A mass m hangs on the end of a cord around a pulley of

In Problems 1 through 6, express the solution of the given initial value problem as a sum of two oscillations as in Eq. (8). Throughout, primes denote derivatives with respect to time t. In Problems 14, graph the solution function x.t / in such a way that you can identify and label (as in Fig. 3.6.2) its period.

In Problems 1 through 6, express the solution of the given initial value problem as a sum of two oscillations as in Eq. (8). Throughout, primes denote derivatives with respect to time t. In Problems 14, graph the solution function x.t / in such a way that you can identify and label (as in Fig. 3.6.2) its period.

In Problems 1 through 6, express the solution of the given initial value problem as a sum of two oscillations as in Eq. (8). Throughout, primes denote derivatives with respect to time t. In Problems 14, graph the solution function x.t / in such a way that you can identify and label (as in Fig. 3.6.2) its period.

In Problems 1 through 6, express the solution of the given initial value problem as a sum of two oscillations as in Eq. (8). Throughout, primes denote derivatives with respect to time t. In Problems 14, graph the solution function x.t / in such a way that you can identify and label (as in Fig. 3.6.2) its period.

mx00 C kx D F0 cos !t with ! !0; x.0/ D x0, x0 .0/ D 0

mx00 C kx D F0 cos !t with ! D !0; x.0/ D 0, x0 .0/ D v0

In each of Problems 7 through 10, find the steady periodic solution xsp.t / D C cos.!t / of the given equation mx00 C cx0 C kx D F .t / with periodic forcing function F .t / of frequency !. Then graph xsp.t / together with (for comparison) the adjusted forcing function F1.t / D F .t /=m!. x00 C 4x0 C 4x D 10 cos 3t

In each of Problems 7 through 10, find the steady periodic solution xsp.t / D C cos.!t / of the given equation mx00 C cx0 C kx D F .t / with periodic forcing function F .t / of frequency !. Then graph xsp.t / together with (for comparison) the adjusted forcing function F1.t / D F .t /=m!. x11 + 3x0 C 5x = 4 cos 5t

In each of Problems 7 through 10, find the steady periodic solution xsp.t / D C cos.!t / of the given equation mx00 C cx0 C kx D F .t / with periodic forcing function F .t / of frequency !. Then graph xsp.t / together with (for comparison) the adjusted forcing function F1.t / D F .t /=m!.2x00 C 2x0 C x D 3 sin 10t

In each of Problems 7 through 10, find the steady periodic solution xsp.t / D C cos.!t / of the given equation mx00 C cx0 C kx D F .t / with periodic forcing function F .t / of frequency !. Then graph xsp.t / together with (for comparison) the adjusted forcing function F1.t / D F .t /=m!. x00 C 3x0 C 3x D 8 cos 10t C 6 sin 10t

In each of Problems 11 through 14, find and plot both the steady periodic solution xsp.t / D C cos.!t / of the given differential equation and the actual solution x.t / D xsp.t / C xtr.t / that satisfies the given initial conditions.

In each of Problems 11 through 14, find and plot both the steady periodic solution xsp.t / D C cos.!t / of the given differential equation and the actual solution x.t / D xsp.t / C xtr.t / that satisfies the given initial conditions.

In each of Problems 11 through 14, find and plot both the steady periodic solution xsp.t / D C cos.!t / of the given differential equation and the actual solution x.t / D xsp.t / C xtr.t / that satisfies the given initial conditions.

In each of Problems 11 through 14, find and plot both the steady periodic solution xsp.t / D C cos.!t / of the given differential equation and the actual solution x.t / D xsp.t / C xtr.t / that satisfies the given initial conditions.

Each of Problems 15 through 18 gives the parameters for a forced massspringdashpot system with equation mx00 C cx0 C kx D F0 cos !t. Investigate the possibility of practical resonance of this system. In particular, find the amplitude C.!/ of steady periodic forced oscillations with frequency !. Sketch the graph of C.!/ and find the practical resonance frequency ! (if any) . m D 1, c D 2, k D 2, F0 D 2

Each of Problems 15 through 18 gives the parameters for a forced massspringdashpot system with equation mx00 C cx0 C kx D F0 cos !t. Investigate the possibility of practical resonance of this system. In particular, find the amplitude C.!/ of steady periodic forced oscillations with frequency !. Sketch the graph of C.!/ and find the practical resonance frequency ! (if any) . m D 1, c D 4, k D 5, F0 D 10

