Videos
The expression for the steady state displacement x(t)for a cam.
The steady stat response for the given Fourier series is given by
Given:
The displacement produced by the cam is given by the following figure
Figure 1 Figure 2
Where m = 1 Kg, 
and k = 4900N/m.
Fourier series approximation = 
Concept Used:
We first derive an expression for transfer function of the given system and then calculate 
magnitude of the given transfer function, phase angle, bandwidth, and magnitude of phase angles of respective frequencies, and finally calculate the expression for steady state response.
Calculation:
Fourier series approximation = 
From Figure 1 equation of motion is given by,
(1)
Substituting for m = 1 Kg, 
and k = 4900N/m in equation (1)
Applying Laplace transform for the above equation
The transfer function for the given system
T(s) 
(2)
Therefore magnitude of above transfer function is given by
(3)
The phase angle is given by
(4)
From the equation (3) we can observe that 
attains a peak of 1 when 
0. Hence 
= 0 and 
is calculated as follows
Squaring on both sides we get,
On further simplification we get
The roots of the equation are 
Taking only the positive value 
143.42 rad/s
Therefore, the bandwidth lies between 0 and 143.42 rad/s that is 
As 
is greater than 143.42 rad/s which is outside the required bandwidth, we consider only 0, 10 
20 
30 
and 40 
for frequency values.
From equation (3) 
Substituting for 
0, 10 
20 
30 
and 40 
respectively we get
.... (5)
= 
=1.015
1.263
1.038
0.806
From equation (4) we have 
Now calculating the phase angles for corresponding frequencies 
0, 10 
20 
30 
and 40 
respectively we get,
rad
2.247 rad
2.038 rad
The steady state response for the given Fourier series
Substituting the values of 
and 
for corresponding frequencies 
0, 10 
20 
30 
and 40 
respectively we get,
Conclusion:
Therefore, the steady stat response for the given function is given by
Answer to Problem 9.33P
The steady stat response for the given Fourier series is given by
Explanation of Solution
Given:
The displacement produced by the cam is given by the following figure
Figure 1 Figure 2
Where m = 1 Kg, and k = 4900N/m.
Fourier series approximation =
Concept Used:
We first derive an expression for transfer function of the given system and then calculate magnitude of the given transfer function, phase angle, bandwidth, and magnitude of phase angles of respective frequencies, and finally calculate the expression for steady state response.
Calculation:
Fourier series approximation =
From Figure 1 equation of motion is given by,
(1)
Substituting for m = 1 Kg, and k = 4900N/m in equation (1)
Applying Laplace transform for the above equation
The transfer function for the given system
T(s) (2)
Therefore magnitude of above transfer function is given by
(3)
The phase angle is given by
(4)
From the equation (3) we can observe that attains a peak of 1 when 0. Hence = 0 and is calculated as follows
Squaring on both sides we get,
On further simplification we get
The roots of the equation are
Taking only the positive value
143.42 rad/s
Therefore, the bandwidth lies between 0 and 143.42 rad/s that is
As is greater than 143.42 rad/s which is outside the required bandwidth, we consider only 0, 10 20 30 and 40 for frequency values.
From equation (3)
Substituting for 0, 10 20 30 and 40 respectively we get
.... (5)
=
=1.015
1.263
1.038
0.806
From equation (4) we have
Now calculating the phase angles for corresponding frequencies 0, 10 20 30 and 40 respectively we get,
rad
2.247 rad
2.038 rad
The steady state response for the given Fourier series
Substituting the values of and for corresponding frequencies 0, 10 20 30 and 40 respectively we get,
Conclusion:
Therefore, the steady stat response for the given function is given by
Want to see more full solutions like this?
