IMPORTANT EQUATIONS - CHAPTER SUMMARY - 2. 4 OTHER SOURCES OF POTENTIAL ENERGY

2. 4 OTHER SOURCES OF POTENTIAL ENERGY

2.15 CHAPTER SUMMARY

2.15.2 IMPORTANT EQUATIONS

Force-displacement relation for a linear spring

(2.4) Potential energy developed in a linear spring

(2.6) Stiffness of a helical coil spring

(2.11) Stiffness of longitudinal bar

(2.16) Stiffness of a simply supported beam at its midspan

(2.18) Stiffness of a cantilever beam at its end

(2.21) Torsional stiffness of shaft

(2.25) Equivalent stiffness of nsprings in parallel

(2.28) k_eq = a

n i=1

k_i k_t =

JG L k = 3EI

L³ k = 48EI

L³ k = AE

L k = GD⁴

64Nr³ V = 1

2 kx² F = kx

(πM_A)_ext = (πM_A)_eff

(πF)_ext = (πF)_eff πM_O = I₀a

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Equivalent stiffness of nsprings in series

(2.31)

Determination of equivalent stiffness for arbitrary combination of springs

(2.32) Potential energy due to gravity

(2.34) Force developed in viscous damper

(2.37) Work done by viscous damping forces

(2.47) Equivalent mass when linear displacement is used as generalized coordinate

(2.50) Equivalent moment of inertia when angular coordinate is used as generalized coordinate

(2.51) Equivalent mass of a system including approximation of inertia effects in springs

(2.57) Work done by external sources

(2.64) Small angle assumption

(2.71) (2.73) (2.74) Differential equation governing equivalent mass-spring and viscous-damper system

(2.84) Differential equation governing equivalent system when chosen generalized coordinate is an angular coordinate

(2.85) I_equ

+ c_{t, eq}u

+ k_{t, eq}u = M_eq(t) m_eqx$

+ c_eqx#

+ k_eqx = F_eq(t) tanu L u

cosu L 1 sinu L u U₁:2 = -

t₂ t₁

F_eqx# dt m_eq = m +

3 T =

1 2I_equ

T = 1 2m_eqx#

U₁:2 = - L

x 0

c_eqx# dx F =cv

V =mgh V = 1

2k_eqx² k_eq =

1 a

n i=1

1 k_i

P

ROBLEMS

SHORT ANSWER PROBLEMS

For Problems 2.1 through 2.15, indicate whether the statement presented is true or false.

If true, state why. If false, rewrite the statement to make it true.

2.1 The differential equation governing the free vibrations of a sliding mass-spring and viscous-damper system (without friction) is the same as the differential equation for a hanging mass-spring and viscous-damper system.

2.2 The differential equation governing the motion of a SDOF linear system is fourth order.

2.3 Springs in series have an equivalent stiffness that is the sum of the individual stiffnesses of these springs.

2.4 The equivalent stiffness of a uniform simply supported beam at its middle is 3EI/L³.

2.5 The term representing viscous damping in the governing differential equation for a system is linear.

2.6 When the equivalent systems method is used to derive the differential equation for a system with an angular coordinate used as the generalized coordinate, the kinetic energy is used to derive the equivalent mass of the system.

2.7 The equivalent systems method can be used to derive the differential equation for linear SDOF systems with viscous damping.

2.8 The inertia effects of a simply supported beam can be approximated by placing a particle of mass one-third of the mass of the beam at the midspan of the beam.

2.9 The static deflection of the spring in the system if Figure SP2.9 is mg/k.

2.10 The springs in the system of Figure SP2.10 are in series.

Slender bar of mass m L

FIGURE SP 2.09

m k₁

k₂

FIGURE SP 2.10

2.11 A shaft can be used as a spring of torsional stiffness JG/L.

2.12 Energy dissipation is used to calculate the equivalent viscous-damping coefficient for a combination of viscous dampers.

2.13 The added mass of a fluid entrained by a vibrating system is determined by calculating the potential energy developed in the fluid.

2.14 If it is desired to calculate the reactions at the support of Figure SP2.14, the effects of the static spring force and gravity cancel and do not need to be included on the FBD or in summing forces on the FBD.

2.15 Gravity cancels with the static spring force, and hence, the potential energy of neither is included in potential energy calculations for the system of Figure SP2.15.

Problems 2.16 through 2.25 require a short answer.

c L k 2

L 2

FIGURE SP 2.14

2.16 What is the small angle assumption and how is it used?

2.17 When are the free-body diagrams of a system drawn when they are used to derive the differential equation of a linear SDOF system?