Each of Problems 15 through 18 gives the parameters for a forced massspringdashpot system with equation mx00 C cx0 C kx D F0 cos !t. Investigate the possibility of practical resonance of this system. In particular, find the amplitude C.!/ of steady periodic forced oscillations with frequency !. Sketch the graph of C.!/ and find the practical resonance frequency ! (if any) m D 1, c D 6, k D 45, F0 D 50

Each of Problems 15 through 18 gives the parameters for a forced massspringdashpot system with equation mx00 C cx0 C kx D F0 cos !t. Investigate the possibility of practical resonance of this system. In particular, find the amplitude C.!/ of steady periodic forced oscillations with frequency !. Sketch the graph of C.!/ and find the practical resonance frequency ! (if any) . m D 1, c D 10, k D 650, F0 D 100

A mass weighing 100 lb (mass m D 3:125 slugs in fps units) is attached to the end of a spring that is stretched 1 in. by a force of 100 lb. A force F0 cos !t acts on the mass. At what frequency (in hertz) will resonance oscillations occur? Neglect damping.

A front-loading washing machine is mounted on a thick rubber pad that acts like a spring; the weight W D mg (with g D 9:8 m=s2) of the machine depresses the pad exactly 0:5 cm. When its rotor spins at ! radians per second,

Figure 3.6.10 shows a mass m on the end of a pendulum (of length L) also attached to a horizontal spring (with constant k). Assume small oscillations of m so that the spring remains essentially horizontal and neglect damping. Find the natural circular frequency !0 of motion of the mass in terms of L, k, m, and the gravitational constant g

A mass m hangs on the end of a cord around a pulley of radius a and moment of inertia I , as shown in Fig. 3.6.11. The rim of the pulley is attached to a spring (with constant k). Assume small oscillations so that the spring remains essentially horizontal and neglect friction. Find the natural circular frequency of the system in terms of m, a, k, I , and g.

A building consists of two floors. The first floor is attached rigidly to the ground, and the second floor is of mass m D 1000 slugs (fps units) and weighs 16 tons (32,000 lb). The elastic frame of the building behaves as a spring that resists horizontal displacements of the second floor; it requires a horizontal force of 5 tons to displace the second

A mass on a spring without damping is acted on by the external force F .t / D F0 cos3 !t. Show that there are two values of ! for which resonance occurs, and find both.

Derive the steady periodic solution of mx00 C cx0 C kx D F0 sin !t: In particular, show that it is what one would expectthe same as the formula in (20) with the same values of C and !, except with sin.!t / in place of cos.!t /.

Given the differential equation mx00 C cx0 C kx D E0 cos !t C F0 sin !t with both cosine and sine forcing termsderive the steady periodic solution xsp.t / D q E2 0 C F 2 0 p.k m!2/2 C .c!/2 cos.!t /; where is defined in Eq. (22) and D tan1.F0=E0/. (Suggestion: Add the steady periodic solutions separately corresponding to E0 cos !t and F0 sin !t (see Problem 25).) 27

According to Eq. (21), the amplitude of forced steady periodic oscillations for the system mx00 C cx0 C kx D F0 cos !t is given by C.!/ D F0 p.k m!2/2 C .c!/2 : (a) If c = ccr= p2, where ccr D p4km, show that C steadily decreases as ! increases. (b) If c

As indicated by the cart-with-flywheel example discussed in this section, an unbalanced rotating machine part typically results in a force having amplitude proportional to the square of the frequency !. (a) Show that the amplitude of the steady periodic solution of the differential equation mx00 C cx0 C kx D mA!2 cos !t (with a forcing term similar to that in Eq. (17)) is given by C.!/ D mA!2 p.k m!2/2 C .c!/2

Apply the result of Problem 26 to show that the amplitude C of the resulting steady periodic oscillation for the car is given by C D a pk2 C .c!/2 p.k m!2/2 C .c!/2 : Because ! D 2v=L when the car is moving with velocity v, this gives C as a function of v.

Figure 3.6.12 shows the graph of the amplitude function C.!/ using the numerical data given in Example 5 (including c D 3000 Ns=m). It indicates that, as the car accelerates gradually from rest, it initially oscillates with amplitude slightly over 5 cm. Maximum resonance vibrations with amplitude about 14 cm occur around 32 mi=h, but then subside to more tolerable levels at high speeds. Verify these graphically based conclusions by analyzing the function C.!/. In particular, find the practical resonance frequency and the corresponding amplitude.