Chapter 9 Solutions
EBK SYSTEM DYNAMICS
- (read image, answer given)arrow_forward6/86 The connecting rod AB of a certain internal-combustion engine weighs 1.2 lb with mass center at G and has a radius of gyration about G of 1.12 in. The piston and piston pin A together weigh 1.80 lb. The engine is running at a constant speed of 3000 rev/min, so that the angular velocity of the crank is 3000(2)/60 = 100л rad/sec. Neglect the weights of the components and the force exerted by the gas in the cylinder compared with the dynamic forces generated and calculate the magnitude of the force on the piston pin A for the crank angle 0 = 90°. (Suggestion: Use the alternative moment relation, Eq. 6/3, with B as the moment center.) Answer A = 347 lb 3" 1.3" B 1.7" PROBLEM 6/86arrow_forward6/85 In a study of head injury against the instrument panel of a car during sudden or crash stops where lap belts without shoulder straps or airbags are used, the segmented human model shown in the figure is analyzed. The hip joint O is assumed to remain fixed relative to the car, and the torso above the hip is treated as a rigid body of mass m freely pivoted at O. The center of mass of the torso is at G with the initial position of OG taken as vertical. The radius of gyration of the torso about O is ko. If the car is brought to a sudden stop with a constant deceleration a, determine the speed v relative to the car with which the model's head strikes the instrument panel. Substitute the values m = 50 kg, 7 = 450 mm, r = 800 mm, ko = 550 mm, 0 = 45°, and a = 10g and compute v. Answer v = 11.73 m/s PROBLEM 6/85arrow_forward
- Using AutoCADarrow_forward340 lb 340 lb Δarrow_forward4. In a table of vector differential operators, look up the expressions for V x V in a cylindrical coordinate system. (a) Compute the vorticity for the flow in a round tube where the velocity profile is = vo [1-(³] V₂ = Vo (b) Compute the vorticity for an ideal vortex where the velocity is Ve= r where constant. 2πг (c) Compute the vorticity in the vortex flow given by Ve= r 2лг 1- exp ( r² 4vt (d) Sketch all the velocity and vorticity profiles.arrow_forward
- In the figure, Neglects the heat loss and kinetic and potential energy changes, calculate the work produced by the turbine in kJ T = ??? Steam at P=3 MPa, T = 280°C Turbine Rigid tank V = 1000 m³ Turbine Rigid tank V = 100 m³ V = 1000 m³ V = 100 m³ The valve is opened. Initially: evacuated (empty) tank O a. 802.8 Initially: Closed valve O b. 572 O c. 159.93 Od. 415 e. 627.76 equilibriumarrow_forwardPlease find the torsional yield strength, the yield strength, the spring index, and the mean diameter. Use: E = 28.6 Mpsi, G = 11.5 Mpsi, A = 140 kpsi·in, m = 0.190, and relative cost= 1.arrow_forwardA viscoelastic column is made of a material with a creep compliance of D(t)= 0.75+0.5log10t+0.18(log10t)^2 GPA^-1 for t in s. If a constant compressive stress of σ0 = –100 MPa is applied at t = 0, how long will it take (= t1/2) for the height of the column to decrease to ½ its original value? Note: You will obtain multiple answers for this problem! One makes sense physically and one does not.arrow_forward
- A group of 23 power transistors, dissipating 2 W each, are to be cooled by attaching them to a black-anodized square aluminum plate and mounting the plate on the wall of a room at 30°C. The emissivity of the transistor and the plate surfaces is 0.9. Assuming the heat transfer from the back side of the plate to be negligible and the temperature of the surrounding surfaces to be the same as the air temperature of the room, determine the length of the square plate if the average surface temperature of the plate is not to exceed 50°C. Start the iteration process with an initial guess of the size of the plate as 43 cm. The properties of air at 1 atm and the film temperature of (Ts + T)/2 = (50 + 30)/2 = 40°C are k = 0.02662 W/m·°C, ν = 1.702 × 10–5 m2 /s, Pr = 0.7255, and β = 0.003195 K–1. Multiple Choice 0.473 m 0.284 m 0.513 m 0.671 marrow_forwardA 40-cm-diameter, 127-cm-high cylindrical hot water tank is located in the bathroom of a house maintained at 20°C. The surface temperature of the tank is measured to be 44°C and its emissivity is 0.4. Taking the surrounding surface temperature to be also 20°C, determine the rate of heat loss from all surfaces of the tank by natural convection and radiation. The properties of air at 32°C are k=0.02603 W/m-K, v=1.627 x 10-5 m²/s, Pr = 0.7276, and ẞ = 0.003279 K-1 The rate of heat loss from all surfaces of the tank by natural convection is The rate of heat loss from all surfaces of the tank by radiation is W. W.arrow_forwardA 2.5-m-long thin vertical plate is subjected to uniform heat flux on one side, while the other side is exposed to cool air at 5°C. The plate surface has an emissivity of 0.73, and its midpoint temperature is 55°C. Determine the heat flux subjected on the plate surface. Uniform heat flux -Plate, € = 0.73 Cool air 5°C 7 TSUIT Given: The properties of water at Tf,c= 30°C. k=0.02588 W/m.K, v=1.608 x 10-5 m²/s Pr = 0.7282 The heat flux subjected on the plate surface is W/m²arrow_forward
Elements Of ElectromagneticsMechanical EngineeringISBN:9780190698614Author:Sadiku, Matthew N. O.Publisher:Oxford University Press
Mechanics of Materials (10th Edition)Mechanical EngineeringISBN:9780134319650Author:Russell C. HibbelerPublisher:PEARSON
Thermodynamics: An Engineering ApproachMechanical EngineeringISBN:9781259822674Author:Yunus A. Cengel Dr., Michael A. BolesPublisher:McGraw-Hill Education
Control Systems EngineeringMechanical EngineeringISBN:9781118170519Author:Norman S. NisePublisher:WILEY
Mechanics of Materials (MindTap Course List)Mechanical EngineeringISBN:9781337093347Author:Barry J. Goodno, James M. GerePublisher:Cengage Learning
Engineering Mechanics: StaticsMechanical EngineeringISBN:9781118807330Author:James L. Meriam, L. G. Kraige, J. N. BoltonPublisher:WILEY