2.18 What is meant by “quadratic forms”?

2.19 The inertia effects of the spring in a mass-spring and viscous-damper system can be approximated by adding a particle of what to the mass?

2.20 What is the same in each spring for a combination of springs in parallel?

2.21 In general, how is the equivalent stiffness of a combination of springs calculated?

2.22 Draw a FBD showing the spring forces applied to the system of Figure SP2.22 at an arbitrary instant. Label the forces in terms of .u

L 3 2L

FIGURE SP 2.15

k k

k 3 L

3 L

FIGURE SP 2.22

2.23 Draw a FBD showing the forces developed in the viscous dampers acting on the bar of Figure SP2.23 at an arbitrary instant. Label the forces in terms of .u#

2.24 Describe the equivalent systems method.

2.25 When are static spring forces not drawn on the FBD of external forces?

2.26 Can the equivalent systems method be used to derive the differential equation of a nonlinear SDOF system? Explain.

Problems 2.27 through 2.44 require short calculations.

2.27 What is the equivalent stiffness of springs of individual stiffnesses k₁and k₂ placed in series?

2.28 What is the equivalent stiffness of the springs in the system of Figure SP2.28?

2.29 What is the equivalent torsional stiffness of the shafts in Figure SP2.29?

θ c

L 2

L 6 L 2

L 3

L 2 Rigid bar

c c

FIGURE SP 2.23

x k k

2k 4k

3k FIGURE SP 2.28

50 cm

Aluminum r = 20 mm

Steel r = 15 mm

60 cm

FIGURE SP 2.29

2.30 When a tensile force of 300 N is applied to an elastic element, it has an elongation of 1 mm. What is the stiffness of the element?

2.31 What is the potential energy developed in the elastic element of Short Problem 2.30 when a 300 N tensile force is applied?

2.32 What is the potential energy in the elastic element of Short Problem 2.30 when a 300 N compressive force is applied?

2.33 A spring of torsional stiffness 250 N m/rad has a rotation of 2° when a moment is applied. Calculate the potential energy developed in the spring.

2.34 What is the torsional stiffness of an annular steel shaft (G 80 10⁹N/m²) with a length of 2.5 m, inner radius of 10 cm, and outer radius of 15 cm?

2.35 What is the torsional stiffness of a solid aluminum shaft (G 40 10⁹N/m²) with a length of 1.8 m and a radius of 25 cm?

2.36 What is the longitudinal stiffness of a steel bar (E 200 10⁹N/m²) with a length of 2.3 m and a rectangular cross section of 5 cm 6 cm?

#

2.42 Evaluate without using a calculator. The argument of the trigonometric function is in radians.

(a) sin 0.05 (b) cos 0.05 (c) 1-cos 0.05 (d) tan 0.05 (e) cot 0.05 (f ) sec 0.05 (g) csc 0.05

2.43 Evaluate without using a calculator.

(a) sin 3° (b) cos 3°

(c) 1-cos 3° (d) tan 3°

2.44 Calculate the equivalent moment of inertia of the three shafts of Figure SP2.44 when is used as the generalized coordinate. Assume the gears mesh perfectly and their moments of inertia are negligible.

u₂

2.37 What is the transverse stiffness of a cantilever steel beam (E 200 10⁹N/m²) with a length of 10 m and a rectangular cross section with a width of 1 m and height of 0.5 m?

2.38 Calculate the static deflection in a linear spring of stiffness 4000 N/m when a mass of 20 kg is hanging from it.

2.39 A spring of unstretched length of 10 cm has a linear density of 2.3 g/cm. The spring is attached between a fixed support and a block of mass of 150 g. What mass should be added to the block to approximate the inertia effects of the spring?

2.40 What is the kinetic energy of the system of Figure SP2.40 at an arbitrary instant in terms of x, which is the downward displacement of the block of mass m₁? Include an approximation of the inertia effects of the springs. The mass of each spring is ms.

