Vibrations. Homework 6: shock. 10 to a few. atomic force. range. of slinky. AFM. to have. cantilever. below. Use. mode.

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1 EN4: ynamics and Vibrations Homework 6: Forced Vibrations ue Friday April 5th School of Engineering Brown University 1. Springs are used for many different engineering and science applications. Typical shock absorbers for passenger cars have the spring constant approximately 1 to a few hundred kn/m while a physics laboratory slinky has its typical spring constant in the range of.1 1. N/m. As shown in the following chart, the contact-mode atomic force microscope (AFM) cantilever has very soft spring constant which is close to a very soft slinky. In AFM community, researchers use effective masss m eff to model the vibration of the AFM cantilever for the first mode natural frequency (or resonant frequency) n1 to have, where k is the spring constant of the cantilever. n1 km eff Find the effective masses for the threee different AFM cantilevers listed below. Use (3kHz, 48N/m) for the short and (19kHz, 4N/m) for the long cantilevers of ynamic mode / tapping mode AFM, and (8kHz,.N/m) for the short and (1kHz,.1N/m) for the long cantilevers of the contact mode AFM. ompare the effective masses to the real mass of the cantilever (density times the volume). The cantilever is made of silicon and 3 the density is.65 g cm. Standard cantilever types for common applications: ynamic mode / tapping mode AFM Available as a long (5 µm) or short (15 µm) cantilever Force Modulation mode AFM ontact mode AFM Available as a long (45 µm) or short (5 µm) cantilever Length: Typically 15-5 µm Width: Typically 4 µ m Thickness: 4-8 µm Resonant Freq: 19-3 KHz Spring onstant: 4-48 N/m Length: Typically 5 µm Width: Typically 45µm Thickness:.5 µm Resonant Freq: 6 KHz Spring onstant: 3 N/m Length: Typically 5-45µm Width: Typically 8-4 µm Thickness: 1- µm Resonant Freq: 1-8 KHz Spring onstant:.1 -. N/m

2 [Both sets are correct for grading: 1 point each for the total of 5 points] m eff k n1 Answer based on original data set: Answer based on revised data set:. A passenger car has front and rear suspensions composed of identical four spring dashpot shock absorbers, one for each wheel. The weight of the car is 19kg and the initial sag (shortening of the spring by the weight) of the shock absorber is 14cm. (1) Find the spring constant of the shock absorber. 19 kg 9.8 m/sec / 4.14 m 1 33 kn m k [ points] () In lecture 4, we defined the damping ratio km, where is the damping coefficient which is measured in a unit of kg/s and sometimes denoted by c in other text books. It was defined that the damping is critical if the damping ratio 1. For most of passenger cars the shock absorbers are designed underdamped to have a typical value of.56. Find the damping coefficient of this passenger car. [ points] km kg s 19 4 kg 44 kg s

3 3. Here we will see the energy absorption characteristics of the above shock absorber. (1) onsider an elongation displacement cycle of the shock absorber as shown below. B Elongation (inch) A E Time (sec) (1.1) raw the force - displacement diagram for the force applied to the spring during the displacement cycle. Mark the corresponding points of A, B,, and E on the drawing. What is the work dissipated during this cycle? rawing [ points] Force (kn) (.5,.85) B. A, E isplacement (m).4 (.5,.85) Work dissipated =. [ points]

4 (1.) raw the force - displacement diagram for the force applied to the dashpot during the displacement cycle. Mark the corresponding points of A, B,, and E on the drawing. What is the work dissipated during this cycle? rawing [ points] F x 44 kg s.5 m s 1. kn for A B. F x 44 kg s.5 m s 1. kn for B. F x 44 kg s.5 m s 1. kn for E..5 + Force (kn) A, E B isplacement (m) B +.5 Work dissipated =.kn.5m4.kj [ points] (1.3) raw the force - displacement diagram for the force applied to the whole shock absorber during the displacement cycle. Mark the corresponding points of A, B,, and E on the drawing. What is the work dissipated during this cycle? rawing [ points]

5 1.5 Force (kn) 1.5 A, E B B isplacement (m) Work dissipated = same as ().kj [ points] () onsider an elongation displacement cycle of the shock absorber x t sint as shown below. Here B 1 inch and the time t is measured in second. Elongation (inch) A E Time (sec) (.1) raw the force - displacement diagram for the force applied to the spring during the displacement cycle. Mark the corresponding points of A, B,, and E on the drawing. What is the work dissipated during this cycle? rawing [ points] Same as (1.1). Work dissipated = same as (1.1) =. [ points]

6 (.) raw the force - displacement diagram for the force applied to the dashpot during the displacement cycle. Mark the corresponding points of A, B,, and E on the drawing. What is the work dissipated during this cycle? rawing [ points] F x 44 kg s.5 m s cost 1.35cost kn x.5sint m Force (kn).4 A.3 E..1 isplacement (m) B Work dissipated =.35kN.5m.7kJ [ points] (.3) raw the force - displacement diagram for the force applied to the whole shock absorber during the displacement cycle. Mark the corresponding points of A, B,, and E on the drawing. What is the work dissipated during this cycle? rawing [ points].83 sint.35 costkn.5sintm F kxx33 kn m.5sint m 44 kg s.5 m s cost 1 x

7 Force (kn) A, E B isplacement (m) cycle tdt Work dissipated = Fdx.83sin t.35cos t.5cos t dt.35.5cos kj.35.5 kj.7 kj : Same as (.). [ points] 4. A shock absorber is driven by steady-state elongation cycles of x t sint, and the corresponding steady-state net force is measured to be F t Asint Bcost. Express the spring constant k and the damping coefficient in terms of A, B, and. F kx x k sint cost Asint B cost [1 point] This implies k and A [ points] B. [ points] Therefore, if you actuate a linear viscoelastic shock absorber with x t sint and measure the force F t Asint Bcost, you can measure the spring constant and the damping coefficient.

8 5. The -kg variable-speedd motorized unit is restrained in the horizontal direction by two springs, each of which has a stiffness of.1 kn/m. Each of the two dashpots has a viscous damping coefficient 58 kg/s. In what range of spin speeds N (revolution/sec) can the motor run for which the magnification factor M will not exceed? The net spring constant is k and the net damping coefficient is. [1 point] [1 point] 1 M 1 n n 1 1 n n x x 4 x with 4 x n or x 4 3 x 4 [ points] x n n 4 4 3

9 or n n Therefore, we have km N [ points] or km N [ points] with for the double springs and double dashpots. [ points] km Inserting the values, 58 kg/s, k =.1 kn/m and m = kg, we have N 1.8 revolution/sec [1 point] or N.56 revolution/sec. [1 point]