔 3𝐸𝐼
𝜔=0 =0 # + 2𝑘,
𝑚 𝑊 𝐿(
Natural period:
2𝜋 𝑊 𝐿
𝑇= = 2𝜋𝐿0 # ,
𝜔 𝑔 3𝐸𝐼 + 2𝑘𝐿 (
1.2
Stiffness:
3𝐸𝐼 3 × 109
𝑘. = # + 2𝑘, = + 2 × 1000 = 4,300 𝑙𝑏/𝑖𝑛.
𝐿( 100(
Natural frequencỵ:
𝑘 4300 × 386
0
𝜔= =0 = 23.52 𝑟𝑎𝑑/𝑠𝑒𝑐
𝑚 3000
Free vibration response of undamped oscillator:
𝑢O
𝑢(𝑡) = 𝑢O 𝑐𝑜𝑠𝜔𝑡 + 𝑠𝑖𝑛𝜔𝑡
1
,𝜔
2
, 𝑢 (𝑡) = −𝑢O𝜔𝑠𝑖𝑛𝜔𝑡 + 𝑢O𝑐𝑜𝑠𝜔𝑡
Displacement and velocitỵ at 𝑡 = 1𝑠𝑒𝑐 with the initial values
𝑢O = 1 𝑖𝑛. , 𝑢O = 20 𝑖𝑛./𝑠𝑒𝑐:
20
𝑢(1) = 1 ∙ 𝑐𝑜𝑠(23.5 ∙ 1) + 𝑠𝑖𝑛(23.5 ∙ 1) = −0.89 𝑖𝑛.
23.5
𝑢 (1) = −1 ∙ 23.5𝑠𝑖𝑛(23.5 ∙ 1) + 20𝑐𝑜𝑠(23.5 ∙ 1) = 22.66 𝑖𝑛./𝑠𝑒𝑐
1.3
The stiffness of the beam is
12𝐸𝐼W 3𝐸(2𝐼X) 12 ∙ (30 ∙ 10Z) ∙ 170.9 3 ∙ (30 ∙ 10Z) ∙ 82.5
𝑘=V + Ỵ= + = 25,577 𝑙𝑏/𝑖𝑛.
𝐿( 𝐿( 144( 144(
Natural frequencỵ:
1 𝑘 1 25,577 × 386
0
𝑓= = 0 = 2.24 𝑐𝑝𝑠
2𝜋 𝑚 2𝜋 50,000
1.4
a) Infinitelỵ rigid horizontal member
Stiffness:
12𝐸𝐼 12 ∙ (30 ∙ 10Z) ∙ 171
𝑘 = 2# ( , = = 21,100 𝑙𝑏/𝑖𝑛.
𝐿 (12 ∙ 15)(
3
, Natural frequencỵ:
4
If the weight w is displaced bỵ amount, ỵ, the beam and the springs will exert a total force
on the mass of
3𝐸𝐼
𝑃=# + 2𝑘, 𝑢
𝐿(
The beam and springs act in parallel. The equivalent stiffness is:
𝑃 3𝐸𝐼
𝑘 = =# + 2𝑘,
.
𝑢 𝐿(
Natural frequencỵ:
𝑘 𝑔 3𝐸𝐼
𝜔=0 =0 # + 2𝑘,
𝑚 𝑊 𝐿(
Natural period:
2𝜋 𝑊 𝐿
𝑇= = 2𝜋𝐿0 # ,
𝜔 𝑔 3𝐸𝐼 + 2𝑘𝐿 (
1.2
Stiffness:
3𝐸𝐼 3 × 109
𝑘. = # + 2𝑘, = + 2 × 1000 = 4,300 𝑙𝑏/𝑖𝑛.
𝐿( 100(
Natural frequencỵ:
𝑘 4300 × 386
0
𝜔= =0 = 23.52 𝑟𝑎𝑑/𝑠𝑒𝑐
𝑚 3000
Free vibration response of undamped oscillator:
𝑢O
𝑢(𝑡) = 𝑢O 𝑐𝑜𝑠𝜔𝑡 + 𝑠𝑖𝑛𝜔𝑡
1
,𝜔
2
, 𝑢 (𝑡) = −𝑢O𝜔𝑠𝑖𝑛𝜔𝑡 + 𝑢O𝑐𝑜𝑠𝜔𝑡
Displacement and velocitỵ at 𝑡 = 1𝑠𝑒𝑐 with the initial values
𝑢O = 1 𝑖𝑛. , 𝑢O = 20 𝑖𝑛./𝑠𝑒𝑐:
20
𝑢(1) = 1 ∙ 𝑐𝑜𝑠(23.5 ∙ 1) + 𝑠𝑖𝑛(23.5 ∙ 1) = −0.89 𝑖𝑛.
23.5
𝑢 (1) = −1 ∙ 23.5𝑠𝑖𝑛(23.5 ∙ 1) + 20𝑐𝑜𝑠(23.5 ∙ 1) = 22.66 𝑖𝑛./𝑠𝑒𝑐
1.3
The stiffness of the beam is
12𝐸𝐼W 3𝐸(2𝐼X) 12 ∙ (30 ∙ 10Z) ∙ 170.9 3 ∙ (30 ∙ 10Z) ∙ 82.5
𝑘=V + Ỵ= + = 25,577 𝑙𝑏/𝑖𝑛.
𝐿( 𝐿( 144( 144(
Natural frequencỵ:
1 𝑘 1 25,577 × 386
0
𝑓= = 0 = 2.24 𝑐𝑝𝑠
2𝜋 𝑚 2𝜋 50,000
1.4
a) Infinitelỵ rigid horizontal member
Stiffness:
12𝐸𝐼 12 ∙ (30 ∙ 10Z) ∙ 171
𝑘 = 2# ( , = = 21,100 𝑙𝑏/𝑖𝑛.
𝐿 (12 ∙ 15)(
3
, Natural frequencỵ:
4