2.41 Calculate an equivalent torsional-damping coefficient for the system of Figure SP2.41 when , which is the clockwise angular rotation of the bar, is used as the generalized coordinate.

u m

m m

x I

m_s m_s

No slip

Thin disk of mass m₂

r₂

m₁ r₁

FIGURE SP2.40

L 3 L 3 L

c c

FIGURE SP2.41

J₁ θ1

θ2

θ3

J₂

J₃ Gear with

n₂ teeth

Gear with n₄ teeth

Gear with n₃ teeth Gear with

n₁ teeth

FIGURE SP 2.44

2.45 Match the quantity with the appropriate units

(a) spring stiffness, k (i) N m

(b) torsional stiffness, k_t (ii) rad

(c) damping coefficient, c (iii) N m/rad

(e) torsional damping coefficient, c_t (iv) N m/s

(f ) potential energy, V (v) kg m²

(g) power delivered by external force, P (vi) N/m

(h) moment of inertia, I (vii) N m s/rad

(i) angular displacement ␪ (viii) N s/m

CHAPTER PROBLEMS

2.1–2.8 Determine the equivalent stiffness of a linear spring when a SDOF mass-spring model is used for the systems shown in Figures P2.1 through P2.8 with xbeing the chosen generalized coordinate.

#

x 20 kg

E = 200 × 10⁹ N/m² I = 1.15 × 10^–4 m⁴

1 m 1 m

FIGURE P 2.1

x k

k L

E, I

Massless beam

m FIGURE P 2.2

E, I

20 kg Massless beam

40 cm

60 cm 40 cm

FIGURE P 2.3

E = 210 × 10⁹ N/m² I = 6.1 × 10^–6 m⁴ L = 2.5 m

8 × 10⁴ N/m 6 × 10⁴ N/m 1 × 10⁵ N/m

m L 2

3L 2

L 2

7L 15 k L 3 L 5 x θ

3k k

x k

3k 2k

r 3r

k k

L k 3

L 3

L 2 2L

Rigid link L 2 x

k 3k r

No slip x

FIGURE P 2.4 FIGURE P 2.5

FIGURE P 2.6 FIGURE P 2.7

FIGURE P 2.8

2.9 Two helical coil springs are made from a steel (E 200 10⁹N/m²) bar with a radius of 20 mm.

One spring has a coil diameter of 7 cm; the other has a coil diameter of 10 cm. The springs have 20 turns each. The spring with the smaller coil diameter is placed inside the spring with the larger coil diameter. What is the equivalent stiffness of the assembly?

x u θ

65 cm

r = 10 mm E = 200 × 10⁹ N/m² G = 80 × 10⁹ N/m² FIGURE P 2.10

2.10 A thin disk attached to the end of an elastic beam has three uncoupled modes of vibration. The longitudinal motion, the transverse motion, and the torsional oscillations are all kinematically independent. Calculate the following for the system of Figure P2.10.

(a) The longitudinal stiffness (b) The transverse stiffness (c) The torsional stiffness

3 × 10⁵ N/m

5 × 10⁵ N/m 4 × 10⁵ N/m

45°

x 30° 45°

FIGURE P 2.11

20 µm

0.2 µm 1 µm

x Each layer is 0.1 µm thick FIGURE P 2.12

2.11 Find the equivalent stiffness of the springs in Figure P2.11 in the x direction.

2.12 A bimetallic strip used as a MEMS sensor is shown in Figure P2.12. The strip, has a length of 20 m. The width of the strip is 1 m. It has an upper layer made of steel (E 5 210 3 109 N/m2) and a lower layer made of aluminum (E 5 80 3 109 N/m2) . Each layer is 0.1 m thick. Determine the equivalent stiffness of the strip in the axial direction.

m m

2.13 A gas spring consists of a piston of area Amoving in a cylinder of gas. As the piston moves, the gas expands and contracts, changing the pressure exerted on the piston. The process occurs adiabatically (without heat transfer), so

where pis the gas pressure, is the gas density, is the constant ratio of specific heats, and Cis a constant dependent on the initial state. Consider a spring when the initial pressure is p₀and the initial temperature is T₀. At this pressure, the height of the gas column in the cylinder is h.let F ₀A+ Fbe the pressure force acting on the piston when it has displaced a distance xinto the gas from its initial height.

d r g

r p = Cr^g

(a) Determine the relation between Fand x.

(b) Linearize the relationship of part (a) to approximate the air spring by a linear spring. What is the equivalent stiffness of the spring?

(c) What is the required piston area for an air spring (␥1.4) to have a stiffness of 300 N

#

m for a pressure of 150 kPa (absolute) with h 30 cm.

2.14 A wedge is floating stably on an interface between a liquid of mass density , as shown in Figure P2.14. Let xbe the displacement of the wedge’s mass center when it is disturbed from equilibrium.

(a) What is the buoyant force acting on the wedge?

(b) What is the work done by the buoyant force as the mass center of the wedge moves from x_lto x₂?

(c) What is the equivalent stiffness of the spring if the motion of the mass center of the wedge is modeled as a mass attached to a linear spring?

r d

Length of wedge = L Mass density of wedge = ρw

FIGURE P 2.14

2.15 Consider a solid circular shaft of length Land radius cmade of an elastoplastic material whose shear stress–shear strain diagram is shown in Figure P2.15(a). If the applied torque is such that the shear stress at the outer radius of the shaft is less than ␶p, a linear relationship between the torque and the angular

displacement exists. When the applied torque is large enough to cause plastic behavior, a plastic shell is developed around an elastic core of radius r c, as shown in Figure 2.15(b). Let be the applied torque which results in an angular displacement of

(a) The shear strain at the outer radius of the shaft is related to the angular displacement .The shear strain distribution is linear over a given cross section. Show that this implies

(b) The torque is the resultant moment of the shear stress distribution over the cross section of the shaft,

Use this to relate the torque to the radius of the elastic core.

(c) Determine the relationship between Tand .

(d) Approximate the stiffness of the shaft by a linear torsional spring. What is the equivalent torsional stiffness?

u d d T =

c 0

2ptr²dr u =

Lt_P rG

u =

g_cL c

u =

t_pL cG + du T =

pt_p c² 2 + dT

c r

Plastic shell Elastic

core

γ G

τp

(a) (b)

FIGURE P 2.15

⑀ σp E

σ = f(E)

(a) FIGURE P 2.16

2.16 A bar of length Land cross-sectional area Ais made of a material whose stress- strain diagram is shown in Figure P2.16. If the internal force developed in the bar is such that , the bar’s stiffness for a SDOF model is

Consider the case where Let be the applied load which results in a deflection of .

(a) The work done by the applied force is equal to the strain energy developed in the bar. The strain energy per unit volume is the area under the stress–strain curve. Use this information to relate Pto .

(b) What is the equivalent stiffness when the bar is approximated as a linear spring for s 7 s_p?

d¢ d

¢ =

s_pL E + d¢

P = s_pA + dP s 7 s_p.

k = ^AE

L. s 6 s_p

2.17 Calculate the static deflection of the spring in the system of Figure P2.17.

2.18 Determine the static deflection of the spring in the system of Figure P2.18.

m₁ m₂

r₂ r₁

FIGURE P 2.17

m = 20 kg

Spring is stretched 20 mm when bar

is vertical 5 × 10³ N/m

1.2 m

0.4 m FIGURE P 2.18

2.19 A simplified SDOF model of a vehicle suspension system is shown in Figure P2.19. The mass of the vehicle is 500 kg. The suspension spring has a stiffness of 100,000 N/m. The wheel is modeled as a spring placed in series with the suspension spring. When the vehicle is empty, its static deflection is measured as 5 cm.

(a) Determine the equivalent stiffness of the wheel

(b) Determine the equivalent stiffness of the spring combination.

2.20 The spring of the system in Figure P2.20 is unstretched in the position shown.

What is the deflection of the spring when the system is in equilibrium?

2.21 Determine the static deflection of the spring in the system of Figure P2.21.

2.22 Determine the static deflections in each of the springs in the system of Figure P2.22.

Suspension spring Wheel stiffness k_s

k_w

FIGURE P 2.19

2.23 A 30 kg compressor sits on four springs, each of stiffness 1 10⁴N/m. What is the static deflection of each spring.

2.24 The propeller of a ship is a tapered circular cylinder, as shown in Figure P2.24.

When installed in the ship, one end of the propeller is constrained from longitudinal motion relative to the ship while a 500-kg propeller mass is attached to its other end.

(a) Determine the equivalent longitudinal stiffness of the shaft for a SDOF model.

(b) Assuming a linear displacement function along the shaft, determine the equivalent mass of the shaft to use in a SDOF model.

150 kg 3 m

2000 N/m I = 8.2 × 10^–7 m⁴ E = 210 × 10⁹ N/m²

FIGURE P 2.20

m k

E, I L

L 2

FIGURE P 2.21

40 cm 20 cm

4 kg

1 × 10⁵ N/m 2 × 10⁵ N/m

FIGURE P 2.22

r₀ = 30 cm r₁ = 20 cm E = 210 × 10⁹ N/m² ρ = 7850 kg/m³

r₀ r₁

10 m FIGURE P 2.24

2.25 (a) Determine the equivalent torsional stiffness of the propeller shaft of Problem 2.24.

(b) Determine an equivalent moment of inertia of the shaft of Problem 2.24 to be placed on the end of the shaft for a SDOF model of torsional oscillations.

2.26 A tightly wound helical coil spring is made from an 1.88-mm diameter bar made from 0.2 percent hardened steel (G 80 10⁹N/m², 7600 kg/m³).

The spring has a coil diameter of 1.6 cm with 80 active coils. Calculate (a) the stiffness of the spring,

(b) the static deflection when a 100 g particle is hung from the spring, and (c) the equivalent mass of the spring for a SDOF model.

2.27 One end of a spring of mass m_s1and stiffness k₁is connected to a fixed wall, while the other end is connected to a spring of mass m_s2and stiffness k₂. The other end of the second spring is connected to a particle of mass m. Determine the equivalent mass of these two springs.

2.28 A block of mass m is connected to two identical springs in series. Each spring has a mass m and a stiffness k. Determine the equivalent mass of the two springs at the mass.

2.29 Show that the inertia effects of a torsional shaft of polar mass moment of inertia Jcan be approximated by adding a thin disk of moment of inertia J/3 at the end of the shaft.

2.30 Use the static displacement of a simply supported beam to determine the mass of a particle that should be added at the midspan of the beam to approximate inertia effects in the beam.

2.31–35 Determine the equivalent mass or equivalent moment of inertia of the system shown in Figures P2.31 through P2.35 when the indicated generalized coordinate is used.

x m 2r No slip

Sphere of mass m k

FIGURE P 2.31

2.36 Determine the kinetic energy of the system of Figure P2.36 at an arbitrary instant in terms of including inertia effects of the springs.x#

θ m

L m/2

L/2

A B C

AB and BC are slender bars

L Slender rod of mass m

Slender rod of mass m

m 4L

5 L 3

Rigid massless connector

θ Gear with

n₂ teeth

Gear with n₁ teeth

Gear with n₄ teeth

J_G

J₁ θ1

J_G

J_r

J₃ Gear with n₃ teeth FIGURE P 2.33

FIGURE P 2.34 FIGURE P 2.35

c 2m

2r r

x No slip

r_D

I_p

k, m_s

k, m_s m

FIGURE P 2.36 L 3

2m Slender rod 3

of mass m x

FIGURE P 2.32

2.37 The time-dependent displacement of the block of mass mof Figure P2.36 is x(t) 0.03e^–1.35tsin (4t) m. Determine the time-dependent force in the viscous damper if c125 N s/m.

2.38 Calculate the work done by the viscous damper of Problem 2.37 betweent 0 and t1 s.

2.39 Determine the torsional viscous-damping coefficient for the torsional viscous damper of Figure P2.39. Assume a linear velocity profile between the bottom of the dish and the disk.

#

Disk of radius r

Oil of density ρ, viscosity µ Depth of oil = h

θ⋅

FIGURE P 2.39

2.40 Determine the torsional viscous-damping coefficient for the torsional viscous damper of Figure P2.40. Assume a linear velocity profile in the liquid between the fixed surface and the rotating cone.

h d

θ˙

Oil of density ρ, viscosity µ

Come of base radius r, height h

Gap width, d

FIGURE P 2.40

2.41 Shock absorbers and many other forms of viscous dampers use a piston moving in a cylinder of viscous liquid as illustrated in Figure P2.41. For this

configuration the force developed on the piston is the sum of the viscous forces acting on the side of the piston and the force due to the pressure difference between the top and bottom surfaces of the piston.

(a) Assume the piston moves with a constant velocity v_p. Draw a free-body diagram of the piston and mathematically relate the damping force, the viscous force, and the pressure force.

(b) Assume steady flow between the side of the piston and the side of the cylinder. Show that the equation governing the velocity profile between the piston and the cylinder is

(c) Assume the vertical pressure gradient is constant. Use the preceding results to determine the velocity profile in terms of the damping force and the shear stress on the side of the piston.

(d) Use the results of part (c) to determine the wall shear stress in terms of the damping force.

(e) Note that the flow rate between the piston and the cylinder is equal to the rate at which liquid is displaced by the piston. Use this information to determine the damping force in terms of the velocity and thus the damping coefficient.

(f ) Use the results of part (e) to design a shock absorber for a motorcycle that uses SAE 1040 oil and requires a damping coefficient of 1000 N m/s.

#

dx = m 0v² 0r²

x h

r D

V_p

Oil of viscosity µ, density ρ FIGURE P 2.41

2.42–51 Derive the differential equation governing the motion of the one degree-of- freedom system by applying the appropriate form(s) of Newton’s laws to the appropriate free-body diagrams. Use the generalized coordinate shown in Figures P2.42 through P2.51. Linearize nonlinear differential equations by assuming small displacements.

Dalam dokumen mechVib theory and applications (Halaman 137-157)